Efficient Energy Consumption Scheduling: Towards Effective Load Leveling

Transcription

1 eergies Article Efficiet Eergy Cosumptio Schedulig: Towards Effective Load Levelig Yua Hog 1, *, Shegbi Wag 2 ad Ziyue Huag 3 1 Departmet of Iformatio Techology Maagemet, Uiversity at Albay, SUNY, 1400 Washigto Ave, Albay, NY 12222, USA 2 Departmet of Marketig Trasportatio & Supply Chai, North Carolia A&T State Uiversity, 1601 E Market St, Greesboro, NC 27411, USA; swag@catedu 3 Departmet of Iformatio & Supply Chai Maagemet, Uiversity of North Carolia at Greesboro, 1400 Sprig Garde St, Greesboro, NC 27412, USA; z_huag4@ucgedu * Correspodece: hog@albayedu Academic Editors: Pierluigi Siao ad Miadreza Shafie-khah Received: 7 November 2016; Accepted: 12 Jauary 2017; Published: date Abstract: Differet agets i the smart grid ifrastructure (eg, households, buildigs, commuities) cosume eergy with their ow appliaces, which may have adjustable usage schedules over a day, a moth, a seaso or eve a year Oe of the major objectives of the smart grid is to flatte the demad load of umerous agets (viz cosumers), such that the peak load ca be avoided ad power supply ca feed the demad load at aytime o the grid To this ed, we propose two Eergy Cosumptio Schedulig (ECS) problems for the appliaces held by differet agets at the demad side to effectively facilitate load levelig Specifically, we mathematically model the ECS problems as Mixed-Iteger Programmig (MIP) problems usig the data collected from differet agets (eg, their appliaces eergy cosumptio i every time slot ad the total umber of required i-use time slots, specific prefereces of the i-use time slots for their appliaces) Furthermore, we propose a ovel algorithm to efficietly ad effectively solve the ECS problems with large-scale iputs (which are NP-hard) The experimetal results demostrate that our approach is sigificatly more efficiet tha stadard bechmarks, such as CPLEX, while guarateeig ear-optimal outputs Keywords: smart grid; schedulig; load levelig; demad respose; demad side maagemet 1 Itroductio The smart grid itegrates the commuicatio etwork ito the existig power grid ad provides operatioal itelligece via aalyzig data collected from differet agets o the grid [1], icludig power suppliers (eg, utilities) ad eergy cosumers (eg, households, factories, uiversities ad hospitals) For istace, smart meters are istalled at the power cosumer ed to moitor the eergy usage i a real-time fashio; the meter readigs are cotiuously trasmitted to the electric utility with a time iterval as frequet as 15 mi [1] Aalyzig such fie-graied meter readigs (viz the eergy demad) fuctios for may applicatios for electric utilities ad cosumers o the grid, such as load forecastig [2], billig [3], regioal statistics [4] ad eergy theft detectio [5] The time series power usage of differet cosumers directly geerates the demad load (also time series) of the power supply I reality, the demad load of both residetial ad commercial buildigs highly fluctuates at differet times [6], eg, peak vs off-peak times Such fluctuatio would result i may issues o the power grid For istace, it makes it difficult for the utilities to always balace their power supply ad demad load withi a tight margi, the the eergy trasmissio ad productio might ot be optimal Furthermore, power blackouts may occur if the power supply caot feed the demad load over time At the same time, the power quality (eg, volts) of idividual cosumers might Eergies 2016, xx, x; doi:103390/ wwwmdpicom/joural/eergies

2 Eergies 2016, xx, x 2 of 27 be affected at peak times Therefore, the smart grid has begu to develop techiques that ca flatte the demad load of differet buildigs, commuities or geographical areas Specifically, may utilities try to icetivize a flatteed demad load of the cosumers by adoptig dyamic pricig plas for the eergy cosumptio times (eg, time-of-use pla lower price at off-peak, higher at o-peak ad highest at critical peak) Furthermore, the ABB Group (Automatio ad Power Techologies) ( provides a power storage-based solutio (eg, a eergy bak or battery) to store the excessive eergy durig periods of light load ad deliver it at peak times More recetly, a series of itelliget load maagemet techiques [7 10] was proposed for smart homes to automatically implemet the dyamic schedules for appliaces, eg, turig off uecessary lights, chagig the time for washig clothes, such that the peak electricity demad ca be flatteed to some extet However, most of the existig load levelig techiques have some drawbacks or limitatios First, the dyamic pricig plas highly rely o the behavioral respose from each cosumer to flatte the demad load I the case that the cosumers do ot care about the high price of eergy cosumptio at peak times, the pla would ot be effective for load levelig Sice the respose is somewhat radom from the cosumers, it is also challegig to quatify the effectiveess of load levelig Secod, the eergy storage-based techiques require additioal devices or facilities to implemet the scheme ad extra maiteace cost Fially, the itelliget load maagemet techiques [7 10] oly relatively flatte the peak demad towards a optimal goal; however, the proposed solutios either quatitatively measure the optimum or coverge towards the optimal objectives i their approaches Meawhile, their implemetatio is limited for oly oe household (a sigle smart home), rather tha multiple cosumers To address the above cocers, we propose a ovel aget-based approach to flatte the demad load by optimally schedulig the usage times for appliaces held by a sigle or multiple aget(s) (aget refers to a eergy cosumer i this cotext), motivated as follows Agets (eg, smart homes [7,10]) may have may appliaces with adjustable usage schedules For istace: 1 Whe to use some appliaces or machies is ot strictly tied to fixed time slots everyday For istace, the air coditioer ca be programmed to ru at differet times; washig clothes ca be postpoed to a certai time [10]; i a factory, machies may have adjustable schedules to maufacture ad assemble parts i the morig or afteroo of a day or differet days 2 A rechargeable battery is attached with a icreasig umber of appliaces such as electric vehicles, laptops, cordless vacuum cleaers, cell phoes, tablets, etc The batteries ca be charged at ay time, whereas the appliaces ca be used at other differet times I these cases, the battery chargig time will be recorded as power cosumptio time by the meters More specifically, we focus o a sigle or multiple agets appliaces, each of which has a set of possible i-use time slots, ad optimize their schedules to alig the time series power usage i the specified time slots (ivolved i schedulig) to a flatteed or fixed amout For istace, Figure 1 presets a real-world household s time series power cosumptio over 3 h [11] (which fluctuates with several peak loads) We formulate a mathematical model to produce a optimal eergy cosumptio schedule for all of the appliaces (ote that some appliaces have adjustable usage schedules, while some other appliaces may have fixed usage schedules; we cosider both i our model, as discussed i Sectio 71), which lays the time series aggregated eergy cosumptio (of all of the appliaces) close to a ideal fixed amout, ie, the horizotal lie i Figure 1 Notice that, although the optimal solutio of the schedulig problem may ot be able to get the exact horizotal lie, the overall deviatio is miimized as a small umber close to zero, as show i the experimets Give p agets (each aget holds some appliaces), the schedulig problem ca be applied to: (1) each aget s appliaces; or (2) all of the agets appliaces together For Case (1), each aget locally formulates ad solves the schedulig problem where p = 1 The, each aget s demad load at differet times ca be flatteed with its ow schedulig solutio, ad the overall demad load of

3 Eergies 2016, xx, x 3 of 27 all p agets (eg, households i a commuity) ca be automatically flatteed to a stable aggregated amout For Case (2), the overall demad load ca be directly flatteed usig the schedulig solutio (joitly derived from all p agets) Thus, while demadig a fixed amout of load from the grid at differet times, our schedulig-based load levelig techique ca greatly improve the reliability of power supply from the electric utility Power Cosumptio (Watts) TV Microwave Ove Rage Hood Without Schedulig Ideal Schedulig 5:00 PM 5:04 PM 5:08 PM 5:12 PM 5:17 PM 5:21 PM 5:25 PM 5:29 PM 5:34 PM 5:38 PM 5:42 PM 5:46 PM 5:51 PM 5:55 PM 5:59 PM 6:03 PM 6:08 PM 6:12 PM 6:16 PM 6:20 PM 6:25 PM 6:29 PM 6:33 PM 6:37 PM 6:42 PM 6:46 PM 6:50 PM 6:54 PM 6:59 PM 7:03 PM 7:07 PM 7:11 PM 7:16 PM 7:20 PM 7:24 PM 7:28 PM 7:33 PM 7:37 PM 7:41 PM 7:45 PM 7:50 PM 7:54 PM 7:58 PM Kettle Dryer Figure 1 Eergy cosumptio without schedulig vs ideal schedulig I this paper, we defie such a schedulig problem for load levelig as the Eergy Cosumptio Schedulig (ECS) problem Notice that the ECS problem ca be itegrated ito both the Demad Respose (DR) programs [9,12] ad the Demad Side Maagemet (DSM) programs [13 17] i shiftig load to off-peak times so as to beefit both electricity cosumers ad utilities More specifically, if the automated cotrol systems are i place, our ECS problem ca be implemeted as a DR program that ecourages eergy cosumers to make short-term reductios i eergy demad ad cosume the electricity at the off-peak times Similarly, our ECS problem ca be also implemeted as a DSM program to pursue eergy efficiecy from a log-term poit of view with possible facility upgrade, such as buildig automatio upgrades The ECS problem will be formulated amog oe or multiple agets (eg, households), ay of which (or a trusted-third party) ca be the problem solver ad ows the schedulig facility to derive the optimal schedule with iputs from all of the agets Specifically, all of the agets are expected to sed the iformatio of their appliaces (eg, cosumptio rate) ad the possible time slots of their appliaces to the problem solver, which the formulates ad solves the optimizatio problem to obtai the optimal cosumptio times of all of each aget s ivolved appliaces After solvig the problem, the problem solver will distribute the optimal solutio to the correspodig agets for ruig their appliaces per their optimal schedules I the real-world Iteret of Thigs (IoT), formulatig ad solvig the ECS problem, as well as the commuicatio amog agets ca be implemeted i the curret smart grid ifrastructure [1], which eables commuicatio amog etities o the existig power grid (viz agets with computatio capacity, such as smart homes [7,10]) The optimal solutio ca be implemeted i two differet ways: maual ad automatic For the former oe, each aget will receive a message of the optimal ruig times of its appliaces The, each aget ca maually tur o their appliaces per the scheduled times For the latter oe, each aget s share of the optimal solutio (the scheduled times of its appliaces) ca be automatically implemeted i the existig DR or DSM devices (eg, a load cotrol switch) I additio, the ext geeratio smart grid would eable households or buildigs (ie, a smart home [7,10]) to automatically switch their appliaces o ad off at pre-scheduled time slots; the optimal solutios of our ECS problem ca be directly implemeted i such a smart home eviromet Thus, our schedulig problem ca be smoothly itegrated ito the IoT to fuctio as load levelig at the demad side Note that the ECS problem is a fudametal eergy cosumptio schedulig problem, which outputs a specific time to tur o each appliace The, the output schedules ca be implemeted i differet power grid ifrastructures (eg, Asia, Europe ad North America), as log as differet agets ca commuicate with each other Although the layouts ad cofiguratios i the load coectio ad

