Marie Ouvrard

Valérie Campillo

SOCIOLOGY OF TRANSLATION AND ANTHROPOLOGICAL ANALYSIS OF COMMUNICATION FOR ICT SUSTAINABILITY An "on-the-road communication " project in Ghana

Keywords: sustainability, information accessibility, translation steps, anthropological analysis of communication Sustainability and Information Accessibility

Based on the actants (Latour) concept as well as the situated action (Suchman), the sociology of translation (Latour) and the anthropology of communication (Hymes) compared to the recommended sharing values of international project management, the implementation and the sustainability of a programme dedicated to socially or geographically isolated populations are requiring specific methods and conditions for reaching informational and cultural accessibility. Combining permanent intention, action and retroaction for moving human mind boundaries through ICTs induce a complex communicational process based on the need of partners, facilitators, locally-recognised leaders or respected individuals as well of the use of updated high technology as the 5th screen which gathers the technical advantages of the four previous screens and the people around an event where every one is acting.

As technological means have been developed for reaching geographically isolated populations and as commitments and recommendations have been adopted for socially isolated populations, the general purpose of this study is to consider how these populations could reach information in a sustainable way. Throughout an anthropological analysis of communication, we are searching to clarify the conditions for information accessibility through communicational concepts. We will illustrate the conceptual aspects as well as the necessary conditions for reaching sustainability and information accessibility through an « on-the-road communication » project planed in Ghana for enabling community participation in Highly Pathogenic Avian Influenza (HPAI) risk reduction, combining movies entertainment with the HPAI campaign.

Figure 1 & 2: The "on-the-road communication" -A 4x4 lorry at work for an AIDS prevention campaign in Yemen

This information emergency transmission for developing countries is on the way to be implemented by European Organisations, NGOs and United Nations Organisations through the use of the 5 th screen (Skacan 1 ). In fact, the such called 5 th screen gathers broadcasting on a wide screen through satellite access as well as mobile computing abilities. The use of a 4x4 lorry reaches geographically and socially isolated populations. All the messages can be broadcasted in vernacular language by using an instantaneous dubbing system. The Nomadic Dream Machines Team can train technicians all over the world to use the 5th screen 4x4 lorry and maintain the High Technical system updated and working.

Figure 3: The "on-the-road communication" and 5th screen technical concept

As said before, technical aspects have been almost overcome. From our point of view, we need to watch at several concepts which are supporting a specific anthropological analysis [START_REF] Ouvrard | Regards des Sciences de l'information et de la comnunication sur une organisation -Partenariat Universités/Entreprises dans le domaine des TIC -Master Recherche en sciences de l'information et de la communication-Aix Marseille Universités[END_REF]) in order to reach sustainability and information accessibility. These concepts also take into account indeterminate factors of diversity that are present among the various actors involved in the project. We will investigate the "actants" concept (Latour) as well as the situated action (Suchman), the sociology of translation (Latour) and the anthropology of communication (Hymes) compared to the recommended sharing values of project management.

fascinating way. Actants include human and non-human actors. The actors included in a network, as for instance in the « on-the-road communication » project, take shape by virtue of their relations with one another. Nobody and nothing can be excluded from the network of relations. Therefore, in the project, we will include at the same level of impact, influence, ability and acting: the 4x4 lorry, the 5 th screen, the movies, all the humans as technicians, local population, local and international experts, the HPAI itself, the chicken ritual, the techniques as well as the information contents, and so on. It is interesting to notice that as soon as an actor-network is engaged in the web of relations, it becomes part of « entelechy », that is a vital force leading to self-fulfilment. If we consider all the relations that could be drawn using the actants concept: the relation between the 4x4 lorry and the local population, the 4x4 lorry and the local or international organisations, the 4x4 lorry and the HPAI, the 4x4 lorry and the chicken ritual, etc. or the HPAI and the local population, the HPAI and the local and international organisations, the HPAI and the chicken ritual, etc., all these humans and non-humans are involved in the project web of relationships. Through their shape and acts they all become part of a situation that constrains them from expressing their abilities in one way or another. This idea becomes a very precious data for the communicational facilitators, whose part is to establish the relationships between all actants, and for success.

Associate humans and non-humans

The fact of associating humans and non-humans gives a specific role to what some could consider as an object. Bruno Latour has shown how Pasteur, in his text presenting his discovery of specific yeast obtained from lactical acid, gave a important role to the ferment and as if the ferment could exist outside of him (Latour). For instance, in our case, we could consider the 5 th screen or the 4x4 lorry as actors. As soon as we look at these objects in that way, a simple fact becomes an event: because the 5 th screen or the 4x4 lorry are acting as well as human actors, they reveal their roles as non-human actors. In fact when the local population or the international experts are talking about these tools they give them a role even outside them. The technicians and the drivers have called the 4x4 lorry: Ramses II because of the wrappings they had used for fixing broken parts.

The 5th screen concept

The 5th screen gathers the competences of the first 4 screens. We consider the 1st screen as the cinema screen, which is a big screen that can be watched collectively. The 2nd screen is the TV screen that can be watched by a person or a family with broadcast images. The 3rd screen is the computer screen that is mainly individual but gathers computer competencies. The 4th screen is the mobile screen of mobile telephone, which is watched individually with all the satellite connexion devices. Therefore, the 5th screen is a movie screen that can be watched collectively with all broadcast, computer and satellite mobile devices included. As well, it includes a dubbing system for translation in vernacular languages.

The 5th screen associated to the 4x4 lorry becomes a symbolic and real actor. People, who see it, use it or watch at it, can talk about it under different names, according to their beliefs. As we said before the technician call it: Ramses II. The UN communication experts call it: "the on-the-road communication". In the mean time, the audience is just fascinated by looking at the moving broadcast images in the middle of nowhere. Each time it creates "events". How a tool like this becomes a "fetish" is a phenomenon well described by Latour (Latour). In a project associating human and non-human actants, action is not only a competency from human actors, it is also a competency obtained by an association of actants (including human actors, TIC, innovation, training, partnerships). This association is a kind of composition as in the conception of the tool, as in its elaboration and as in its use.

Maintaining relations with actors from different horizons

In "Pandora's hope", Latour [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF] shows how every researcher needs others. Frédéric Joliot, as a nuclear physician, was the first to elaborate the possible production of an artificial nuclear chain reaction. From 1939, this motivation to produce this reaction came to correspond to a polytechnician's motivation: Raoul Dautry, who had also become Army Minister at a time when it was necessary to conceive new army tools for insuring national security. The concomitancy of both motivations, a scientific one and a political one, was the basis for demonstrating that "to maintain the minister and the neutrons that where engaged in the same project, to make them act together and discipline them, were not distinct tasks. Both were absolutely necessary." [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF].

Added to data and arguments, alliances allow professionals and various disciplines to share interests one to each other and motivation to act together. It allows, the acceleration of the circulation of arguments, data and discoveries. These alliances are situated where means and actors are mixed up. Coming back to our "on-theroad communication" programme, means as the 5th screen 4x4 lorry and actors as local and international experts, local leaders, technicians, national movie association artists and technicians, the village population, and so on, are all mixed up at a time or at another. It becomes interesting to put together different motivations and wishes in order to be engaged in the same project, for acting together and disciplining each other and becoming necessary one to each other.

Maintaining the goal of each actor

At the same time, it does not mean that all actors and actants should have the same goal. In Sociology of Organisations, Bernoux [START_REF] Bernoux | La sociologie des organisation[END_REF]) takes into account the fact that "men do not accept to be treated as means at the service of goals that managers give to the organisation. Each one has got his own objectives, his own goals". He considers that every actor, whatever is his responsibility level has got specific strategies which are more or less conform as well as antagonist sometimes to other leaders of the organisation. Crozier and Freidberg (1981) look at an organisation an "a human build-up or a structured human group". Organisation is in perpetual movement and its members, in the mean time, develop their own strategies and weave "regular relationships that are submitted to the moving constraints of the environment", and structure it. Through this analysis, these authors go against the Taylorian idea that aims to master all decisional elements in a known and stable model. In organisations, no decision-making comes from this model because "a human group is in a perpetual evolution, the members' strategies are changing, their behaviours are often not predictable". These elements allow us to take into account the "strategically value" that offer a communication programme and all that is produced through this means: the 5th screen 4x4 lorry and the local, national and international organisations. In that way we give back to each actor participating in the programme, his own involvement, value, motivation. For the anthropologist facilitators, this viewpoint can be very useful in the Highly Pathogenic Avian Influenza (HPAI) risk reduction on-the-road programme in order to observe, to respect and to find the way to maintain relations with actors of different horizons and to facilitate their action together.

The actants exchange and their enrichment together

According to Latour's words, in the Pasteur's story, Pasteur realised a scenario with "characters who have got a competency which define their performance". This means that "the ferment" become an actant. Pasteur has to study laboratory "proofs" in which "the ferment-actant" will show what it is able to: "its performancethe lactic fermentation". In order to know what its actant/ferment is able to do, Pasteur (as a director in a play) is going to build up "not edited and unexpected scenarios" in his laboratory for plunge his actant (the ferment, non-human actor) in the unexpected scenarios. In this way, Pasteur shows that:

1) The yeast is a living organism 2) The ferment has got a new competency because it can provoke a specific lactical fermentation 3) All this contributes to convince his pairs (as they were doubtful regarding Pasteur's discovery at the beginning).

Pasteur's discovery goes through three proofs:

• The first comes from language and the way of telling facts which transforms them into events as in fairly tails and in myth [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF];

• The second comes from non-verbal and non-linguistic components (laboratory tools as: glasses... or yeast, Pasteur himself or his assistants); • The third comes from the evidence showing that the ferment competence is due to ferment itself and not to scientific ability: Pasteur has just allowed the ferment to reveal itself.

These three proofs constitute, according to Latour, an experience: "an experience is a text about a non-textual situation [...] that authors will submit to the questioning to determinate if [….] there is a true situation behind […] and if the actors and his authors are really doted with a new competence: Pasteur has proved that the ferment is a living thing which owns the ability for creating a specific fermentation as different as the one made in the beer yeast" [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF].

