Real-time pricing in environments with shared energy storage systems

Transcription

1 ORIGINAL ARTICLE Real-tme prcng n envronments wth shared energy storage systems Konstantnos Sterots & Georgos Tsaousoglou & Nkolaos Efthymopoulos & Prodromos Makrs & Emmanouel (Manos) Varvargos Receved: September 207 /Accepted: 0 August 208 # Sprnger Nature B.V. 208 Abstract A major challenge n modern energy markets s the utlzaton of energy storage systems (ESSs) n order to cope up wth the dfference between the tme ntervals that energy s produced (e.g., through renewable energy sources) and the tme ntervals that energy s consumed. Modern energy prcng schemes (e.g., realtme prcng) do not model the case that an energy servce provder owns a shared ESS that ts customers could take advantage of, even though a shared ESS s more effcent than the operaton of many ndvdual ESSs (.e., personal ESS case). Thus, we propose a shared ESS aware real-tme prcng model that acheves a very attractve tradeoff between the servce provder s and end user s nterests. We also compare our system wth ts predecessors and we wtness ts superorty. The proposed scheme allows energy consumers to use shared ESS and have cost-effcent energy servces and n the same tme they protect ther nterests (far bllng). Keywords Smart grds. Demand response program. Shared energy storage system. Real-tme prcng. Energy consumpton schedulng K. Sterots (*): G. Tsaousoglou : N. Efthymopoulos : P. Makrs : E. (. Varvargos Insttute of Communcaton and Computer Systems, Department of Electrcal and Computer Engneerng, Natonal Techncal Unversty of Athens, Iroon. Polytechnou Str. 9, Zografou, 5773 Athens, Greece e-mal: konsterots@mal.ntua.gr G. Tsaousoglou e-mal: geotsaousoglou@mal.ntua.gr N. Efthymopoulos e-mal: nkoseft@mal.ntua.gr P. Makrs e-mal: prodromosmakrs@central.ntua.gr E. (. Varvargos e-mal: vmanos@central.ntua.gr E. (. Varvargos Department of Electrcal and Computer Systems Engneerng, Monash Unversty, Melbourne, Australa Introducton The projected growth n global electrcty demand (USEIA 206), the global concerns for the envronment, and the condtons n the global economy (USEPA 207) combned wth the ncreasng use of nformaton and communcaton technologes (ICT) are motvatng the transton from a centralzed grd to a decentralzed, dynamc, and flexble grd (Smart grd). The Smart Grd concept (Farhang 200), whch ncludes the modernzaton of the exstng power network through the ICT contrbuton, ams at the effcent energy usage and the reducton of the total energy cost. In ths context, the use of energy storage technologes to address producton and consumpton fluctuatons can provde an alternatve soluton to the costly conventonal thermal generators and lead to a more flexble and reslent grd. The mplementaton of energy storage systems (ESSs) can brng varous advantages, such as grd

2 operatonal cost savngs, relablty mprovement, CO 2 emssons reducton, and job creaton. Currently, the applcaton of ESSs n Smart Grds s recevng more attenton than ever from system operators, snce ESSs are experencng a substantal growth n ther performance and operatonal effcency, along wth a contnuous CAPEX reducton durng the last years (Deutsche Bank 205; World Energy Councl 206), whle the ntegraton of renewable energy sources makes producton more dstrbuted, dynamc, and unpredctable. Varous battery models (TESLA 206; LG Chem 206; Eaton Nssan 207; SonnenBattere 207) are manufactured for resdental and commercal use. However, due to both hgh purchase/nstallment costs and space lmtaton, the soluton of ntegratng a shared ESS to serve an energy communty s more effcent than ndvdual ESS operaton. The concept of Bshared^ ESS ncludes two cases. In the frst case, a company, such as an energy servce provder (ESP), nstalls at ts premses (or leases) and operates a bulk ESS. In the second case, the ESP nstalls and centrally controls small ESSs at users premses. In the context of the frst case, HENBO energy storage (Henbo 205) has already nstalled a shared ESS n Upperlands, Northern Ireland, n order to demonstrate the hgh value of combnng renewable energy generaton wth a shared ESS. Moreover, Tesla s plannng to buld the world s largest shared ESS n South Australa wth the objectve beng to ncrease local electrcty grd relablty and stablty (Edelsten 207). In ths paper, we adopt the case n whch a bulk ESS s nstalled and controlled by an ESP n such a way that ESS benefts are farly dstrbuted among the users. It s proved by Rahbar et al. (206) that the nstallment of a shared ESS ncreases the end-users profts even by 0% n comparson wth the ndvdual ESSs case. Moreover, our recent work n (Tsaousoglou et al. 206) demonstrates that a vrtual storage bank, consstng of several ndvdual storage capactes, can ncrease the proft by up to % n comparson wth dstrbuted ESSs operatng ndependently. Thus, a shared ESS can reduce the energy costs of the ESP and at the same tme beneft the users n terms of convenence and energy blls. The man challenge for the ESP s how to operate optmally the shared ESS. Thus, ntellgent ESS control strateges are needed so as not to rsk the smooth grd operaton, especally at peak hours (or harmonze producton wth consumpton n case of hgh RES penetraton). A method toward ths goal (modfy the aggregate energy consumpton curve (ECC) n order to reduce the energy cost) s demand sde management (DSM). DSM ncludes, among others, smart energy prcng schemes n order to ncentvze the energy consumers toward a consumpton pattern that provdes a good trade-off between ther desred pattern and the one that s cost-effcent for the energy system (Palensky and Detrch 20). A great deal of recent research has focused on developng new, smarter prcng models wth the purpose of arousng the users demand response (DR) capabltes (Khan et al. 206). The most common flat prce model provdes no monetary motvaton for the users to re-shape ther consumpton curve, whle the follow up model of nclnng block rates, although t drves users toward a lower total consumpton, t stll does not guarantee that t s able to gve back to the energy producers the exact cost of energy (Mohsenan- Rad and Leon-Garca 200). Consequently, the ntroducton of tme of use (ToU) models motvates consumers to shft loads nto low prcng hours; however, they are stll statc and the prces do not reflect the realtme state of the grd (prces are calculated a pror through estmatons and users are just prce takers). Thus, t often results n congeston problems durng the low-prce hours. Recently, real-tme prcng (RTP) models (Meng and Zeng 203; Samad et al.200; L et al. 20) were proposed n order to drectly connect the generaton, transmsson, and dstrbuton costs to the retal prce and essentally reduce the peak-to-average rato of a set of users. Toward the realzaton of RTP, the frst step s the development of a two-way communcaton system between the ESP and the end-users. Then, through a lmted number of message exchanges, prces are derved n real tme resolvng the trade-off between the electrcty cost mnmzaton and the users comfort maxmzaton (Samad et al. 200; Letal.20). A prcng model has to fulfll several requrements (by achevng an attractve trade-off) that s related wth () the end user s satsfacton, () the stablty of the energy producton/transmsson/consumpton system, and () the fnancal proftablty of the company that offers energy servces accordng to ths prcng model. The frst one s denoted as user s welfare and s formulated accordng to the dfference between a utlty functon that expresses how much an end user values a specfc consumpton pattern and the actual cost of energy that t consumes. In the context of the comparson of two prcng schemes, user s welfare expresses whch prcng scheme leads to more compettve servces n the open market. The second requrement s