4 Eergies 2016, xx, x 4 of 27 power distributio of such systems are quite differet, each aget (eg, a household) i such systems ca maually or automatically tur o its appliaces accordig to the received schedulig time i the optimal solutio I summary, the mai cotributios of this paper are give as follows: We propose a ovel Eergy Cosumptio Schedulig (ECS) problem for a sigle or multiple (eergy demad) agets to flatte their demad load We exted the ECS problem to a more geeralized form (the GECS problem) by eablig each aget to specify a rage of usage time for each of their appliaces This ca complete the schedulig as the appliaces required usage times are ukow at the schedulig stage Both ECS ad GECS problems are mathematically modeled as Mixed-Iteger Programmig (MIP) problems We develop a ovel effective algorithm (temporal decompositio) for efficietly solvig them Note that the algorithm ca retur a ear-optimal solutio for the ECS/GECS problem with 1,000,000 variables i reasoable time The experimetal results demostrate that our algorithm is sigificatly more efficiet tha the stadard bechmarks (eg, IBM ILOG CPLEX 122), while esurig ear-optimal solutios The remaider of the paper is orgaized as follows Sectio 2 reviews the relevat literature Sectios 3 ad 4 preset the Eergy Cosumptio Schedulig (ECS) problem ad the Geeralized Eergy Cosumptio Schedulig (GECS) problem, respectively The, Sectio 5 illustrates our ovel algorithm for solvig the ECS ad GECS problems Sectio 6 demostrates the experimetal results, ad Sectio 7 gives some discussios Fially, Sectio 8 cocludes the paper ad discusses the future work 2 Related Work The smart grid overlays the power distributio etwork with a commuicatio etwork [1], collects massive sesor data ad develops automatio techologies to improve the grid performace [9] As a critical compoet i smart grid ifrastructure, demad respose maagemet [18] aims at optimizig the power cosumptio at the demad side: electricity cosumers Specifically, Demad-Side Maagemet (DSM) eables operatioal itelligece at the sigle home level ad utilizes the home area etwork to iteract with the power grid For istace, the power usage of each appliace ca be moitored i the smart home [7,10] A icreasig umber of appliaces, such as lights, HVACs (Heatig, Vetilatig ad Air Coditioig) ad refrigerators, ca be programed to optimize the eergy cosumptio, cut dow the utility bill (eg, automatically shuttig dow heatig as temperature is high) ad support the Iteret of Thigs (IoT) The proposed aget-based eergy cosumptio schedulig i this paper pursues load levelig by further optimizig the i-use schedule of the appliaces Besides our aget-based model, some other approaches were adopted i the idustry ad academia, icludig the time-of-use eergy pricig pla, storig electricity at light load ad deliverig it at peak times (eg, ABB Group), ad home automatio (eg, a least slack first policy/algorithm proposed by Barker et al [10] i the SmartCap applicatio) I additio, load levelig problems are also solved i some specific eviromets, such as reducig T & Dlie losses [19], fault-tolerat distributed computig systems [20] ad system-wide demad respose maagemet [21] As discussed i Sectio 1, our aget-based eergy cosumptio schedulig is differet from the prior work, primarily i two ways First, our schedulig solutio could coverge towards a optimal or ear-optimal load levelig from the global poit of view (multi-agets) while beig subjected to the costraits derived from the i-use schedule of the appliaces Secod, with the participatio of multiple agets, the demad load ca be flatteed to a stable amout with reduced deviatio of the origial routie schedule of the appliaces (as discussed i Sectio 7) Furthermore, schedulig problems have bee studied for differet applicatios i the smart grid ifrastructure For example, Li ad Tsai [22] proposed a home eergy maagemet system facilitated by o-itrusive load moitorig techiques to save o electricity bills via schedulig Lu et al [23] proposed a multi-objective eergy cosumptio schedulig to miimize the total eergy cosumptio

5 Eergies 2016, xx, x 5 of 27 cost ad maximize the social utility Paterakis et al [24] mathematically formulated the problem of distributio etwork recofiguratio to determie the optimal radial cofiguratio by miimizig the active power losses ad a set of commoly-used reliability idices wrt the umber of customers Chetto [25] studied the schedulig for real-time jobs, which are executed o a uiprocessor system supplied by a reewable eergy source Wag et al [26] studied the eergy-aware data allocatio ad task schedulig problem o multiprocessor system for real-time applicatios Li et al [27] studied the problem of schedulig co-desig for reliability ad eergy by miimizig total eergy while guarateeig reliability costraits Ahmed et al [28] proposed a hybrid Lightig Search Algorithm (LSA)-based Artificial Neural Network (ANN) to predict the optimal o/off status for residetial appliaces, which ca provide iputs (eg, estimated some cadidate ruig time slots for appliaces) to fuctio as our ECS problems Fially, sice the curret cetralized model of productio ad trasmissio is icredibly iefficiet ad places the grid uder great pressure [29,30], ovel multi-aget systems [31] have sigificatly advaced the developmet of the smart grid recetly, especially demad respose [21,28,32] For example, Cha et al [32] proposed a multi-aget system to perform schedulig for maximum beefit i respose to the electricity prices Agrawal et al [33] studied the decetralized power supply restoratio problem i the case that lie failure occurs The proposed multi-aget scheme ca optimally cover differet sub-regios ad i tur lead to the aget-based decetralized cotrol [33] Cerquides [29] preseted a multi-aget framework for microgrids o the power grid to trade their local electricity o the eergy market A etwork of households with solar paels or other distributed eergy resources are able to sell the excess electricity to other households, which demad extra eergy from the mai grid To deal with ucertaity i the microgrids eergy geeratio, Strawser et al [34] developed a multi-aget power market for sellig electricity, which ca price reliability 3 Eergy Cosumptio Schedulig We first study the ECS problem where each appliace s overall eergy cosumig ( i-use ) time is predetermied, eg, the washer should be cotiuously ruig for 2 hours (specified by the aget for schedulig) Some frequetly-used otatios are give i Table 1 Table 1 Frequetly-used otatios p m i i [1, p] j [1, m i ] k [1, ] e ij c ij i umber of agets umber of equally-divided time slots for schedulig aget i s umber of appliaces aget idex aget i s appliace idex time slot idex the eergy cosumptio of aget i s j-th appliace i ay time slot (if i-use) aget i s j-th appliace s umber of i-use time slots aget i s overall cosumptio aget i s j-th appliace is o or off i time slot k 31 Objective Fuctio As show i Table 1, we deote the umber of time slots as, ad aget i s j-th appliace s umber of i-use time slots as c ij where c ij, i [1, p] ad j [1, m i ] Lettig e ij be aget i s j-th appliace s cosumed eergy i a sigle time slot, i [1, ], all of aget i s appliaces overall cosumptio i all of the time slots is a costat: i = m i j=1 (e ijc ij ), where c ij out of time slots are o for aget i s j-th appliace The, we ca aggregate all of the p agets overall cosumptio i time slots as = p i=1 i I additio, all agets overall average eergy cosumptio withi ay sigle time slot (averagig by time) is Recall that the goal of our model is to allocate all of the agets appliaces eergy