Through the rhetoric, there is a mutual enrichment of human and non-humans actors.

At the ICT 21 in Valenciennes, in March 2006, where the European experts for sustainable ICT solutions gathered, the 5th screen 4x4 lorry got the "Special price of the Jury". For obtaining this prize, the 5th screen 4x4 lorry was exposed at the entrance showing the Aids prevention programme in Yemen. The developer invited people to see the technological abilities of the "on-the-road communication" tool. The audience and specifically the experts in sustainability realised the technological interest of such a tool for reaching geographically and socially isolated people. The price was obtained on the three proofs described above. The DVD presentation shown was transforming the fact into event, the technological components (specifically the good sound quality) were talking by themselves and the AIDs prevention campaign shown on the screen was revealing the impact of such a tool in countries where the collective sharing of information is part of the culture. And therefore this kind of event gathered thousands of people.

The DVD presentation has been the chosen means for maintaining relationships with the Sustainable Development experts by translating the technical and experimental abilities of the Nomadic Dre@m Machines team in the ICT 21 Sustainable Development language. For reaching that, as Latour shows it with Joliot's example [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF], the epistemological and ontological questions are carefully and intimately mixed up. This mixing produces effects, as in our case the enthusiasm, awarded by a prize, gives more emphasis to the speech. For Latour, this step of convincing other people and disciplining them is fundamental. At the beginning the experts where looking at the Nomadic Dre@m Project without paying much attention, with scepticism, in an undisciplined and detached way but composing a group without whom nothing can be done. The Highly Pathogenic Avian Influenza (HPAI) risk reduction programme has still not got the entire adhesion from the International Organisations, HPAI and Communication experts groups. A previous AIDS programme in Yemen obtained easily the UNESCO's clearances and had been able to develop for months in this part of Africa allowing not only men but also women and youngsters to reach to AIDS information, also because, for the first time, Nomadic Dre@m Machines had been able to technically show movies in daylight on a wide screen. This technical progress gives the opportunity to women and children to watch an information programme as they are constrained to go back home at sunset. Here we can admit the importance of convincing by disciplining the men (the UNESCO programme deciders) and mobilising the things (the 4x4 audio-visual lorry). The same process is used in scientific research as shown in the Pasteur's discovery.

In conclusion, throughout the event, the proof of experimentation and the process of convincing, we could say that actors are reaching a better definition of themselves and an added value is created (figure 4). We can also notice the symmetry between things produced by humans (the 5th screen 4x4 lorry, the itinerant dubbing system, the DVD presentation for ICT 21 in Valenciennes, etc.) and the way the humans exist in relation with what makes them act (when the Nomadic Dre@m Team works for including the instantaneous and itinerant dubbing system or creates the e DVD presentation for giving legibility and understanding to others as value, etc.). For the actants, the fact of going through the process of convincing gives meaning to their actions and abilities and makes them exist through their actions (figure 4). The situated action

On the other hand, Suchman (1987) founded the "situated action" on observations of interactions between human practice and technical artefact, specifically studying how humans use machines as a photocopier or deal with a constraint like using a canoe. From her hypothesis that these observations are interesting and founded [START_REF] Relieu | Introduction au numéro spécial Activité et Action/Cognition Située, @ctivités[END_REF], she highlights:

• The importance of taking into account the material, social and cultural context where contextual factors and mechanisms producing mutual intelligibility between actants have got a fundamental part; • The important part of social interaction as a privileged coordination support for actions with objects (without taking too much into account the spatial organisation); • The fact of taking into account context elements and action characteristics (with possible errors of categorisation); • Observation as a methodological tool (with video use for example) and how "the language is intertwined in socially organised activities" implying notions of "internal representations" and "hidden social structures"; • The actors' commentary gathering which is considered as a culturally interesting production even though it does not enlighten the situated organisation of activities.

This methodology gives a fine-grained analysis of the situated action without providing systematic recommendations or prescriptions for conceiving new tools. Now, the question is how to transfer the observation results and the fine-grained analysis of the situated action in order to allow actants, in an "arid" context:

• To achieve their respective goals even though there are distinct;

• To acquire an added value;

• To access a communicative value through a situated action.

The setting of an international program is based on the hypothesis that the program should get sustainable advantages for populations and partners. The fact of having this way of observing, thinking and acting promulgated by "situated action" concept and method, through video registration of interactions between actants (to get, at least, audiovisual documentation on "what has happened" and how actants have acted before, during and after the event) produces four levels of informational results:

• Feedback for technical and human relationships improvement after analyzing (first level);

• Valorization for convincing and proving to experts, founders and deciders (second level);

• New material that can be used for informing in other places, other populations, through new events (third level); • Audio-visual material for anthropological research or other disciplines (fourth level).

The first NDM experimentation regarding information programme for socially isolated people has given a good basis and material to build up the Pathogenic Avian Influenza (HPAI) risk reduction programme. Gathering a team's abilities, field knowledge and observations, the developer made a first NDM prototype in 2003. This machine, called Infobooth 1, made for United Nations (and financed by DFID) was officially at work in June 2004 before leading a programme on preventive health and on Aids risks all over Yemen. Many videos were taken during this campaign and showing the success of the Yemenite experience. After few modifications, the Infobooth1 is still working for the Health Ministry of Yemen. Even though the Infobooth1 represented a qualitative jump regarding the traditional means for informational campaign, the team had to reflect on the vehicle's bad road abilities. The first level: feedback for technical and human relationships improvement after analysing is reached. The second level can be illustrated as follows: the enormous crowd attracted by the Infobooth1 as shown on videos give a valorisation for convincing and proving to experts, founders and deciders the impact of such a tool.

At the third level, the Yemenite campaign has been the basis to create NDM's added value thanks to the situated action it represents. From this experience, new material as the web sites (in French, English and Arabic), have been created. Many reports have been written in order to sustain different other projects as HPAI risk reduction programme that had to start in Ghana. The 5 th screen attended the 2007 International Aid and Trade Exhibition in Geneva (that gathered aid agencies, UN Agencies, Government offices, international development banks) and created an event.

From the Yemenite campaign, no audiovisual material has been directly made and used for anthropological research or other disciplines (fourth level), following the L. Suchman's idea that has founded the "situated action" on observations of interactions between human practice and technical artefacts, specifically studying how humans use machines. This step could have been done in the HPAI risk reduction programme. Getting this rigorous audiovisual material could give an interesting source of information for NDM project and provide a scientific basis for the 5 th screen use adapted to socially and geographically isolated people.

The translation steps

Among the links that will allow a group of actants and actions to exist in a situation and to attract the valuable recognition of their actions, the links created with other groups of actants increase understanding outside their own group. These links should be privileged because they help the group of actants to find the way of being understandable outside their own group. In a project management as an international communication programme, the ones that think that actants and actions are recognised only because there are producing contents and internal actions, even though there are "innovative", seem to go nowhere. The other reasoning that consists of thinking that, whatever the programme or the technology are effectively doing, the most important is to negotiate with the various social groups that are involved and to reach a wide extension in the programme, do not go further.

In reality, we could establish that both are necessary at the same time. For being able to talk outside about an ICT programme such as the Pathogenic Avian Influenza (HPAI) risk reduction programme and about its actions as well as reflecting on what it is going on between all the involved partners, it is necessary to go through translation steps and interpretations. The NDM Project Manager in charge of HPAI risk reduction programme had to gather the NDM team's motivation and agreement around this project by translating the different partners' goals and procedures (in that case: United Nations and European organisations). On the other side, she had to go to different meetings and exchanges and to write down various reports finding the words to be understood by all the different United Nations and European organisations partners. According to Latour, the chain of translations has to be done in both ways: esotherical and exoterical and vice versa [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF].

This translation of the situation as an experience does not make the fact more visible but, may be more notable, translating it in an event. It gets a peculiar character in our demonstration because it is going to underline each time that an actant does not get the impression of acting but more to be surprised by what he is doing. As Latour underlines, action is not any more a question of mastering or construction but of "crossings, events, and circumstances". Actants are continuously asking themselves what they are doing (concrete measurable actions), how they are doing (competences, performances that they are developing action after action) and why they are acting (the meaning and the cohesion that they give to their actions as well as the acquired performances). They also give specific attention to the credibility they have regarding themselves through the common references they are using inside the programme, which represent the esoteric translation. On the other side, actants are given a specific attention to the use of "universally know references" in order to be recognised outside. This is considered as the exoterically translation. The earning is the external recognition.

As described in a previous paragraph, the way of telling facts and the questioning from others produces a necessary actants exchange and mutual enrichment. Throughout this rhetorical process, a mutual enrichment of the situation is obtained (that it is transforming in an event and from which actants are earning expertise through the transformation that happened by translating and by questioning the situation). This is the way it becomes a notable event.

Latour defines the translation in this way: "Instead of opposing words to the world, the anthropology of science, by its insistence on practice, has multiplied intermediaries who attract attention on characteristic transformations of sciences... In its linguistically and material connotations, the anthropologist goes back to all the steps and displacements done by other actants who absolutely need the translation for acting. Instead of opposing context and content, the translation steps go back to the work through which the actants modify, displace or translate their various and contradictory interest". [START_REF] Latour | Pandora's Hope: Essays on reality of Science Studies[END_REF].

It is interesting to see that in the HPAI risk reduction programme the chain of translation has not been properly done. Few members of the NDM team have not been convinced enough to concentrate on actions and to convince other group of actants. On the other side, more or less at the same time, NDM developers attended too many worldwide meetings even though they were very much involved in giving a wide extension of the 5 th screen use.

Shared values or anthropology of communication?