3 denoted as behavoral effcency and expresses the capablty of a prcng scheme to acheve the objectves (e.g., load curtalments and shfts) whch motvates. Intutvely, behavoral effcency of a prcng scheme expresses how frendly t s to a TSO/DSO (ssues relevant wth energy network stablty, effcency, and costs) and mplctly affects several fnancal metrcs (e.g., nvestments n RES, energy storage, and network upgrades). Farness (n terms of rewardng the users that modfy ther consumpton) s a necessary condton toward behavoral effcency. Usually, t s also lnked wth mnmzng the system s energy cost. The thrd requrement s denoted as proft dynamcs and t represents the proft percentage per energy unt and the total revenues of the ESP. In other words, t expresses the fnancal growth potental of the ESP that deploys a specfc prcng scheme. Whle RTP motvates energy consumpton changes that reduce energy cost, the shared ESS (wthout an approprate prcng scheme) may act as a substtute and some of energy cost reducton DR operatons (shft or curtalment of devces power consumpton) are not adequately ncentvzed (Jang et al. 204; Koutsopoulos et al. 20; Codemo et al. 203; Nguyen and Le 204). In ths paper, we jontly desgn the ESS optmal operaton strategy along wth an ESS-aware RTP model and thus we make the better of both avalable tools (ESS operaton and DR actons) toward a more effcent energy schedulng. The proposed model s also able to acheve hgh farness (behavoral effcency) wthout sacrfcng at all users welfare and proft dynamcs. The rest of the paper s organzed as follows. In BRelated work^ secton, we brefly dscuss on the related work and we hghlght the contrbutons of ths paper. In BSystem model^ secton, we present the way that we model the proposed system agan accordng to the recent lterature. In BProblem formulaton and proposed algorthm^ secton, we propose our nnovatve ESS aware RTP scheme. In BPerformance evaluaton^ secton, we evaluate our proposed system and we compare t wth exstng systems n recent lterature. Fnally, n BConclusons and future work^ secton, we conclude and dscuss future work. Related work There exsts a sgnfcant amount of work on energy management solutons that ntegrate ESSs. In some cases, ESSs are operated by ndvduals and n other works users are consdered to take advantage of a shared ESS that s usually offered by ther ESP. In the former case, each user operates the ESS accordng to her nterest whle n the latter, the total cost s reduced due to the multplexng of the ESS s use.ths necesstates the development of a far prcng scheme that wll allow consumerstousesharedessandnthesametmeprotect ther nterests. In both cases, users could also optonally partcpate n DR programs (e.g., ToU, RTP) and could be thus motvated to modfy ther desred consumpton. There are systems that manly focus on the personalzed use of an ESS. In more detal, researchers (L et al. 20; Jang et al. 204; Omowunm et al. 207; Atzen et al. 203a, b; Nguyen et al. 205) consder systems where energy consumers who own resdental ESSs partcpate n DR programs. In the work of Omowunm et al. (207), users schedule ther flexble loads and ther personal ESS n order to maxmze ther proft under a ToU prcng scheme. Atzen et al. (203a, b) proposed a non-cooperatve and a cooperatve game among energy prosumers, who schedule ther ESS chargng/dschargng and ther prosumpton pattern. The objectve n the frst case s the mnmzaton of the ndvdual energy cost and n the second case, the mnmzaton of the total energy cost. Researchers n Nguyen et al. (205) developed a game-theoretc system n whch users (who possess ESSs) exchange nformaton wth the ESP n order to optmally schedule ther consumpton and ther ESS operaton by maxmzng ther welfare. L et al. (20) and Jang et al. (204) consdered users ownng resdental ESSs. An ESP sets a prce (accordng to a RTP model) and users respond by selectng ther consumpton curve and operatng ther ESS n a way that both maxmze ther welfare. Fnally, researchers n (Alam et al. 207) propose a combnaton of a DSM program and a Peer-to-Peer (P2P) energy tradng model, where users leverage dstrbuted renewable energy generaton, resdental ESSs, and ther demand flexblty n order to mnmze the total system cost. Takng nto consderaton the ESS market prces, a reasonable soluton s the sharng of an ESS among several energy users. Rahbar et al. (206) demonstrated that a shared ESS can potentally ncrease the total users proft compared to the case of personal (dstrbuted) ESS. ESS s supervsed by a central controller, who dstrbutes the fnancal beneft of the ESS equally among the partcpatng users (n terms of energy cost reducton). Moreover, researchers n (Koutsopoulos et al. 20;

