MODEL AND ALGORITHMS FOR THE REAL TIME MANAGEMENT OF RESIDENTIAL ELECTRICITY DEMAND. A. Barbato, G. Carpentieri

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1 MODEL AND ALGORITHMS FOR THE REAL TIME MANAGEMENT OF RESIDENTIAL ELECTRICITY DEMAND A. Barbao, G. Carpenieri Poliecnico di Milano, Diparimeno di Eleronica e Informazione, barbao@ele.polimi.i, giuseppe.carpenieri@mail.polimi.i ABSTRACT Household Demand Side Managemen (DSM) sysems play a key role in improving he efficiency of he enire elecrical sysem. An efficien managemen of he energy resources can indeed allow spreading domesic energy loads in a smar way in order o reduce he peak power of he overall demand. To achieve his goal, home appliances and energy sorage sysems have o be conrolled hrough he definiion of energy plans for fuure periods (offline) and he real ime conrol of energy resources (online). In his paper, we propose an opimizaion model and a se of heurisics for he online demand side managemen, o properly reac in real ime o evens which were unexpeced in a previous offline sage of he home managemen sysem, such as wrong weaher forecass or user s misbehaviours. The final goal is o reschedule he energy consumpion and producion coherenly wih he energy plan defined in he offline phase, aking ino accoun he real producion of renewable energy sources and he capaciy of sorage devices. Along wih he heurisics, we show he numerical resuls obained by applying hem on realisic insances of he problem. Index Terms Energy Demand Managemen, Residenial Buildings, Smar Grid, Heurisics; 1. INTRODUCTION Demand side managemen sysems are receiving a growing aenion in he developmen of he fuure smar grid. Infac, he applicaion of a funcional DSM sysem can significanly improve he efficiency of he energy producion and disribuion, in paricular in residenial conexs. The residenial secor represens one of he mos promising scenarios for using DSM capabiliies. Domesic users, infac, are one of he major conribuors o he naional energy balances, and an even more criical siuaion is forecas for he near fuure since he home energy consumpion will probably exceed 40% of he oal yearly consumpion in mos of he wesern counries. A DSM can be used o modify or reduce residenial users energy demand hrough an offline and/or online approach. In he offline approach, he DSM defines he opimal energy plan for fuure periods by scheduling loads, deciding when o buy or sell energy and so on, according o forecass (PV producion, devices usage, ec.). In he online case, he DSM defines in real-ime he energy plan of households based on he acual siuaion of he energy marke and on users needs. In his paper, we propose a par of he DSM sysem developed wihin he BEE Projec [1]. This sysem has been designed for managing he elecriciy consumpion of residenial users, wih he objecives of minimizing he energy bills and improving he efficiency of he elecriciy grid. To his purpose, we combine he offline and online approaches. The offline demand opimizaion model [2] has been designed o schedule he nex day house appliances aciviies and power exchanges wih he grid. In order o define he fuure energy plan, he opimizaion model requires predicions on he phoovolaic panels producion and he users preferences abou he home devices fuure usage. The oucome of his offline DSM is he 24 hour profile represening he energy ha is planned o be purchased\sold from\o he grid for he nex day. This informaion can be used by he energy reailer o opimize he grid managemen. Unforunaely, predicions are naurally subjec o uncerainy, so he real energy demand and supply are ofen differen from he planned profiles. For his reason, he advanages obained by adoping he offline mehod are nullified. In his paper, we propose an online DSM model, which deals wih real ime evens ha were unexpeced or wrongly forecas. The final goal is o opimally reschedule home devices aciviies and decide when o sore\buy\sell energy o he grid, coherenly wih he offline plan. The remainder of his paper is organized as follows. In Secion 2 we review previous works on offline and online demand side managemen. In Secion 3 we describe he basic characerisics of he offline model designed wihin he BEE projec. In Secion 4 we presen an online model and wo heurisics developed o quickly reac o real ime evens. Secion 5 repors he numerical resuls ha we have obained by esing he algorihms based on realisic daa. Finally, he paper is concluded in Secion 6.