6 Eergies 2016, xx, x 6 of 27 cosumig time slots, such that their total power cosumptio withi every time slot is equal to or close to at all times I other words, the sum of deviatios betwee all of the agets overall eergy cosumptio withi each time slot ad should be miimized I additio, we defie biary variables i [1, p], j [1, m i ], k [1, ], {0, 1} as whether aget i s j-th appliace is o or off i time slot k If = 1, aget i s j-th appliace is o i time slot k; otherwise, it is off Thus, we have the objective fuctio: mi : p i=1 m i j=1 ( e ij ) (1) Note that the legth of time slots ca be short to make every appliace be completely o or off i every time slot, the the overall power cosumptio would be relatively stable i every time slot If the above objective value equals zero, the demad profile will be exactly a horizotal lie (as show i Figure 1) I most cases, the optimal objective value caot reach zero sice the variables are discrete I those cases, we will pursue a miimum deviatio close to zero, such that the demad profile would be flatteed towards the horizotal lie 32 Costraits 321 Fixed Number of I-Use Time Slots Essetially every aget s each appliace ( i [1, p] ad j [1, m i ]) has a equality costrait: the correspodig total umber of i-use time slots ( o ) i the schedule is specified as c ij : st Here, we have p i=1 m i equality costraits i total 322 Ruig Appliaces i Cotiuous Time Slots = c ij (2) I the real world, may appliaces might be o i cotiuous time slots, especially i the case where the time slot is very short (eg, 15 mi or less) The, meetig the followig costraits is equivalet to makig all of the appliaces ruig i cotiuous time slots (details are give i Appedix A) i [1, p], j [1, m i ] : x ij1 c ij c ij x ij2 (2c ij 1) c ij + c ij+1 x ij3 (3c ij 3) c ij + c ij+1 + c ij+2 x ij4 (4c ij 6) c ij + c ij+1 + c ij+2 st k=3 k=3 x ij( 2) (3c ij 3) 2 k= c ij k= c ij + k= c ij +1 x ij( 1) (2c ij 1) 1 k= c x ij ijk + k= c ij +1 x ij c ij k= c ij +1 (3) 323 Agets Prefereces of the I-Use Time Slots If ay aget has prefereces for the usage time of their appliaces, eg, a aget iteds to use the washer for oe hour i the morig istead of the whole day (schedulig is doe for all of the appliaces over oe day), the some local costraits ca be derived for such partially adjustable schedule For istace, the ECS problem schedules time slots, ad aget i s j-th appliace has a

7 Eergies 2016, xx, x 7 of 27 partially adjustable schedule (ruig c ij cotiuous time slots betwee time slot 1 ad time slot 2 where 0 < 1 < 2 < ad c ij 2 1 A set of costraits ca be give by lettig k [1, 1 1] ad k [ 2 + 1, ], = 0 Similarly, = 1 ca be also specified by aget i based o its prefereces Ideed, such costraits could reduce the complexity of the ECS problem We ca deote this kid of costrait as = 0 or 1 if specified by aget i i the mathematical models 33 Problem Formulatio As a result, after combiig the objective fuctio (Equatio (1)) ad costraits (Equality Costraits (2) ad Iequality Costraits (3)), we ca mathematically formulate our ECS problem as below: mi : p m i i=1 j=1 ( e ij ) i [1, p], j [1, m i ] : = c ij x ij1 c ij c ij x ij2 (2c ij 1) c ij + c ij+1 x ij3 (3c ij 3) c ij + c ij+1 + c ij+2 x ij4 (4c ij 6) c ij + c ij+1 + c ij+2 st k=3 k=3 x ij( 2) (3c ij 3) 2 k= c ij k= c ij + k= c ij +1 x ij( 1) (2c ij 1) 1 k= c ij + k= c ij +1 x ij c ij k= c ij +1 k [1, ], {0, 1}( = 0 or 1 if specified by aget i) where costat = p i=1 m i j=1 (c ije ij ) Lettig k [1, ], y k = p i=1 m i j=1 (e ij ) (overall deviatio i time slot k), the problem ca be trasformed ito the followig Mixed-Iteger Programmig (MIP) problem (details are give i Appedix B): mi : y k i [1, p], j [1, m i ], = c ij k [1, ], p i=1 m i j=1 (e ij ) y k k [1, ], p i=1 m i j=1 (e ij ) + y k k [1, ], y k 0 i [1, p], j [1, m i ] : x ij1 c ij c ij x ij2 (2c ij 1) c ij + c ij+1 st x ij3 (3c ij 3) c ij + c ij+1 + c ij+2 x ij4 (4c ij 6) c ij + c ij+1 + c ij+2 x ij( 1) (2c ij 1) 1 k= c x ij ijk + k= c ij +1 x ij c ij k= c ij +1 i [1, p], j [1, m i ], k [1, ], {0, 1} ( = 0 or 1 if specified by aget i) k=3 k=3 x ij( 2) (3c ij 3) 2 k= c ij k= c ij + k= c ij +1 (4) (5)

8 Eergies 2016, xx, x 8 of Measures of Evaluatig Schedulig Solutios The objective value of the ECS problem would ot be zero i geeral, ad solvig ay ECS problem usig differet algorithms may result i differet optimal solutios (due to the NP-hard ature of the mixed iteger programmig problem) Therefore, we defie a measure to evaluate the accuracy of the optimal schedulig solutios obtaied by differet algorithms, the deviatio ratio, which is defied as the miimum overall deviatios (i the optimal solutio) divided by the overall cosumptio i all of the time slots: DevRatio = y k = y k p i=1 m i j=1 (c ije ij ) (6) Hece, we ca use the above measure to compare the accuracy of differet algorithms used to solve the ECS problems I additio, the ECS problem miimizes the overall deviatio Thus, we ca ormalize the overall deviatio with the iitial overall deviatio to examie how much deviatio has bee reduced i the optimal solutio or other solutios i the problem solvig process To this ed, we ormalize the overall deviatio ito [0,1] usig the followig formula: Normalized Deviatio = y k(ay Solutio) y k(iitial Solutio) (7) I Sectio 6, we demostrate the experimetal results usig the above two measures 4 Geeralized ECS Problem I this sectio, we exted the ECS problem to a more geeral form i which the overall cosumptio of each appliace (ie, aget i s j-th appliace) is a variable i rage [a ij, b ij ] rather tha fixig it as c ij This extesio works i the case that some appliaces total umber of required i-use time slots is still ukow at the schedulig stage If we let i [1, p], j [1, m i ], a ij = b ij = c ij, the ew problem is the reduced to the origial ECS problem Thus, we ame this more geeral problem as the GECS problem I the GECS problem, give c ij ad z ij, we deote aget i s j-th s appliace s total umber of i-use time slots as x ij Thus, we have x ij =, ad the overall eergy cosumptios are: = p i = i=1 p i=1 m i j=1 (e ij x ij ) = p m i i=1 ij ) (8) j=1(e where is a variable rather tha a costat i the GECS problem Similar to the ECS problem, we ca derive the objective fuctio ad costraits as below 41 Objective Fuctio Replacig i the ECS problem s objective fuctio (Equatio (1)), we thus have the objective fuctio of the GECS problem: mi : p m i i=1 j=1( e ij ) 1 p m i i=1 ij ) (9) j=1(e 42 Costraits Similar to the ECS problem, we derive the costraits for the umber of i-use time slots, ruig appliaces i cotiuous time slots ad agets prefereces of the i-use time slots (if available)

9 Eergies 2016, xx, x 9 of Number of Ruig Time Slots Note that i the GECS problem, the total umber of i-use time slots for each appliace is a variable x ij [a ij, b ij ] The, we have the followig two groups of costraits: 422 Ruig Appliaces i Cotiuous Time Slots { i [1, p], j [1, mi ], a ij i [1, p], j [1, m i ], b ij (10) Similar to the ECS problem, we ca obtai all of the iequality costraits for ruig appliaces i cotiuous time slots as below (details are give i Appedix C): i [1, p], j [1, m i ] : x ij1 c ij x ij2 (2 1) + +1 x ij3 (3 3) k=3 x ij4 (4 6) k=3 st x ij( 2) (3 3) 2 k= k= + k= +1 x ij( 1) (2 1) 1 k= x + ijk k= +1 x ij k= +1 (11) where i [1, p], j [1, m i ], x ij = 43 Problem Formulatio Similar to the optimizatio model of the ECS problem (Equatio (4)), we ca mathematically formulate the GECS problem as below: a ij mi : p i=1 m i j=1( e ij ) 1 p i=1 m i i [1, p], j [1, m i ] : ij ) j=1(e b ij x ij1 c ij x ij2 (2 1) + +1 x ij3 (3 3) k=3 st x ij4 (4 6) k=3 x ij( 2) (3 3) 2 k= k= + k= +1 x ij( 1) (2 1) 1 k= + k= +1 (12) x ij k= +1 k [1, ], {0, 1}( = 0 or 1 if specified by aget i) where i [1, p], j [1, m i ], x ij = Similar to the ECS problem, we ca deote the sum of absolute values by additioal variables y 1,, y Therefore, after removig the absolute values (similar to Equatio (5)), the GECS problem ca be trasformed to:

10 Eergies 2016, xx, x 10 of 27 i [1, p], j [1, m i ], b ij mi : y k i [1, p], j [1, m i ], a ij k [1, ], p i=1 m i j=1 (e ij 1 (e ij )) y k 0 k [1, ], p i=1 m i j=1 (e ij 1 (e ij )) y k 0 k [1, ], y k 0 i [1, p], j [1, m i ] : a ij b ij x st ij1 c ij x ij2 (2 1) + +1 x ij3 (3 3) k=3 x ij4 (4 6) k=3 x ij( 2) (3 3) 2 k= k= + k= +1 x ij( 1) (2 1) 1 k= x + ijk k= +1 x ij k= +1 i [1, p], j [1, m i ], k [1, ], {0, 1}( = 0 or 1 if specified by aget i) (13) where i [1, p], j [1, m i ], x ij = 5 Algorithms I this sectio, we first describe the overview of the eergy cosumptio schedulig ad the preset a efficiet algorithm for oe or multiple agets to effectively solve the ECS (or GECS) problem ad implemet the optimal schedulig solutio for load levelig 51 Overview As described i Sectio 1, all of the agets first sed their appliaces cosumptio rates ad estimated ruig times to the problem solver, which ca be ay aget or a trusted-third party If the appliaces ruig times are give as a fixed umber of time slots, the a ECS problem will be formulated (as show i Sectio 3); if the appliaces ruig times are give as rages of time slots, the a GECS problem will be formulated (as show i Sectio 4) For the ECS problem, we propose a ovel algorithm deoted as the Temporal Decompositio (TD) to let the problem solver efficietly solve it The details of the TD algorithm are give i Sectio 52 For the GECS problem, the problem solver first utilizes Liear Programmig (LP) relaxatio to fid out the optimal cosumptio amout of each appliace The, cosiderig the optimal solutio of the LP problem as the fixed cosumptio amout for each appliace, the GECS problem ca be coverted to a ECS problem, which ca be solved usig the TD algorithm by the problem solver The details of solvig the GECS problem are give i Sectio 53 Fially, the problem solver distributes the shares of the optimal solutio to the correspodig agets, which are their appliaces specific ruig time slots As a result, all of the agets ca tur o their appliaces i the specific ruig time slots i the optimal schedulig solutio: for all = 1, aget i turs o its j-th appliace i time slot k Figure 2 demostrates the flow diagram of the eergy cosumptio schedulig for load levelig (both ECS ad GECS problems)

11 Eergies 2016, xx, x 11 of 27 Aget 1 (m 1 Appliaces): Each Appliace s cosumptio rate, ad ruig time (time slots #, or a rage of time slots #) Solve the Problem usig TD ECS Problem (time slots #) Aget 1 (m 1 Appliaces): Tur o each appliace i the specific ruig time slots i the Optimal Solutio Aget 2 (m 2 Appliaces): Each Appliace s cosumptio rate, ad ruig time (time slots #, or a rage of time slots #) Problem Solver (Ay aget or a trustedthird party) Optimal Solutio Aget 2 (m 2 Appliaces): Tur o each appliace i the specific ruig time slots i the Optimal Solutio Aget p (m p Appliaces): Each Appliace s cosumptio rate, ad ruig time (time slots #, or a rage of time slots #) Solve the Problem usig LP Relaxatio ad TD GECS Problem (a rage of time slots #) Aget p (m p Appliaces): Tur o each appliace i the specific ruig time slots i the Optimal Solutio Figure 2 Overview of the eergy cosumptio schedulig for load levelig 52 Solvig the ECS Problem with Temporal Decompositio The ECS problem ivolves 2 global costraits with respect to p i=1 [m i( + 1)] variables shared by all p agets, while i [1, p], aget i holds m i local equality costraits ad m i local iequality costraits with respect to m i variables Sice such a Mixed-Iteger Programmig problem (MIP) icludes a overwhelmig majority of biary variables with a umber of p i=1 (m i), the commercial solvers, such as CPLEX or GUROBI [35], caot produce a optimal solutio withi reasoable time as i [1, p], m i, p ad/or are large Thus, we desig a efficiet heuristic algorithm to effectively ad efficietly geerate optimal or ear-optimal solutios Specifically, we decompose the ECS problem ito subproblems for differet time slots i the schedulig The algorithm begis with solvig the subproblem regardig the first time slot k = 1, ad solves all of the subproblems i a temporal sequece Thus, we deote the algorithm as temporal decompositio Sice the costat represets the optimal overall cosumptio amout for every time slot, the process of optimizig all p agets appliaces overall cosumptio i differet time slots is relatively idepedet I each subproblem, k [1, ], the objective fuctio is simply give as y k, which is time slot k s share i the ECS problem s origial objective fuctio, the deviatio betwee all agets appliaces cosumptio i time slot k ad the costat optimal amout More specifically, i the ECS problem, pairs of global costraits are give for represetig the deviatio of differet time slots cosumptio ad, respectively, which are idepedet of each other; pairs of global costraits do ot have ay overlapped variables Thus, k [1, ] the k-th subproblem oly eeds to ivolve a pair of global costraits (correspodig to time slot k) We ca first formulate decomposed subproblems with oly global costraits k [1, ], the k-th subproblem is: p i=1 m i mi : y k j=1 (e ij ) y k p i=1 st m i j=1 (e ij ) y k y k 0, i [1, p], j [1, m i ], {0, 1} ( = 0 or 1 if specified by aget i) (14) where the variables i the k-th subproblem are a subset of variables i the ECS problem, which correspod to time slot k We deote its optimal solutio as i [1, p], j [1, m i ], xijk, the optimal values idicatig whether all of p agets appliaces are o or off i time slot k I the meatime, deviatio y k is miimized to y k

12 Eergies 2016, xx, x 12 of 27 Furthermore, we also have to take ito accout the ECS problem s local costraits Recall that each local costrait is created with two criteria: 1 aget i s j-th appliace s total umber of i-use time slots is c ij i the schedulig 2 all c ij i-use time slots are cotiuous The, we ca simplify all of the local costraits accordig to two groups of rules that assig values for variables i [1, p], j [1, m i ], k [1, ], i all decomposed subproblems i which represets the o/off status of aget i s j-th appliace i time slot k Specifically, i time slot k, 1 k s subproblem (ote that k 1 subproblems have bee solved): Rules of the umber of i-use time slots For aget i s j-th appliace: Rule 11: if k 1 u=1 x iju = c ij, the all of the biary variables u [k + 1, ], x iju i the remaiig ( k + 1) subproblems (icludig the curret subproblem) must be zero This rule meas that if a appliace has bee o for c ij time slots i the past (k 1) time slots, it must be off i all of the remaiig time slots Rule 12: if c ij k 1 u=1 x iju = (k 1), the all of the biary variables u [k, ], x iju i the remaiig ( k + 1) subproblems (icludig the curret subproblem) must be oe This rule meas that if a appliace has bee off for ( c ij ) time slots i the past (k 1) time slots, it must be o i all of the remaiig time slots Rule 13: if 0 < c ij k 1 u=1 x iju < (k 1), the all of the biary variables u [1, ], x iju i the remaiig ( k + 1) subproblems (icludig the curret oe) ca remai as either zero or oe This rule meas that if a appliace has either bee o for c ij time slots or bee off for ( c ij ) time slots i the past (k 1) time slots, it ca be either o or off i the followig time slots Rules of cotiuous i-use time slots For aget i s j-th appliace: Rule 21: if k 1 u=1 x iju = c ij, the the biary variable i the curret subproblem ad the biary variables u [k + 1, ], x iju i all of the remaiig subproblems must be zero (the same as Rule 11) This rule meas that if a appliace has bee cotiuously o for c ij time slots i the past (k 1) time slots, it must be off i all of the remaiig time slots Rule 22: if k 1 u=1 x iju < c ij ad xij(k 1) = 1, the the biary variable i the curret subproblem, ad the biary variables u [k + 1, k + c ij k 1 u=1 x iju 1], x iju i the followig (c ij k 1 u=1 x iju 1) subproblems must be oe(the biary variables i c ij cotiuous subproblems are oe) This rule meas that if a appliace is o i the most recet time slot ad has ot bee o for c ij time slots i the past (k 1) time slots yet, it must be o i the followig (c ij k 1 u=1 x iju 1) time slots Rule 23: if k 1 u=1 x iju < c ij ad xij(k 1) = 0 (o i-use time slot yet), the the biary variable i the curret subproblem ca be either zero or oe This rule meas that if a appliace has ot bee o for c ij time slots i the past (k 1) time slots, it must be off i all of the past time slots (due to the characteristics of cotiuous ruig) The, it ca be either o or off i the followig time slots Note that all six rules will be applied to decomposed subproblems from a global perspective Sice the first group of rules esures that aget i s j-th appliace s total umber of cosumptio time slots equals c ij ad the secod group of rules esures that such c ij i-use time slots are cotiuous, the compliace of the above two groups of rules is equivalet to meetig all of the local costraits i the ECS problem After solvig the subproblems (otice that: (1) without loss of geerality, ay aget or a cetralized site ca be the solver; (2) k [1, ], the k-th subproblems is joitly formulated by all the p agets; i [1, p], aget i iputs its share of the problem correspodig to time slot k; (3) each aget