4 Codemo et al. 203; Parso and Glelmo 20; Pardar et al. 205) consdered an ESP operatng a shared ESS. In Koutsopoulos et al. (20) and Rahbar et al. (205), the optmal control of an ESS s studed n order to mnmze the cost of energy that s purchased from the man grd. Furthermore, Codemo et al. (203) studed the energy cost mnmzaton problem n the presence of a shared ESS and proposed four dfferent algorthmc solutons. In all the above works, users do not partcpate n a DR program and therefore they are not motvated to alter ther desred consumpton pattern. In order to further reduce energy costs and consumpton curve fluctuatons, DR programs have been also consdered n the lterature. In Parso and Glelmo (20), Bahramrad et al. (202), and Zhang et al. (203), the mcrogrd (MG) scheduler operates a shared system of generaton unts and a shared ESS n order to optmze the operaton of the MG. In order to further reduce MG s costs,the MG operator apples drect control on consumers loads, wth the dsadvantage of substantal reducton of users welfare. Fnally, Pardar et al. (205) consdered a shared ESS, whch s controlled by an aggregator wth the am of mnmzng the aggregated energy cost. The aggregator s DR strategy s to fnancally ncentvze the users n order to modfy ther consumpton. In addton to the aforementoned works, n ths paper, we propose a comprehensve model to jontly optmze the users flexble loads and the ESS operaton, along wth an ESS-aware far RTP scheme (F-RTP-S) that nteract (users pay the exact cost of energy that they consume). Thus, the major nnovatons of ths paper are: & & & The desgn of an archtecture that ntegrates advanced prcng schemes that ncentvze behavoral changes (.e., RTP) and a shared ESS. The creaton of an nnovatve prcng model (F- RTP-S), whch dsposes hgh farness (end users pay the exact energy cost that they consume) wthout sacrfcng at all users welfare and ESP s proft dynamcs. The comparson of the proposed prcng model (F- RTP-S) wth the conventonal RTP and the RTP n envronments wth shared ESS (RTP-S). To the best of our knowledge, ths s the frst work that co-desgns an optmal shared ESS operaton and an ESS-aware real-tme prcng scheme, so that t makes the best out of both the ESS operaton and the users DR capabltes. System model We assume an ESP that partcpates n the energy wholesale market to purchase energy. The model and the operaton of the market that we envsage s descrbed and modeled theoretcally by Roozbehan et al. (200) and Kothar and Nagrath (2003). In addton, ESP leverages a shared ESS, nstalled at ts premses, n order to offer to ts clents (energy consumers) servces wth better qualty (lower energy cost). Ths means that the ESP s able to buy energy from the wholesale market n a specfc tme nstant and store t n order to provde t to ts users n a future moment. In ths work, we do not consder the partcpaton of the ESP n flexblty markets (balance and congeston market). We consder t an orthogonal problem and we refer to Makrs et al. (208) and Tsaousoglou et al. (207) for a more detaled study n ths ssue (Table ). We consder a dscrete-tme model wth a fnte horzon H that models a day. Each day s dvded nto T tmeslots of equal duraton. The proposed archtecture s depcted n Fg., whereasetn = {,..,N} of users (ESP s clents) are connected wth the man grd drectly and through the ESP s ESS. The ESP partcpates n wholesale markets durng slot k to purchase g k unts of electrcty from generators, and then sell x k r k unts to ts N customers n the retal market or charge (the word charge n ths paper exclusvely means energy transfer to the ESS and not bllng) the ESS wth r k > 0 unts (r k < 0 corresponds to dschargng the ESS). In order to mnmze the total cost, ESP optmally operates the ESS and calculates a prce vector ρ ={ρ,..,ρ k,..,ρ T }(seebproblem formulaton and proposed algorthm^ secton), whch t communcates to ts clents through a communcaton network lyng on top of the electrc grd. Every user possesses a smart meter and an energy schedulng unt, whch schedules her flexble applances by maxmzng her welfare (cf. BProblem formulaton and proposed algorthm^ secton). Then, each user nforms ESP about her decded schedule x ¼ x ;::;xk ;::;xt and ESP responds to users sgnal wth the updated prces and adjusts the chargng/ dschargng ESS schedule (cf. BProblem formulaton andproposedalgorthm^secton). Ths teratve process contnues, untl the system reaches a state of convergence. In the rest of ths secton, the model of each component of the aforementoned archtecture s presented.

5 Table Notaton Notaton Table (contnued) Notaton k k _ ch k _ ds j N H gh s; N T et a et b t s t l B S mn S max S k r k V k η c η d g k x x k x k ex ex k x k x k x k ;RTP ee E Enduserndex Tme slot ndex Index of tme slots n whch ESS s charged Index of tme slots n whch ESS s dscharged Algorthm teraton ndex Set of end users Set of tme slots (Schedulng Horzon) Desred load tme schedule of user Total number of end users Total number of tme slots Desred load splugntmeslotofuser Desred load s task fnshng tmeslot of user Earlest acceptable tmeslot for user load to start operatng Latest acceptable tmeslot for user load to fnsh operatng Storage capacty Mnmum amount of energy stored n the ESS allowed Maxmum amount of energy stored n the ESS allowed Amount of energy stored n the ESS at the begnnng of tme slot k The amount of energy charged from (r k 0) or dscharges to (r k <0) the grd at tme slot k Sunk cost of the stored energy n ESS at the begnnng of tme slot k Chargng effcency parameter Dschargng effcency parameter Total energy drawn from the man grd n tme slot k Energy consumpton schedule of user Energy consumpton of user, at tme slot k Aggregate energy demand at tme slot k Desred energy consumpton schedule of user Desred energy consumpton of user, at tme slot k Predcted energy consumpton of user, attmeslotk Lower bound of user s consumpton at tme slot k Energy consumpton of user, at tme slot k n case of RTP Desred cumulatve energy consumed by user s Btype B load Lower bound of cumulatve energy consumed by user s Btype B^ load U k ω k a k Utlty functon of user, at tme slot k n case of loads x k of Btype A^ U max, δ C c π ρ ρ k ρ k RTP G TW AUW EP TF (k) UF () UB UFB UB o, SB SoD CoC SBD Elastcty parameter of user, at tme slot k n case of loads of Btype A^ Predetermned parameter of utlty functon n case of loads of Btype A^ Maxmum utlty attaned by user from the operaton of loads of Btype B^ Elastcty parameter of user, at tme slot k n case of loads of Btype B^ Conventonal generaton cost functon Parameter of the cost functon ESP proft percentage Vector of prces through the schedulng perod Prce per unt of energy at tme slot k Prce per unt of energy at tme slot k n case of RTP Total energy cost Total welfare Aggregate users welfare Proft of ESP Tme farness User farness User s energy bll User s far energy bll User s energy bll wth no shared ESS nstalled The beneft that an ESS produces Savngs of ESS Dscharge Cost of ESS Charge User s dvdend of the beneft that an ESS produces Energy storage model We consder an ESS of capacty B nstalled by ESP. We denote as S k the amount of the energy stored n the ESS at the begnnng of tme slot k (.e., ntalzaton phase). In order to avod overchargng, there s an upper bound S max. A lower bound S mn s used to avod deep dschargng (Sh et al. 205; Huang et al. 204). Hence: S mn S k S max ; k ¼ ; 2; ; T ðþ For the energy balance of the ESS the energy S T that s stored nsde the ESS at the end of the schedulng perod H s set the same as the ntally stored energy S (Vytelngum et al. 200; Chen et al. 20; Atzen et al. 203a, b; Abbey and Joos 2009). Thus:

