Édouard Amouroux
email: edamouroux@gmail.com

Thomas Huraux
email: thomas.huraux@edf.fr

Francois Sempé
email: sempe.francois@gmail.com

Nicolas Sabouret
email: nicolas.sabouret@upmc.fr

Yvon Haradji
email: yvon.haradji@edf.fr

Simulating human activities to investigate household energy consumption

Keywords: Agent-Based Modelling, Social Simulation, Energy Control, Inhabitants' Dynamic Behaviour

come

Introduction

Energy efficiency and consumption reduction is a major challenge for our society due to awareness raising to the greenhouse effect and growing tensions on the energy market. European Union (EU) set the ambitious objective to divide by four greenhouse gas emissions by 2050. Yet, a large part of the final energy consumed in Europe, 26.6%, is used within residential sector, according to the EEA (European Environment Agency, 2008). For this reason, several programmes in the residential sector have been proposed such as building renovation, definition of low energy building (LEB) and smart energy controller [START_REF] Freire | Predictive controllers for thermal comfort optimization and energy savings[END_REF], consumer awareness to cost and carbon emissions thanks to real-time feedback [START_REF] Rogers | Adaptive home heating control through gaussian process prediction and mathematical programming[END_REF]. At a larger scale, smart grid have been proposed [START_REF] Massoud | Toward a smart grid: power delivery for the 21st century[END_REF] in order to optimise the production and the distribution of electricity depending on consumption.

Nevertheless, individuals' behaviour also strongly influences energy consumption. In France, according to [START_REF] Poquet | Pour les ménages, la recherche du confort prime encore sur les économies d'énergie[END_REF], despite a moderate increase of household electricity consumption (8% from 1973 to 2003), a strong increase of the specific uses of electricity (i.e. electricity used except for heating) occurred (85% for the same period). In this context, the study of propositions to diminish specific uses of electricity becomes a crucial issue.

One aspect of the problem is to study the con-sumption of electrical appliances in real-life situations. That aspect is at the core of the REMODECE1 european project, that provides knowledge on electricity waste due to inefficient appliances and standby consumption. A complementary approach is to study household activities. Several works have been proposed in this direction such as (Duy [START_REF] Ha | A home automation system to improve household energy control[END_REF][START_REF] Mahdavi | Toward empiricallybased models of people's presence and actions in buildings[END_REF][START_REF] Kashif | Simulating the dynamics of occupant behaviour for power management in residential buildings[END_REF], but they usually focus on electricity demand prediction, so as to optimise energy control systems. Nevertheless, one might want to consider influencing people's consumption behaviour through incentives such as variable electricity pricing, awareness campaigns, etc. For instance, [START_REF] Newsham | The effect of utility time-varying pricing and load control strategies on residential summer peak electricity use: A review[END_REF] suggests that a critical peak price program might reduce by 30% the peak load. Several models allows to take into account human activities and their resulting electricity demands. Many of them rely on statistical approaches such as [START_REF] Widén | A high-resolution stochastic model of domestic activity patterns and electricity demand[END_REF] and [START_REF] Yamaguchi | Occupant Behavior Model For Households To Estimate High-temporal Resolution Residential Electricity Demand Profile[END_REF] that use hidden Markov chains to generate human activities related to electricity demands. Despite their statistical validity, they focus on representing an "average" familly and scenario and, thus, do not allow to investigate on the activity dynamics related, for instance, to specific events. Actually, agent-based models are best fitted for such analysis, as recently exhibited by the model proposed in [START_REF] Kashif | Simulating the dynamics of occupant behaviour for power management in residential buildings[END_REF]. This model relies on the Brahms language [START_REF] Sierhuis | Modeling and simulating human activity[END_REF] to allow explicit representation and organisation of human activities. Nevertheless, the parametrization of this model appears to be very complex. They are hardly accessible to field experts and do not allow simulation of complex and long situations.

In this paper, we present a meta-model and a simulator, called SMACH, that allows experts to model, simulate and study the household activities and their relation with electrical consumption depending on specific pricing policies or appliance use. This system can be used to evaluate possible incentives to diminish peak hours electricity demand.

In numerous modelling efforts, expertise collection is combined with definition of the technical architecture which often limits the scalability of the model. In contrary, our approach, incremental and situated modelling [START_REF] Sempé | Incremental and situated modeling for multi-agent based simulations[END_REF] , clearly separated the two processes: (1) we propose a meta-model generic enough to (2) let field experts integrate their knowledge and evaluate their hypotheses by defining scenarios (for instance, in the present application : a household of two adults, two kids, one infant, their activities their habits, house environment and specific events). Once a new question emerges or an incoherence is pointed out by domain experts, the cycle starts again by making amendment to the previous meta-model. In practice, our meta-model and the dedicated simulator (SMACH) allow experts to define fine-grain activities (ranging from few minutes to hours) and to study their evolution over long period of time (one year). Using an advanced and intuitive user interface, the simulator allows experts to understand the respective influence of activities and energy consumption, and the co-influence of activities.

In this paper, we focus on the presentation of the meta-model of human activities within the household (presented in Section 2) and the dynamics of this model which includes communication, beliefs, action selection and behaviour adaptation (detailed in Section 3). Section 4, introduces the simulator GUI and provides two distinctive examples of household emergent adaptation forced by action competition and variable electricity pricing. Section 5 concludes the paper and presents the ongoing work.

Meta-Model of human activity

Our meta-model of human activity follows two purposes that are tightly connected. First, it aims at describing human behaviours in the context of household electricity consumption. This description must remain easy to manipulate, even for non-computer specialists. To this purpose, we followed an individual-centered agent-based approach in which human activity is decomposed into tasks. Second, it serves as agent description model for our multi-agent simulator (section 3). This simulator allows to study the evolution of human behaviour depending on the characteristici.e.s of energy pricing and household appliances, that are defined in the environment.

Environment

Let [d s , d e ] be the considered time period. Energy prices can be specified to different time periods {[d i ; d i+1 ], price i } with d 0 = d s and price i ∈ R + , the kilowatt hour price (kWh) for the i th sub-period.

The house is composed of several rooms where electrical appliances are spread. Let R be the set of rooms and E the set of appliances. For each room r (r ∈ R), we note E r ⊂ E the set of electrical appliances in this room. In our model, each appliance can only be in one room, fixed for the duration of the simulation.

Appliances

Each appliance e ∈ E is characterised by its electrical consumption θ i . We note power : E → R + the function that returns the current electrical consumption of an appliance. In our model, we consider two different kinds of appliances: State-based appliances are defined as a tuple {θ o , θ s , st} where θ o and θ s are the electrical consumptions (in R + ) when e is running or in standby mode, and st ∈ {off, standby, on} is the state of the appliance, modified by individuals in the house during their activities. Heaters, TVs, fridges... can be represented by state-based appliances.