13 Eergies 2016, xx, x 13 of 27 utilizes all six rules ad its local costraits to examie the possible values of their variables i every subproblem), the optimal solutios of all of the subproblems ca directly form the optimal solutio of the origial ECS problem: optimal value y1 + y y ; optimal solutio xijk The details of the temporal decompositio are preseted i Algorithm 1 ad Figure 3, ad the accuracy of the temporal decompositio algorithm is validated i Sectio 6 Algorithm 1: Temporal decompositio 1 forall time slot k [1, ] do 2 retrieve the optimal values i the previously solved k 1 subproblems: i [1, p], j [1, m i ], u [1, k 1], x iju 3 check two groups of rules: Rules 11, 12, 13 ad 21, 22, 23 with the followig values to decide the available biary values for the variables i the curret subproblem (k-th): i [1, p], j [1, m i ], c ij i [1, p], j [1, m i ], u [1, k 1], x iju solve the k-th subproblem with formulatio show i Equatio (14) to obtai the optimal solutio i [1, p], j [1, m i ], x ijk ad y k 4 retur all of the optimal solutios i all k subproblems Optimal values:, x ij1 * x ij2 * * x ij * Time Slots/Subproblems: 1 2 k I the kth subproblem 1 Load the optimal values i the previous (k-1) subproblems ad check two groups of rules (6 i total) to determie the available values for its biary variables 2 Solve the curret (kth) subproblem Figure 3 Temporal decompositio 53 Solvig the GECS Problem with Liear Programmig Relaxatio ad Temporal Decompositio Sice the costat optimal cosumptio amout of ay time slot i the ECS problem has bee chaged ito 1 p i=1 m i j=1 (e ij ) i the GECS problem (which is ot a costat), Algorithm 1 caot be directly applied to solve the GECS problem To tackle this issue, we propose a two-phase approach to solve the GECS problem: (1) fid the optimal cosumptio amout for each of the appliaces; ad (2) solve the schedulig problem with the optimal cosumptio amouts (fixed) I Phase (1), the problem solver relaxes k [1, ], from biary variable {0, 1} to cotiuous rage [0, 1] i the GECS problem ad defies ew iteger variables i [1, p], j [1, m i ],

14 Eergies 2016, xx, x 14 of 27 ω ij = to approximate each appliace s total umber of i-use time slots (similar to c ij i the ECS problem) The, the problem solver ca formulate ad solve the followig LP relaxatio problem: mi : y k m 1 j=1 [x 1j1 1 ω 1j]e 1j + + m p j=1 [x pj1 1 ω pj]e pj y 1 0 m 1 j=1 [x 1j1 1 ω 1j]e 1j m p j=1 [x pj1 1 ω pj]e pj y 1 0 st m 1 j=1 [x 1j 1 ω 1j]e 1j + + m p j=1 [x pj 1 ω pj]e pj y 0 m 1 j=1 [x 1j 1 ω 1j]e 1j m p j=1 [x pj 1 ω pj]e pj y 0 i [1, p], j [1, m p ], ω ij = (15) i [1, p], j [1, m p ], a ij ω ij b ij y k 0, i [1, p], j [1, m i ], 0 1 After solvig the above LP relaxatio problem, i [1, p], j [1, m i ], ω ij ca be fixed as costats with the optimal values i the LP relaxatio problem ωij (which are rouded to itegers ω ij ) Therefore, the optimal value ωij ca serve as the (optimal) total eergy cosumptio of aget i s j-th appliace As a result, the GECS problem is trasformed ito a ECS problem I Phase (2), the problem solver applies temporal decompositio (Algorithm 1) to solve the trasformed GECS problem with the optimal cosumptio amouts of the appliaces (derived from Phase (1)) The GECS problem (viz a ECS problem) is formulated as below: mi : p i=1 i [1, p], j [1, m i ], = ωij m i j=1 ( e ij ) i [1, p], j [1, m i ] : x ij1 ωij ω ij x ij2 (2 ωij 1) ω ij + ω ij +1 x ij3 (3 ωij 3) ω ij + ω ij +1 + ω ij +2 k=3 st x ij4 (4 ωij 6) ω ij + ω ij +1 + ω ij +2 k=3 x ij( 2) (3 ωij 3) 2 k= ωij k= ωij + k= ωij +1 (16) x ij( 1) (2 ωij 1) 1 k= ωij + k= ωij +1 x ij ωij k= ωij +1 i [1, p], j [1, m i ], k [1, ], {0, 1}( = 0 or 1 if specified by aget i) where costat = p i=1 m i j=1 (e ij ωij ) Note that ω ij will be loaded ito the two groups of rules as aget i s i-th appliace s overall cosumptio time i time slots (viz c ij i the ECS problem) for satisfyig all of the local costraits Similar to the ECS problem, after solvig the problem, the problem solver distributes the optimal solutio to each aget For all = 1 i the optimal solutio, aget i turs o its j-th appliace i time slot k

15 Eergies 2016, xx, x 15 of 27 6 Experimetal Results I this sectio, we coduct experimets to compare our temporal decompositio algorithm with the commercial software IBM ILOG CPLEX 122 o solvig the ECS/GECS problems 61 Dataset Richardso et al [36] collected 22 dwelligs power cosumptio over two years i East Midlads, UK I the real dataset, each of the 22 smart meters is associated with 33 appliaces with 1,051,200 readigs (oe readig per miute) We radomly select p [1, 10] smart meters (each of which is a aget) ad aggregate the readigs for every 10 mi withi two moths to geerate our experimetal data: = 8640 time slots ca be geerated (10 mi each) The, our experimets are coducted o mixed sets of real ad sythetic data Each appliace s cosumptio amout withi ay sigle time slot e ij ad total umber of i-use time slots c ij are available i the dataset O the other had, we radomly select 30% 50% of the appliaces as appliaces with a adjustable schedule, where the schedule rages are radomly geerated from all of the time slots For istace, a appliace (which has adjustable schedule) rus c ij = 2 6 time slots (2 h) out of = 24 6 time slots (24 h) We radomly geerate a subset of time slots (rather tha all of the time slots i 24 h), as the appliace s i-use time slots rage specified by the agets with their prefereces, eg, the time slots i the first 8 h Note that such a sythetic schedule rage is expaded from the appliace s i-use time (radomly for 5 10 times) i the real data I the GECS problem, the rage of i-use time slots umber [a ij, b ij ] is radomly geerated with the criterio i, j, a ij c ij b ij, where c ij is available i the dataset 62 Settigs We implemeted the Mixed-Iteger Programmig (MIP) solver, IBM ILOG CPLEX (Versio 122), usig MATLAB (versio R2015a) I additio, we coded our TD algorithm usig the same software MATLAB ad also ivoked CPLEX while solvig ay decomposed MIP subproblem Thus, the compariso betwee the pure CPLEX solver ad our TD algorithm was doe i the same codig eviromet We deote directly solvig the MIP problems usig CPLEX as CPLEX ad our temporal decompositio algorithm as TD (i which all of the decomposed subproblems are solved by CPLEX), respectively I both ECS ad GECS problems, each variable has three differet dimesios: p [1, 10] (umber of agets), m 1,, m p [1, 33] (umber of appliaces held by each aget) ad [1, 8640] (umber of time slots) We test the results i three groups of experimets as below: Group 1 (small/medium): testig the accuracy i case that CPLEX ca fid the exact optimal solutios withi reasoable time The, (p, m i, ) is specified as (1, 20, 12), (1, 20, 24), (2, 20, 12), (2, 20, 24), (5, 5, 12), (5, 5, 24), (5, 5, 48), (5, 10, 12), (5, 10, 24) ad (5, 15, 12), respectively Group 2 (large): testig the efficiecy/scalability ad accuracy o varyig umber of time slots ad the umber of appliaces per aget Fixig p = 3 agets, each aget has 11, 12, 13,, 30 appliaces, ad the umber of time slots varies from The largest MIP problem i this group icludes 540,000 biary variables Group 3 (large): testig the efficiecy/scalability ad accuracy o the varyig umber of time slots ad the umber of agets Fixig the umber of appliaces held by each aget as 20, the umber of agets varies i the rage [1,10], while the umber of time slots varies from The largest MIP problem i this group icludes 1,200,000 biary variables We ru each test for five times ad average the results Notice that oly the testig cases i Group 1 could fid the optimal solutio withi 1 h, which is a meaigful stoppig poit adopted i may other computatioal studies [37] Ideed, i Groups 2 ad 3, CPLEX failed to fid the optimal solutio withi 5 h For such cases, we cosider the best feasible CPLEX solutios obtaied withi the 5-h time