6 Fg. System model S Tþ ¼ S ð2þ Addtonally, we denote as r k the quantty of energy that s charged (f r k 0) toordscharged(f r k <0)from the ESS at tme slot k. In practce, there are energy losses durng both the chargng and the dschargng procedures, whch can be specfed by the chargng (η c )and dschargng (η d ) effcency parameters, where 0 η c and 0 η d. We assume that battery power leakage s neglgble and power converter losses are lumped wth battery (ds) chargng effcency. Accordng to these, we use the followng equaton n order to model the storage dynamcs by respectng the aforementoned constrants: ( S k þ η S kþ c *r k ; f r k 0 ¼. S k þ rk f r k ð3þ < 0 η d ; In each tme slot k, the dscharged energy s used by the users. Accordng to t, the ESS cannot dscharge more energy than the users need. Thus: r k N ¼ x k ; n case that rk < 0 ð4þ In Eq. (4), x k s the predcted amount of energy that user wllconsumeattmenstantk. Thswork assumes that users state ther demands and ths consttutes them perfectly predcted. The case of naccurate predctons s analyzed n Tsaousoglou et al. (207), Sterots et al. (208), Makrs et al. (208), and Chapman and Verbc (207). Moreover, n our prevous work (Tsaousoglou et al. 207), we analyze the robustness of our proposed model wth respect to naccuraces n the predctons. The proposed algorthms of ths work are orthogonal to the solutons analyzed n the references above and thus the latter can be appled on top of the former ones. Therefore, herenafter we wll exclusvely use varable x k when we refer to user s demand at tmeslot k. In practce, there are costs related to the ESS, such as nstallment, operatonal, and agng costs, whch should be accounted for the long-term battery management. However, these factors are not taken nto account for our nvestgaton of real-tme energy storage schedulng over a relatvely short tme horzon. We wll ncorporate these factors n our future studes as they do not have a qualtatve mpact n the exstng work. Fnally, we assume that there exst no delays n the energy transactons between the grd and the ESS.

7 Load n systems wth ESS In each tme slot k, then energy consumers have total energy demand x k ¼ N ¼ xk. The energy drawn from the man grd at tme k s g k. Snce the total demand has to be met from the proposed system n every tme slot, we have: g k ¼ x k þ r k Conventonal energy cost ð5þ ESP supples ts customers wth energy by purchasng t from the man grd (wholesale market). The cost of the energy g k that s drawn from the man grd s gven by functon C (g k ), whch s usually assumed n the lterature (Samad et al. 200; Koutsopoulos et al. 20; Rahbar et al. 205; Mohsenan-Rad et al. 200) to be quadratc: Cg k ¼ ð þ πþ c g k 2 k H ð6þ In Eq. (6), c s a cost parameter, C (g k ) represents the ESP s cost to buy an amount of energy equal to g k,and varable π represents the percentage of proft that ESP has. Energy consumers utlty functon Each consumer N ndependently chooses her consumpton pattern, takng nto account the electrcty prce vector ρ that ESP communcates to her. However, not all consumers value consumpton n the same way. In order to evaluate our proposed system, we utlze the concept of utlty functon from the feld of Mcroeconomcs (Mas-Colell et al. 995) to model a consumer s satsfacton (expressed n monetary value) regardng her electrcty consumpton. For the sake of our study s completeness, we consder two scenaros: (A) users flexble loads fall nto the BType A^ category of devces that users care about how much power they consume at each tmeslot (e.g., HVAC systems, lghtng, etc.); (B) users operate flexble applances of BType B^ such as electrc vehcles (EVs), for whch they care about the cumulatve energy that these devces consume through the entre schedulng perod. Wthout loss of generalty, we consder users operatng one flexble load. Ths work can be easly extended n order to nclude users operatng more than one flexble applance. Loads of Btype a^ In order to model users ownng flexble loads of Btype A^ and evaluate our proposed system, we adopt an analytc expresson U k x k ; ωk for the utlty functon that s wdely used n the lterature (Samad et al. 200; Ma et al. 204; Zhang et al. 204) and s flexble enough to model the behavor of dfferent users (Samad et al. 200) n terms of responsveness to monetary ncentves. In the aforementoned works, the utlty functon s general form s the same for every user N and n each tme slot k H and s gven by U k 8 x k ; >< ω k *xk ak ωk ¼ ω k 2 >: 2* a k 2 *xk 2 ; f 0 < x k < ωk a k ; f x k > ωk a k ð7þ In Eq. (7), ω k s a parameter that expresses the degree to whch user s affected by fnancal ncentves to alter (shed) her consumpton n tmeslot k. The lower the value of ω k s, the more responsve (flexble) the user s to prce n tme k. Fgure 2 depcts utlty of user at tme slot k as a functon of x k for three dfferent values of ω k (ω < ω 2 < ω 3 ). The above mathematcal expresson of U k x k ; ωk satsfes two desred propertes:. User s satsfacton ncreases wth energy consumpton. 2. After a certan level of consumpton, user s satsfacton gets saturated. Fg. 2 Consumer s utlty as a functon of consumpton for dfferent values of ω (ω < ω 2 < ω 3 )