Program-based appliances are defined as a tuple < P e , st > where P e is a set of operating programs characterised by load curves. Each program p is a couple (τ p , φ p ) where τ p is the program duration and φ p : [0, τ p ] → R + gives the appliance consumption at each time of the program. The status st of the appliance is then defined by a couple st = (p c ,t c ) where p c is the currently selected program and t c is the time since the beginning of this program. Thus, power(e) = φ p c (t c ). Ovens can be represented by this kind of appliances. Note that state-based appliances can be seen as specific cases of program based appliances (with three programs and constant load curves).

All energy consumption profiles comes in our model from real data from the REMODECE project.

Household members and their behaviours

Individuals represent the household members. Each individual i ∈ I is characterised by its responsibility level res i and the set of actions it can perform A i (see below).

The responsibility level allows us to distinguish between children, youngsters, adults and elderlies in the simulation (see Section 3). Moreover, at each time of the simulation, we note:

• room(i) ∈ R the room in which the individual is located;

• action(i) ∈ A i the individual's current activity;

• B i the individual's beliefs set (see Section 3.1.1);

• C i the individual's communications set (see section 3.1.2);

Tasks

A task t ∈ T represents a generic activity in the house, from which individual actions are derived. A task is a tuple < τ min , τ max , bene, visi, coll, E t , T pre > where :

• τ min , τ max the minimum and the maximum task duration;

• bene ∈ {alone, collective} describes whether this task, once achieved, can be used by other individuals or not (see preconditions below);

• visi ∈ {none, individual, room, house} describes the visibility (by other individuals) during and after the task realisation (see visibility of actions in the following subsection);

• coll ∈ {none, allowed, mandatory} describes whether this task cannot, can or must be performed alone or with other individuals.;

• E t = {< e, p >} | e ∈ E & p ∈ {e.
Pe} the set of appliances used during this task realisation and the associated programs (for program-based appliances);

• T pre ⊂ T is the set of pre-conditional tasks of t.

An individual can perform a given task if and only it has the information that all pre-conditional tasks have been achieved by himself (if bene = alone) or by others (if bene = collective). The precondition relation maps into a graph of tasks;

Actions, rythms and events

An action a ∈ A represents the instanciation of a task by an individual. We denote actor(a) the unique i ∈ I

such that a ∈ A i . 2
An action is defined by a tuple a = t, w, R a , st where t ∈ T is the associated task, w ∈ W is the action rhythm (see below), R a ∈ R is the set of rooms in which this action can be performed and st ∈ undone, done is the action status, whose dynamics is described in section 3.2.2. When an action has been achieved (st = done), we note rooms(a) the set of rooms in which it has been done.

Rhythms

In order to express habits of humans, the concept of rhythm has been introduced in the metamodel. A rhythm w (w ∈ W ) allows to define, for each action, a frequency and preferred periods. The preferred period allows to specify the month, day of the week and the time slots in which an agent may perform a task. To each of these specifications is associated a force indicating if the system should more or less take into account these periods. More formally, a rhythm is a tuple per w , f req w , var w , Pp w where :

• per ∈ {day, week, month, year} is the period considered; • f req ∈ N + is the frequency for this period;

For instance, per = day and f req = 2 means "twice a day". • var ∈ {weak, medium, strong} is the frequency variability; • PP is a set of preferred sub-periods in the considered period. Each preferred period pp i is a tuple with [d 1 , d 2 ], strength where d 1 and d 2 define the sub-period, strength ∈ {weak, medium, imperative} the strength of the preferred period; For instance, let us consider the task "eating". We might want to define the default rhythm for this task as being performed three times a day in most situations (per = day, f req = 3, var = weak) with preferred periods pp1 = < [7 : 00, 9 : 00], imperative >, pp 2 =< [12 : 00, 14 : 00], weak > and pp 3 =< [19 : 00, 22 : 00], strong >, which means that our individual will eventually eat in the morning, possibly at lunch and very probably at dinner. Note that the number of preferred periods does not need to be equal to f req (in which case some executions of the task will remain completely free over the period).

Events Events represents exceptional situations (such as holidays, sickness, etc.) during which usual actions cannot be performed. An event q ∈ Q can be defined as a couple ([d s q , d e q ],W q with W q a set of couples (a, w). Concretely, an event associates new rhythms w to some actions a for a given time period [d s q , d e q ]. For instance, during holidays, the rhythm associated with actions "work" is null.

Collective aspects

Our meta model was designed so as to integrate the collective aspects in family organisation, such as helping the children, sharing domestic tasks or spending time together during certain activities. The coll and bene parameters in tasks play a key role in leading to a cooperation mechanism among individuals, following the simulation model that is presented in the next section. As an illustration, there is a synchronisation mechanism between agents when coll = mandatory so that individuals wait for each other to begin the task. Similarly, the fact that some tasks with collective benefits can be used by other agents allows to really describe complex cooperation in the house.

Simulation of Human Activities

Based on the meta-model presented in the previous section, human activity is simulated in a multiagent simulator called SMACH. In this simulator, agents correspond to the human individuals in the model and their procedural loop is presented in the next two The adaptation of this loop over time to the scenario constraints is presented in subsection 3.3.

Multi-Agent model

To each individual i ∈ I corresponds an agent in SMACH. All agents run synchronously in the platform and each step consists into selecting the best current action based on the agent's beliefs. In this subsection, we present the agent beliefs and the communication model. Next subsection presents the agent's dynamics.

Beliefs

A belief b of an individual is a tuple b = d, a, s such that d is the creation date, a ∈ A an action and s ∈ {undone, running, done} is the believed status of this action. < d, a, s >∈ B i means that, at time d, individual i had an information about action a being in status s (with s = running when ∃i ∈ I such that action(i) = a). This belief is kept in i's beliefs base as long as no observation or deduction contradicts it.

The initial beliefs set B i of an agent i is that all actions have the status : undone. We also consider individuals know from the start which one can perform which task. The dynamics of the beliefs set is presented in section 3.2.1.

Communications

In addition to individual actions, people in the household must communicate to exchange information or request the participation of others in tasks. This is represented using a multi-agent communication approach such as FIPA-ACL (FIPA consortium, 2003), based on speech-act theory by Searle [START_REF] Searle | Speech Acts[END_REF].

A conversation c ∈ C i is an ordered set of n messages, msg 1 , ..., msg n with msg i =< d,to, f rom, per f ∈ Per f , c > where d is the date of the message, to and f rom are the sender and the receiver of the message, per f is the message performative and c its content.