16 Eergies 2016, xx, x 16 of 27 limit as a surrogate for the optimal solutio Note that we did try to let CPLEX ru loger i may cases, but the solutio quality provided by CPLEX after 6 10 h had o sigificat differece i its result i 5 h If we let CPLEX ru eve loger, say 48 h, it occasioally could offer a slightly better TD result CPLEX result solutio, but the empirical error gap [37] (simply defied as CPLEX result ) is still maily withi ±10% More importatly, i practice, it might be uecessary to wait for CPLEX to provide us with a slightly better solutio with greatly icreased rutime 63 Accuracy I practice, we ca cosider the optimal solutio obtaied by the commercial tool CPLEX as the exact optimal solutio ad the compare the optimal solutio retured by our algorithm to that of CPLEX However, CPLEX ca oly retur the optimal solutio for a small or a medium size of the ECS problems (eg, the experimetal Group 1) withi reasoable time We the first look at the deviatio ratios (defied i Equatio (6)) of the exact solutios (by CPLEX) preseted i Table 2 I these cases, our algorithm (TD) ca retur optimal solutios extremely close to the exact optimal solutios obtaied by CPLEX: out of 10 pairs of results, seve pairs are idetical, ad CPLEX performs slightly better i three pairs For large-scale ECS problems, we plot two algorithms deviatio ratios i the experimetal Groups 2 ad 3 i Figure 4 ad coclude the followig observatios The deviatio ratio decreases as the problem size icreases with greater p ad/or greater m 1,, m p ad/or greater, sice it is more likely to further level the overall cosumptios i differet time slots whe more agets ad/or more appliaces (with adjustable ruig schedules) ad/or more time slots are ivolved i the schedulig I these two groups of experimets, the umber of variables falls ito [30,000,1,200,000] Sice CPLEX could ot fid the optimal solutio i 5 h, the retured best feasible solutio by CPLEX is slightly worse tha the ear-optimal solutio retured by our TD algorithm, as show i Figure 4 Covergece of deviatio: As show i Figure 5, we plot the ormalized deviatio (defied i Equatio (7)) of some selected iteratios i our TD algorithm, applied to three ECS problems with differet sizes (small: 240 biary variables; medium: 3600 biary variables; large: 24,000 biary variables) We ca observe the covergece of the deviatio miimizatio process from oe to a small umber close to zero as below Accuracy vs total umber of appliaces: Notice that, i the ECS problems, if multiple agets are ivolved i the schedulig (p > 1), they ca commuicate with each other to schedule their appliaces to flatte the overall power cosumptio Therefore, give a fixed umber of time slots for schedulig, the performace of the accuracy is depedet o the umber of overall appliaces, regardless of the umber of agets ad the umbers of appliaces held by each aget This also applies to the GECS problems Table 2 Small/medium-scale ECS problem (Group 1): TD vs CPLEX (p, m i, ) # of Biary Variables Deviatio Ratio (%) Rutime (s) CPLEX TD CPLEX TD (1, 20, 12) (1, 20, 24) (2, 20, 12) (2, 20, 24) (5, 5, 12) (5, 5, 24) (5, 5, 48) (5, 10, 12) (5, 10, 24) (5, 15, 12)

17 Eergies 2016, xx, x 17 of 27 Optimal Deviatio Ratio Optimal Deviatio Ratio CPLEX (=1500) CPLEX (=3000) CPLEX (=6000) TD (=1500) TD (=3000) TD (=6000) Number of Appliaces per Aget (3 Agets) (a) CPLEX (=1500) CPLEX (=3000) CPLEX (=6000) TD (=1500) TD (=3000) TD (=6000) Number of Agets (20 Appliaces per Aget) (b) Figure 4 ECS problem (Groups 2 ad 3): TD vs CPLEX (accuracy) (a) Group 2 (up to 540,000 variables); (b) Group 3 (up to 1,200,000 variables) Normalized Deviatio Variables 3600 Variables 240 Variables Selected Iteratios i the Temporal Decompositio Algorithm Figure 5 Normalized deviatio vs iteratios (temporal decompositio) 64 Case Study Besides the experimetal results of deviatio, we also coduct a case study to demostrate the effectiveess of load levelig via our eergy cosumptio schedulig approach The power cosumptio data are selected from the dataset collected by Richardso et al [36] i the UK We select a sample house from the 22 houses i the dataset, which icludes 30 electric appliaces Out of all of the appliaces, we study the load levelig i two cases: (1) all of the appliaces ca have adjustable schedules; ad (2) oly 20 appliaces ca have adjustable schedulig The power cosumptio i the dataset has bee aggregated from 1 mi per readig to 15 mi per readig, the we have 15 mi as the time slot legth for schedulig I the experimets, we study the schedulig problems over two time rages [12:00 am 12:00 pm] (midight to oo) ad [12:00 am 12:00 pm] (oo to midight), respectively Thus, we have 4 12 = 48 time slots i each schedulig problem After solvig the four ECS problems

18 Eergies 2016, xx, x 18 of 27 by our TD algorithm (two time rages ad two cases of appliaces schedules), we demostrate the results i Figure 6a,b, respectively I both figures, ECS (30) ad ECS (20) represet the ECS problem (solved by the TD algorithm) with all 30 appliaces ivolved i the schedulig ad with oly 20 appliaces ivolved i the schedulig, respectively Origial ad ideal meas the origial cosumptio without schedulig ad the ideal schedulig (cosumptio equals the average amout all of the time), respectively As a result, we ca have two observatios: (1) the demad load (eergy cosumptio) ca be flatteed by our ECS problem at differet times: for both ECS (30) ad ECS (20) ad (2) if more appliaces have adjustable schedules for the schedulig (eg, ECS (30)), the demad load (eergy cosumptio) curve ca be flatteed closer to the ideal case 6 Power Cosumptio (kw) Origial ECS (30) ECS (20) Ideal Power Cosumptio (kw) :00 AM 12:00 PM 1:00 AM 1:00 PM 2:00 AM Origial ECS (30) ECS (20) Ideal 2:00 PM 3:00 AM 3:00 PM 4:00 AM 4:00 PM 5:00 AM 5:00 PM (a) (b) Figure 6 Load levelig (power cosumptio with schedulig vs power cosumptio without schedulig) (a) 12:00 am 12:00 pm (48 time slots) (b) 12:00 pm 12:00 am (48 time slots) 6:00 AM 6:00 PM 7:00 AM 7:00 PM 8:00 AM 8:00 PM 9:00 AM 9:00 PM 10:00 AM 10:00 PM 11:00 AM 11:00 PM 65 Efficiecy ad Scalability Figure 7 shows the rutime for two algorithms (TD ad CPLEX) with varyig p, m 1,, m p ad to solve large-scale ECS problems CPLEX fails to provide optimal solutios i all of the cases withi 5 h Istead, our algorithm remarkably outperforms CPLEX o efficiecy ad scalability sice it ca retur a ear-optimal solutio i sigificatly less time i almost all of the cases As the problem size icreases alog three differet dimesios (by icreasig values of p, m i, ), the rutime of temporal decompositio icreases extremely slow with a liear tred O the cotrary, the rutime of CPLEX icreases expoetially as p, m 1,, m p ad/or icreases, as show i Table 2 (ote that the rutime of CPLEX i Figure 7 exceeds 10,800 s: 5 h i all of the cases i experimetal Groups 2 ad 3, the CPLEX is termiated at that poit) I our TD algorithm, for each testig case, the umber of times of ivokig IBM ILOG CPLEX is fixed (which is the umber of time slots ad also the umber of subproblems) I other words, the covergece agaist the solvig step umber (based o ivokig CPLEX) is fixed for every schedulig problem I the meatime, each time whe CPLEX is ivoked, the overhead time of iterfacig is less tha s Eve with large-scale problems, such as = 8640, the total overhead time of iterfacig

19 Eergies 2016, xx, x 19 of 27 is still less tha 01 s, which ca be egligible comparig to the overall solvig time i those testig cases Furthermore, the rutime required for solvig each subproblem i sequece decreases as the time slot umber k icreases from 1 Take the testig case (p = 5, m i = 10, = 12) as a example, solvig 12 subproblems usig CPLEX requires 592, 478, 389, 338, 279, 245, 229, 202, 185, 154, 148 ad 133 s, respectively Note that the overall iterfacig time takes oly s Rutime (sec) Rutime (Sec) CPLEX (=1500) CPLEX (=3000) CPLEX (=6000) TD (=1500) TD (=3000) TD (=6000) Number of Appliaces per Aget (3 Agets) CPLEX (=1500) CPLEX (=3000) CPLEX (=6000) TD (=1500) TD (=3000) TD (=6000) (a) Number of Agets (20 Appliaces per Aget) (b) Figure 7 ECS problem (Groups 2 ad 3): TD vs CPLEX (rutime) (a) Group 2 (up to 540,000 variables); (b) Group 3 (up to 1,200,000 variables) The highly efficiet ad scalable feature of our TD algorithm is more practical ad accessible o the smart grid sice schedulig should be implemeted olie amog multiple agets ad accomplished i a very short time Ideed, olie schedulig caot wait for a couple of days to derive the optimal solutio by CPLEX 66 Experimetal Results for the GECS Problem For the GECS problem, we coducted aother group of experimets usig a similar dataset ad obtaied a similar set of experimetal results As show i Table 3, we ca draw similar observatios for our temporal decompositio algorithm as the ECS problem For large-scale problems (eg, 900,000 biary variables), our algorithm oly takes 13,42381 s to obtai a accurate ear-optimal solutio However, if we use CPLEX to solve the same GECS problem, the feasible solutio obtaied after 5 h is still worse tha TD s ear optimal solutio (deviatio ratio 0049% vs 0043%)