8 The above assumptons are satsfed by the gven choce of U k x k ; ωk beng ncreasng and concave, respectvely. In Eq. (7), a k s a predefned parameter. We consder the utlty functon s saturaton pont as the user s desred consumpton, whch s denoted as e x k. Thus, parameter a k s calculated through Eq. (8)below: a k ¼ ωk e x k ð8þ The aggregate utlty that user attans through the schedulng perod s U ¼ T k¼ U k. In the case of Btype A^ flexble loads, consumers declare to ther EMSs the upper (desred) and the lower bounds or ther loads power consumpton at each tmeslot: x k xk x ek ; k H ð9þ Loads of Btype B^ In ths scenaro, an electrcty consumer sets her desred n operatng schedule ex ¼ x e ;::; x ek ; ::; x e o T ; k gh s;,where h gh s; ¼ et a ; t e b s a tme nterval where et a s the tmeslot at zwhch t s desrable for the load to start and t eb s the tmeslot at whch t normally fnshes ts task f t starts operaton at et a.addtonally,shesetsatmeconstrant t s, whch s the earlest tme n whch her EMS s allowed to set the load to start ts task and a deadlne t l, whch s the latest tme by whch the task of her load should be completed. Furthermore, she declares the upper (desred) and lower bounds of the load s power consumpton at each tmeslot along wth the upper (desred) and the lower bounds of the load scumulatve energy consumpton along wth: x k xk ee ; k t s ; tl ð0þ E k¼tl k¼t s x k ee ðþ Therefore, regardng user s load, we can defne a feasble schedulng set X that s: X ¼ n x je k¼tl k¼t s x k ee ; h x k xk ee ; k t s ; tl ;d x k ¼ 0; k Hn t s ; tl ð2þ We assume that each user s fully satsfed when her load consumes cumulatve energy equal to ee.the overall utlty that she obtans from her load soperaton depends on ts total energy consumpton over the entre schedulng perod. Therefore, we adopt a quadratc utlty functon as Samad et al. (202) that s mathematcally expressed as follows: 2 U ¼ U max; δ * ee k¼tl k¼t x k s ð3þ Wthout loss of generalty, n order for U (0)= 0, we set U max; ¼ δ * ee : Each user declares how much she values the energy consumed by her load through parameter δ.hghervaluesofδ mean hgher utlty value,.e., less flexblty. Fnally, t s hghlghted that the aforementoned utlty functons do not consttute novelty of ths work. On the contrary, they are wdely adopted for the evaluaton of prcng schemes (Samad et al. 202;Maetal.204; Zhang et al. 204). The proposed prcng schemes are open to varous types of users and cost models. Problem formulaton and proposed algorthm In ths secton, we present the method that ESP uses n order to calculate the vector of prces ρ.addtonally,n order to model the problems of schedulng the operaton of ESS and the consumpton of users, we formulate a blevel optmzaton problem. In the upper level, ESP mnmzes ts energy cost, whle n the lower level each user maxmzes her welfare accordng to her preferences (utlty functon) and the prces set by the ESP. Wthout harm of generalty, the objectve of the ESP s to mnmze ts energy cost by optmally operatng ESS. ESP receves the users energy schedules x and calculates the ESS chargng/dschargng schedule by solvng the followng optmzaton problem: mn r T k¼ Cgk ð4þ Subject to (), (2), (3), (4), (5). In ths case, Eqs. () and(3) express physcal constrants that ESS sets. Addtonally, n order to avod

9 havng an energy surplus or defct durng the schedulng horzon H and to take nto consderaton the requrements of the followng schedulng horzon, we mpose the constrant whch presented n Eq. (2). Alternatvely, we could set S T + to be larger than a mnmum value. Addtonally, constrant n Eq. (5) expresses that () n the case of ESS chargng, the energy drawn from the man grd s equal to the energy charged to the ESS plus the energy demand, and () n the case of ESS dschargng, the energy drawn from the grd plus the energy that ESS provdes s equal to total demand. Thus, we have a typcal convex optmzaton problem that can be solved accordng to Bertsekas (205) and/or Boyd and Vandenberghe (2004). The queston that s answered n the remander of ths secton s how ESP wll put n the blls of ts customers the cost of energy that s charged to or dscharged from the ESS n a far way n order to avod harmng the motvaton for users DR actons. The approach appled throughout the lterature s captured n Eq. (5). Users are blled wth the total energy cost of the ESP n each tme slot k. The problem wth ths approach s that the cost of chargng the ESS at tme k s pad by the users who happen to consume at that tme slot (off-peak hours). Therefore, the users that consume durng peak hours (when the ESS dscharges) get a dscount on ther prce, at the expense of the former users. ρ k ¼ Cgk ð5þ x k In order to acheve a far ESS-aware RTP scheme, we denote as V k the sunk cost of the energy that s stored n ESS at the begnnng of tme nterval k. In other words, V k represents the cost that ESP has pad to the wholesale market n order to buy and store energy. Thus, n tme nstant k ESP dsposes n ts ESS k j¼0 rk energy unts. Then, we can calculate V k+ as a functon of V k through Eqs. 6 and 7. V kþ ¼ V k þ Cg k Cx k ; f r k 0 ð6þ V kþ ¼ Vk* þ rk S k ; f r k < 0 ð7þ Accordng to Eq. (6), n the case of ESS chargng durng nterval k, V k+ (whch s the value of energy that s stored n ESS at the begnnng of tme nstant k+)s ncreased from V k (whch s the value of energy that s stored n ESS at the begnnng of tme nstant k) byan amount equal to the cost of the total energy drawn from the man grd (second term of the sum n Eq. (6)) mnus the cost of energy drawn to cover the total users demand durng tme slot k (thrd term of the sum n Eq. (6)). In the case of ESS dschargng, V k+ s lower than V k, whch s decreased by a factor that s equal to the fracton of the ESS energy that s dscharged as depcted n Eq. (7). In more detal, n Eq. (7), the multplcand of V k (value of energy n ESS at the begnnng of tme k) s equal wth the rato between the energy remanng n ESS after the dscharge and the energy n ESS before the dscharge (S k ). Through the use of the sunk cost of stored energy, we are able to determne the real-tme energy prce ρ k,.e., the prce per unt of energy consumpton at tme slot k, as ρ k ¼ Cxk x k ; f r k 0 ð8þ ρ k ¼ Cxk þ V k V kþ x k ; f r k < 0 ð9þ Auser, as a response to prce sgnal ρ, adapts her consumpton accordng to her preferences. Each user s objectve s to maxmze her total utlty mnus her electrcty bll, or n other words her Welfare. Thus, each user N receves the prce vector ρ from the ESP and calculates her energy schedule x by solvng the followng optmzaton problem: max x U ðx Þ T k¼ ρk* xk ð20þ In the above optmzaton problem, soluton x should satsfy ether Eq. (9) n case users own loads of Btype A,^ or Eq. (2) n case users operate loads of Btype B.^ Customers (end users) and ESP exchange messages, as n every real-tme prcng archtecture (e.g., Samad et al. 200; Zhang et al. 204), untl the convergence of the system to the optmal prce vector and energy schedule of the partcpatng users. The proposed algorthm s noted as F-RTP-S and t s summarzed n Table 2. In Tables 2, 3, and 4, ndex j expresses the teraton of the algorthm and the term desred accuracy expresses the condton of the termnaton of the algorthm. Fnally,