In our model, the main considered performatives are : I do ..., Are you ready to ... ?, What are you doing?, Did you ... ?, Can you ... ? and their following replies No, Yes, Later. The content of the message, when required, is always a single task t ∈ T .

The selection of messages, based on the agents beliefs, is presented in the next subsection.

Agents' dynamics

This section first presents how beliefs are maintained in the agents base. Then, we present how actions are selected, based on individual's preferences, beliefs and communications.

Beliefs update

Beliefs in the agent's base B i are added or removed at each turn, based on the agent's perception (either direct perception in the visibility zone of an action, or via a message) and on automatic inferences on actions status and individual's possible rooms.

More formally, let d 0 be the current time stamp and let us note + i b andi b the addition and removal of belief b in B i .

We first define the visibility zone of an action a ∈ A, depending on its status and task's visibility, as follows (all cases are exclusive):

• if a.t.visi = none, then it is not visible: visi(a) = / 0;
• if a is active (i.e. ∃i ∈ I such that action(i) = a) and has visibility a.t.visi ∈ {individual, room}, then visi(a) = {room(i)} ∀i|action(i)=a ;

• if a is active and a.t.visi = home, then visi(a) = R

• if a is finished (i.e. a.st = done) and a is visible on individuals (a.t.visi = individual), then visi(a) = room(actor(a));

• if a is finished and a.st.visi ∈ {room, home}, then visi(a) = rooms(a);

The agent belief base B i is updated when observing actions from their visibility zone:

∀a|room(i) ∈ visi(a), d , a, status(a) ∈ B i ⇒ + i d 0 , a, status(a)
Similarly, when receiving a message with performative I do or an answer to a message Did you in a conversation, a new belief is added: + i d 0 , a, status(a) , with a the action corresponding to the sender and the task (content) of the message.

Moreover, a coherence mechanism removes all previous incompatible belief:

• An action can only have one status:

+ i d 0 , a, status(a) ⇒ ∀d < d 0 , -i d , a, s = status(a)
• An agent can only do one action at a time: • Two actions associated with one single "exclusive" task (a 1 .t = a 2 .t t and t.coll = none) cannot be active simultaneously:

+ i d 0 , a,
i d , a 2 , running and

+ i d 0 , a 2 , done
Note that when a previous belief is updated, its status is moved to done (actions that can no longer be performed are considered as achieved by default ).

Action dynamic

The status of an action st (from undone to done) is actually more complex than a simple done/undone boolean. It is controlled by four internal variables:

• real number ∈ N + is the number of realisation already conducted within the current day. The value of a.st ∈ {done, undone} used in beliefs is directly derived from a.real number ≥ 1;

• real max ∈ N + is the planned maximum number of realisation for the current day (which directly depends on w);

• τ ≤ a.t.τ max is the elapsed duration when a is running;

• pr ∈ [-1; 1] is the action priority.

The priority a.pr evolves in time (see below, section 3.2.5) and the agent selects the action with the highest priority amongst all possible actions. An action is possible if and only if all the following criteria are met (see also Table 1):

• Every precondition tasks have been conducted; Note that a.st cannot be computed directly by the agent. On the contrary, the agent will use its belief base to decide whether actions are done or undone and whether they can be performed. In the algorithm in table 1, a.st = x means ∃ < d, a, done >∈ B i . However, when starting an action, the simulator will check all the actual preconditions and an agent will not be able to perform an action that it believed possible if table 1 is not checked. In this case, all inconsistent beliefs are removed from the agent base.

Interruption and realisation

The running action of an individual (a = action(i)), can be interrupted for two reasons: 1) another (possible) action, a , has a higher priority or 2) a became impossible. When this occurs, another action will be selected and the status of a is modified depending on its duration time :

• If a has been executed for a duration τ ≥ τ min , its number of realisation a.real number is incremented and a.τ is reset to 0. The action status a.st switches do done.

• If a.τ < τ min The action is put in a "standby mode": a.τ is blocked at its current value, its number of realisation is not increased. For instance, a child is interrupted during homework, (s)he does not loose the benefit of having started working.

Daily reset

The action's state evolves over time depending on the individual behaviour and his/her environment. It is reset at midnight except for the sleep action which is reset at noon. In practice, a.st is reset to undone, a.real number , a.τ and a.pr are reset to 0, and the maximum number of realisation is re-computed depending on the action rhythm if the new day contains a preferential period, ppi, and if some realisation are to be done (w.real < w.real max ).

Action priority

An action priority takes into account the individual's internal state: preference 3 , commitment to the current action (see table 2 for details) and the influence of the associated rhythm (a.w, see also 3.3). In addition to the individual's internal state, the priority of an action may be influenced by external factors: energy price level (variability of the energy price is summarised as 3 levels), other individuals invitations (through messages) and events influence. Table 2 describes the computation of this raw priority, called pr raw :

pr raw = pre f + in f real + in f pp + in f eng + in f ener + in f inci + in f eve
All the coefficient values have been determined empirically, only their relative order is significant. For instance, the individual preference is much less important than its commitment/lassitude to his/her current action.

In order to allow the representation of sequential behaviour (e.g. leave for work requires to be suit up which requires to have taken a shower), the action priority must also consider priority of the most urgent depending action, i.e. the maximum of priorities of all actions for which a is in precondition: pr dep = max a.t∈a .T pre {a .pr} Thus, the action priority a.pr is the maximum of these two values: a.pr = max{a.pr raw , pr dep }

Behaviour adaptation

Adaptation is essential in our model as we want to study the possible reactions of a household to variable energy pricing, habits emergence and other single events (e.g. holidays, sickness, etc.). Adaptation is conducted trough re-organisation of the household, i.e. when tasks are made and who does which collective task.

We implemented two adaptation mechanisms during a simulation: (1) everyday life obligations or strong habits are represented through rhythms with imperative frequencies and/or preferred period (PP) and ( 2) habits may emerge thanks to the controlled mobility of actions.

Everyday life strong constraints

Some everyday life actions may require to be conducted at specific time. Such constraint is controlled 3 minimal factor enabling complete differentiation between actions with similar priorities (a) pre f ∈ [-0, 01; +0, 01] is the individual preference for the action;

(b) in f real ∈ {-0, 2; 0} is the penalty if the action already reached its maximum number of realisation (a.real max );

(c) in f pp ∈ {penalty, 0, bonus} is the action's rhythm influence, with penalty and bonus computed as described below in section 3.3. in f pp = 0 if a.pp = / 0 and is negative (penalty) when the current date is outside a preferred period and positive (bonus) otherwise.