20 Eergies 2016, xx, x 20 of 27 Table 3 The Geeralized Eergy Cosumptio Schedulig (GECS) problem: TD vs CPLEX (p, m i, ) # of Biary Variables Deviatio Ratio (%) Rutime (s) CPLEX TD CPLEX TD (5, 10, 100) , (5, 20, 100) 10, , (5, 10, 200) 10, , (5, 20, 200) 20, , (10, 20, 400) 80, , (10, 30, 400) 120, , (10, 20, 800) 160, , (10, 30, 800) 240, , (15, 20, 1000) 300, , (15, 30, 1000) 450, , (15, 20, 2000) 600, , (15, 30, 2000) 900, ,000 13, Discussios 71 Appliace Categories i Schedulig Ciabattoi et al [38] has categorized the residetial appliaces based o their usage patters: (1) cotiuous use appliaces (eg, refrigerator); (2) periodical use appliaces without huma iteractio (eg, ove ad microwave); (3) periodical use appliaces with huma iteractio (eg, vacuum); (4) multimedia appliaces; ad (5) lightig I the real world, differet appliaces ad their usage patters may ifluece the schedulig process, as well as implemetig the optimal schedules First, some appliaces may have o-adjustable schedules (eg, cotiuous use appliaces, such as refrigerator) I this case, whe we formulate the ECS or GECS problems, the biary variables derived for the appliaces will be fixed as costats based o the give rage of time slots The, the overall cosumptio of all of the appliaces held by differet agets i every time slot will be optimized towards Notice that, sice our approach is based o schedulig the usage times for appliaces with adjustable usage schedules, if there are too may appliaces with fixed usage schedules, the optimal demad profile (time series cosumptio) may ot be extremely close to the perfectly flatteed cosumptio amout (but still better tha the demad profile without schedulig) This is a possible limitatio of our proposed approach Secod, some appliaces have adjustable schedules (eg, periodical use appliaces with huma iteractio, such as a vacuum), but their available time slots i the schedulig are ukow beforehad For istace, if the optimal schedulig is derived for the ext day, the eergy cosumer does ot kow whe to use the vacuum i the ext day I such a case, o-itrusive load moitorig [39] ca help idetify the usage patters of the appliace ad suggest a rage of ruig time for the appliace (the, a GECS problem should be formulated) I our studied problem, we assume that the usage time or a rage of usage time should be specified prior to the schedulig ad try to flatte the demad load based o schedulig the existig appliaces that have a estimated ruig time If ay aget would like to tur o a appliace that has ot bee ivolved i the schedulig, we assume that such a appliace does ot affect the load sigificatly i this paper If such a appliace ca lead to a extremely high demad load, all of the agets ca commuicate with each other ad implemet the schedulig agai immediately, sice our algorithm is highly efficiet to solve the optimizatio problem 72 Number of Biary Variables ad Load Levelig Performace Recall that the aget-based ECS ad GECS problems are formulated by p agets with m 1,, m p appliaces, respectively (eg, each aget represets a household) I geeral, as more

21 Eergies 2016, xx, x 21 of 27 appliaces with a adjustable schedule are ivolved i the schedulig (each aget holds more appliaces with a adjustable schedule or more agets are ivolved i the schedulig), the performace of load levelig would become better with less aggregated deviatio betwee the actual cosumptio ad the ideal amout i all time slots This is simply because more appliaces with adjustable schedule (held by differet agets) could elarge the feasible regio of optimizatio problem (which ca be trasformed to a mixed iteger programmig problem with liear costraits) I the experimets, we will validate this observatio usig both temporal decompositio algorithm ad the stadard solver CPLEX 73 Ruig Multiple Times I reality, each time slot ca be either log or short, eg, as log as oe day ad as short as 5 mi [1] The proposed ECS ad GECS problems assume all of the appliaces are cotiuously o i a specified umber of time slots Occasioally, a aget may ited to tur o a appliace multiple times out of the etire time slots; for istace, i a 12-h schedulig ( = 12 ad 1 h each time slot), if aget i plas to ru the washer (its j-th appliace) for 4 h i total, but 2 h i the morig (k [1, 4]) ad 2 h i the eveig (k [9, 12]) We ca formulate two sets of costraits for such appliace i the schedulig as k [1, 4], 4 = 2 ad k [9, 12], 4 = 2 Such additioal costraits ca be formulated without affectig the efficiecy of solvig the ECS ad GECS problems 74 Short-Term Schedulig Through tuig the legth of each time slot to a short time (eg, 1 mi), our ECS/GECS problems ad the TD algorithm ca also efficietly idetify the optimal schedule for appliaces i a real-time maer sice the performace of the algorithm is depedet o the umber of time slots rather tha the legth of the time slots Therefore, i short-term schedulig with real-time requiremets, we ca specify a reasoable umber of time slots i each schedulig ad iteratively execute the TD algorithm ad schedulig to implemet load levelig i real time 8 Coclusios ad Future Work I this paper, we studied the aget-based ECS problem o the smart grid, which flattes the demad load for a sigle aget or multiple agets via schedulig the usage of agets appliaces at the demad side We also exteded the ECS problem to a more geeral format, the GECS problem i which each every aget s appliace ca have a ukow umber of i-use time slots at the schedulig stage After mathematically modelig these two ECS problems as Mixed-Iteger Programmig (MIP) problems, we proposed a ovel decompositio algorithm to efficietly ad accurately solve them We compared our algorithm with the stadard bechmark CPLEX i experimets As demostrated i the experimetal results (eg, deviatio ratios i the optimal solutios, rate of covergece ad rutime), our algorithm is prove to be sigificatly more practical ad accessible (highly efficiet ad accurate) tha CPLEX I the future, we will exted the studies of eergy cosumptio schedulig (ECS) problems for load levelig i two ways O the oe had, we will try to study the boud of our temporal decompositio algorithm ad theoretically examie the accuracy of our proposed efficiet solver O the other had, while solvig the ECS ad GECS problems, multiple agets o the grid have to share their iput data (eg, each aget s appliaces, total umber of i-use time slots) ad output (ie, specific aget s usage schedule of their appliaces i the optimal solutio) to joitly formulate ad solve the MIP-based ECS/GECS problems Such iformatio disclosure would explicitly compromise the cosumers privacy o the power grid [40,41] We will explore privacy-preservig schemes [42,43] to effectively formulate ad efficietly solve the ECS/GECS problem amog multiple agets o the smart grid with limited disclosure [44,45] Furthermore, we will exted the aget-based ECS problems for appliaces to the etities with reewable eergy sources [46,47], cosiderig each aget ad its appliaces as a microgrid (which both cosumes ad geerates electricity) Two categories of research

22 Eergies 2016, xx, x 22 of 27 problems will be ivestigated by itegratig the schedulig ad microgrids First, oe or multiple agets ca schedule ot oly cosumptio, but also geeratio for differet applicatios, such as load levelig [21], power flow aalysis ad optimizatio [48,49] Secod, the faults i power flow ad the distributio etwork [50 53] may lead to specific costraits i the schedulig problem After icorporatig such costraits i the ECS problems, we ca propose the fault-tolerat schedulig problems for both eergy cosumptio ad geeratio i the cotext of microgrids Ackowledgmets: This work is partially supported by the Natioal Sciece Foudatio uder Grat No CNS ad the FRAP-B Grat i the Uiversity at Albay, SUNY We thak the aoymous reviewers for their costructive commets Author Cotributios: Yua Hog formulated the optimizatio models for the ECS ad GECS problems; Shegbi Wag ad Yua Hog desiged the TD algorithm for efficietly solvig the schedulig problems; Yua Hog ad Shegbi Wag coceived ad desiged the experimets; Shegbi Wag ad Ziyue Huag performed the experimets; Ziyue Huag aalyzed the data; Shegbi Wag cotributed reagets/materials/aalysis tools; Yua Hog ad Shegbi Wag wrote the paper Coflicts of Iterest: The authors declare o coflict of iterest Appedix A Ruig Appliaces i Cotiuous Time Slots (ECS) Per Equatio (2) (the equality costraits), aget i s j-th appliace should be i-use for c ij time slots out of time slots where c ij Recall that = 1 meas the appliace is o i time slot k while = 0 meas the appliace is off The, we ca represet all of the possibilities of the c ij cotiuous time slots i Table A1: Table A1 c ij cotiuous time slots: ( c ij + 1) differet possibilities Time Slots c ij c ij + 1 c ij + 1 Possibility 1 x ij1 x ij2 x ij3 x ijcij Possibility 2 x ij2 x ij3 x ijcij x ij(cij +1) Possibility 3 x ij3 x ijcij x ij(cij +1) x ij(cij +2) Possibility ( c ij + 1) x ij( cij +1) x ij As show i Table A1, there are ( c ij + 1) differet possibilities for c ij cotiuous time slots (oe possibility per row) The, there are a set of costraits: if ay variable = 1, the all of the variables i exactly oe out of all of the possibilities (oe out of ( c ij + 1) rows i Table A1) should be equal to oe More specifically, If x ij1 = 1, the c ij = c ij must hold (oly oe possibility) This esures that such a appliace is cotiuously o i the first c ij time slots This costrait is equivalet to: x ij1 c ij c ij (A1) If x ij1 = 0, the iequality always holds (o costrait); otherwise x ij1 = 1, we have c ij = c ij (caot be greater tha c ij ) If x ij2 = 1, the c ij = c ij or c ij+1 = c ij (exactly oe out of two equalities) should hold sice there are exactly two possibilities to form c ij cotiuous time slots This esures that such a appliace is cotiuously o i time slots [1, c ij ] or [2, c ij + 1] Ideed, if c ij = c ij