10 Table 2 Proposed algorthm for the prce and the energy consumpton calculaton n F-RTP-S Algorthm (F-RTP-S) Intalzaton: set j =,x k; j N, k H = x ek, rk, j =0,ρ k, j = C N ¼ xk; j ð Þ N ¼ xk; j 2 Repeat 3 for each user N 4 Calculate x k; j ; k H by solvng (20) 5 Update ρ k, j, k H usng (6),(7),(8),(9) 6 end for 7 Calculate r k, j k H by solvng (4) 8 k; jþ Calculate dvergence = maxx x k; j N, k H 9 Untl dvergence < desred accuracy 0 End Table 4 Algorthm for the prce and the energy consumpton calculaton n RTP-S Algorthm 2 (RTP-S) Intalzaton: set j =,x k; j N, k H = x ek, rk, j =0,ρ k, j = C N ¼ xk; j ð Þ N ¼ xk; j 2 Repeat 3 for each user N 4 Calculate x k; j ; k H by solvng (20) 5 Update ρ k, j, k H usng (5) 6 end for 7 Calculate r k, k H by solvng (4) 8 k; jþ Calculate dvergence = maxx N, k H 9 Untl dvergence < desred accuracy 0 End x k; j the Appendx presents the proof of the convergence of F-RTP-S algorthm. In order to compare the proposed F-RTP-S (wth respect to the requrements that we analyzed n the ntroducton), we also mplemented RTP as analyzed n L et al. (20) n case that there s no ESS at all. In ths case (RTP), the prce ρ k s calculated from Eq. (5) and the prcng algorthm s summarzed n Table 3. Addtonally, n order to compare F-RTP-S, we put RTP n cases that there s an ESS (RTP-S) but wthout Table 3 Algorthm for the prce and the energy consumpton calculaton n RTP Algorthm 2 (RTP) Intalzaton: set j =,x k; j = x ek,, ρk, j = C N ¼ xk; j ð x k; j N ¼ xk; j 2 Repeat 3 for each user N 4 Calculate x k; j ; k H by solvng (20) 5 Update ρ k, j, k H usng (5) 6 end for 7 k; jþ Calculate dvergence = maxx N, k H 8 Untl dvergence < desred accuracy 9 End Þ N, k H usng Eqs. (6) (9) whch preserve farness to RTP n case that there s ESS. In ths case, the prce ρ k s calculated from Eq. (5) and the algorthm s summarzed n Table 4. Pease note that n ths case, the problem s that the cost of chargng the ESS at tme k s pad by the users who happen to consume at that tme slot and not by those who actually consume the ESS dschargng energy. Fnally, we consder the case where ESP optmally schedules the functon of a shared ESS; however, users are not motvated to alter ther consumpton pattern (as n RTP, RTP-S, F-RTP-S) and thus, they consume electrcty accordng to ther desred energy schedules. Ths s noted as (S). In other words, n ths case, there s no DR program and x k ¼ x ek. In ths case, the ρk s calculated agan from Eq. (5) as n RTP but the consumpton of each user s a prory set (prce ndependent). Ths algorthm s summarzed n Table 5. Intutvely, the advantage of F-RTP-S over RTP-S s that the cost of chargng the ESS wth r k at tme slot k s Table 5 Algorthm for the prce calculaton n S Algorthm 4 (S) Intalzaton: set x k ¼ e x k, rk =0, N, k H 2 Calculate r k, k H by solvng (4) 3 Calculate ρ k, k H by solvng (5)

11 not pad by the users that happen to consume n k,butby the users that actually consume ths amount of energy at a later tme slot k > k. Fnally, Table 6 below summarzes the features of the four prcng algorthms that we present above. Performance evaluaton In ths secton, we set up a smulaton testbed n order to llustrate the performance of the proposed prcng model (F-RTP-S) and we present the results of the smulatons that we conducted. We consder a system of an ESP and N = 50 users, each of them operatng one flexble load that desrably consumes an energy amount ee ½4; 30 kwh. Wthout loss of generalty, we set x k ¼ 0 for each user at every tmeslot k. The desred operaton tme schedules of users loads (H s,, N) vary through the schedulng perod and the aggregate electrcty consumpton curve of the system s flexble loads s depcted n Fg. 3. We consder an average flexblty case, n whch users are nether reluctant to change ther consumpton pattern, nor hghly enthusastc to do so. We select the parameters of utlty functons (Eqs. (7) and (3)) accordngly. The schedulng perod conssts of T = 24 tmeslots of equal duraton of h. Energy cost functon parameter c (Eq. (6)) s set to 0.02 and ESP proft percentage (π) s set to Fnally, as far as the ESS parameters are concerned, we set S max = B, S mn =0.2*B, S =0.5*S max, η c and η d to. We are aware that the ESS cost ncreases wth B and ths factor affects the decson of the ESP concernng the sze of ESS that t wll nstall. However, storage szng s out of the scope of ths work and ths problem wll be addressed n a future study. As we dscussed earler, n order to derve a comparatve study, we consder the followng four cases: & Case (RTP): RTP scheme wth no ESS nstalled. A set of users s served by a sngle ESP. The partcpants decde ther consumpton schedule by Table 6 Prcng schemes and avalable features Feature/prcng scheme DR ESS Far ESS bllng across tme RTP Yes No No S No Yes No RTP-S Yes Yes No F-RTP-S Yes Yes Yes & & & maxmzng ther welfare and takng nto consderaton the prce that ESP sets. Case 2 (S): A shared ESS s nstalled n the premses of ESP whle users do not partcpate n a DR scheme (.e., there s no RTP). ESP utlzes the shared ESS n order to mnmze ts energy cost. Users consumpton schedule s ther desred regardless the energy cost. : A shared ESS s nstalled n the premses of the ESP and an RTP scheme s mplemented. Users solve the same optmzaton problem as n case. ESP makes use of the ESS, n order to further reduce the energy cost. In every tme slot, customers are blled wth the cost of the total energy that s produced and not wth the energy that s ntended to cover ther demand (.e., unfar prcng scheme). : The proposed prcng model, n whch the ESP farly allocates ts energy costs among the tme slots (.e., n every tmeslot, consumers pay for the energy they consume and not for the total energy that ESP purchases, part of whch s used to charge the ESS). Furthermore, n ths secton, t s shown that our proposed prcng model also farly allocates the ESP s costs among ts clents (see Fgs. 0 and 4). For evaluaton purposes, we defne the followng key performance ndcators (KPIs):. Energy cost (G): The cost that ESP has to pay n order to purchase the energy that needs to cover the energy demands of ts customers G ¼ T k¼ Cgk ð2þ 2. Total welfare (TW): The summaton of the aggregate users welfare and ESP profts TW ¼ AUW þ EP Where, AUW ¼ N ¼ U N ¼ T k¼ ρk*xk and EP ¼ N ¼ T k¼ ρk*xk T k¼ Cg k ð22þ ð23þ ð24þ

