21 IEEE 4th Annul Conference on Decision nd Control (CDC December 1-18, 21. Os, Jpn Demnd response for ggregted residentil consumers with energy storge shring Kveh Pridri, Alessndr Prisio, Henri Sndberg nd Krl Henri Johnsson Abstrct A novel distributed lgorithm is proposed in this pper for networ of consumers coupled by energy resource shring constrints, which ims t minimizing the ggregted electricity costs. Ech consumers is equipped with n energy mngement system tht schedules the shiftble lods ccounting for user preferences, while n ggregtor entity coordintes the consumers demnd nd mnges the interction with the grid nd the shred energy storge system (ESS vi distributed strtegy. The proposed distributed coordintion lgorithm requires the computtion of Mixed Integer Liner Progrms (MILPs t ech itertion. The proposed pproch gurntees constrints stisfction, coopertion mong consumers, nd firness in the use of the shred resources mong consumers. The strtegy requires limited messge exchnge between ech consumer nd the ggregtor, nd no messging mong the consumers, which protects consumers privcy. Performnce of the proposed distributed lgorithm in comprison with centrlized one is illustrted using numericl experiments. Index Terms Demnd response, mixed integer liner progrmming, distributed scheduling lgorithms. I. INTRODUCTION Residentil res re responsible for nerly 4% of the energy consumption in developed countries, which re nown to hve significnt potentil for energy nd cost svings, s well s for lod shifting, compred to industry nd trnsporttion [1]. To te dvntge of this potentil, Demnd Response (DR hs received incresed ttention in recent yers since it cn efficiently support lod blncing nd economicl/environmentl cost reduction [2]. DR is commonly defined s chnges in electricity use by consumers in response to chnges in the electricity price over time [2]. Effective DR policies nturlly require smrt pplinces, which cn be switched on or off in response to specific DR signls, e.g., price signls. Severl wors hve proposed lod mngement strtegies nd scheduling smrt pplinces, ccounting for price informtion (e.g., see [2], [3], [4], [], [6], [7], [8], nd [9]. The forementioned wors do not consider electricl storge systems (ESS, while it would be more flexible nd efficient for consumers to mnge their energy use in response All the uthors re with the ACCESS Linneus Center nd the Automtic Control Lb, School of Electricl Engineering, KTH Royl Institute of Technology, Sweden. Emils: pridri@th.se, prisio@th.se, hsn@th.se nd llej@th.se. This wor is supported by the Europen Institute of Technology (EIT Informtion nd Communiction Technology (ICT Lbs, the EU 7th Frmewor Progrmme (FP7/27-213, grnt greement n 68224, the Swedish Energy Agency, Swedish Foundtion for Strtegic Reserch through the ICT-Psi project, the Swedish Governmentl Agency for Innovtion Systems (VINNOVA, nd the Knut nd Alice Wllenberg Foundtion. to time-vrying electricity prices nd networ congestion, ting dvntge of the cpbility of these devices to store energy nd relese it when it is more convenient [1], [11], [12], [13], nd [14]. Since storge devices re still expensive, resonble solution to fford the expense nd benefit from the use of n ESS would be to shre it mong severl consumers. Therefore, the households should be coordinted by n ggregtor/coordintor. Aggregtors re new entities in the electricity mret tht ct s meditors between users nd the utility opertor, nd possess the technology to perform DR signls nd communicte with both users nd utilities [1]. In [16], n lgorithm is built on the lternting directions method of multipliers (ADMM, focusing on decentrlized lgorithms for Electric Vehicles chrging. In ddition, coordintion frmewor bsed on ADMM is proposed in [17] to negotite mong the households nd coordintor, with the min gol being to minimize the imblnce mong communities, while including objectives nd constrints for ech community nd ting into ccount ech user s qulity of life/ctivities. The min contribution of this pper is to propose novel distributed optimiztion lgorithm for scheduling of smrt pplinces in residentil res shring n ESS. Here, in ddition to the detiled modeling of lods nd ESS, stte of helth of the ESS is ten into ccount. The scheduling problem is formulted s MILP problem with the im of minimizing the ggregted electricity cost. In this study, n ESS is integrted