In its article "La cartographie des incidences: la théorie", The International Development Research Center (Canada) which has studied the development of international program based on Sustainable Development quotes: « les ententes et les partenariats fondés sur des valeurs communes ont été remplacés par une confiance bureaucratique fondée sur des plans, des budgets et des comptes » [START_REF] Roche | Impact Assessment for Development Agencies[END_REF]. They criticize bureaucracy in the sense that they admit it influences the choice of partners and the way relationships are developed with them as well as the conception of the chosen programmes. In that sense, they are contesting the fact that well-written planning documents and well-established reports increase the quality of development as well as managers' influence on results. Therefore, they encourage vision and shared values, involvement, taking risks, innovation and looking for new partners instead of bureaucratic instruments arguing that these instruments are lowering enthusiasm, provoking a loss of goals and avoiding sustainable results.

On our side, we are not discussing the "bad" effects of bureaucracy but we are questioning the fact that we need sharing values and visions. In our example on Pathogenic Avian Influenza (HPAI) risk reduction programme, we are wondering how the international experts that want to eradicate HPAI from earth can share the same vision and values with the Nomadic Dre@m Machines technical team as well as the local people who are killing poultry for ritual and drinking the infected blood. It might be that there are common values or a shared aimed result. But what kind of social and reliable enquiry would be necessary to know if this is really shared?

If we suggest the Hymes' approach [START_REF] Hymes | Vers la compétence de communication -traduction F. Mugler[END_REF]) that works on the talk ethnography it is because it is based on the fact that language is action, studying its use and variability inside any kind of collectives. He observes and analyses the relationships between speech use, speech as an action and social structures. The language use as well as the communication modes as gesture, mimic, non-verbal expression... "all that we know or do from a language take a place in the spacious whole of knowing and abilities that take parts in all kind of communication" [START_REF] Hymes | Vers la compétence de communication -traduction F. Mugler[END_REF]. From our point of view, to observe and take into account these communication modes that are created between the various partners in the Pathogenic Avian Influenza (HPAI) risk reduction programme, as creating each time a new speech action that can be audio-visually registered and valued could be the way of linking the culturally different groups acting together in respect with their different goals for reaching the HPAI risk reduction.

In conclusion

Information campaign could become accessible and sustainable, for socially and geographically isolated people, considering that:

• Human and non-human (as the ITC tool as the 5th screen) are intimately intertwined;

• Each actor from various horizons can maintain his own goal as the relationships are maintained;

• The enrichment of the actants altogether occurs in the exchange and the construction of scenarios;

• There is added communicational value given by the situated action;

• The esoterically and exoterically translation steps are necessary;

• The communicative non-verbal modes are the added communicative value created by group acting together.

We have focalised more on advantages and limits of such considerations for HPAI risk reduction campaign in Ghana at an informational level of accessibility as conditions for sustainability. Another main objective could be to focalised more on the impact of combining movies entertainment with informational campaigns and to study the implementation of combined campaigns in « arid » contexts where it is necessary to improve disaster mitigation.

Figure 4 :

 4 Figure 4: Troughout the event, the proof of experimentation and the process of convincing, an added value is created


Nathanael Beeker
email: nathanael.beeker-adda@edf.fr

Paul Malisani
email: paul.malisani@edf.fr

Nicolas Petit
email: nicolas.petit@mines-paristech.fr

Modeling populations of electric

Keywords: Water heating, Electric Hot Water Tank, Smart piloting, Fokker-Planck equation

published or not. The documents may come

INTRODUCTION

The increasing share of intermittent renewable electricity sources in the energy mix (European Commission [2011], [START_REF] Edenhofer | IPCC special report on renewable energy sources and climate change mitigation[END_REF]) raises new difficulties in management of the electricity production and equilibrium in distribution networks. Demand Side Management (DSM), which is a portfolio of smart piloting techniques aiming at modifying consumers' demand, is a promising solution for such concerns [START_REF] Palensky | Demand side management: Demand response, intelligent energy systems, and smart loads[END_REF]). A key factor in developing DSM is the ability to find energy storage capacities. In this context, the large pools of electric hot water tanks (EHWT) have a well recognized potential.

An EHWT is a domestic electric appliance which heats a volume of water with an electric thermo-plunger that can be controlled. The home user drains hot water from the EHWT at various times of the day. The literature [START_REF] Blandin | Modélisation et validation expérimentale de nouveaux concepts de ballons solaires à forte stratification[END_REF], [START_REF] Zurigat | Influence of inlet geometry on mixing in thermocline thermal energy storage[END_REF] and the references therein) models EHWT as vertical cylindrical tanks driven by thermo-hydraulic phenomena: heat diffusion, buoyancy effects and induced convection and mixing, forced convection induced by draining and associated mixing, and heat loss at the walls.

To model EHWT, one-dimensional distributed parameter models of the temperature profile in the tank have been developed (see Beeker et al. [2015a,b]). The observed temperature profiles are increasing with height. For smart piloting applications, one can define three variables of interest providing a simplified representation of the temperature profile, under the form of three amounts of energy. From this, advanced control designs can be studied. Among them, optimal control strategies are particularly appealing as they address topics of direct interest both for endusers and electricity producers: cost reduction, comfort constraints, and yield management. Yet, the situation is not that straightforward. In truth, a real challenge lies in solving optimal control problems for large numbers of such EHWT. Coordinated individual control of each tank is feasible for pools of moderate sizes (typically from 2 to 10 EHWT). However, for large pools of EHWT (tens of thousands), this approach is a stalemate. Unfortunately, the real stakes and industrial expectations belong to this range.

Interestingly, it is possible to recast this problem into a distributed parameters approach. This is the path we explore in this article. Following the works of Malhamé and Chong [1985], we consider that the local (individual) control variables of EHWT are each defined according to stochastic processes. Then, we combine i) this randomness, ii) the diversity in the distribution of the states of the EHWT, iii) the randomness of the water consumptions, and we develop a partial differential equation (PDE) for a large pool of EHWT. This takes the form of Fokker-Planck equations (see [START_REF] Risken | The Fokker-Planck Equation[END_REF]) governing the probability distributions of the population of EHWT. The work of Malhamé and Chong was originally focused on a mitigated load represented by a single state, which we need to extend for the smart piloting applications under consideration here. This extension results into a rich system of PDE, which constitutes the main contribution of this paper.

The paper is organized as follows. In the pool, a single EHTW is a macroscopic but small subsystem described by state variables. These variables of interest are defined in Section 2. They have hybrid dynamics by construction. To account for the randomness of water consumption, we propose a single EHWT model as a Markovian stochastic process in Section 3. Then, we introduce probability density function of the population of EHWT and derive the Fokker-Planck equations in Section 4. A summary of the obtained input-output description of the EHWT pool insulation thermostat heating elements hot water cold water is reported in Section 5. Conclusions and perspectives are given in Section 6.

VARIABLES OF INTEREST IN A EHWT

General description, stratification, and effects of heating

A typical EHWT is a vertical cylindrical tank filled with water. A heating element is plunged at the bottom end of the tank (see Fig. 1). The heating element is poleshaped, and relatively lengthy, up to one third of the tank. Cold water is injected at the bottom while hot water is drained from the top at exactly the same flowrate (under the assumption of pressure equilibrium in the water distribution system). In the tank, layers of water with various temperature coexist (see Fig. 2). At rest, these layers are mixed only by heat diffusion which effects are relatively slow compared to the other phenomena [START_REF] Han | Thermal stratification within the water tank[END_REF]). Existence of a non uniform (increasing with height) quasi-equilibrium temperature profile in the tank is called stratification [START_REF] Dincer | Thermal Energy Storage: Systems and Applications[END_REF], [START_REF] Lavan | Electric Load Model Synthesis by Diffusion Approximation of a High-Order Hybrid-State Stochastic System[END_REF]). In practice, this effect is beneficial for the user as hot water available for consumption is naturally stored near the outlet of the EHWT, while the rest of the tank is heated (see Fig. 2(b)). Due to this effect and the cylindrical symmetry of the system, one can assume that the water temperature in the tank is homogeneous at each height.

Following the description above, the temperature T of the tank is a continuously increasing function of height (see Fig. 2). The constant inlet temperature T in constitutes a lower bound of the temperature profile. The heating process is driven by turbulence generated by buoyancy effects, which is the cause of a local mixing in the bottom of the tank. We consider that this mixing is perfect on a spatial zone, refereed to as the plateau (see Beeker et al. [2015a,b]), and does not affect the temperature profile in the upper part of the tank (see Fig. 2(b)). During the heating process, the plateau grows and gradually covers the whole tank. The user specifies a temperature T max at which the heating has to be stopped to prevent overheating. As a result of the heating process, if the temperature at the bottom of the tank is T max , then the temperature in the tank is uniformly at T max once the heating is finished.

The user can also specify a comfort temperature T com . Water having temperature higher than T com can be blended with cold water to reach T com and is therefore useful, while water having temperature lower than T com is useless.

Consumption, control and objectives

Each EHWT has two inputs: water consumption and heating power. The user consumes certain quantities of energy each day. For this reason, consumption of hot water is an (uncontrolled) input of our problem. On the other hand, the heat injected via the heating element in the tank is a control variable.

The control design can have various objectives. The most obvious one is individual cost reduction for each single unit in response to a price signal. At larger scales, one can naturally consider a pool of tanks, and aim at reaching a load profile for the aggregate consumption.

Variables of interest: Available, delay and reserve energies

Describing the exact temperature profile inside the tank is unnecessary for the applications discussed above. Instead, a few (3) variables of interest can be considered.

The available energy a is defined as the energy contained in the zones having temperature greater than the comfort temperature T com . This constitutes a direct comfort index for the user. If a reaches 0 and a water drain is applied, then the comfort constraints is violated.

The delay energy τ is defined as the energy required by the plateau to reach the temperature T com . When the tank is heated at constant maximum power, in the absence of drains and heat losses, τ is simply proportional to the time necessary to reach a state from which a can effectively be positively impacted by the heating process.