23 Eergies 2016, xx, x 23 of 27 holds, the c ij+1 = c ij 1 ad vice versa Hece, we ca combie them together: c ij + c ij+1 = 2c ij 1; ad we formulate the costrait as: x ij2 [c ij + (c ij 1)] c ij + c ij +1 (A2) Similarly, if x ij3 = 1, the c ij = c ij, c ij+1 = c ij or c ij+2 k=3 = c ij (exactly oe out holds) This esures that such a appliace is cotiuously o i time slots [1, c ij ], [2, c ij + 1] or [3, c ij + 2] The, we derive the costrait as: x ij3 [c ij + (c ij 1) + (c ij 2)] c ij + c ij +1 + c ij +2 k=3 (A3) For time slot s = 4,, 1, the correspodig costrait ca be formulated as: x ijs [sc ij s (k 1)] c ij + c ij c ij +s 1 k=s (A4) If x ij = 1 (last time slot), the k= c ij +1 = c ij holds This esures that such a appliace is cotiuously o i the last c ij time slots The, we ca formulate the costrait as: x ij c ij k= c ij +1 (A5) I summary, there are ( c ij + 1) ew iequality costraits derived to esure ruig aget i s j-th appliace i c ij cotiuous time slots The ECS problem should iclude all of the costraits (Iequalities (A1) (A5)) for aget i s appliace j Thus, for all of the agets i [1, p], each of their appliaces j [1, m i ] has such a set of costraits (Iequalities (A1) (A5)) to esure ruig such a appliace i cotiuous time slots (c ij i total) Appedix B Problem Trasformatio (ECS) First, the objective fuctio ca be coverted (replacig every p i=1 m i j=1 (e ij ) with y k i the objective fuctio): mi : p m i i=1 j=1( e ij ) mi : y k (B1) The, some additioal costraits must be added (esurig that miimizig y k ca also miimize p i=1 m i j=1 (e ij ) ): which are equivalet to: k [1, ], p i=1 m i j=1 (e ij ) y k st k [1, ], y k 0 i [1, p], j [1, m i ], k [1, ], {0, 1} (B2)

24 Eergies 2016, xx, x 24 of 27 k [1, ], p i=1 m i j=1 (e ij ) y k k [1, ], p i=1 st m i j=1 (e ij ) + y k k [1, ], y k 0 i [1, p], j [1, m i ], k [1, ], {0, 1} (B3) Fially, combiig the origial costraits (Equatio (4)) ad extra costraits required for problem trasformatio (Equatio (B3)), we ca obtai the Mixed-Iteger Programmig (MIP) problem form for our ECS problem (Equatio (5)) Appedix C Ruig Appliaces i Cotiuous Time Slots (GECS) For aget i s j-th appliace, agai its total umber of i-use time slots x ij = will replace c ij i the ECS problem Similar costraits (similar to the discussios i Appedix A) ca be derived as below: If x ij1 = 1, the x ij = x ij must hold (oly oe possibility) This esures that such a appliace is cotiuously o i the first x ij time slots This costrait is equivalet to: x ij1 x ij x ij (C1) If x ij1 = 0, the iequality always holds (o costrait); otherwise x ij1 = 1, we have x ij = x ij (caot be greater tha x ij ) If x ij2 = 1, the x ij = x ij or x ij+1 = x ij (exactly oe out of two equalities) should hold sice there are exactly two possibilities to form x ij cotiuous time slots This esures that such a appliace is cotiuously o i time slots [1, x ij ] or [2, x ij + 1] Ideed, if x ij = x ij holds, the x ij+1 = x ij 1 ad vice versa Hece, we ca combie them together: x ij + x ij+1 = 2x ij 1, ad formulate the costrait as: x ij2 [x ij + (x ij 1)] x ij x ij +1 + (C2) Similarly, if x ij3 = 1, the x ij = x ij or x ij+1 = x ij or x ij+2 k=3 = x ij (exactly oe out of three equalities holds) This esures that such a appliace is cotiuously o i time slots [1, x ij ], [2, x ij + 1] or [3, x ij + 2] The, we ca formulate the costrait as: x ij3 [x ij + (x ij 1) + (x ij 2)] x ij + x ij +1 + x ij +2 k=3 (C3) for time slot s = 4,, 1, the correspodig costrait ca be formulated as: x ijs [sx ij s (k 1)] x ij + x ij x ij +s 1 k=s (C4) if x ij = 1 (last time slot), the k= x ij +1 = x ij holds This esures that such a appliace is cotiuously o i the last x ij time slots The, we ca formulate the costrait as: x ij x ij k= x ij +1 (C5)

25 Eergies 2016, xx, x 25 of 27 Similar to the ECS problem, there are ( x ij + 1) ew iequality costraits derived to esure ruig aget i s j-th appliace i x ij cotiuous time slots The GECS problem should iclude all of the costraits (Iequalities (C1) (C5)) for aget i s appliace j Thus, for all of the agets i [1, p], each of its appliaces j [1, m i ] has such a set of costraits (Iequalities (C1) (C5)) to esure ruig such a appliace i cotiuous time slots (x ij i total) Refereces 1 Fag, X; Misra, S; Xue, G; Yag, D Smart Grid The New ad Improved Power Grid: A Survey IEEE Commu Surv Tutor 2012, 14, Aug, Z; Toukhy, M; Williams, JR; Sachez, A; Herrero, S Towards Accurate Electricity Load Forecastig i Smart Grids I Proceedigs of the 4th Iteratioal Coferece o Advaces i Databases, Kowledge, ad Data Applicatios, Sait-Gilles, Belgium, 29 February 5 March Li, HY; Tzeg, WG; She, ST; Li, BSP A Practical Smart Meterig System Supportig Privacy Preservig Billig ad Load Moitorig I Proceedigs of the 10th Iteratioal Coferece, ACNS 2012, Sigapore, Jue 2012; pp Chu, CK; Liu, JK; Wog, JW; Zhao, Y; Zhou, J Privacy-preservig Smart Meterig with Regioal Statistics ad Persoal Equiry Services I Proceedigs of the 8th ACM SIGSAC Symposium o Iformatio, Computer ad Commuicatios Security, ASIACCS 2013, Hagzhou, Chia, 8 10 May 2013; pp Salias, S; Li, M; Li, P Privacy-preservig Eergy Theft Detectio i Smart Grids: A P2P Computig Approach J Sel Areas Commu 2013, 31, Masters, GM Reewable ad Efficiet Electric Power Systems, 2d ed; Wiley: Hoboke, NJ, USA, Davidoff, S; Lee, MK; Yiu, C; Zimmerma, J; Dey, AK Priciples of Smart Home Cotrol I Proceedigs of the 8th Iteratioal Coferece o Ubiquitous Computig, Orage Couty, CA, USA, September 2006; pp Ullah, MN; Mahmood, A; Razzaq, S; Ilahi, M; Kha, RD; Javaid, N A Survey of Differet Residetial Eergy Cosumptio Cotrollig Techiques for Autoomous DSM i Future Smart Grid Commuicatios arxiv 2013, arxiv: Lee, J; Kim, H-J; Park, G-L; Kag, M Eergy Cosumptio Scheduler for Demad Respose Systems i the Smart Grid J If Sci Eg 2011, 27, Barker, S; Mishra, A; Irwi, D; Sheoy, P; Albrecht, J SmartCap: Flatteig peak electricity demad i smart homes I Proceedigs of the 2012 IEEE Iteratioal Coferece o Pervasive Computig ad Commuicatios, PerCom 2012, Lugao, Switzerlad, March 2012; pp Barker, S; Mishra, A; Irwi, D; Cecchet, E; Sheoy, P; Albrecht, J Smart*: A Ope Data Set ad Tools for Eablig Research i Sustaiable Homes I ACM 2012 Workshop o Data Miig Applicatios i Sustaiability, Beijig, Chia, Maharja, S; Zhu, Q; Zhag, Y; Gjessig, S; Basar, T Depedable Demad Respose Maagemet i the Smart Grid: A Stackelberg Game Approach IEEE Tras Smart Grid 2013, 4, Albadi, M; El-Saaday, E Demad Respose i Electricity Markets: A Overview I 2007 IEEE Power Egieerig Society Geeral Meetig, Tampa, FL, 2007, pp Logethira, T; Sriivasa, D; Shu, TZ Demad Side Maagemet i Smart Grid Usig Heuristic Optimizatio IEEE Tras Smart Grid 2012, 3, Paterakis, NG; Erdic, O; Bakirtzis, AG; Catalao, JPS Optimal Household Appliaces Schedulig Uder Day-Ahead Pricig ad Load-Shapig Demad Respose Strategies IEEE Tras Id If 2015, 11, Paterakis, NG; Tascikaraoglu, A; Erdic, O; Bakirtzis, AG; Catalao, JPS Assessmet of Demad-Respose-Drive Load Patter Elasticity Usig a Combied Approach for Smart Households IEEE Tras Id If 2016, 12, Liu, W; Wu, Q; We, F; Ostergaard, J Day-Ahead Cogestio Maagemet i Distributio Systems Through Household Demad Respose ad Distributio Cogestio Prices IEEE Tras Smart Grid 2014, 5, Sarker, MR; Ortega-Vazquez, MA; Kirsche, DS Optimal Coordiatio ad Schedulig of Demad Respose via Moetary Icetives IEEE Tras Smart Grid 2015, 6,

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