2 2. RELATED WORK Demand side managemen has been deeply sudied by he scienific communiy because of he possible advanages achievable hrough his kind of mechanism, such as he peak shaving and load shifing. The execuion of many home appliances, for example, may be posponed wih only a lile discomfor for households. Even jus opimizing his procedure may lead o significan peak reducions. In lieraure, wo kinds of demand side managemen are proposed: offline and online. In he offline scenario, DSM mechanisms [3], [4], [5] are designed o define he opimal energy plans of households for fuure periods, based on daa forecass. In he online scenario, a real ime energy plan is defined according o evens ha occur during he day. In [6] an opimizaion model is presened o adjus he hourly load level of a given consumer in response o elecriciy prices. The objecive of he model is o maximize he uiliy of he consumer, subjec o several consrains such as minimum daily energy consumpion levels and limis on hourly loads. Real ime managemen is also discussed in [7]. In his case a mulilevel opimizaion framework for demand-side load managemen of a group of houses is proposed. The conrol algorihm provides predicions on energy consumpion and he possibiliy of adjusing he energy allocaion in real ime is inroduced, o cope wih errors in he forecas energy consumpion. Finally, in [8] he auhors propose a sysem which works as an online smar demand response soluion and is based on an explici demand-based power supply conrol. By applying he mehod i is possible o reduce he energy consumpion wihou damaging he comfor of domesic users. Moreover, energy reailer companies are allowed o se and modify power ceiling values based on conracs wih consumers. Wih respec o mos of hese papers, our online model is based on he oucome provided by an offline demand side managemen model and is goal is o reac o real ime evens by rescheduling he energy consumpion coherenly wih he offline plan. Moreover, more aenion has been spen on modeling household conexs: we ake ino accoun a realisic domesic scenario (domesic loads, PV generaors, baeries, energy prices) as well as he user requiremens. 3. BEE PROJECT FRAMEWORK The BEE (Brigh Energy Equipmen) Projec [1] is a research aciviy of Poliecnico di Milano. The purpose of he projec is o provide advanced ools o residenial users in order o make hem an acive par of fuure Smar Grids. In he considered scenario, houses can be equipped wih PhooVolaic (PV) panels, baeries and sensors (e.g. power meers for he monioring of he energy consumpion of home devices).the core of he framework is he BEE box, a smar processing uni which manages and opimizes he home energy plan and exchanges daa wih he oher acors of he elecrical sysem. The home energy managemen is performed in wo differen seps, offline and online. The offline DSM opimizaion model [2] is inroduced wih he ask of scheduling he house appliances aciviies and power exchanges wih he elecriciy nework for he nex day. This model uses he predicions on PV producion and devices fuure usage. For he PV plan, we defined an ad-hoc learning mehod which predics he panels producion based on he weaher forecas. Similar algorihms have also been inroduced o predic he house load demand [9] (i.e. which home appliances will be used and a wha ime of he day) based on daa provided by he power meer sensors. The offline opimizaion model designed in he BEE Projec [2] is shorly presened in he following subsecion Offline Opimizaion Model The problem is modelled as a Mixed Ineger Linear Programming (MILP) model. The 24 hour dayime is divided ino 96 ime slos of 15 minues each (se T ). In order o schedule house appliance aciviies (se A), a se of binary variables is defined for each aciviy a A and for each ime slo T, equal o 1 if he aciviy a sars in he ime slo, 0 oherwise. Besides, he coninuous non-negaive variables y OF F and z OF F represen he amoun of bough and sold energy, respecively, in each ime slo. x OF F a Objecive Funcion The objecive of he model is o minimize he daily energy bill. Denoing wih c and g respecively he cos of bough and sold energy in he ime slo, he objecive funcion can be defined as: min T ( c y OF F g z OF F ) Aciviy scheduling For every aciviy a A, associaed wih he execuion of a house appliance, he devices predicion algorihm presened in [9] auomaically compues is earlies saring ime, ST a, and is laes saring ime, ET a, ha define a ime window in which he aciviy can be launched. Consrains: ET a (1) x OF F a = 1 a A (2) =ST a guaranee ha he aciviy a is carried ou in he required inerval (ST a, ET a ). Baery consrains The charge and discharge raes are represened by he coninuous non-negaive variables cr OF F and dr OF F. Such variables are bounded, for each T, according o he following consrains, where τ max and ϑ max are he maximum charge and discharge raes, respecively: cr OF F τ max, dr OF F ϑ max T (3)