The reserve energy µ is defined as the energy contained in the tank that is currently unavailable for consumption, i.e. the energy contained in the water under T com . When, thanks to the heating process, τ reaches the value 0, the energy µ becomes available for consumption: this generates an immediate (discontinuous) increase of a, and µ is reset to 0.1 Fig. 2 illustrates the dynamics of these variables. A drain (pictured in (a)) is mainly characterized by a decrease of a and an increase of τ , with a slight raise of µ due to an energy transfer from a. On the other hand, in the heating pictured in (b), τ decreases at the same rate as µ rises, until the plateau reach T com . An energy transfer from µ to a takes place then. This is pictured in (c). A case where comfort constraint is violated is pictured in (d). Tmax-Tinj and let m be the maximal energy that can be contained in the tank under the temperature T max . Then, by definition, a, τ, µ are subject to the following inequalities :

0 ≤ a ≤ m, 0 ≤ τ ≤ λm, 0 ≤ µ ≤ λm, a + 1 λ (τ + µ) ≤ m, λm ≤ a + τ + µ (1)
from which we define the following open polyhedron of R 3 and its faces (see Fig. 3):

Ω 0 ={(a, τ, µ)|a, τ, µ > 0, λm < a + τ + µ and a + 1 λ (τ + µ) < m} F 1 = Ω0 ∩ {(a, τ, µ)|µ = 0} F 2 = Ω0 ∩ {(a, τ, µ)|τ = 0} F 3 = Ω0 ∩ {(a, τ, µ)|λm = a + τ + µ} F 4 = Ω0 ∩ {(a, τ, µ)|a + 1 λ (τ + µ) m}.
(2)

The following edges and vertices are considered

E 1 =F 1 ∩ F 2 E 2 =F 3 ∩ F 4 V 1 ={(λm, 0, 0)} V 2 ={(m, 0, 0)} V 3 ={(0, λm, 0)} V 4 ={(0, 0, λm)}.
(3)

In practice, x (a, τ, µ) can only belong to E 2 (low energy, e.g. Fig. 2 (d)), Ω 0 (medium energy, e.g. Fig. 2 (a) and (b)) and E 1 (high energy, e.g. Fig. 2 (c)). Faces F 1 to F 4 , and vertices V 1 to V 4 constitute boundaries of these three domains. Note that uniformly cold tank can also stack in V 3 . In the sequence, we note Ω = Ω 0 ∪ E 1 ∪ E 2 ∪ V 3 . Any index i = 0, 1, 2, 3 will refer to these domains, respectively.

Transient behavior

Heating mostly induces a continuous variation of x.

If τ, µ > 0 (i.e. x ∈ Ω 0 ), then the heating yields a decrease of τ and an increase of µ. Under certain conditions, a threshold effect can be observed: when τ reaches 0 (i.e. when x reaches F 2 ), then suddenly, all the unavailable energy µ becomes available, µ takes the value 0 and all its energy is transferred into a. This induces a discontinuity, transferring x to E 1 in which the heating has again a continous effect on x, increasing the available energy a.

Heat losses also mostly induce a continuous variation of x, where a and µ decrease while τ increases. The reverse threshold effect can also be observed: when x reaches F 3 , an entire layer of water goes under the temperature T com , which causes a to take the value 0 and x to jump in E 2 in which it will again vary continuously. The drains have a very short duration compared to heating and heat loss. They yield quasi-instantaneous decrease of a and µ and increase of τ , having time of occurrence and magnitude depending on the user's habits. We model them as jumps only. Very large decreases can cause x to jump from E 1 to Ω 0 (or even to E 2 ), or from Ω 0 to E 2 .

λm V 1 d τ µ m λm λm 0 F 3 Ω F 2 E 1 F 1 F 4 E 2 V 3 V 4 V 2
The transients are pictured in Fig. 2. A summary of jump dynamics is pictured in Fig. 4.

EHWT AS A HYBRID-STATE STOCHASTIC PROCESS

Stochastic process representation

The times of occurence and magnitude of the drains are user-dependent. We model this dependency under the form of a stochastic process. This results in a model similar in spirit to the single-dimensional representation originally proposed by Malhamé and Chong [1985].

In our model, each EHWT is defined by the state vector

X t = a t τ t µ t ∈ Ω (4)
and its heating status S t ∈ {r, h}, which is also of stochastic nature. From now-on, the indexes {r, h} refer to "rest" and "heating", respectively.

The dynamics of X t is governed by the phenomena described in Section 2.5. In each domain Ω 0 , E 1 , E 2 , the state vector X t continuously (due to heating and heat loss) and discontinuously (due to drains) changes, with respect to a stochastic differential equation, constituting the flow map, that will be stated below. A jump of hybrid nature appears when

X t reaches F 2 , F 3 , V 1 or V 4 .
This discontinuity, according to the terminology of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], constitutes the jump map. A correspondence between phenomena and stochastic/hybrid representation can be found in Table 1.

Flow map: Stochastic process dynamics on each domain

The stochastic differential equations of X t and S t are

dX t = v(X t )dt + dJ t + σ(X t )dW t dS t = dN t (5)
where

• v(X t )dt is the drift component which represents the heat losses and/or heating effects. • dJ t is the infinitesimal integration with respect to a 3dimensional compound Poisson process J t representing the jump effects of drains on a t , τ t and µ t . • Uncertainties are lumped into a variance term σ(X t ), integrated with respect to a Wiener process (or standard Brownian motion) W t . • dN t is the infinitesimal integration with respect to a 1-dimensional Poisson process N t representing the status switch between h and r.

Expressions of v, σ and of the random characteristics of J t depend on the domain and the status of the EHWT. We now detail them.

Heat loss and heating modeling as a drift

Under the assumption that the ambient temperature T a is equal to T inj , the heat loss per unit of time for a t is equal to -k Sρcp a t , where S is the cross-section of the tank, ρ and c p are the density and heat capacity of water, and k is the heat loss coefficient of the tank per unit of height (see Fig. 2).

Likewise, for µ t , the heat loss is equal to -k Sρcp µ t . This heat loss generates a positive effect on τ t which therefore increases with the rate k Sρcp µ t . In summary,

v r (X t ) = k Sρc p -1 0 0 0 -1 0 0 1 0 X t . (6)
To this heat loss drift is added a drift v h (X t ) due to power injection for tanks subject to heating. In Ω 0 and E 2 , power injection p lowers τ t and increases µ t , so that one has v h (X t ) = p [0 -1 1] T . In E 1 , the injected power only affects a t , and therefore v h (X t ) = [p 0 0] T . Finally, when the tank is heating

v(X t ) = v r (X t ) + v h (X t ) (7)
and otherwise v(X t ) = v r (X t ).

Drain as a Poisson process

The drains appear as a sequence of quasi-instantaneous events of various magnitudes, so that we choose to model them as a compound Poisson process J t (see e.g. [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF]).

Then, the time between jumps follows an exponential law of parameter θ(t), and the magnitude of jumps is characterized by a probability density function ω : Ω 2 × R + → R + . In words, a jump from y at time t maps to Ω with distribution characterized by ω(y, •, t).

EHWT variability as a Wiener process

Some phenomenon are not taken into account in the description above (e.g. diffusion). We choose to lump these into into a variance parameter σ(X t ) acting through a Wiener process. However, the closer X t is to the boundary of Ω, the smaller the uncertainty should be on some of variables. Thus, certain components of σ(X t ) vanish at the boundaries of domains. Additionally, the heating increases uncertainty on the dynamics. Therefore, the variance takes the form σ(X t ) = σ r (X t ) + σ h (X t ) (8) for heating tanks, and σ(X t ) = σ r (X t ) otherwise.

On/off heating switch modeled with a Poisson process

The switching between the two statuses h and r constitute a sequence of instantaneous events that can be controlled. We choose to also model it with a Poisson process of intensity α(X t , t, S t ) = α St (X t , t) (indexed on S t for transition from S t to the opposite one). This means that instead of exactly setting the switching times, two functions α r , α h define a probability to switch from one status to another, depending on the state X t and time t.

Jump map: Hybrid modeling of the domain switch

The threshold jump when heating or with heat losses constitute hybrid deterministic jumps. When reaching a certain boundary, it maps a domain to another, depending on the status. The transition, in the framework of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], gives x + (the value after the jump) as a function of x (the value before the jump). For the sake of simplicity, a summary is given in Table 2. The maximal energy m that can be contained in the tank is attained at point V 2 , when heating. We assume that the heating automatically switches off at this point for security reasons, which is characterized by a hybrid jump for the status from h to r. This jump is also presented in the jump map.

FOKKER-PLANCK PDE FOR A LARGE POOL OF EHWT

EHWT pool population representation

Representing a large pool of tanks each having a 3dimensional state leads to an unnecessarily large finitedimensional system, which can be difficult to design controllers for. Rather, a probability density functions representation can be employed.

The main idea is to define seven function

f r 0 , f h 0 , f r 1 , f h 1 , f r 2 , f h 2 , f r
3 (for each domain Ω 0 , E 1 , E 2 and V 3 , one for each status r or h at the exception of V 3 in which only resting tanks can stack) which represent the population density of the tanks of a given status in a certain domain. These positive functions are subject to the balance consistency

Ω0 (f h 0 +f r 0 )+ E1 (f h 1 +f r 1 )+ E2 (f h 2 +f r 2 )+f r 3 = 1. (9)
The dynamics governing these probability functions are obtained from the preceding dynamics. We now detail them.

Fokker-Planck equation for a stochastic process

A useful tool for the population distribution computation is the Fokker-Planck equation (see [START_REF] Risken | The Fokker-Planck Equation[END_REF]). For a set of independent Markov process in a state space Ω following the same generic stochastic equation

dX t = v(X t , t)dt + σ(X t , t)dW t + dJ t (10)
where v(X t , t), σ(X t , t) ∈ R 6 , dW t is the integration with respect to a 1-dimensional Wiener process W t , and dJ t is the integration with respect to a compound Poisson process of intensity θ(t) and whose compound distribution is represented with the probability density function ω (i.e. when a jump occurs on state x, the probability density function of transition to state y is represented with ω(x, y, t) at time t), the probability density function (when the number of stochastic process tends to infinity) is given by the Fokker-Planck equation (see e.g. [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF])

∂ t f (x, t) = -∇ x • [v(x, t)f (x, t)] + ∇ x • [D(x, t)∇ x f (x, t)] + θ(t) Ω (f (y, t) -f (x, t))ω(x, y, t)dy (11) for (x, t) ∈ Ω × R + , where D(x, t) = 1 2 σ(x, t) • σ t (x, t) ∈ R 3×3 . ( 12
)

Detailed expression of the PDE

Several observations can be made on the stochastic model presented in Section 3. First, for each tank, the stochastic process X t defined by (5) constitutes a Markov process. Moreover, given a pool of tanks, the independence of the stochastic process of each tank appears as a reasonable hypothesis, given the assumption that hot water consumptions of distinct households are not related.