12 3. Tme farness at tme nterval k (TF(k)) s a KPI that expresses the rato between the cost of the energy that s consumed at tme nterval k and the ncome of the ESP at the same tme nterval k. For clarty (demonstraton purposes), n ths rato, we subtract one n order to have zero as the desred value of TF(k) UFB ¼ UB o; SB N ; N Where, UB o; ¼ T k¼ ρk RTP *xk ;RTP ; N SB ¼ SoD CoC ð28þ ð29þ ð30þ TFðÞ¼ k Cgk ρk* N ¼ xk ; k H ð25þ 4. Farness of user (UF()) s a KPI that expresses the way that ESS s added value s dstrbuted n a far way among all the partcpatng users j UFðÞ¼ UB UFB j ; N ð26þ UFB Where the bll that each user pays s: UB ¼ T k¼ ρk*xk ; N ð27þ UFB s the bll that user must pay n a far system and s determned from the subtracton between the bll that user would pay n case of no ESS and the beneft (added value) that ESS brngs dvded by the number of users. In Eqs. (28), (29), (30), (3), and (32), the far bll of user s defned. Where, ρ k RTPThe prce that ESP sets n tme slot k n case (RTP wth no ESS) x k ;RTPEnergy consumpton of user n tme slot k n case CoC ¼ k ch Cg k ch C k ch x k ch ð3þ SoD ¼ k ds C k ds x k ds Cg k ds ð32þ In Eq. (3), g k _ ch denotes the energy that ESP purchases n the wholesale market at tme nstants when the ESS s charged and x k _ ch denotes the total users consumpton n the same tme ntervals. In Eq. (32), x k _ ds denotes the total users demand n tme ntervals when the ESS s dscharged, whle g k _ ds denotes the energy that s purchased n the wholesale energy market by the ESP durng these tme slots. Users own loads of Btype a^ In ths scenaro, ω (Eq. (7)) s unformly selected for each consumer and every tmeslot k wthn the nterval Fg. 3 Users desred aggregate energy consumpton curve Aggregate Power Consumpton (kw) Tme (h)

13 Fg. 4 Energy cost rato of cases 2, 3, and 4 (S, RTP-S, F-RTP-S) as a functon of ESS capacty (B).2. Case 2 (S) Energy Cost Rato ESS Capacty (kwh) [, 5]. In order to compare the four algorthms n terms of energy cost (G) and total welfare (TW), we ntroduce the terms energy cost rato and total welfare rato: Energy cost rato ¼ Total welfare rato ¼ energy cost n case m energy cost n case total welfare n case m total welfare n case ; m ½2; 3; 4 ð33þ ; m ½ 2; 3; 4 ð34þ Fgure 4 depcts the energy cost rato of all three algorthms that utlty an ESS. An RTP scheme mplementaton (case RTP) manages an 0% reducton n the energy cost (G) n comparson wth a conventonal energy retal market, n whch users smply consume ther desred energy demand wthout partcpatng n a DR program (energy cost n case 2 wth B = 0 s larger than energy cost n case by.5%). We hghlght here that ths reducton hghly depends on users flexblty. The energy cost reducton could be larger n case of hgher flexblty, but t could also be neglgble n the case of nelastc users demand. In order to acheve smaller energy ESP costs, an alternatve to RTP s the nstallment of a shared ESS (cases 2, 3, and 4). In ths case, the energy cost reducton ncreases wth the ESS Fg. 5 Total welfare rato of cases 2, 3, and 4 (S, RTP-S, F-RTP-S) as a functon of ESS capacty (B) Total Welfare Rato Case 2 (S) ESS Capacty (kwh)

14 Fg. 6 Energy cost rato of cases 2, 3, and 4 (S, RTP-S, F-RTP-S) as a functon of ESP proft percentage (π) Case 2 (S) Energy Cost Rato ESP Proft Percentage capacty. The utlzaton of an ESS alone (wthout a DR program mplementaton) can reduce the energy cost up to 30% (B = 500 kwh) more than what RTP alone can do. In ths paper, we propose a RTP scheme combned wth the nstallment of a shared ESS, whch as we can see n Fg. 4, acheves even better results, as cases 3 and 4 (RTP-S and F-RTP-S) reduce the ESP s energy cost by 33 and 36% respectvely n comparson wth Case. In Fg. 5, total welfare rato s depcted. We observe that a shared ESS deployment reduces the total energy cost wthout dmnshng socal welfare. On the contrary, an ESS nstallaton can margnally ncrease TW by up to 2.4%. In order to prove that the superorty of the proposed prcng model s ndependent from the market condtons (proft percentage π), we present the followng two fgures. In Fg. 6, the energy cost rato n each of the three algorthms that nclude an ESS operaton as a functon of the ESP proft percentage (π) s depcted. We observe that an ESS of suffcent capacty (B = 200 kwh) acheves lower energy cost (G) regardless of the value of π that ESP selects. Fgure 7, whch Fg. 7 Total welfare rato of cases 2, 3, and 4 (S, RTP-S, F-RTP-S) as a functon of ESP proft percentage (π).5. Case 2 (S) Total Welfare Rato ESP Proft Percentage

15 Fg. 8 CDF of TF (k) n cases 3 and 4 (RTP-S, F-RTP-S). B = 200 kwh Probablty Average TF - Case 3 Average TF - Case Tme Farness (TF(k)) depcts the total welfare rato as a functon of the ESP proft percentage (π), ensures that the energy cost reducton comes wth no TW decrease. Fgures 8 and 9 depct the cumulatve dstrbuton functon (CDF) of TF (k). As analyzed n the begnnng of ths secton, TF (k) s a KPI that ndcates whether the ESP s ncome n each tme nterval reflects the electrcty s cost(how Bfar^ s the prcng n each tme slot k). Our proposed F-RTP-S s the farest possble. As we observe n Fg. 8 (B = 200 kwh), n the case of RTP- S, there s a wde dstrbuton of TF (k) n varous tme slots around the average TF (k). RTP-S s less far than F-RTP-S by 46% on average. Ths gap ncreases wth ESS capacty as we can see n Fg. 9, whereess capacty s set to 300 kwh and F-RTP-S s farer than RTP-S by 66% on average. The nstallment of a shared ESS should be equally benefcal for every user among the ESP s customers. Ideally, users blls should be equal to what they would pay n case (RTP) mnus ther ESS beneft dvdend, whch s: SBD ¼ SB N ; N ð35þ Fg. 9 CDF of TF (k) n cases 3 and 4 (RTP-S, F-RTP-S). B = 300 kwh Probablty Average TF - Case 3 Average TF - Case Tme Farness (TF(k))