(d) in f eng ∈ [-0.1; +0, 1] is the commitment or lassitude influence. Its value is set to 0 for all non running actions. Otherwise, it is defined by a linear interpolation from 0.1 when the action just began, to 0 when it reaches τ min and -0.1 when at τ max .

(e) in f ener is the electricity price influence, defined by:

-in f ener = 0 if a.t.cons = 0 or price = medium; -in f ener = -0.1 if price = high; -in f ener = +0.02 if price = low.
Tasks that consume energy are strongly penalised when the energy price is high, but only marginally favored when it is low.

(f) in f inci ∈ [0; +0, 1] computes the influence of other individuals, through messages with performative Are you ready to and Can you, also called invitations. When a.t.coll = none, in f inci = 0. Otherwise, it is increased by +0, 02 per invitation (with a maximum of 0.1).

(g) in f eve ∈ [-0, 1; +0, 1] is the events' influence.

For each q ∈ Q such that d current ∈ [d s q , d e q ] and ∃(a, w) ∈ W q , in f eve is modified by ±0.05, depending on the new rythm w's properties compared to the action's initial rythm a.w. through the PP influence (see Table 2): positive during a PP and negative outside of any PP, in order to favour within PP and penalise outside PP realisation respectively. Unfortunately, as in the everyday life,i.e. several actions may compete for a realisation during an overlapping timeslot (e.g. have breakfast and take shower both should be done between 7 to 8am). To lessen such competition between imperative rhythms, each action at the end of the day reevaluates its bonus and penalty values according to the following policy:

bonus must remain between 0 and 0.15; penalty must remain between -0.15 and 0;

When the action could not be executed during its PP this day, bonus is increased 0.02;

When the action was achieved during its PP for this day, the bonus is decreased by 0.01. This ensures a slow re-organisation capacity.

When the action was executed outside a PP, bonus is increased by 0.01;

When the system could not reach the target realisation number for this action (i.e., real number < real max ), penalty is decreased by 0.02, since we want to give priority to action achievement over PP constraints.

In other words, actions that could be done during their PP during day n will be less favoured on their PP in day n + 1 and vice-versa. Moreover, actions that missed some executions will be encouraged. Although this policy cannot guarantee the satisfaction of all realisation constraints (which may not be possible at all), it provides modeller more freedom regarding rhythms definition.

Everyday life habits

Outside the competitive timeslot, habits may emerge as in everyday life. This is achieved in SMACH via the automatic creation of non-imperative rhythms during simulation. In practice, if an action has been triggered at a given period, this period will be favoured the following days (using a limited memory window)4 . As a consequence, all actions with no specified PP are allowed to find the most suitable periods for their realisations and to keep them as long they are suitable.

The maximum values for bonus and penalty are lesser than those of imperative PP, so this kind of actions can change its PP more easily. The next section will give an example of such adaptation: in conjunction with the priority penalty during periods with expensive energy, the household organisation is altered.

Implementation and evaluation

The SMACH simulator has been implemented in Java and several example scenarios have been proposed to validate our approach. We evaluated different generic modelling and simulation platforms (i.e. Repast [START_REF] North | A declarative model assembly infrastructure for verification and validation[END_REF], Netlogo [START_REF] Wilensky | Netlogo. center for connected learning and computer based modeling[END_REF] and GAMA [START_REF] Taillandier | Gama: A simulation platform that integrates geographical information data, agent-based modeling and multi-scale control[END_REF]) but none would meet our two most important requirements. First, a tailored modelling helps to ease definition of simulation scenario by field ex-perts. Programming-like modelling language are out of the question for non-computer-scientists. Second, a domain-related simulation analysis tool is also required by field experts and the implementation of such a tool in a constrained environment can be very costly. For these two main reasons, we developped our own agent platform for the SMACH simulator, which offer a detailed simulation analysis GUI. This GUI is detailed in the following subsection. We will also discuss two examples that illustrate the ability of the model to express rhythm constraints and, at the same time, and the actions reorganisation process. The first example focuses on action competition whereas the second one demonstrates how is handled household adaptation in the situation of a change in electricity pricing policy.

Simulation analysis GUI

The SMACH simulation analysis GUI shown in figure 1 can be detailed as follows. The overall electricity consumption is shown in panel (1). Each color represents a different appliance's consumption (the central gap correspond to holidays). Panel (1') presents some global variable (total electricity consumption, associated price and social indicators). Below, panel (2), details individuals activities over the week selected in panel 1 (S45) whereas panels (3 and 3') presents a zoom for even more details (one rectangle representing one minute). Finally, panel (4) allows to investigates actions actual rhythms (only some actions are selected). Concretely, the horizontal axe represent the passing of days whereas the vertical axe represen the time of the day. For instance, we can see that the work actions (blocks centered horizontally) occurs only during week days.5

Dealing with competing actions

Our first example considers a family of 3 with a conflict situation: they have to make diner, have diner and have shower within the 7 to 9 pm time slot. This time slot imposes a constraint due to the actions durations and the exclusiveness of the have shower action (one at a time). The adaptation process will organise progressively the actions to ease their execution.

In the simulation first week, individuals usually fails to have shower as shown in Fig. 2 (a). This diagram represents the individuals' acitvities as colored bar over time. In particular, the sleep action (strawberry pink) and the work action (brown) can serve as reference points. These actions have regular schedule since the beginning (no adaptation needed) as they have no strong competitors and large preferred periods (around 8 hours). In the conflict timeslot, dinerrelated actions are occuring whereas have shower is not (only the mother do it on monday). The advantage of diner-related actions is due to two factors: (1) make dinner is a collective benefit task (one individual can cook for 3) and ( 2) have dinner is a coordinated one which imply strongly incentive invitations.

The conflict is resolved through adaptation of the preferred periods (PP) and results in the following working-day pattern showed in Fig. 2 (b). After work, the mother takes her shower while the father and Liz make diner. Once diner is ready, the father or Liz invite the others to have diner. After diner, the father takes a shower followed by Liz.

This example demonstrates how the adaptation process can organize the actions in relation with PP constraints. Although, this organization is not fully stable, due to the continuous adaptations of the PP, it can last several weeks.