Therefore, one can directly follow the work of Malhamé and Chong [1985], and derive the Fokker-Planck equations.

In our case, for each domain and each status, this equation takes the form of a parabolic PDE. The hybrid character of the stochastic process appears in the boundary conditions and yields an additional integral source term.

On each domain i = 0, 1, 2 (for Ω 0 , E 1 , and E 2 ), f r i and f h i are driven by a system of the form

∂ t f r i + ∇ x • [v r f r i ] = ∇ x • [D r i ∇ x f r i ] -(α r i + θ)f r i + α h i • f h i + S r i (f (•, t), x, t) ∂ t f h i + ∇ x • [(v r + v h i )f h i ] = ∇ x • [(D r i + D h i )∇ x f h i ] -(α h i + θ)f h i + α r i • f r i + S h i (f (•, t), x, t) (13)
while f r 3 follows an ordinary differential equation that will be stated later.

Parameters specification on each domain

Each term of the PDE derives from the infinitesimal generator defining the parabolic operator used in the Fokker-Planck equation. Each term in the stochastic differential equation has a matching term in the partial differential equation (see e.g. [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF], [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]). For the sake of clarity, we present in Table 1 each stochastic term in our case and its corresponding term in the PDE. The necessary steps of computations are omitted for brevity.

The heat loss drift in the PDE has the same form as the one in the stochastic equation, i.e.

v r (x) = k Sρc p -1 0 0 0 -1 0 0 1 0 x ∈ R 3 . ( 14
)
The drift

v h i ∈ R 3 , the diffusion D r i (x, t), D h i (x, t) ∈ R 3×3
+ , and the source terms S s i (f (•, t), x, t) ∈ R + for s = r, h have to be defined in each domain. The term caused by the Poisson process leads to source terms of various integral forms, depending on the probability density function ω i on each domain (e.g. ω 0 = ω |Ω0 ). Source terms can also appear due to hybrid transfer from other domains in the form of an integral flow. For that purpose, on F 2 and F 3 , we introduce the functions η 2 and η 3 s.t.:

η 2 (y, x) = 1 if [a x τ x µ x ]
T = [a y + µ y τ y 0] T and 0 otherwise; and

η 3 (y, x) = 1 if [a x τ x µ x ] T = [0τ y a y + µ y ]
T and 0 otherwise.

Details are reported in Table 3. Finally, the exchange terms α s i (x, t) can be chosen as they are control-dependent, while θ(t) does not depend on space.

Boundary conditions

The domain Ω 0 has 4 boundaries (F 1 to F 4 ), while E 1 and E 2 have 2 boundaries, each in the form of vertices. Boundary conditions stems from the behavior of the stochastic process. A special case is the boundary V 3 where uniformly cold tanks stack.

On the borders, we have :

• d τ (x), d µ (x) → 0 when x → F 1 or F 2 , • d a (x) → 0 when x → F 3 or F 4 , • d 1 (x) → 0 when x → V 1 or V 2 , • d 2 (x) → 0 when x → V 3 or V 4 .
This allows to define boundary conditions of the Dirichlet or free boundary types, except in V 3 . Their definitions stem from exchange between the domains: free boundary corresponds to the case where tanks flowing outside the domain flow inside another domain as a source term (hybrid jumps). On the contrary, the zero Dirichlet boundary conditions correspond to the fact that no new tank can enter the system (the population is fixed). A summary is presented in Table 4.

The vertex V 3 constitutes a free boundary for f r 2 , in which the population of completely cold tanks can stack. The population f r 3 (t) at this point can be heated and constitutes the input flow for f h 2 through the boundary condition

(v h 2 + v τ 2 (V 3 ))f h 2 (V 3 , t) = α r 2 (V 3 , t)f r 3 (t). There- fore, f r 3 (t) is driven by ḟ r 3 (t) = -α r 2 (V 3 , t)f r 3 (t) + v τ 2 (V 3 )f r 2 (V 3 , t) + E2 f r 2 (y)ω 2 (y, V 3 , t)dy (15)

INPUT-OUTPUT DESCRIPTION

The input of the system is the set of functions (over space and time

) α = [α r 0 , α h 0 , α r 1 , α h 1 , α r 2 , α h 2 ]
T , which determine the intra-domain migration between the populations of heating and resting tanks.

To control the pool of tanks, several indicators can be interesting. They are the output of the proposed model. Among them, the mass (number) of tanks breaking the comfort constraints is

CB(t) = E2 (f h 2 + f r 2 )(x, t)dx + f r 3 (t) (16)
and the total power demand is

P W (t) = 2 i=0 Ω f h i (x, t)dx (17)
They constitute valuable performance indexes for the system. The system can be seen with the input/output representation depicted in Fig. 6.

A natural goal is then to design controls (α(•, t) : Ω → R 6 + ) s.t. CB is as low as possible, while the total power demand P W follows a given objective function.

As a mean of illustration, the probability density functions on E 1 , E 2 , and a representative segment of Ω 0 from the middle of E 2 to V 2 are shown in Fig. 5. Two profiles are shown. A fictional initial one, and the one subsequent to the following heating policy. We choose to promote heating (i.e. α r i high and α h i low) on E 1 , E 2 , and V 3 , and let the tanks rest (i.e. α r i low and α h i high) on Ω 0 . The profile varies as is shown in the figure, and tends to spread due to diffusion and integral drains. After some time, due to diffusion effects, a stationary profile representative of the cycle E 2 → E 1 → Ω 0 → E 2 should take place.

CONCLUSION AND PERSPECTIVES

In this article we have explained the derivation of a model for a large pool of EHWT. The input is a parameter defining the stochastic process of heating of each indidividual EHWT in the pool. The outputs are the overall comfort variable defined in (16) and the total power demand (17). The dynamics are a collection of Fokker-Planck partial differential equations.

V 3 V 4 f 2 V 3 V 4 f 2 V 1 V 2 f 1 middle of E 2 V 2 f 0 V 1 V 2 f 1 middle of V 2 V 2 f 0 to E1 from E2 to Ω0 to E2 from V3 to V3 f 3 : high f 3 : low at t 0 at t 1 > t 0 from E1 from Ω0
Fig. 6. Evolution of probability density function, given a heating policy (red: heating, blue: resting)

The next steps should address the control problems based on this input-output description. A question to be solved can be formulated as follows: how to design α(•, t) so that the power demand matches some desirable history while limiting or minimizing the discomfort? This problem belongs to the class of optimal control (tracking) of distributed parameters systems with in-domain actuation and is the topic of current investigations. F 1 f r 0 (x, t) = 0 and f h 0 (x, t) = 0 F 2 f r 0 (x, t) = 0 and f h 0 (x, t) free F 3 f r 0 (x, t) free and f h 0 (x, t) free F 4 f r 0 (x, t) = 0 and f h 0 (x, t) = 0 E 1 V 1 f r 1 (x, t) free and f h 1 (x, t) = f h 2 (V 4 , t) V 2 f r 1 (x, t) = f h 1 (V 2 , t) and f h 1 (x, t) free E 2 V 3 f r 2 (x, t) free and (v h 2 + v r (x))f h 2 (x), t) = α r 2 (x), t)f r 3 (t) V 4 f r 2 (x, t) = 0 and f h 2 (x, t) free

Fig. 1 .

 1 Fig. 1. Schematic view of an electric hot water tank.

Fig. 2 .

 2 Fig. 2. Temperature profile, and corresponding states in various situations 2.4 Domains of definition of state variables

Fig. 3 .

 3 Fig. 3. Domains of definition of the state variables

Fig. 5 .

 5 Fig. 5. Input/output representation of the pool of EHWT.

Table 1 .

 1 Correspondance between: phenomenon, hybrid stochastic counterpart, and PDE term

	Phenomenon			Stochastic representation		Fokker-Planck equation
	Heat loss				Drift					Convection
	Heating				Drift					Convection
	Drain			Poisson process				Integral source term
	Threshold effect on the comfort		Hybrid jump		Free boundary condition, and source term
	Control			Poisson process				Coupling source term
	Uncertainty			Brownian motion					Diffusion
	In			Ω 0						E 1			E 2
	If reaches	F 2 × {h}	F 3 × {r, h}		V 1 × {r}	V 2 × {h}	V 4 × {h}
	Jump to	E 1 × {h}	E 2 × {r, h}		E 2 × {r}	V 2 × {r}		E 1 × {h}
		a +		a + µ	a +		0	a +		0		a +		a + µ
	According to the transition	τ +	=	τ	τ +	=	τ	τ +	=	τ	s + = r	τ +	=	τ
		µ +		0	µ +	a + µ	µ +		a + µ		µ +		0

Table 2 .

 2 Jump map

	Domain Drift v h
	Ω 0	p	0 -1	da(x) 0	0 dτ (x)	0 0	S r 0 = θ(t)	Ω 0	f r 0 (y, t)ω 0 (y, x, t)dy + θ(t) E 1	f r 1 (y, t)ω 1 (y, x, t)dy
			1 1	0		0 1 0 0 dµ(x)	S h 0 = θ(t)	Ω 0	f h 0 (y, t)ω 0 (y, x, t))dy + θ(t) E 1 S r 1 = 0	f h 1 (y, t)ω 1 (y, x, t))dy
	E 1	p	0	d 1 (x)	0 0 0
			0			0 0 0			S h 1 =	F 2	η 2 (y, x)(v 0 (y) + v h 0 )f h 0 (y, t)dy
	E 2	p	0 -1 1	1 2 d 2 (x)	0 0 0 0 1 -1 0 1 1	S r 2 = θ(t) S h 2 = θ(t)	Ω 0 Ω 0 f h f r 0 (y)ω 0 (y, x, t)dy + θ(t) E 2 η 3 (y, x)v 0 (y)f h f r 2 (y)ω 2 (y, x, t)dy 0 (y, t)dy 0 (y)ω 0 (y, x, t)dy + F 3 +θ(t) E 2 f h 2 (y)ω 2 (y, x, t)dy

i Diffusion D r i (x) and D h i (x) Source terms S r i (f (•), x, t) and S h i (f (•), x, t)

Table 3 .