16 Fg. 0 CDF of UF () n cases 3 and 4 (RTP-S, F-RTP-S). B = 200 kwh 0.8 Probablty Average UF - Case 3 Average UF - Case User Farness (UF()) In order to assess ths, we have ntroduced n the begnnng of ths secton UF () as a KPI. We ntend to prove that n F-RTP-S, users bll vector s closer to the deal one (whch s zero as derved from Eq. (26)), than n RTP-S.Thus,nFg.0, we present the CDF of UF () for RTP-S and F-RTP-S cases. As we can observe, the CDF of F-RTP-S and the average value of UF () nf- RTP-S are much closer to zero (desred value) than the CDF of RTP-S and the average value of UF ()nrtp-s. Ths fact ndcates that the benefts that an ESS nstallment provdes are more farly dstrbuted when F-RTP-S s appled. Ths s explaned by the fact that, n case RTP-S scheme s mplemented, users who consume energy at tmes when the ESS s charged, they pay the whole ESS chargng cost, whle users who consume energy at ESS dscharge tme slots reap the total ESS beneft. On the contrary, the F-RTP-S scheme mplementaton s farer toward the users, snce they pay what they actually consume. Users own loads of Btype B^ In ths scenaro, δ s randomly selected wthn the nterval [, 5] n order to consder users of medum Fg. Energy Cost rato of cases 2, 3, and 4 (S, RTP-S, F- RTP-S) as a functon of ESS capacty (B).4.3 Case 2 (S) Energy Cost Rato ESS Capacty (kwh)

17 Fg. 2 Total welfare rato of cases 2, 3, 4 (S, RTP-S, F-RTP-S) as a functon of ESS capacty (B).2. Total Welfare Rato 0.8 Case 2 (S) ESS Capacty (kwh) flexblty. Addtonally, we consder that each consumer sets her earlest and latest load operatng tmes one to three hours before and after gh s;. Also, we set E ¼ 0. In Fg., the energy cost rato of all three algorthms that utlze an ESS s depcted. The RTP scheme mplementaton reduces energy cost by 26% n comparson wth the base case n whch users consume ther desred energy demand n lack of any DR program (energy cost n case 2 wth B = 0 s larger than energy cost n case by 35.7%). In order for algorthm (S) to acheve better results than algorthm (RTP), the nstallaton of an ESS of capacty more than 200 kwh s necessary. Eventually, for B = 300 kwh, algorthm (S) manages a 7.5% larger cost reducton than algorthm (RTP). The combnaton of RTP mplementaton and an ESS nstallment provdes ESP wth the capablty to further decrease ts energy cost. More specfcally, algorthm (RTP-S) accomplshes a larger cost reducton than RTP by.2%, whle F-RTP-S s even more effcent as t reduces energy cost by 6% more than RTP. In Fg. 2, we can see that both RTP-S and F-RTP-S are effcent n reducng energy cost, whle they ncrease TW comparng to RTP (case ). In fact, algorthm (RTP-S) mplementaton results n a hgher TW than RTP by.4%, whle F-RTP-S s even Fg. 3 CDF of TF (k) n cases 3 and 4 (RTP-S, F-RTP-S). B = 200 kwh Probablty Average TF - Case 3 Average TF - Case Tme Farness (TF(k))

18 Fg. 4 CDF of UF () n cases 3 and 4 (RTP-S, F-RTP-S). B = 200 kwh Probablty Average UF - Case 3 Average UF - Case User Farness (UF()) more effectve as t acheves a 6% ncrease n TW comparng to RTP. Fgure 3 demonstrates that RTP-S s less far than F- RTP-S by 4.3% n terms of TF (k) on average when ESP nstalls an ESS of capacty B = 200kWh. Fgure 4 demonstrates a huge dscrepancy n terms of UF () on average between F-RTP-S and RTP-S, as n F-RTP-S users average bll dffers from UFB by 26%, whle the equvalent percentage n RTP-S s 400%. Ths s explaned by the fact that there are consumers, who should pay near to zero blls (accordng to UFB ); however, n RTP-S, they stll pay a normal electrcty bll. Hence, F- RTP-S s greatly farer than RTP-S on average regardng the bllng of consumers. Moreover, ESS szng algorthms n conjuncton wth dfferent prcng models need to be studed, n order for the ESP to optmally desgn ts busness strategy. Fundng The work presented n ths paper receved fundng from the European Unon s Horzon 2020 research and nnovaton programme under grant agreement No n the context of the SOCIALENERGY project. Complance wth ethcal standards The authors declare that they have no con- Conflct of nterest flct of nterest. References Conclusons and future work Energy storage systems are becomng a substantal component of the Smart Grd as they facltate ESPs and the consumers n varous ways. Due to the hgh costs of ESS technologes, t s reasonable to assume that a progressve utlty company would deploy a shared ESS. In ths paper, we proposed a RTP scheme adopted by an ESP, whch also leverages a shared ESS n order to further mnmze ts energy costs. As a future work, we ntend to extend the results of ths paper to more advanced system models that nclude renewable energy generatng unts, more advance DSM strateges, sophstcated ESS degradaton and deprecaton cost models. Abbey, C., & Joos, G. (2009). A stochastc optmzaton approach to ratng of energy storage systems n wnd-desel solated grds. IEEE Transactons on Power Systems, 24(), Alam, M. R., St Hlare, M., & Kunz, T. (207). An optmal P2P energy tradng model for smart homes n the smart grd., 0, /s Atzen, I., Ordonez, L. G., Scutar, G., Palomar, D. P., & Fonollosa, J. R. (203a). Noncooperatve and cooperatve optmzaton of dstrbuted energy generaton and storage n the demand-sde of the smart grd. IEEE Transactons on Sgnal Processng, 6(0), Atzen, I., Ordonez, L. G., Scutar, G., Palomar, D. P., & Fonollosa, J. R. (203b). Demand-sde management va dstrbuted energy generaton and storage optmzaton. IEEE Transactons on Smart Grd, 4(2),