3 In each ime slo, he baery energy depends on he energy in he previous ime slo, and on he charge and discharge raes, according o he following consrains: e OF F = e OF ( 1) F + crof F dr OF F T (4) Finally, he energy charge level of he baery can exceed is capaciy γ max : e OF F γ max T (5) Balancing consrains These consrains force he balance beween he acquired energy (firs member) and he consumed energy (second member) in each imeslo: y OF F + π OF F + dr OF F = z OF F + p OF a F + cr OF F a A (6) where π OF F represens he prediced PV producion in he ime slo and p OF a F is he energy consumed by he device a in he slo, depending on is saring imeslo. Based on daa forecass (PV panels producion and load demand) and energy ariffs, he opimizaion model defines an energy plan for he nex day which minimizes he daily bill. The model, in paricular, schedules when o buy and sell energy (i.e. y OF F and z OF F, see Figure 1) and when o sar home appliances. The oupu is an energy plan which can be used by he energy reailer o opimize is producion/disribuion. Power [W] Energy Demand Energy Supply Demand\Supply Profiles Day Time [h] Fig. 1. Example of offline profiles (y OF F, z OF F ). 4. ONLINE OPTIMIZATION MODEL The offline mehod presened in Secion 3.1 requires he predicion of some parameers which are naurally subjec o uncerainy, so he user energy demand and supply may acually be differen from he planning. For his reason we propose a new opimizaion model which reschedules home aciviies (appliance execuions, sorage operaions) in real-ime, wih he goal of obaining a demand and supply profiles (i.e. y OF F and z OF F ) which are as close as possible o he offline profiles. Since in he offline model he 24 hour dayime is divided ino 96 imeslos of 15 minues each (se T ), his model is conceived o be solved a he beginning of each imeslo. In real-ime, anyway, we no longer have inaccurae daa bu precise daa. In paricular, a any ime slo, we know he se of devices which have been used or are sill running (x ON a(1... 1) ), he overall consumpion of he devices in he curren ime slo ( a A pon a ), he energy charge level of he baery e ON 1 and he real energy producion of he phoovolaic panel (π ON ). Based on hese daa, he online model has o define he value of he following variables for he curren ime slo, saisfying he consrains presened in Secion 3.1: baery charge or discharge raes cr ON and dr ON, amoun of energy o buy and sell, respecively y ON and z ON and he schedule of he appliances for he curren and fuure ime slos x ON a(, ). The final objecive of he model is o keep he demand and supply profiles as close as possible o hose defined by he offline phase (i.e. y OF F and z OF F ). We call Profile Error (P E ) for each imeslo he difference beween he real profile and he scheduled one: P E = y ON y OF F + z ON z OF F (7) Every 15 minues, he energy plan is rescheduled according o he following objecive funcion: 96 P E min P E = (8) 96 =1 subjec o he same consrains as he offline model, where we subsiue he OFF variables wih he respecive ON variables defined above. Noe ha hese variables are reaed as parameers for he imeslos prior o he acual. Minimizing PE allows avoiding high peaks due o unexpeced demands of high amouns of energy. As a new schedule is required almos insanly, in he following we will presen wo heurisics o efficienly solve his problem. The heurisics differ in he ime scale used o reschedule he energy plan, which can be single slo or muli slo. The single slo heurisic reschedules aciviies only for he curren ime slo, wih he goal of minimizing P E, while he muli-slo heurisic minimizes he value of P E by reopimizing hrough he remaining porion of he day. In his case, he day ahead forecass are reused for he imeslos subsequen o he acual one Single-Time Slo Online Heurisic In he single ime slo scenario, in every ime slo, he online algorihm has o define he energy demand and supply profile in (i.e. y ON and z ON ) wih he aim of minimizing he profile error P E. In order o achieve his goal, he algorihm can boh charge or discharge he baery and reschedule he devices ha have sill o be used. The algorihm used for minimizing P E performs he following four seps: Sep 1) No Baery and Appliances Scheduling The algorihm verifies if i is possible o use he same demand and