 3 Definition if distributed parameters equation term on each domain

	Domain Boundary	Boundary condition
	Ω 0

Table 4 .

 4 Boundary conditions

The rationale behind these definitions is that to plan the heating, we account for the time left before the energy reserve embodied by a (in the total energy a + µ) is consumed, and the time necessary to provide new hot water, embodied by τ .


Nathanael Beeker
email: nathanael.beeker-adda@edf.fr

Paul Malisani
email: paul.malisani@edf.fr

Nicolas Petit
email: nicolas.petit@mines-paristech.fr

An optimization algorithm for load-shifting of large sets of electric hot water tanks

Keywords: Energy storage, Electric water heating, Load shifting, Smart operation, Optimal scheduling, Load curve

Large sets of Electric Hot Water Tanks (EHWT) found in numerous countries appear as very relevant for load-shifting applications. An EHWT heats water over relatively long periods of time, for later consumption. The simple time-of-use pricing strategy applied to most consumers increases the overall electricity consumption in the beginning of the night, while hot water is used much later, in the next daytime. With the development of home automation, more advanced strategies could enable improved cost-reductions for both users and the electricity producer. An efficient and robust improvement is to reschedule each EHWT heating, while keeping the heating periods undivided (to minimize the malicious impacts of complex dynamic effects taking place inside each EHWT), so that an objective load curve for the whole set of EHWT is reached. The first contribution of this article is to formulate the rescheduling of EHWT heating with comfort constraints as an optimization problem. The second contribution is a heuristic to solve this problem with less than 1% optimality loss for representative test cases with large sets of EHWT (typically hundreds of thousands).

Introduction

A key factor in the development of demand side management techniques [START_REF] Palensky | Demand side management: Demand response, intelligent energy systems, and smart loads[END_REF] is the availability of energy storage capacities [START_REF] Aguilar | Domestic water heat and water heater energy consumption in Canada[END_REF][START_REF]Energy road map[END_REF]. In this context, the large sets of Electric Hot Water Tanks (EHWT) found in numerous countries appear as very relevant for load-shifting applications and are the subject of various optimization studies. An EHWT heats water over relatively long periods of time, for later consumption [START_REF] Beeker | A distributed parameters model for electric hot water tanks[END_REF][START_REF] Beeker | Dynamical modelling for electric hot water tanks[END_REF]. While numerous advanced pricing policies have been studied and developed, the time-of-use pricing policy remains of dominant importance for most consumers, e.g. in France. Such policies define broad blocks of hours (for instance on-peak from 6:00 am to 10:00 pm, and off-peak from 10:00 pm to 6:00 am) during which a predetermined fixed rate is applied. The starting times of these periods being known, straightforward heating policies are commonly applied to each individual EHWT which is turned-on immediately after the start of the off-peak period until it is fully heated. This simple strategy increases electricity consumption in the nighttime (one period when market electricity prices are low), while hot water is used in the next daytime. At large scales, the result is that the overall consumption of the set of EHWTs rapidly decreases to a low level in the middle of the night, when the electricity production costs are the lowest, unfortunately. This negative effect has to be addressed. With the development of home automation, more advanced strategies are believed to enable improved cost-reductions for both users and electricity producers [START_REF] Beeker | Discrete-time optimal control of electric hot water tank[END_REF]. An efficient and robust improvement is to reschedule each EHWT heating in the night time, while keeping the heating period undivided (to minimize the malicious impact of complex dynamic effects taking place inside each EHWT, combining convection, diffusion and stratification [START_REF] Dincer | Thermal Energy Storage: Systems and Applications[END_REF][START_REF] Han | Thermal stratification within the water tank[END_REF]), to attain a desired load curve for the whole set of EHWTs, while ensuring individual users comfort. The first contribution of this article is to formulate the rescheduling of a set of EHWT heatings with comfort constraints as an optimization problem. The optimization problem considers hot water consumptions, production objectives and comfort constraints. In view of applications, a large set of EHWTs is considered, each being characterized by its electric power and heat loss coefficient. Initially, a certain heating starting time is assigned to each EHWT having a uniquely defined duration for the subsequent indivisible heating period. Each duration is scaled according to (next day) hot water consumptions. The sum of all the power consumptions defines the initial load curve, as a function of time. The optimization problem that we wish to solve is the rescheduling of each heating to minimize a given criteria (discussed in Section 2). It is the first contribution of the article.

Fig. 1. Schematic view of the cross-section of an electric hot water tank (EHWT)

A heuristic specifically designed for the discrete-time quadratic formulation is proposed to solve this problem. This is the second contribution of the article. The design of any such heuristic strongly depends on the distribution of the durations of the undividable heating. Here, a reference distribution obtained from data measured in a vast set of French households is presented and studied. The data set contains heating periods lasting up to 8 hours. The definition of the proposed heuristic stems from the following considerations. First, one notes that the scheduling of EHWT with long heating durations is the most likely to generate high consumption in on-peak period. Secondly, a rigid individual scheduling for a high number of EHWT is prone to generating singularities resulting in high consumption peaks in the load curve. The heuristic is designed to reduce singularities while generating a high diversity in the distribution. For this purpose, we consider a stochastic heuristic. Its governing principle is to sort the EHWT by decreasing order of duration times, and, then, to randomly schedule them one-at-the-time according to an adaptive distribution law. Each duration times is compensated to account for heat losses. For a given EHWT, this distribution law depends on the residual load curve, which is obtained by constructing the objective load curve minus the power consumption of the tanks already scheduled. To maximize diversity, the distribution law favors scheduling within time periods containing only few previously rescheduled EHWT. Optimization results produced for real data are given. Objective load curves have been provided by the French utility EDF, and several distribution laws have been tested. They lead to an optimality loss of less than 1%. The paper is organized as follows. In Section 2, the optimization problem is formulated. Then, in Section 3, the heuristic is presented. Simulation results are reported in Section 4. Conclusions and perspectives are given in Section 5. insulation thermostat heating elements hot water cold water

Formulation of the problem 2.1. Electric water heating

In the following, we consider a set of electric hot water tanks. An EHWT is a vertical cylindrical tank filled with water [START_REF] Beeker | A distributed parameters model for electric hot water tanks[END_REF][START_REF] Beeker | Dynamical modelling for electric hot water tanks[END_REF]. A heating element is plunged at the bottom of the tank, and the water is thermally insulated from the ambient using some insulation material, e.g. thermal foam (see Fig. 1). We represent each tank (labelled with ) using the following parameters and state variables:

 Energy of the water in the tank at time (defined with respect to cold water). This energy takes value between 0 and a maximum constant . It is to be noticed that depends on the volume of the tank, and also on a maximal acceptable temperature in the tank (this value is set by the user). In the sequence, the unit used to express energetic contents will be the watt hour (Wh).  We lump energy losses to the ambient into a heat loss coefficient , representing the heat loss per unit of time (as a percentage of the total energy). The units of is in h -1 .  We note the power of the heating element, expressed in watt (W). We assume that this power cannot be modulated, i.e. that if the tank heating is on, then the injected power is constant and is equal to . We focus on a time interval in which the tanks are heated, usually during the night. For each tank , we define a time interval , in which heating of tank is allowed. For instance, in France, households benefit in the night-time of block of hours in which the electricity price is reduced to promote electricity consumption when the prices are low, and EHWT heating has to be confined to these off-peak periods. The state-of-the-art heating policy is to switch-on each EHWT at the beginning of its off-peak period until it is fully heated. This strategy leads to an aggregated load curve that rapidly decreases in the middle of the night, when the prices of electricity are the lowest. Depending on the season and the market prices, various load curves can bring substantial savings for the electricity producer (see Fig. 2 where an example of load curve is represented along with market prices during a typical night).

Fig. 2. SPOT market price of electricity (orange) on the night from 2/8/16 to 2/9/16 and typical reference load curve (blue)

The development of home automation opens the way to dependable estimation of the variable and precise setting of the time of heating of each tank. In a general way, we formulate the following problem: given a reference load curve corresponding to the heating of the tanks from the energy to ( ) , how can the heating be rescheduled to approach an objective load curve ?

Preliminary: Time of heating rescaling and load curve admissibility

Any rescheduling of the heating of any EHWT has to take into account comfort constraints. Due to heat loss, the duration of heating has to depend on the starting time. If the EHWT is heated sooner (respectively later), heat losses are increased and more (respectively less) energy has to be injected in the EHWT. For any EHWT, a change in heating time can be analytically computed as a function of its heating starting time. We address the effect in the following way.

Fig. 3. Heating starting times and durations

For any given tank that, in the so-called reference scenario, starts its heating at time for a period of length , we wish to estimate heating duration that is needed if one chooses to start the heating at time instead, without altering the energy of the tank obtained at final time (see Fig. 3). A simple integration yields

) )

Then, if , we have , and if , we have (see Fig. 3). This delay effect has an impact on the global load curve. Indeed, to reschedule the time of heating of tanks from a reference load curve to an objective load curve , one needs to ensure that the energy injection is compensated. If we consider that the heat loss coefficients of the tanks are all close to their average , the same reasoning as before leads to

∑ ∑ ∫ ∑
for and then to the following "admissibility" condition for : ∫

Problem formulation

We can now define our optimization problem. We propose the following formulation. Given EHWT characterized by their heat loss coefficients , their powers , their allowed time intervals , their initial starting times and heating durations , define

∑

Given an objective load curve verifying (6), we desire to solve

∫ (∑ )

s.t. , the couple satisfies and .