Adaptation of habits in relation with a new pricing

The second example is based on similar parameters but shows how PP modifications work with agents sensitivity to the electricity price. In this example, the simulation spans over 10 weeks from January to March. In order to evaluate the capabilities of adaptation to variable electricity price we make the following hypotheses. In January, the electricity price is fixed to a medium level. The following month, we introduce a variable price policy: high from 6 to 8 pm and low for the rest of day (everyday). Finally, in March, the price is switched back to the original policy (fixed price at medium level). Several actions, i.e. watch TV, use computer, garden, take shower, play chess, without specified PP are also added to the ones presented in the previous example. Some of them consume electricity and are, thus, sensible to electricity price. The variation of electricity pricing induce actions mobility as show on the long term activity diagrams (Fig. 3). In Fig. 3 (a), we focus on the watch TV action (in black).Although, two actions have been added as time landmark : have breakfast (in the bottom) that takes place between 6:30 and 8:15 and have diner (in the top) between 19:30 and 20:30. In this simulation, individuals do watch TV during 4 periods: (1) After breakfast, (2) during the afternoon on week-ends and wednesday (3) In January, individuals mostly watch TV before the diner though a slow change from after diner to after breakfast habit is appearing all along this month. When February comes, with its special pricing, the before diner period is abandoned to the benefit of the after breakfast and after-diner periods due to the energy price penalty. One may notice that a slim black line remains before diner in February. Indeed only 2 agents have completely quit watching TV before diner. The third one continues to do it but for a shorter duration. It means that the other influences, mainly the strength of the before-diner PP, have balanced the energy-price penalty. It is interesting to notice that the effect of the penalty is progressive (it may completely stop or just shorten an action) and vary from one agent to another. In March, when the special pricing is stopped, we can observe a motion from afterbreakfast and after-diner periods to the before-diner period as there is no electricity price penalty. Nevertheless, the last two weeks of simulation still bear this penalty influence (as adaptations are a progressive process): the before-diner period happens later and lasts shorter. This evolution demonstrates the capacity both to create and to modify habits thanks to the adaptation of the PP.

Note that, the movement of the watch TV action implies that other action have moved too. On the second long-term diagram (3 (b)), we track the evolution of the do homework action (red). As the time slot occupied by the watch TV action in January is free in February, the action do homework happens more often during the before-diner period. We observe that the return of an uniform pricing does not push back the do homework action: Energy special pricing effects persists even after its end.

Conclusion and current work

In this paper, we presented the model and implementation of SMACH, a multi-agent simulator of human activity. This simulator supports the investiga-tion of activity adaptation and energy consumption evolution in response to different appliances or pricing policies. We could not present all the details of this simulator that results from a 5-year collaboration between the French EDF energy company and AI researchers, but we outlined its main features: accessibility of the model to non-computer specialists and intelligent interface for activity analysis and explanation. SMACH also comprises a machine learning algorithm for automated management of temperature in the house (Gil-Quijano and Sabouret, 2010) and an interactive modelling system that helps refining the agent's behaviour [START_REF] Sempé | Incremental and situated modeling for multi-agent based simulations[END_REF]. This model has been evaluated in several example scenarios. The two examples presented in this paper illustrated the ability of our model to represent both the constraints and some of the "degrees of freedom" of everyday life. Furthermore, with the help of the two adaptation processes (actions competition and habits emergence), agents are able to explore new organisation and to discover pattern of actions in relation with time constraints and energy price for saving purpose. In addition to this evaluation, our simulator is provided with a participatory-simulation user interface (introduced in [START_REF] Haradji | Human Activity and Social Simulation[END_REF] ) that allow to give control of one or several agents to users. Students in our lab "played their own role" in our test scenarios, which allowed us to validate the believability of the model. In most situations, students could not distinguish between artificial agents and humancontrolled ones.

We are currently extending the SMACH model to study the activity of groups of families in different environments, over long period of time (one year) and taking into account external temperature and building's thermodynamical properties. This lead us to reconsider the action rhythm model and to use multilevel agent systems for individuals, families and activities. Our long-term goal is to allow energy companies to be able to investigate incentive to reduce or to have a better prediction of consumption peaks using simulation of human activity.

  status(a) ⇒ ∀a |actor(a) = actor(a ),i d , a , s and + i d 0 , a , done ; • A task "must" be over after its maximum duration: if ∃b ∈ B i such that b.d + b.a.t.τ max > d 0 , theni b and + i d 0 , b.a, done ;

Figure 1 :

 1 Figure 1: SMACH GUI (analysis mode)

Figure 2 :

 2 Figure 2: Activity diagram: (a) first week and (b) working day pattern

Figure 3 :

 3 Figure 3: Long term activity diagram: (a) watch Tv and (b) homework

Table 2 :

 2 Raw priority external factors.

http://remodece.isr.uc.pt

When different individuals can perform the same task, each one is associated with a different instance.

The implementation detail of this mechanism is not detailed in this paper but it is very similar to the one presented in the previous subsection

In addition, another simulation analysis GUI mode is also available in order to compare several simulations: overall evolution and specific period of time can be compared with side by side diagrams.


Jérôme Rouquette

Julien Haines

Véronique Bornand

Philippe Philippe

Pa Pa

Pa Papet

Piezo-Ferroelectrics: Pressure-induced phase transitions in Pb(Zr 1-x Ti x )O 3 (PZT)

) octahedral tilting leading to unit cell doubling. Future studies as a function of pressure, temperature and electric field on single crystals of heterovalent relaxor ferroelectric isotypes of PZT are proposed in order to to better understand these potentially technologicallyimportant piezo-ferroelectric materials.

Introduction

Lead zirconate titanate solid solutions Pb(Zr 1-x Ti x )O 3 (PZT) represent one of the most important classes of piezoelectric materials because of their excellent properties. These materials are used in different systems such as piezoceramics (piezoelectric transducers and actuators) and thin films (MEMS system or high-frequency components). The highest piezoelectric constants and electromechanical coupling coefficients have been obtained for compositions near the morphotropic phase boundary (MPB) between rhombohedral (R3m) and tetragonal (P4mm) phases (x ≈ 0.48) 1 . Recently the MPB has attracted particular attention due to the discovery of a monoclinic phase (Cm) in this composition range at temperatures below 300K [START_REF] Noheda | [END_REF] . In this monoclinic form, the polarization can be modulated for example as a function of an external applied electric field between the rhombohedral and tetragonal directions based on a polarization rotation mechanism thus providing a possible explanation for this high piezoelectric response 3 . In addition, a doubling of this monoclinic unit cell was reported at low temperature (T ≤ 210 K) for PbZr 0.52 Ti 0.48 O 3 due to antiferrodistortive tilting of the (Zr,Ti)O 6 octahedra based on the appearance of superlattice reflections in electron 4 and neutron diffraction 5 data. This complex phase diagram around the MPB does not take into account further local structural distortions arising from the static disorder of the zirconium and titanium ion displacements 6,7 which are, in addition, at the origin of the well-known ferroelectric domain structure. Furthermore, recent experiments based on electron diffraction 6 and neutron diffraction studies 8 , Raman scattering results 9 , as well as theoretical calculations 10 have indicated the existence of an intrinsic short range dynamic disorder over nearly the entire PZT solid solution predominantly due to offcenter lead displacements 8,11 . Therefore, the boundaries between the different phases, in particularly close to the MPB are still subject to debate.