4 supply profile defined by offline model (i.e. y ON and z ON = z OF F = y OF F ), wih no changes in he schedule of he devices and wihou using he baery. For his reason cr ON and dr ON are boh se o 0 and he appliance execuion for he fuure ime slos is unchanged, so ha x ON x OF a(, ) F a(,+1,...96) =. The balancing consrain (6) is updaed according o he real ime value of he phoovolaic panel producion and of he energy consumed by home devices in he curren ime slo : y OF F + π ON = z OF F + a A p ON a (9) If equaion (9) is verified, hen i is possible o use he same demand and supply defined by offline model, P E is equal o 0 and he algorihm sops. Noice ha in his case, since cr ON and dr ON are boh equal o 0, he baery consrains (3), (4), (5) are saisfied as well as he appliances scheduling consrain (2), since he devices scheduling has no been modified. If equaion (9) is no verified, he algorihm goes o sep 2 o balance he inpu and oupu energy of he sysem described in (9) using he baery; Sep 2) Baery The algorihm readaps he charge and discharge raes. Two differen cases may occur: (a) If he inpu energy (i.e. he firs erm of equaion (9)) is greaer han he oupu energy, he baery can be charged o is se o he difference beween he inpu and he oupu energy, so ha he updaed balancing consrain (10) is verified: minimize P E. In his case, dr ON y OF F + π ON = z OF F is sill se o 0 and cr ON + a A p ON a + cr ON (10) If he baery consrains (3), (4), (5) are saisfied for he new values of cr ON and dr ON hen i is possible o use he same demand and supply defined by he offline model (i.e. y ON = y OF F and z ON = z OF F ), P E is equal o 0 and he algorihm sops. Oherwise he algorihm compues he maximum value of cr ON which saisfies he baery consrains. For he new charge rae value, he difference, D, beween he inpu and oupu energy in he consrain (10) is compued and he algorihm goes o sep 3. (b) If he inpu energy of equaion (9) is less han he oupu energy, he same kind of procedure described for he above case a) is performed excep for he fac ha in his case he baery has o be discharged in order o minimize P E. Sep 3) Appliances Scheduling The algorihm changes he schedule of appliances. In his procedure, we sar, in he curren ime slo, one or more among he devices ha sill have o be used. Once again, wo possible scenarios can occur: (a) The difference, D, beween he inpu and oupu energy in he balance consrain is posiive: in his case, i is required o increase he oupu energy in he balance consrain (hence, he energy consumed by devices a A pon a ) by he amoun D. For his purpose, we group he appliances which sill have o be launched (we call his se B A) and we define a subse of devices C B which may be sared in he curren ime slo wihou violaing he consrain (2): C = {b B : [ST b ; ET b ]} Each of hese devices is characerized by he corresponding power consumpion p c. The problem is o decide which devices belonging o C have o be sared in he ime slo o increase he devices oal consumpion by he amoun D. This is a paricular case of he 0-1 knapsack problem, a well know problem in combinaorial opimizaion: given he se of iems C, each wih a weigh and a value (in his case boh equal o p c ), deermine which iem o include in a collecion so ha he oal weigh is less han or equal o a given limi (in his case D) and he oal value is maximized. The knapsack problem is an NP-hard problem bu i can be efficienly solved by a number of algorihms [10]. Afer having defined he opimal subse of appliances o sar in he ime slo, he balancing consrain (10) is updaed wih he new value of he amoun of energy consumed by he devices in he curren ime slo (i.e. a A pon a ). If he difference D beween he inpu and oupu energy is 0, i is possible o use he same demand and supply defined by he offline model, P E is equal o 0 and he algorihm sops. Oherwise he algorihm goes o sep 4. (b) The difference, D, beween he inpu and oupu energy in he balancing consrain is negaive: he algorihm goes sraigh o sep 4. Indeed, in his case, i would be required o decrease he oupu energy and hence he amoun of energy consumed by he devices. Unforunaely, in our model appliances aciviies are non preempable so ha i is no possible o sop devices ha have already sared. Sep 4) New Energy Demand and Supply The algorihm he difference D in he balance consrain beween inpu and oupu energy. In his case he value of he profile error P E is equal o D. equally disribues on y ON and z ON 4.2. Muli-Time Slo Online Heurisic In he muli ime slo scenario, in every ime slo, he online model has o define he energy demand and supply profile in (i.e. y ON and z ON ) wih he aim of minimizing he mean value of he profile error, P E, compued hrough all he day long. In order o achieve his goal, he sysem can boh charge/discharge he baery and reschedule he devices ha sill have o be used. The heurisic used for minimizing P E performs he following wo seps: Sep 1) Single-Time Slo Mechanism The algorihm presened in Secion 4.1 is applied o he curren ime slo in order o define he values of y ON and z ON ha minimize he profile error P E based on he real ime value of he PV producion and he energy consumed by home devices. The same algorihm is also applied o every fuure ime slo h (i.e. for h = +1, ) based on he PV panel and devices usage forecas also considered in he offline sep (real ime daa are no ye available for hese slos) as o minimize P E h. A his