with the ggregtor to increse level of comfort nd profit for the users. The role of the ggregtor is thus to provide DR services to the grid opertor nd economicl incentives to the users to reshpe their demnd profiles, gurnteeing tht the benefits of using shred resources re firly llocted to ll the users. The ggregtor negotites with the home users to shift their lods nd increse the ESS benefits, providing incentives to the users to shift their consumption. The rest of the pper is structured s follows. Section II describes the system nd the connections between the two lyers (ggregtor nd prtment of the system. Section III proposes distributed scheduling lgorithm for smrt pplinces nd EESs to cope with pe-demnd shving, nd firly devoting the monetry profits to the prtments (bsed on their flexibility in lod shifting. Section IV presents simple motivtion exmple to show the coupling between ESS nd prtment consumption. In ddition, preliminry simultion results in this section, illustrte the performnce of the proposed distributed lgorithm in comprison with centrlized one. Finlly, Section V provides conclusions nd 978-1-4799-7886-1/1/$31. 21 IEEE 224
ESS Aggregtor Distributed Optimiztion Algorithm Solving MILP Electric Utility Opertor Grid be implemented in hierrchicl fshion where the problems t prtment levels re executed in prllel nd the updtes of ggregted demnd profile is crried out by the high-level ggregtor. In the following we first describe the pplince nd the ESS modeling, bsed on the study [14]. HEMS HEMS HEMS Aprtment #1 Aprtment #i Aprtment #N Fig. 1. Schemtic of interconnected prtments nd ggregtor. suggestions for future studies. II. SYSTEM DESCRIPTION In this pper, we consider ctive prtments, i.e., houses where effective DR policies re enbled through the integrtion of smrt pplinces, scheduling lgorithms, energy mngement systems, nd informtion exchnge over wireless communiction technologies. As depicted in Figure 1, we consider smll-scle community which cn rnge from the prtments in one building to smll district of city. The overll system forms microgrid, with one single point of common coupling (PCC with the distribution grid. Ech prtment is equipped with Home Energy Mngement System (HEMS, which is responsible for loclly operting end-user smrt pplinces. Ech HEMS is connected to n ggregtor entity vi communiction networ, which ims t coordinting the prtments, scheduling the ESS nd mnging the interction with the distribution grid. The prtments re independent from ech other nd coupled only through the shred ESS nd the PCC power limits. In this structure, the ggregtor coordintes the prtments through energy shift request signls. In this negotition, the ggregtor provides economicl incentives to home users ccepting to modify their energy pttern. In the next section we describe the distributed pproch to the energy mngement of the system under considertion. III. PROBLEM FORMULATION In this section we describe distributed pproch to solve the problem of coordinting set of smrt pplinces locted in N prtments shring n ESS such tht ech prtment cn profit from the use of the ESS while technicl nd opertionl constrints, s well s user preferences, re stisfied. In order to mnge lrge set of pplinces we propose n itertive two-lyer hierrchicl pproch. The pproch is bsed on distributed lgorithm with problems formulted t the prtment nd t the ggregtor levels being MILP problems. The locl HEMSs re coordinted by the ggregtor in order to come up with n greement throughout negotition itertions nd provide fesible solution to the centrlized problem. We remr tht the proposed lgorithm is suitble for model predictive control frmewors: it cn PCC A. Applinces nd ESS constrints This section is bsed on the mthemticl formultion illustrted in [14]. The scheduling horizon is discretized into T uniform time slots. The number of pplinces in the prtment is denoted by N, nd n i denotes the number of un-interruptible energy phses for ech pplince. The energy ssigned to energy phse j of pplince i for prtment during the whole period of time slot is denoted by p ij. Binry decision vribles (xij re required to indicte whether prticulr energy phse is being processed or not. Moreover, two other sets of binry decision vribles re needed to model the decision problem. One is denoted s s ij, with vlue of one indicting tht, in pplince i, energy phse j is lredy finished by time