Due to their high polarizability, ferroelectric materials exhibit large dielectric constants and excellent piezoelectric properties. These characteristics are extremely dependent on three variables (for any given composition): the applied electric field (E), the mechanical stress (σ ij ) and the temperature (T) [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF] . In addition to the ongoing investigation of the relatively well-studied, highly complex, composition-temperature phase diagram of PZT, the effect of hydrostatic pressure (σ ij = P for i = j) is therefore of great importance in order to better understand the dependence of both structural and dielectric properties on the two fundamental variables P and T for such a ferroelectric system. In the present review, recent investigations of PZT as a function of pressure and temperature by neutron and x-ray diffraction, Raman spectroscopy and dielectric measurements to characterize the multi-scale behaviour of this material at the nano-micro-macro-scopic range are presented.

Study of morphotropic PbZr 0.52 Ti 0.48 O 3

Atmospheric pressure neutron diffraction data as a function of the temperature were consistent with the presence of the ambient-temperature (Cm) and low-temperature (Cc) monoclinic forms [START_REF] Rouquette | [END_REF] in PbZr 0.52 Ti 0.48 O 3 . Interestingly, the associated temperature phase transition sequence observed by diffraction 4,5 and dielectric measurements 14,15 could also clearly be distinguished by resonance Raman spectroscopy 16 . In the literature, Raman measurements reported for composition close to the MPB showed only minor spectral changes in the temperature-composition range consistent with the static and dynamic disorder associated with the PZT solid solution 17,18,19,20 , Fig. 1a. Using an excitation line of 647.1 nm with an energy close to a self-trapped level exciton energy deficient complex (Ti Ti' -V O .. ) 21,22,23 of PZT powder, a resonance effect was induced. In order to illustrate this difference in behaviour, Raman spectra obtained as a function of temperature with both 647.1 nm and 514.5 nm excitation lines are represented on Fig. 1. Resonance Raman spectroscopy clearly allows the structural changes of PZT to be identified as a function of P and T [START_REF] Rouquette | [END_REF]16,24,25,26 .

Early high pressure studies of the morphotropic composition PbZr 0.52 Ti 0.48 O 3 were restricted to the determination of the phase boundary between the ferroelectric tetragonal and cubic paraelectric forms at high temperatures for pressures up to 0.8 GPa 27,28 . Neutron diffraction data indicate that at close to room temperature, the cubic form is not obtained at high pressure but instead a series of low-symmetry phases are obtained. Superlattice reflections, arising from a doubling of the elementary pseudo-cubic perovskite unit cell induced by antiferrodistortive tilting of the (Zr,Ti)O 6 octahedra. are found to appear at low pressure (P ≥ 1.1 GPa) and are characteristic of the monoclinic Cm-Cc phase transition, the same transformation that is observed at low temperature and ambient pressure 5 . Note the importance of the use of neutron diffraction, which enables to distinguish superlattice reflections to be detected. These would be extremely weak in x-ray diffraction. Neutron diffraction data obtained at 288 K and 3.8 GPa are consistent with a triclinic structure, space group F1 (Fig. 2) 30 . The superlattice reflections further increase in intensity in this phase.

The structural data obtained from neutron diffraction experiments can be used to calculate the spontaneous polarization (P s ) and the polarization rotation angle (θ) with respect to the pseudo-tetragonal z-axis in the monoclinic phase (Fig. 3).

Additionally, the high-pressure distorted perovskite F1 phase exhibits a second rotation angle φ (calculated from P s(x) and P s(y) ) to describe the orientation of the spontaneous polarization. The spontaneous polarization calculated from the structural data at 3.8 GPa and 288 K is 17 µC/cm 2 with an in plane (110) component of rotation angle (θ) of 30°. With increasing pressure, a transformation to a pseudo-cubic paraelectric phase ( 1 F ) has been identified. In this centrosymmetric structure, Fig. 4, the Zr, Ti cations are located on inversion centers and the constituent octahedra are tilted about directions parallel to a, b and c. Such a space group is consitent with the persistence of the observed highpressure Raman spectra in the paraelectric state [START_REF] Rouquette | [END_REF]16,24,25,26 . In contrast to ambient-pressure paraelectric cubic m Pm3 structure, the first-order Raman signal is not forbidden by group theory for the 1 F phase.

The results obtained enable a P-T phase diagram to be construct for PbZr 0.52 Ti 0.48 O 3 , Fig. 5. In this diagram, data obtained from Raman spectroscopy, dielectric measurements and x-ray diffraction are also reported. Additionally, the spontaneous polarization vector calculated from neutron diffraction studies are represented for each P-T point. The polarization rotation mechanism and the associated reduction in polarization as a function of P and T are clearly visible.