5 poin we compue he mean value, here called P E ref, of he profile error. If P E is less han P E ref he algorihm goes o sep 2 o ry and increase he error experimened in he curren ime slo wih he aim of decreasing P E ref, oherwise i sops. In fac, if P E is greaer han he error mean value, i would be useless o furher increase is value: we know for sure ha in ime slo we would make a very high error, while he nex ime slos may sill be subjec o errors, because of he uncerainy of he forecass used for planning he energy demand for hese slos. Sep 2) Slo Selecion The algorihm selecs he firs ime slo i (i = + 1, + 2,..., 96) in which he error P E i is greaer han P E ref. Le I = P E i P E. We apply he single-ime slo heurisic o he curren ime slo wih he goal of obaining a Profile Error Value (PEV) equal o: { P E + I if P E + I P E ref P EV = P E ref (11) oherwise where P E was he error obained in he curren ime slo in he firs sep of he algorihm. Subsequenly, he single slo mechanism is reapplied o every fuure ime slo h (i.e. for h = + 1, + 2,..., 96) using he phoovolaic panel and devices usage forecas also considered in he offline sep o minimize P E h. Finally, he new mean value of he profile error, P E new, is compued. If i is less han he one previously obained P E ref, hen P E ref is se equal o P E new and he algorihm performs again he sep 2 saring from i = + 1. Oherwise P E ref is no updaed and i is increased by one before execuing again he sep NUMERICAL RESULTS The online heurisics have been implemened in C++ and have been esed on several insances. We considered a configuraion obained from daa relevan o he Ialian sandard user [11]. I consiss of a residenial household having 11 home appliances, whose load consumpion profiles have been defined based on lieraure [12]. Moreover, he house is equipped wih a 1 kwp PV panel and a 3 kwp sorage baery wih a capaciy of 10 kwh or 30 kwh. Wih regard o he dimension of he saring ime window of each appliance (consrains (2)), we considered wo differen flexibiliy levels: low (ET a ST a = 3) and medium flexibiliy (ET a ST a = 5). Finally, we used he dynamic pricing ariff also adoped in [2]. We solved each insance wih he offline opimizaion model, o define he energy plan for he nex 24 hours. Aferwards, he online algorihms have been applied o reschedule he energy plan during he day. As for he phoovolaic panel we have esed wo paricular scenarios: in he firs case he producion of a sunny day is forecas bu a cloudy day occurs in real ime; in he second case, he producion of a sunny day is forecas bu a rainy day occurs in real ime, inducing a larger predicion error. We also evaluaed he performance of he online heurisics when errors in he device sar ime forecas occur. For his purpose we have generaed errors according o he probabiliy disribuion funcion of he predicion error experimenally obained for he devices sar ime forecas algorihm presened in [9]. Table 1 shows he resuls of our ess for he wrong phoovolaic predicion case. For each online algorihm, we repor he decreasing percenage of he mean value of he profiles error P E wih respec o an unmodified plan (i.e. he energy plan defined in he offline sage is sill used during he day even if errors in he predicion occur). Resuls show ha boh he online algorihms are able o reduce he profiles error caused by wrong predicions in he phoovolaic producion. As expeced, he muli-ime slo approach doesn inroduce a major improvemen on he performance of he sysem. In his case, in fac, predicions are sill used for defining he energy plan of fuure ime slos. Therefore, if fuure ime slos are affeced by forecas errors, as happening in our es scenarios, his procedure urns ou o be ineffecive. This resul is also confirmed by he performance evaluaion obained for he second scenario (sunny is he prediced weaher and rainy he real one, Table 1(b)). In minimizing he profile error, a key role is played by he baery ha gives he online mehods he flexibiliy o buy or sell energy (by charging and discharging he baery) jus o use he same demand and supply defined by he offline model. For his reason, he greaer he baery capaciy is, he more he performance of he online mechanisms improves. Moreover, when no baery is available, he performance of he online algorihms are badly affeced hus confirming he imporance of his elemen in aking decisions o fix wrong forecass. (a) Sunny-Cloudy Baery Sorage Capaciy 0 kwh 10 kwh 30 kwh No online W W W Singe Slo % % % Muli Slo % % % (b) Sunny-Rainy Baery Sorage Capaciy 0 kwh 10 kwh 30 kwh No online W W W Singe Slo % % % Muli Slo % % % Table 1. Mean value of he profiles error P E for wo PV forecas error scenarios ((a) prediced sunny\real cloudy, (b) prediced sunny\real rainy) wih a medium flexibiliy level. (a) Sunny-Cloudy Scheduling Flexibiliy Low Medium No online W W Singe Slo % % Muli Slo % % (b) Sunny-Rainy Scheduling Flexibiliy Low Medium No online W W Singe Slo % % Muli Slo % % Table 2. Mean value of he profiles error P E for wo PV forecas error scenarios (case (a) prediced sunny\real cloudy, case (b) prediced sunny\real rainy) wih a 10 kwh baery.