slot. The other set is denoted s t ij. These decision vribles re used to indicte whether t time slot, pplince i is ming trnsition between running phse j 1 to j. The constrint tht is enforced to me sure tht the energy phses fulfill their energy requirement is s =1 p ij = E ij, i, j, (1 where E ij is the energy requirements for energy phse j in pplince i. To determine tht n energy phse is being processed during time slot, while the limittion on lower nd upper power ssignment to the phse re stisfied, the constrint p ij xij p ij p ij xij, i, j,, (2 is enforced, nd the p ij nd p ij re the lower nd upper limits. Also, the power sfety constrint is imposed s N p ij P,, (3 i=1 where P is the upper limit of the totl energy ssigned t time slot, for ech prtment. The limits on energy phses process time re imposed s T ij m =1 x ij T ij, i, j, (4 where the T ij nd T ij re the lower nd upper limits of the number of time slots for the relted energy phse to be processed. To stisfy the sequentil processing of the energy phses of n pplince nd sequentil opertion of pplinces, the following constrints re imposed respectively x ij s i(j 1, i,, j = 2,..., n i, x ij sĩnĩ,, ( 22
with the ĩ being the index of the pplince which must be finished before the pplince with i index cn strt running. To me sure tht the energy phses re un-interruptible the following constrint is imposed. x ij 1 s ij x ij 1 xij s ij s ij 1 s ij i, j, i, j, = 2,..., m i, j, = 2,..., m. To count the number of time slots spent between the energy phses in n pplince nd impose lower nd upper limits on this number, the constrints t ij = s i(j 1 D ij (x ij =1 (6 + s ij i, j, = 2,..., ni, (7 t ij D ij, i, j = 2,..., n i, (8 re considered, where D ij nd D ij re between-phse dely lower nd upper bounds, respectively. Finlly, to meet the household preferences nd finishing prticulr pplince within specified time intervl, the constrint x ij T P i i, j,, (9 is enforced, where T P i is the time preference intervl. To include n EES the following set of constrints is defined [14]. The ESS dynmics re modeled s follows: b s = αb s 1 + η c b c 1 η d b d 1, = 2,..., T, (1 where α is constnt energy degrdtion in ech smpling intervl, b s is the energy level t time slot nd ηc nd η d re efficiencies ccounting for the losses during chrging nd dischrging. The power exchnged with the EES during time slot is denoted by b c (or bd when the EES is chrging (or dischrging. The following limits on the energy level nd the power exchnged with the ESS re enforced: b s b s b s,, (11 where b s nd b s denote the lower nd the upper bounds respectively, nd b c b c x c, b d b d x d,, (12 where the binry decision vribles x c nd xd indicte whether the EES is chrging or dischrging in time slot, respectively. The bounds on the power exchnged with the ESS re b c nd b d for chrging nd dischrging respectively. Further, the constrint x c + x d 1,, (13 hs to be stisfied to rule out the simultneous chrging nd dischrging during the sme time slot. To te the stte of helth of EES into ccount, the totl number of chrging nd dischrging cycles during dy must be limited to determined number N c, nd the constrints x c xc 1 c t, = 2,..., m x d xd 1 d t, = 2,..., m m c t + dt N c, i=1 (14 re imposed, where the uxiliry binry decision vribles c t nd d t determine the trnsition time slots to strt chrging nd dischrging, respectively. Finlly, it is resonble to ssume tht the initil nd the finl energy levels (b s nd b s T respectively in the EES re the sme, since the finl energy level is lso the initil condition for the next dy scheduling. Hence, the following equlity constrint on the initil nd finl SOC is enforced B. Centrlized problem b s = b s T. (1 The centrlized scheduling problem for networ of prtments with shred ESS is formulted s the following MILP: min =1 c s.t. constrints (1 (1 N N =1 i=1 p ij + b c bd = pgrid. (16 When the number of smrt pplinces increses, solving the centrlized problem cn be computtionlly prohibitive. Thus, we proposed the following distributed lgorithm. C. Description of the distributed lgorithm Here we describe the proposed distributed lgorithm. The lgorithm comprises n initiliztion step, nd the definition of MILP problems t the prtment nd ggregtor levels. Prmeters nd vribles involved in the lgorithm: Tble I reports ll the other prmeters nd vribles defined in the lgorithm. TABLE I: Prmeters nd vribles