Study of Ti-rich PZT

The data obtained by neutron diffraction on PbZr 0.40 Ti 0.60 O 3 (which is tetragonal under ambient conditions) indicated the presence of the monoclinic structures Cm and Cc respectively at 2.7 and 4.1 GPa, Fig. 6a and 6b, the same ferroelectric forms which are proposed to be responsible for the high piezoelectric properties characteristic of the MPB composition PbZr 0.52 Ti 0.48 O 3 at ambient pressure 31 . This means that hydrostatic pressure can tune the MPB and its associated exceptional physical properties by stabilizing the monoclinic phases to include compositions that do not normally lie on the MPB. Both phase transitions were also clearly characterized from spectral changes (frequency, intensity and damping) in Raman spectroscopic data and from the changes in compressibility in an x-ray diffraction study. Additionally, the P4mm-Cm transformation was also identified from the change in slope of the inverse of the dielectric constant (1/ε' r ) as a function of pressure. The latter Cm-Cc phase transition can directly be detected by neutron diffraction from the appearance of superlattice reflections, Fig. 6, indicating a doubling of the unit cell due to octahedral tilting. Fig. 7 represents the 111 and 200 pseudocubic reflections of PbZr 0.40 Ti 0.60 O 3 at 9.6 GPa (highest pressure reached from neutron diffraction experiments) obtained from Rietveld refinements using either the rhombohedral R3c or triclinic F1 space group 32 . Note that the Cc structure was also used as a starting model. Out of these three structural models, the worse refinement was obtained using the rhombohedral structure (Rwp = 5.7%, Rp = 8.7%). The difference from experimental and calculated profiles can clearly be seen in Fig. 7. The 111 reflection should be split for a R3c structure a relative intensity ratio of 0.9. This intensity ratio does not correspond to the observed data. Similarly, the 200 reflection is not split in rhombohedral symmetry, which thus cannot reproduce the experimental data. Consequently the space group R3c is not observed at high pressure for these PZT solid solutions. The best refinement is obtained using the triclinic model (Rwp = 4.9%, Rp = 7.3%). In such a structure, rotations angles ω a and ω b which are identical in Cc symmetry become different, Table 1, and the spontaneous polarization is described with a second rotation angle φ. Use of a Paris-Edinburgh high-pressure cell with tungsten carbide cells limits this work to pressures of about 10 GPa. The paraelectric phase 1 F observed for The in-plane rotation angle of the spontaneous polarization (θ) increases from 0° in the P4mm structure to 76° in the F1 form. This value is close to the pseudocubic [112] direction previously observed for the polarization vector in the recent study of the F1 structure in PLZT (θ=70.5°) 33 , indicating that the rotation of the in-plane component of P s principally occurs between the [001] and the [112] directions. This highpressure rotation of P s is found to be a general feature of PZT solid solutions. Finally, the high-pressure compression behavior is dominated by octahedral tilting. This compression mechanism appears at the Cm-Cc phase transition. The Cc phase is characterized by two distinct rotation angles ω a and ω c . These angles increase on compression and can be followed by the increase in intensity of the superlattice reflections. At the Cc-F1 phase transition, ω a and ω b become distinct as expected in triclinic symmetry. The titanium-rich PZT solid solution PbZr 0.20 Ti 0.80 O 3 was studied under high pressure by X-ray diffraction and Raman spectroscopy 26 . Four high-pressure phase transitions were observed as in the case of PbZr 0.40 Ti 0.60 O 3 . Based on the similarity of the Raman spectra and the P-V data obtained by x-ray diffraction for both compositions, it was proposed that the same phase transition sequence was observed (albeit at higher pressures), including monoclinic (Cm, Cc) and triclinic forms (F1 and F1). In particular, high pressure Raman spectra obtained are consistent with a monoclinic (Cc) -triclinic (F1) phase transition involving a soft mode at 9 GPa, in agreement with the change in compressibility detected by X-ray diffraction. The strong Raman spectra observed in the paraelectric phase is also in agreement with triclinic F1 symmetry. Dielectric measurements of Pb(Zr 1-x Ti x )O 3 for x = 0.48, 0.50, 0.52, 0.56 and 0.60 were studied from ambient pressure up to 2 GPa 31,34 , Fig. 8. Two distinct types of behaviour can be observed: 1) the dielectric constant of Pb(Zr 0.52 Ti 0.48 )O 3 is characterized by a hysteresis loop with a change in slope of ε' r consistent with a phase transition that does not have the same value on compression and decompression and which are represented by arrows on Fig. 8; 2) ε' r of PZT for x = 0.52,0.56 and 0.60 have the same trend on compression and decompression. From the change in slope of the inverse of the dielectric constant (1/ε' r ) as a function of pressure, a transformation can also be detected. Effectively, the dielectric constant is known to diverge at the ferroelectric-paraelectric phase transition but also at a ferroelectric-ferroelectric one as a function of temperature, electric field and pressure [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF] . These two kind of tendencies can be linked to two different phase transitions that were previously observed by neutron diffraction, Cm-Cc for Pb(Zr 0.52 Ti 0.48 )O 3 , P4mm-Cm for the rich-Ti compositions. The former could be ideally detected around the same transformation pressure by a change in volume from the compressibility of the material in x-ray diffraction data, from the appearance of new bands in Raman spectra associated to the doubling of the unit cell or by superlattice reflections in neutron diffraction. The latter transformation was characterized by a change in slope of the volume under pressure, by spectral changes in Raman scattering or by fine features like the line shape in neutron diffractogram. Interestingly, Pb(Zr 0.50 Ti 0.50 )O 3 exhibits both phase transition from dielectric measurements, Fig. 8, respectively around 0.8 and 1.6 GPa. This is clearly consistent from our structural data which showed that this composition was tetragonal, space group P4mm, at ambient pressure and temperature 35 .

P (GPA)

T (K) P S (µC/CM 2 ) θ (°) ϕ (°) ω A (°) ω B (°) ω C (°) 0.
The P4mm and Cm phases of PZT are known to exhibit both dynamic and static disorder from Raman spectroscopy 9 , neutron diffraction 8 and electronic diffraction 6,7 . The reproducibility of dielectric measurements in both compression and decompression data associated to the same value of the transformation pressure imply a continuous structural phase transition. This continuous behaviour was already observed by x-ray diffraction (compressibility modulus), neutron diffraction (weak value of the rotation angle of P s for Pb(Zr 0.40 Ti 0.60 )O 3 at 2.7 GPa) or even Raman spectroscopy ( P ∂ ∂ν ). In this way, this P4mm-Cm transformation is closer to a second order rather than a first order phase transition. From group theory [START_REF] Stokes | Isotropy Subgroups of the 230 Crystallographic Space Groups[END_REF] , this phase transition has therefore both a ferroelectric and a ferroelastic order parameter, respectively the rotation of the spontaneous polarization in the (110) plane and the primary order parameter: the spontaneous deformation e 4 (2e23) =-e 5 (2e13). The Cm-Cc transformation is purely first order consistent with a change in volume from x-ray diffraction data due to the absence of the relevant compression mechanism "octahedral tilting" in Cm space group. Such a first order transformation must generate hysteresis phenomenon (thermodynamic equilibrium) and kinetics behaviour which could explain the different transition values obtained by neutron diffraction and dielectric measurements. This event was also clearly observed from the Raman spectral changes as a function of pressure ( P ∂ ∂ν ) which were associated to a coexistence of phases. This could also explain electronic diffraction data from Noheda et al. which proposed the conservation of the Cm phase down to 10 K with a small amount of Cc proportion (30%) [START_REF] Noheda | [END_REF] . These findings could be explained by a monoclinic Cm-Cc solid state transformation associated to low kinetics. From group theory this phase transition is nonferroïc, its order parameter is therefore the tilting in antiphase of the oxygen octahedral.