6 Table 2 shows he imporan role of he devices scheduling flexibiliy: he greaer he flexibiliy is, he more he sysem is able o reac o incorrec forecass by advancing or delaying he devices aciviies so o mach he energy demand and supply defined by he offline model. Similar consideraions can be done for he numerical resuls, represened in Table 3, obained in esing he performance of he online algorihms when errors occur in he device sar ime forecas. In his case, as expeced, a larger flexibiliy miigaes he effecs of errors in he predicion of he sar ime of appliances. (a) Baery Sorage Capaciy 0 kwh 10 kwh 30 kwh No online W W W Singe Slo % % % Muli Slo % % % (b) Scheduling Flexibiliy Low Medium No online W W Singe Slo % % Muli Slo % % Table 3. Mean value of he profiles error P E wih a medium flexibiliy level (a) and a 10 kwh baery (b). 6. CONCLUDING REMARKS In his paper we proposed a DSM online model developed wihin he BEE Projec [1], along wih wo efficien heurisics. These algorihms can be used o properly reac o errors in he offline energy profiles. The final goal is o plan he household operaions wih he objecive of minimizing he difference beween he real ime demand\supply profiles and he ones defined hrough he offline sage. The proposed mehods have been esed on daa relevan o he Ialian elecric marke in order o correcly appreciae he performance of each algorihm. Tess confirm he benefis of using a proper online DSM model along wih an offline plan. As expeced, a key role is played by he sorage devices, which can be effecively inegraed in he smar grid o guaranee he effeciveness of a priorly scheduled plan. Alhough having discussed he efficacy of he online algorihms, he proposed work represens jus one firs cu analysis and furher invesigaion is herefore required. The model, in paricular, can be exended as o consider he possibiliy of inerruping he execuion of any appliances, or o be inegraed ino a cooperaive muli-house conex. [3] D. Livengood and R. Larson, The energy box: Locally auomaed opimal conrol of residenial elecriciy usage, Service Science, vol. 1, no. 1, pp. 1 16, [4] A. Mohsenian-Rad and A. Leon-Garcia, Opimal residenial load conrol wih price predicion in real-ime elecriciy pricing environmens, IEEE Transacions on Smar Grid, vol. 1, no. 2, pp , [5] A. Molderink, V. Bakker, M. Bosman, J. Hurink, and G. Smi, Managemen and conrol of domesic smar grid echnology, IEEE Transacions on Smar Grid, vol. 1, no. 2, pp , [6] A. Conejo, J. Morales, and L. Baringo, Real-ime demand response model, IEEE Transacions on Smar Grid, vol. 1, no. 3, pp , [7] D. Ha, F. de Lamoe, and Q. Huynh, Real-ime dynamic mulilevel opimizaion for demand-side load managemen, in IEEM IEEE, 2007, pp [8] T. Kao, K. Yuasa, and T. Masuyama, Energy on demand: Efficien and versaile energy conrol sysem for home energy managemen, in IEEE SmarGridComm IEEE, 2011, pp [9] A. Barbao, A. Capone, M. Rodolfi, and D. Tagliaferri, Forecasing he usage of household appliances hrough power meer sensors for demand managemen in he smar grid, in IEEE SmarGridComm IEEE, 2011, pp [10] S. Marello and P. Toh, Knapsack problems: algorihms and compuer implemenaions. John Wiley & Sons, Inc., [11] R. Viadana and L. Croci, Domoic applicaions for he demand side managemen, Final Repor Projec ECORET. [12] Micene Projec. [Online]. Available: hp:// i/index.php?p=progei - MICENE 7. REFERENCES [1] BEE Projec web sie. [Online]. Available: hp: //beeprojec.dei.polimi.i/beeoverview.hml [2] A. Barbao, A. Capone, G. Carello, M. Delfani, M. Merlo, and A. Zaminga, House energy demand opimizaion in single and muli-user scenarios, in IEEE SmarGridComm IEEE, 2011, pp