involved in the lgorithm l itertion number within current time step N number of prtments/single-fmily houses N number of pplinces of prtment,l totl exchnged power with the grid t itertion l p ij,l power feeding into the prtment (per pplince nd per energy phse t itertion l β penlty on the unstisfied shre of energy shift required by the Aggregtor γ rewrd on the redistributed prt of unstisfied energy shift, lower nd upper bounds on the power exchnged with the grid p pp,l ggregted demnd t itertion l E pp totl energy requirements G TOT,l totl profit due to the ESS t itertion l t the end of the horizon G,l profit per prtment t itertion l p AGG,l energy shift required by the Aggregtor t itertion l δp,l ccepted energy shift by prtment t itertion l p,l unstisfied shre of energy shift by prtment t itertion l δp +,l, redistributed energy shift by prtment t itertion l δp,l ("+" for energy increse nd "-" for energy decrese 226
Initiliztion: The following problem is solved for ech prtment, = 1,..., N: ( N min c p ij, (17 =1 i=1 s.t. constrints (1 (9. The ggregted demnd profile, resulting from solving Problem (17 for ech prtment, represents the solution of the centrlized problem (16 without considering ny shred ESS, which is fully seprble in tht cse. The sum of the optiml vlues for ech prtment is n upper bound on the optiml solution of the problem (16 with ( shred ESS. This N N c. sum is computed s G TOT, = T p pp, =1 =1 i=1 p ij, The following problem is solved for initilizing the ggregtor: min c, s.t. constrints (1 (1 + b c bd = pgrid, =1, p pp, = E pp. (18 The ggregted energy profile computed through Problem (18 is the best possible since it ccounts only the totl energy required to run ll the pplinces in the networ of prtments, without considering user preferences nd technicl constrints on the energy ssignment. Thus the optiml vlue of Problem (18 is lower bound on the optiml solution of the problem (16 with shred ESS. Once ll the prtment solve the corresponding Problem (17, they send the computed optiml energy profile to the ggregtor, which clcultes the difference between the ggregted energy profiles obtined t prtment nd ggregtor levels s follows: p AGG, = p pp, N N p ij, =1 i=1. This difference is sent to the prtments s shift request signl nd the lgorithm proceeds ccording the steps described in Algorithm 1. Before describing the itertions of the proposed distributed lgorithm, we formulte the problems to be solved t prtment nd ggregtor levels, to be done fter initiliztion. Problem t prtment level: The problem t prtment level t itertion l is formulted s follows: ( N min c p ij,l + β p,l =1 i=1 s.t. constrints (1 (9 i p ij,l = n p ij,l 1 + δp,l pagg,l N p,l δp,l pagg,l N (19 where the decision vrible δp,l models the differences in the energy profile between two consecutive itertions. The vrible δp,l hs the sme sign of p AGG,l, which is the energy shift request signl sent by the ggregtor t itertion l. Notice tht the unmet shre of the energy shift requested by the ggregtor t time slot, p,l, is penlized in the objective function with fctor greter thn energy prices by t lest 2 order of mgnitude. Unmet energy shift cn be needed minly to void constrint violtion in Problem (19. Problem t ggregtor level: The problem t ggregtor level t itertion l is formulted s follows: min c,l s.t. constrints (1 (1 p pp,l + b c bd = pgrid,l = N N n (2 i p pp,l =1 i=1,l p ij,l. The shift request signl t itertion l is computed s follows: N N p AGG,l = p pp, =1 i=1 p ij,l Computtion of cost benefits due to ESS: The overll profit t the end of the scheduling horizon t ech itertion l is: G TOT,l = G TOT, c,l. The cost benefits re eqully shred mong the prtments, however penlty is ssigned to the unmet energy shift requested by the ggregtor. The profit t prtment level t itertion l is then computed s: G,l = mx( GTOT,l N c p,l,. Steps of the distributed lgorithm : The steps of the proposed lgorithm re detiled in 1. Algorithm 1 Distributed lgorithm 1: Initiliztion nd computtion of p AGG,, 2: for l = 1, 2,..., MxItertion do 3: ech prtment solves Problem (19 4: ech prtment sends to ggregtor the computed power profiles : ggregtor solves Problem (2 6: ggregtor computes G TOT,l 7: ech prtment computes G,l 8: if G,l < G,l 1, prtment ccepts the energy profile, otherwise p ij,l = p ij,l 1, i, j, 9: if ll prtments ccept, stop, otherwise compute nd repet p AGG,l Redistribution strtegy: An improvement in the solution obtined by Algorithm 1 t ech itertion cn be chieved by trying to redistribute the unmet energy shift request from the Aggregtor mong the prtments. Hence, the profiles of the totl positive nd negtive unmet energy per time slot re computed respectively s p +,l = N p,l = N =1 =1 p +,l nd p,l. An dditionl step is to be included in Algorithm 1 between step 3 nd 4. The redistribution is 227