Conclusion and future developments

The effects of pressure and temperature on the most commonly used piezoelectric perovskite solid solution PZT are reviewed in the present work. Based on x-ray and neutron diffraction, dielectric measurements and Raman spectroscopy it is shown that high pressure can tune the morphotropic phase boundary (associated with the exceptional piezoelectric properties) in the composition-pressure plane to include compositions that do not normally lie in the MPB at ambient pressure. Thus, by controlling the appropriate composition and stress state of materials in ceramics and thin films, optimal piezoelectric properties may be obtained for technological applications. This corresponds to a potentially direct application of the pressure parameter to highperformances ferroelectric compounds. Upon increasing pressure, a sequence of low-symmetry structures are identified (Cm, Cc, F1, 1 F ). The high pressure behaviour is consistent with a reduction and a rotation of the spontaneous polarization and the onset of octahedral tilting leading to unit cell doubling. The results obtained have enabled us to construct a pressuretemperature, Fig. 4, and a pressure-composition phase diagram of PbZr 0.52 Ti 0.48 O 3 , Fig. 8. . Dot lines correspond to boundaries proposed from rich-Zr composition. Note that our first results show that R3m is not the correct phase even at ambient pressure. Pc and A OI ~ Pbam were proposed by Woodward et al. 7 . A OI -A OII orthorhombic phase transition is proposed based on high pressure x-ray diffraction and dielectric measurements 40 .

Preliminary results support the extrapolation of the Cm-Cc phase boundary towards rich-Zr compositions 41 . These new data tend to indicate that the rhombohedral R3m phase may not be stable even at atmospheric pressure, as previously suggested by Pandey et al., and that Cm may be the stable phase 42 . This would imply direct group-subgroup relationships at ambient conditions from the P4mm tetragonal space group of PbTiO 3 endmember to the Cm-Cc monoclinic forms of the MPB and then to the R3c supergroup contrary to the originally proposed P4mm-R3m-R3c series of transformations. The boundary with the paraelectric phase is proposed to have a positive slope from the MPB to rich-Zr compositions based on these first preliminary results. This interesting finding implies the existence at least of another phase at high pressure. These features along with the probable complex phase transition sequence for compositions close to lead zirconate make these further investigations particularly necessary and challenging.

As previously mentioned, ferroelectric properties are dependent to the three parameters: stress-electric field-temperature. The present work shows the importance to undertake coupled investigations as a function of P and T in order to determine the influence of two of the three variables on which ferroelectric properties depend. In the future, the next step will be to extend these studies to well-known relaxors-ferroelectric solid solutions (1x)Pb(Zn 1/3 Nb 2/3 )O 3 -xPbTiO 3 + PMN-PT which are heterovalent isotypes of PZT, exhibit a MPB region and shows giant piezoelectric response (superior to PZT) in single crystal form (contrary to PZT). The single crystal form also provides three dimensional information and enables the materials to be polarized in the desired orientation. In this way, it would be possible to characterize these materials as a function of the three fundamental variables, P, T, and E , to have an overall view and a better understanding of this particular family of materials.

Figure 1 .

 1 Figure 1. Raman spectra obtained between 10 and 300 K with a) λ laser = 514.5, and b) λlaser = 647.1 nm. Positions of the Raman peaks at room temperature for PbTiO 3 are indicated 29 .

Figure 2 .

 2 Figure 2. Experimental and calculated profiles from Rietveld refinements of the triclinic F1 phase of PbZr 0.52 Ti 0.48 O 3 at 288 K and 3.8 GPa. The difference profile is on the same scale. Vertical ticks indicate, from the top down, the calculated positions of the F1 phase, tungsten carbide and nickel from the anvils of the pressure cell and lead. Arrows indicate the strongest superlattice reflections.

Figure 3 .

 3 Figure 3. Spontaneous polarization P s (three components: P s(x) , P s(y) and P s(z) ) with the two rotation angles (θ, φ) in triclinic F1 symmetry. In monoclinic phases (space group Cm, Cc), P s has two components (P s(x) = P s(y) and P s(z) ) with one rotation angle (θ).

Figure 4 .

 4 Figure 4. Crystal structure of the triclinic 1 F phase of PbZr 0.52 Ti 0.48 O 3 .

Figure 5 .

 5 Figure 5. P-T phase diagram of PbZr0.52Ti0.48O3. The solid lines represent phase boundaries obtained from X-ray diffraction, Raman spectroscopy and dielectric measurements; Dashed lines correspond to proposed P-T boundaries.

Figure 6 .

 6 Figure 6. Experimental and calculated profiles from Rietveld refinements of (a) the monoclinic Cm phase of PbZr 0.40 Ti 0.60 O 3 at 2.7 GPa and (b) the monoclinic Cc phase of PbZr 0.40 Ti 0.60 O 3 at 4.1 GPa. The difference profile is on the same scale. Vertical ticks indicate the calculated positions of the PZT phase (top) and of tungsten carbide (middle) and nickel (bottom) from the anvils of the pressure cell.

  PbZr 0.52 Ti 0.48 O 3 was not identified in this work for PbZr 0.40 Ti 0.60 O 3 . Extrapolation of P s → 0 using an arbitrary second order polynomial can give a good insight of the ferroelectric-paraelectric pressure transformation. The corresponding value is about 10-11 GPa in accordance with results obtained from x-ray diffraction and Raman spectroscopy measurements. By analogy with PbZr 0.52 Ti 0.48 O 3 , the structural transition 1 1 F F → is proposed for this ferroelectric-paraelectric transformation.

Figure 7 .

 7 Figure 7. Experimental and calculated profiles from Rietveld refinements of (a) the (111) and (b) (200) pseudocubic reflections R3c phase of PbZr 0.40 Ti 0.60 O 3 at 9.6 GPa. (c) (111) and (d) (200) pseudocubic reflectionc F1 phase of PbZr 0.40 Ti 0.60 O 3 at

Figure 8 .

 8 Figure 8. Dielectric constant of PbZr 1-x Ti x O 3 measured at 100 kHz versus pressure for selected compositions (x). Pressure-cycling was performed for each composition. Solid and empty symbols correspond to values obtained on

Figure 8 .

 8 Figure 8. Pressure-composition phase diagram of Pb(Zr 1-x Ti x )O 3 . Solid lines are boundaries from the present study. Dashed lines are extrapolation to lead titanate end member with the paraelectric transition to m Pm3 at 12 GPa from Raman data38 . Note that recent first principles calculations reported different high pressure

Table 1 .

 1 Spontaneous polarization P s and its associated rotation angles (θ,φ), octahedral rotation angles (ω a ,ω b ,ω c ) as a function of P and T.

	0001	280	36.8	0	0	0	0	0
	2.7	295	32.6	4.3	0	0	0	0
	4.1	295	30	42	0	0.8	0.8	2.6
	6.6	295	23	48	0	2	2	4.4
	9.6	295	8	76	24	3	3.1	4.3