Resolution method

We propose here a resolution method for (8), with a discretization in timesteps. The problem defined above can be related to capacity scheduling problems [START_REF] Veinott | Optimal capacity scheduling[END_REF] or cumulative non preemptive scheduling [START_REF] Carlier | Jackson's pseudo-preemptive schedule and cumulative scheduling problems[END_REF]. However the large number of optimization variables, the flexibility yielded by the cumulative nature of power (usually in scheduling problems, the assignment of a job to a machine is required) are not accounted for in classical resolution techniques. Moreover, the problem is also specific due to the time and power scale of the heating. The power consumed in one tank is very small compared to the whole set, because the set it large. By contrast, the duration of the longest heating (up to 7h) is relatively bulky compared to the time horizon (10 to 12h). In discrete-time, determining whether the problem admits an optimum solution is equivalent to the classical "exact cover problem" [START_REF] Jongen | Optimization theory[END_REF], which consists in exactly covering a set with a sub-collection of its subsets. Due to the high complexity of solving this problem (shown to be NP-complete in 1972 by R. Karp [START_REF] Karp | Reducibility among combinatorial problems[END_REF]), and more generally, to the difficulty of minimizing a criteria with a large number of decision variables (the number of tanks can be up to several hundreds of thousands, or millions), we propose in this section an heuristic in the form of a stochastic sequence of rescheduling.

The proposed heuristic is based on the following observations:  The tanks with long heating durations are more difficult to allocate. On the contrary, short durations yield flexibility to our problem.  A rigid rescheduling of heating times according to a deterministic procedure is prone to generate singularities in the resulting load curves. Indeed, depending on the shape of the objective load curve, a deterministic rescheduling is very likely to create patterns which generates undesirable high consumptions peaks. On the contrary, due to the large number of tanks, a stochastic heuristic can take advantage of the natural smoothing induced by introducing diversity in the reallocation. Therefore, we propose the following heuristic. It is based on a residual load curve which is updated at each step , and represents the objective load curve minus all powers from the tanks that have been already rescheduled (see Fig. 4). The steps are:

1. Compensate all durations as if tanks where starting heating at , using equation (4). The choice of and is important. In practice, we give the form of a union of disjoint intervals (in green in Fig. 4 a. and b.), corresponding to starting times in which the duration ) is entirely included in times such that (see Fig. 4 b.). This allows to stop scheduling in already fully loaded periods. Several types of laws can be considered, and have varied efficiency depending on the shape of the objective load curve. If very high slopes are present in the objective load curve, heavy weight should be placed on the boundary of the interval, to promote rescheduling near the boundaries. For this purpose, a parabolic distribution law can be proposed. The law can also directly be defined as a weighted integral of (see Fig. 4 c.).

Simulations results

Dataset

The feasibility of the problem and the efficiency of the heuristic strongly depend on the shape of the objective curve, the time intervals, and the flexibility yielded by the diversity of the durations among the population of the tanks. For this reason, realistic data have been gathered to test the heuristics.

The distribution of rescaled durations has been constructed based on data gathered on a panel of 267 representative households whose EHWT have been equipped with sensors (see Fig 5). The characteristics of the EHWTs (heat coefficient, power of the heating elements) are taken from representative products on the market. The restrictive time intervals have been defined using the distribution of peak hours in France. The 18 distinct peak hours in France each correspond to a different proportion of the households. Finally, 7 realistic reference objective load curves have been considered.

Simulation results

Simulations have been conducted on this dataset. Examples for two different objective load curves are reported in Fig. 6.

a.

b.

Fig. 6. Two examples of objective distribution laws (blue) and final realisations (orange). (a) n=500, p=100 (b) n=5000, p=1000.

To quantify optimality loss, given the obtained load curve , we propose two following indices, respectively corresponding to L1 and L2 norms.

∫ | | ∫ | | √∫ √∫
Results for the various reference objective load curves (with the best results among various probability law) and various values of and are reported in Table 1 and2. The presented algorithm has been implemented in Matlab R15a, and run on a Intel Core i7 (3,3GHz) with 16GB of RAM, using a single core and 25MB of memory. The simulations show satisfactory results and highlight the usefulness of favoring diversity during the rescheduling procedure. Indeed, results with a high number of tanks have better optimality index that the one of the small instances. On the contrary, quality of the results is not increasing with the number of timesteps, meaning that discrete time mesh refinement does not bring any significant additional performance. Interestingly, the proposed heuristic is computed in approx. linear time (with respect to the number of EHWT), and can be applied to very large sets of tanks. However, it is to be noticed that efficiency of a given probability law, and quality of the results depend on the shape of the objective curve and the flexibility yielded by the duration distribution. In our case, with real set and reference objective curve, the flexibility is sufficient.

Conclusion and perspective

In this paper, we have formulated an optimization problem and a resolution method in the form of a heuristic for the optimal rescheduling of heating of large sets of EHWT. Numerical experiments conducted on real data stress the relevance of this method for parameters corresponding to French houses. Similar studies should focus on parameters for other countries. Depending on the load curve, which relates to electricity producers constraints and, the heuristic may require further developments. A main tuning parameters is the choice of the probability density employed in the iterative scheduling procedure. In particular, if sharp transients are to be considered in the load curve, the choice of the probability function may require further investigations.

On the numerical side, the approach could benefit from various classic techniques. In fact, parallelization, semi-lumping are possible ways to explore. This could help speed up the method, which is already reasonably fast, but can be important for large instances. A straightforward implementation of the presented methodology treats a representative set of one million EHTWs in 6000 seconds. A satisfactory rescheduling is obtained after 3 random runs.

Nomenclature

  , and is the total energy consumption on for the tank . Subtracting (2) to (1) leads to ∫ ∫ and, finally,

Fig. 4 .

 4 Fig. 4. Detailed process of the heuristic. (a) at the start of an iteration, a cumulative curve has been determined by scheduling some of the heatings. (b) a new heating is to be scheduled, considering its duration, some intervals of possible starting times are determined from the residual curve. (c) along these intervals, probability density functions are defined, they favor the boundaries of the intervals. (d) randomly, a starting time is selected and a new cumulative curve is computed.

Fig. 5 .

 5 Fig. 5. Typical distribution of durations derived from a representative panel of users, (normalized scale)

Table 1 .

 1 Results for objective load curve 1 Number of tanks n Number of timesteps p

	Computation time (s)

Table 2 .

 2 Results for objective load curve 1 to 8

	Objective load	Number of	Number of		Computation time
	curve	tanks n	timesteps p		(s)
	1	5000	1000	0.0028 0.0029	22.3
	2	5000	1000	0.0036 0.0041	18.6
	3	5000	1000	0.0039 0.0042	18.5
	4	5000	1000	0.0037 0.0044	17.9
	5	5000	1000	0.0032 0.0030	25.6
	6	5000	1000	0.0038 0.0042	15.4
	7	5000	1000	0.0049 0.0065	17.2


Amar Prakash Azad

Sara Alouf

Eitan Altman

come   L'archive ouverte pluridisciplinaire

This note is to correct the authorlist and authors affiliations of [START_REF] Azad | Analysis and optimization of sleeping mode in WiMAX via stochastic decomposition techniques[END_REF]. The submitted version of this paper that appeared in the September 2011 issue of the IEEE Journal on Selected Areas in Communications was missing two co-authors: Sara Alouf and Eitan Altman. The byline and author affiliations should have appeared as in the current correction.

Also, the following information about funding was missing from [START_REF] Azad | Analysis and optimization of sleeping mode in WiMAX via stochastic decomposition techniques[END_REF]:

  Correction to "Analysis and Optimization of Sleeping Mode in WiMAX via Stochastic Decomposition Techniques"

Amar Prakash Azad, Member, IEEE, Sara Alouf, and Eitan Altman, Fellow, IEEE

Stochastic Models, and Journal of Economy Dynamic and Control (JEDC). He received the best paper award in the Networking 2006, in Globecom 2007 and in IFIP Wireless Days 2009 conferences, and is a coauthor of two papers that have received the best student paper awards (at QoFis 2000 and at Networking 2002). His areas of interest include networking, stochastic control and game theory. More information can be found at http://www-sop.inria.fr/members/Eitan.Altman/.

I. ACKNOWLEDGEMENTS

This work was partly supported by the ANR WINEM and by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR), project 4000-IT.


Nathanael Beeker
email: nathanael.beeker-adda@edf.fr

Paul Malisani
email: paul.malisani@edf.fr

Nicolas Petit
email: nicolas.petit@mines-paristech.fr

Statistical properties of domestic hot water consumption

Keywords: energy storage, hot water consumption, stochastic, smart operator

This article studies some of the statistical properties of domestic hot water (DHW) consumption. The study is based on experimental data obtained for a group of households, over a 1 year period. To describe the consumptions, we consider three representative statistical properties of the drain sequence associated to each household: i) the distribution of the magnitude of the drains, ii) a daily pattern of the start times of consumptions, iii) the time between two successive drains. A remarkable outcome of the study is that the time between two successive drains follows a bimodal Weibull distribution. This opens perspectives for piloting applications.

Introduction

The increasing share of intermittent renewable electricity sources in the energy mix [START_REF] Edenhofer | Report on renewable energy sources and climate change mitigation[END_REF][START_REF] Palensky | Demand side management: Demand response, intelligent energy systems, and smart loads[END_REF] raises new difficulties in management of the electricity production and equilibrium in distribution networks. Demand Side Management (DSM), which is a portfolio of smart piloting techniques aiming at modifying consumers' demand, is a promising solution for such concerns [START_REF] Palensky | Demand side management: Demand response, intelligent energy systems, and smart loads[END_REF]. A key factor in developing DSM is the availability of energy storage capacities. In this context, the large pools of electric hot water tanks (EHWT) found in numerous countries appear as a very large potential.