chieved by solving the following problem strting from the prtment level 1: ( N min c p ij,l γ(δp +,l + δp,l where =1 i=1 s.t. constrints (1 (9 i ni p ij,l δp +,l δp,l p ij,l = n p +,l p,l p ij,l + δp +,l δp,l. (21 is the energy per time slot computed t itertion l t step 3. The totl unmet energy per time slot is then updted by subtrcting δp +,l nd δp,l from p +,l nd p,l respectively. Problem (21 is solved then for the next prtments until either there is still unmet energy shift or ll the prtments hve been sed for redistribution. Properties of the distributed lgorithm: Algorithm 1 hs the following desirble properties: fesibility of the solution: t the initiliztion step, bounds on the optiml vlue of Problem (16 re computed. Clerly, the optiml schedules computed t the initiliztion step re not fesible solutions of the centrlized problem. After the initiliztion, during ech itertion of Algorithm 1, fesible solutions re obtined: this is gurnteed by the procedure defined by the lgorithm. Effectively, during generic itertion, the energy profiles sent by the prtments nd included in (2 s given ggregted lod stisfying ll the pplinces constrints nd user preferences, s defined by Problem (19; on the other hnd, the ESS schedule computed by solving (2 fulfills ll the technicl nd opertionl constrints concerning the ESS nd the interction with the distribution grid. Every time the energy profiles t prtment level re computed bsed on energy shift requests from the ggregtor, n updted ESS schedule is computed bsed on the resulting ggregted energy profile. By doing so, the solution computed t ech itertion stisfies ll the constrints formulted in the centrlized problem (16; suboptimlity of the solution: s mentioned bove, t the initiliztion step lower nd n upper bound on the optiml vlue of Problem (16 re computed. Subsequently, t ech itertion of Algorithm 1, the solution steps towrds the optiml solution of the centrlized problem. This is ensured by two spects of the procedure: i n prtment ccepts n updte on it energy use profile only if its locl objective function, which includes lso ESS-relted benefits, decreses; ii the ESS schedule hs to ccount for the energy profiles computed t the prtment level, which certinly leds to vlue of the objective function t the ggregtor level greter thn the one computed t the initiliztion step. However, the lgorithm provides suboptiml solution since there re no gurntees tht the optiml solution is reched when the lgorithm termintes; fir lloction of profits: the ESS-relted profits t the end of the scheduling horizon re eqully divided mong the prtments, so re the energy shift requests. Further, n incentive mechnism is considered: users re penlized for the unmet energy shift request nd rewrded for ting on shre of the totl unmet energy shift requested by the ggregtor. We will include mthemticl proof of this third property in n extended version of this study. We remr tht infesibility cn occur t ggregtor level during generic itertion. This cn be prevented by modifying Problem (2 nd replcing the constrint on p pp,l with the following constrint: p,l where p,l pagg,l = N N =1 i=1 p pp,l p ij,l weighted in the objective function. p,l nd p AGG,l + pagg,l, is opportunely IV. MOTIVATION EXAMPLE AND PRELIMINARY RESULTS Scheduling problem for networ of prtments, which re shring n ESS, is formulted in the (16. From (16 One my notice tht (which is the power exchnge with the grid t PCC is simply the power consumption of the prtments plus power exchnge(chrging/dischrging whit the ESS. In the rel power networ, the is limited within upper nd lower limits, to protect the networ from overlod. In this problem, ggregtors gol is to minimize the electricity consumption cost for whole the system, nd in the most optimistic cse the SHAs in the prtments will be scheduled (while stisfying their constrints when the price of electricity is minimum. Also, ESS will chrge when the price is low nd dischrge when the price is high, to me the most possible profit out of the grid. This optimistic cse will result in the optiml solution for the problem s fr s the is within the power limittion bound for ll the times during the dy, nd in this cse we cn sy tht scheduling of pplinces in the prtments could be done seprtely from the ESS scheduling (the scheduling is decoupled. This is not lwys the cse, nd by scheduling of the SHAs nd chrging of the ESS to be hppened t the sme time (when the price of electricity is low, the violtes the power limittions t some points during the dy. In this cse, the overlod should be shifted to the other times by the ggregtor, in which either users should chnge their desired scheduling or the ESS scheduling should chnge. In this sense, SHAs nd ESS scheduling re coupled with ech other nd the optimiztion lgorithm in the ggregtor level should find n optiml solution, by joint scheduling of SHAs nd ESS. Motivtion exmple: In this exmple, prtments A nd B (number of ptments in Fig. 1 is two shre n ESS, nd whole the system is connected to the grid t PCC. The scheduling of the shred storge, nd lso pplinces in these prtments re shown in Fig. 2, for two different cses,i the boundries on power exchnge with the grid t PCC re not limiting (in the left side of the figure, nd ii the power 228
USD/(Wh Energy (Wh Energy (Wh Energy (Wh x 1 3 2 1 Electricity price 1 1 2 ( 1 1 SHAs consumption of A SHAs consumption of B 1 1 2 (b 1 ESS scheduling 1 1 2 (c Exchnge power t PCC 1 1 (d 2 hour x 1 3 2 1 Electricity price 1 1 2 (e 1 1 SHAs consumption of A SHAs consumption of B 1 1 2 (f 1 ESS scheduling 1 1 2 (g Exchnge power t PCC 1 1 (h 2 hour Fig. 2. Scheduling of pplinces (in the prtments A nd B nd the shred ESS in two different cses, i the boundries on power exchnge with the grid t PCC re not limiting (in the left side, nd ii the power exchnge t PCC is more limited (in the right side. exchnge t PCC is limited in nrrow bound (in the right side. As it is shown in the left side of this figure, by hving n upper limittion on the to be high enough (in this cse 1W, the schedulung of SHAs in the both prtments A nd B nd lso ESS chrging will be scheduled when the electricity price hs the lowest vlue (between 3: nd : m. Therefore, in this cse the ESS chrging nd the SHAs cn be scheduled in decoupled fshion, nd the mximum totl power exchnge (9W between 3: nd : m will not violte the power limittion. SHAs nd ESS scheduling, nd the totl power consumption re illustrted in the prts (b, (c, nd (d,respectively. On the other hnd, in the cse tht the power exchnge limittion (6W is lower enough, ggregtor cnnot eep the sme scheduling for SHA nd ESS, otherwise devition from power limittion will hppen t PCC (between 3: nd : m. In this cse, ggregtor should mnge for overlod shifting from 3:-: m to nother times of the dy, either through negotition with the prtments to shift their SHAs nd incentivise them with monetry profit, or by re-scheduling the ESS chrging/dischrging. In the first solution scenrio, shifting the SHAs consumption from the lowest price time period (3:-: m to the other low price period (16:-21:, will cuse smll increse in electricity bill. This is becuse of the smll difference between the electricity price in these two period. In the second solution scenrio, if the ESS-chrging hppens to shift from 3:-: m time durtion, it will not be ble to dischrge in :-7: m, nd will cuse big effect on profit ming. Thts becuse of the difference between the electricity price in these two period, which is noticeble. Thus, the first solution scenrio for this coupled cse is more money ffordble, nd prts (f nd (g of the figure show the proper scheduling of the SHAs nd ESS. This scenrio cuses no violtion from the power limittion (see prts (h. Therefore, it is necessry for ggregtor to pply n optiml opertion strtegy, through coordinting with prtments, to schedule the SHAs nd the ESS, nd del with the coupling cses. In ddition, by pplying centrlized pproch, the clcultion time would not be resonble when the number of prtments increses. Therefore, essence of hving distributed scheduling pproch is obvious. Preliminry results: In order to evlute the proposed distributed frmewor, we present preliminry results obtined by pplying the Algorithm 1 to microgrid system comprising 4 ctive prtments with 3 smrt pplinces ech: dishwsher, wshing mchine nd dryer. We consider piecewise