An EHWT is a domestic electric appliance which heats a volume of water with a controllable heating element. The user drains hot water from the EHWT at various times of the day, for its comfort. The state of charge of the tank is a function of heating and domestic hot water consumption. Delivery of hot water at all times is a constraint that must be satisfied at best. To design advanced heating control strategies (in view of various objectives such as reducing operation cost for a single tank, or reaching a desirable load profile for the total consumption of a pool of tanks), a model of the demand is a key element. For this reason, we perform some investigations on the dynamics of domestic hot water (DHW) consumption, which has a random nature.

The water drains appear as a sequence of quasi-instantaneous drains in the scale of the day (see Fig. 1). As can be observed on experimental data, the time of occurrence of these drains is not fixed, but is stochastic. In the literature, studies have focused on describing weekly or seasonal consumption patterns, or have presented hour-per-hour mean consumptions [START_REF] Vine | Domestic hot water consumption in four low-income apartment buildings[END_REF][START_REF] Meyer | Domestic hot water consumption in South African houses for developed and developing communities[END_REF][START_REF] Papakostas | Residential hot water use patterns in Greece[END_REF]. One such typical approach can be found in [START_REF] Jordan | Influence of DHW load profile on the fractional energy savings: a case study of a solar combi-system with TRNSYS simulations[END_REF], in which an approach based on aggregation of types of uses allows to generate minute-perminute load profiles, and in [START_REF] Prud'homme | Advanced control strategy of a solar domestic hot water system with a segmented auxiliary heater[END_REF], in which forecast over 2 days are generated using the Kalman filter. Nevertheless, more advanced stochastic modeling of the magnitude of these drains and of the temporal correlation in the sequence, based on data analysis, would represent some valuable ways of improvement to design heating strategy for the tanks. This article aims at establishing such statistical properties. Data have been gathered in 11 distinct households with EHTW equipped with flow meters and temperature sensors, over periods ranging from 292 to 337 days, with a sampling time of one minute. In this paper, using data, we identify statistical characteristics of DHW consumption. This is the main contribution of this article.

The paper is organized as follows. In Section 2, we identify three main statistical features. Then, in Sections 3 to 5, model of these characteristics are presented. Conclusion and perspective are drawn in Section 6.

Characteristics of domestic hot water consumption

Domestic Hot water consumption aggregates various uses in a household: bathing, cleaning, cooking, etc. The durations of resulting drains range from a few seconds to a few minutes. They can be represented as quasi-instantaneous events of various magnitudes in the scale of the day. These drains can be described as a volume of hot water or a quantity of energy taken from the tank (the energy contained in hot water being defined with respect to a cold water temperature reference).

To model the drain sequence, one needs to describe when the drain happens, and how much hot water is consumed during those drains (we make the assumption that two drains occurring within a single minute correspond to a single bigger drain).

The following statistical properties are considered: i) the distribution of the magnitude of the drains, ii) a daily pattern of the start times of consumptions, iii) the time between two successive drains.

This defines a stochastic process for the total hot water consumption at time of the form ∑

where and are the time of occurrence and the magnitude of the drain , resp., and is the Heaviside function: if and if . Classically, the magnitude of the drains can be characterized through frequentist inference [START_REF] Neyman | Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability[END_REF], by estimating a probability density function from the frequency of the data. The results are presented in Section 3.

The times of occurrence of the drains are more complex to describe. The consumption start times are related to the number of persons in the household and their domestic habits. As will be shown in Section 4, they are distributed according to an average daily pattern which is reported. Then, the time between two successive drains follows a bimodal Weibull distribution. This is shown in Section 5.

Frequentist inference for the water drains distribution law

A simple way to model the diversity of magnitude of the drains is to represent the drains as random variables drawn from the same probability law. Since the drains are positive, this probability law can be represented with a probability density function on [0,+∞[ (see Fig. 2).

The probability distribution reported in Fig. 2 has been estimated with a frequentist approach to inference [START_REF] Neyman | Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability[END_REF], for all households and all measurements. The displayed distribution law is the frequency of occurrence of each drain for a large number of measurements.

An interesting observation is that four distinct peaks are visible in the filtered distribution function. This is in accordance with the results presented in [START_REF] Jordan | Influence of DHW load profile on the fractional energy savings: a case study of a solar combi-system with TRNSYS simulations[END_REF], obtained by considering four types of uses (small and medium drains, shower bath and bath tub).

If needed, this result could be refined by taking into account that the measurements are strongly related to the habits of the households (e.g. in term of bathing), the time of the day and the season. For households equipped with additional sensors, the conditional distribution could be estimated online. However, we do not consider this in this article.

Daily pattern for the start times of drains

The hot water consumption is strongly related to the habits of the house occupants. Also, it is clearly visible in the data (as could be expected) that the drains mostly take place at certain times of the day (see Fig. 1). In fact, a daily pattern can be defined. For this, one can simply consider the mean value of the number of drains at a given time of the day, for each household. The procedure is as follows. This procedure is commonly used in statistics (e.g. to determine the intensity function of a non-homogeneous Poisson process [START_REF] Ruwet | Processus de Poisson[END_REF]).

Given a household, we represent each of its set of 24 hours measurements (labeled by =1…n) with a function that represents the number of drains having occurred in this day over the interval , for . In other words, consider any day and the sorted drain times , then is defined as

∑ ( ) (2)
Then, we define the daily pattern as the mean of these functions

∑ . ( 3
)
This function, defined over the time interval , gives, for any household, the average (expected) number of drains that should have occurred at time (see Fig. 3): it is representative of the habits of this household in terms of frequency of the drain. The function (3) obtained with the discussed dataset is reported in Fig. 3. The presence of sharp transients between low slope regions shows that the average function ( 2) is representative of the individual behaviors in each household. High slopes correspond to time periods with frequent drains.

Distribution of the time between two successive drains

The time between two successive drains (during a day ) is also related to the habits of the house occupants. Therefore, we isolate data accordingly.

In the case when the number of drains over time are samples of a nonhomogeneous Poisson process (i.e. the drains are not correlated), then it is possible to construct adjusted increments with the mean value function that must follow an exponential distribution. As will appear, this assumption is not valid in our case, but we still perform the same analysis, yielding different conclusions. The construction is done the following way.

For each period of 24 hours, a set is constructed by taking the image of each drain time , through the function (see Fig. 4). The are then used to define the set of successive increments These variables have an interpretation: for any day , is the average increment of the number of drains from to . Its expected value (for all possible ) is 1. These normalized increments (represented in the ordinate axis in Fig. 4) are assumed to be independent from the choice of the households. They are representative of the correlation between successive drains.

( )    
The are distributed in the interval. Using the whole dataset, an experimental cumulative distribution function (CDF) can be obtained. It is reported in Fig. 5. Remarkably, this function shows a good fit to which corresponds to the CDF of the Weibull distribution of shape parameter and scale parameter [START_REF] Prabhakar Murthy | Weibull models[END_REF]. Among other possibilities (exponential distribution, Gamma distribution, Beta distribution), this is (by far) the best fit.

The Weibull distribution is commonly used to model failure rates over time. In the literature, it is often associated with Autoregressive Conditional Duration (ACD) models [START_REF] Crowley | Point Process Models for Multivariate High-Frequency Irregularly Spaced Data[END_REF] to forecast the distribution of a succession of duration times (e.g. time lapse between two transactions in stock market [START_REF] Tsay | Autoregressive Conditional Duration Models[END_REF]). In Weibull models, the parameter defines the nature of the process. The case corresponds to an exponential distribution. If , then the occurrence of an event raises the probability of a closely following event (cases of immediately consecutive events are frequent). On the contrary, the case corresponds to the case when successive events are spaced out. To establish that the experimental data follow the assumed distribution, a graphical method can be employed. The probability plot consists in plotting two CDF, one against the other: if the two distributions are similar, the points of the probability plot should lie on a straight line. Further, for the Weibull distribution, the shape parameter α can simply be deduced from the slope [START_REF] Prabhakar Murthy | Weibull models[END_REF]. Interestingly, the probability plot in Fig. 6 displays two distinct lines with different slopes. This is a well-identified characteristic of a multimodal or mixed Weibull distribution, indicative of the fact that several shape parameters coexist [START_REF] Doganaksoy | Reliability Analysis by Failure Mode[END_REF][START_REF] Kececioglu | Reliability Engineering Handbook[END_REF]. In our case, we propose the following interpretation. It is likely that two different timescales come into play in the time correlation. Those timescales can correspond to the fact that the user is at home or not, doing an activity that requires hot water. The two modes have parameters , i.e. at all times during the day the occurrence a water drain strongly suggests that another water drain should be expected right after.

Conclusion and perspectives

In this article, we have identified three distinct statistical characteristics present in a dataset of time series of DHW consumption. According to the investigations conducted on the presented dataset, the conclusions are well grounded but they remain preliminary. If these characteristics remain valid for more various cases (e.g. in different countries or with a wider range in the type of households), then this would pave the way to interesting works on stochastic modelling of hot water consumption.

These perspectives are important for smart piloting applications of EHWT. Indeed, the identification of the time lapse between drains with a Weibull distribution appears as particularly useful, since it stresses that if the user is draining water, he is very likely to drain more water shortly. Considering this likely event, piloting strategies can be updated online to take into account the high probability of future drains. Future works should include the definition, if possible, of a global Weibull ACD model that can be fitted online for each household.

At another scale, for a large number of EHWT, stochastic modelling of the consumption opens the way to the development of models for the distribution of the state of charge of the tanks, using e.g. Fokker-Planck equations.

Fig. 1

 1 Fig. 1 An example of sequence of drains for an household over 48h.

Fig. 2

 2 Fig. 2 Experimental probability distribution of drain magnitudes (over the whole recorded sequences).

Fig. 3

 3 Fig. 3 Averaged daily cumulative number of drains

Fig. 4

 4 Fig. 4 Construction of the ajusted increments

Fig. 5

 5 Fig. 5 Experimental CDF for the ajusted increments compared to the Weibull CDF

Fig. 6

 6 Fig. 6 Probability plot of the data against the Weibull distribution

Acknowledgment

EDF Lab is gratefully acknowledged for making available the dataset on domestic hot water consumption.