constnt electricity triff signl extrcted from Nordpool website. We include in the numericl evlution the hourly energy use due to sources of electricity consumption other thn household pplinces. The shred ESS hs the following technicl fetures: Storge cpcity: 2Wh Mximum power exchnge: 8W Mximum Depth Of Dischrge(DOD: 3% Stored energy degrdtion (α: negligible Chrging nd dischrging efficiency: 9% Mximum chrging nd dischrging cycles: (per dy. We then pply both Problem (16 nd Algorithm 1 to the system under considertion nd compute the corresponding schedules of the pplinces, the shred ESS nd the interction with the grid. Figure 3 depicts the comprison between the solution computed by solving the centrlized problem nd the solutions obtined by the proposed distributed lgorithm t itertion 1 nd 6, which is the lst itertion in this prticulr cse study. We cn notice tht, s Algorithm 1 (the distributed pproch is iterted, the solutions get closer to the optiml one (the solution resulting from solving the centrlized problem. The totl electricity cost of the optiml solution is 1.2 USD while the electricity cost resulting from the finl itertion of Algorithm 1 is 1.21 USD, hence only 1.3% higher. On the other hnd, the computtionl time of the centrlized problem (16 ws 74 sec while the proposed distributed lgorithm computes the solution in 7.29 sec, hence the computtionl time hs decresed by two orders of mgnitude. The MILP problems were solved using CPLEX with the YALMIP MATLAB interfce [18]. Further studies will focus on conducting extensive simultions nd investigte the potentil improvements in the lgorithm performnce brought by the redistribution strtegy described in Section III. 229
USD/(Wh Totl consumption (Wh SHAs consumption (Wh Storge schedule (Wh 4 x 1 4 Electricity price 2 1 1 2 Centrlized Distributed (l=1 Distributed (l=6 1 1 2 2 Centrlized Distributed (l=1 Distributed (l=6 1 1 2 2 Centrlized Distributed (l=1 Distributed (l=6 1 1 2 2 hour Fig. 3. Comprison between the optiml solution of the centrlized problem (16 nd the solution computed by Algorithm 1 t itertions 1 nd 6 (the lst itertion. V. CONCLUSION AND FUTURE STUDIES Over the lst decde, storge devices hve become one of the importnt components in smrt grid for pe demnd shving, voltge imblnces mitigtion, nd consumers electricity bill reduction. Due to the high cost of ESSs, it could be convenient to consumers to deploy nd shre them in coopertive mnner. In this wor we propose n itertive distributed pproch to solve the problem of coordinting set of smrt pplinces locted in networ of prtments shring n ESS such tht ech household cn profit from the use of the ESS while technicl nd opertionl constrints, s well s user preferences, re stisfied. The problem of coordinting the shred resources mong the consumers is complicted by firness requirement, i.e., storge will eqully benefit consumers ccording to their flexible lods. The novel distributed scheduling lgorithm proposed in this pper hs the following properties i provides fesible solution to the centrlized scheduling problem; ii lloctes firly ESS-relted profits mong the users; iii requires limited messges to be exchnged between ech consumer nd the ggregtor, nd no messge pssing mong the consumers, to eep consumers privcy, nd (iii is suitble for online optimiztion-bsed control scheme, such s MPC. Numericl results show tht the computed solution is close to the optiml one computed by centrlized problem. Although home pplinces nd EESs re considered in this wor, we point out tht the proposed frmewor cn be lso extended to scenrios considering different uncertinty sources, different storge technologies nd generic progrmmble electricl lods, s well s different optimiztion criteri. As future study, uncertinties on such huge lod shifting by using utomtion system in lrge number of prtments (which cuses the rel-time triff to vry from the dy-hed one will be ten into ccount. REFERENCES [1] Z. Xu, Q.-S. Ji, nd X. Gun, Supply demnd coordintion for building energy sving: Explore the soft comfort, Automtion Science nd Engineering, IEEE Trnsctions on, pp. 1 1, 214. [2] D. O Neill, M. Levorto, A. Goldsmith, nd U. Mitr, Residentil demnd response using reinforcement lerning, in Smrt Grid Communictions (SmrtGridComm, 21 First IEEE Interntionl Conference on, 21, pp. 49 414. [3] M. Song, K. Alvehg, J. 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