Feasblty of Usng Dscrmnate Prcng Schemes for Energy Tradng n Smart Grd Wayes Tushar, Chau Yuen, Bo Cha, Davd B. Smth, and H. Vncent Poor Sngapore Unversty of Technology and Desgn, Sngapore 138682. Emal: {wayes tushar, yuenchau}@sutd.edu.sg. State Key Lab. of Industral Control Technology, Zhejang Unversty, Chna. Emal: chabozju@gmal.com. NICTA, Canberra, ACT, Australa. Emal: davd.smth@ncta.com.au. School of Engneerng and Appled Scence, Prnceton Unversty, Prnceton, NJ, USA. Emal: poor@prnceton.edu. Abstract Ths paper nvestgates the feasblty of usng a dscrmnate prcng scheme to offset the nconvenence that s experenced by an energy user (EU) n tradng ts energy wth an energy controller n smart grd. The man objectve s to encourage EUs wth small dstrbuted energy resources (DERs), or wth hgh senstvty to ther nconvenence, to take part n the energy tradng va provdng ncentve to them wth relatvely hgher payment at the same tme as reducng the total cost to the energy controller. The proposed scheme s modeled through a two-stage Stackelberg game that descrbes the energy tradng between a shared faclty authorty (SFA) and EUs n a smart communty. A sutable cost functon s proposed for the SFA to leverage the generaton of dscrmnate prcng accordng to the nconvenence experenced by each EU. It s shown that the game has a unque sub-game perfect equlbrum (SPE), under the certan condton at whch the SFA s total cost s mnmzed, and that each EU receves ts best utlty accordng to ts assocated nconvenence for the gven prce. A backward nducton technque s used to derve a closed form expresson for the prce functon at SPE, and thus the dependency of prce on an EU s dfferent decson parameters s explaned for the studed system. Numercal examples are provded to show the benefcal propertes of the proposed scheme. Index Terms Smart grd, dscrmnate prcng, dstrbuted energy resources, game theory, energy management. I. INTRODUCTION Energy management (or demand management) s a technque that changes the electrcty usage patterns of end users n response to the changes n the prce of electrcty over tme [1], [2]. Wth the advancement of dstrbuted energy resources (DERs), the technque can also be used to assst the grd or other energy controllers such as a shared faclty authorty (SFA) [3] to operate relably and profcently by supplyng energy to them [4]. The majorty of energy management lterature focuses manly on three dfferent prcng schemes: tme-of-use prcng; day-ahead prcng; and real-tme prcng [5]. Tme-of-use prcng [6] has three dfferent prcng rates: peak, off-peak and shoulder rate based on the use of electrcty at dfferent Ths work s supported by the Sngapore Unversty of Technology and Desgn (SUTD) under the Energy Innovaton Research Program (EIRP) Sngapore NRF2012EWT-EIRP002-045. Davd B. Smth s also wth the Australan Natonal Unversty (ANU), and hs work s supported by NICTA. NICTA s funded by the Australan Government through the Department of Communcatons and the Australan Research Councl through the ICT Centre of Excellence Program. tmes of the day. Day-ahead prcng [7] s, n prncple, determned by matchng offers from generators to bds from energy users (EUs) so as to develop a classc supply and demand equlbrum prce at an hourly nterval. Fnally, realtme prcng [5] refers to tarffed retal charges for delverng electrc power and energy that vary hour-to-hour, and are determned from wholesale market prces usng an approved methodology. Other popular dynamc prcng schemes nclude crtcal peak prcng, extreme day prcng, and extreme day crtcal peak prcng [5]. It s mportant to note that n all of the above mentoned prcng schemes all EUs are charged at the same rate at any partcular tme. Due to government subsdes to encourage the use of renewables [8], more EUs wth DERs are expected to be avalable n smart grd. Ths wll lead to a better completon of a purchasng target for an energy controller, and thus more savng from ts buyng cost. Partcularly, for energy controllers such as an SFA that reles on the man grd as ts prmary source of energy [3], the opportunty for tradng energy wth EUs can greatly reduce ther dependency, and consequently decrease ther cost of energy purchase [9]. Nevertheless, not all EUs would be nterested n tradng energy wth the energy controller f the beneft s not attractve [10]. Ths can precsely happen to EUs wth merely lmted energy capacty, or to EUs that are hghly senstve to the nconvenence caused by the tradng of energy whose expected return could be very small. In ths case, the EUs would store the energy or change ts consumpton schedule rather than sellng t to the energy controller [11]. However, one possble way to address ths s to pay them a relatvely hgher prce per unt of energy, compared to the EUs wth very large DERs, wthout affectng ther revenue sgnfcantly. In fact, allowng dscrmnate prcng not only consderably benefts EUs wth lower energy capacty wthout sgnfcantly affectng others, as we wll see shortly, but also benefts the SFA by reducng ts total cost of energy purchase when adoptng ths flexble prcng. For nstance, consder the numercal example gven n Table I where the SFA buys ts requred 40 kwh energy from EU1 and EU2. EU1 has 50 kwh and EU2 has 10 kwh of energy to sell to the SFA. In case 1, the SFA pays the same prce 20 cents/kwh to each of them, and EU1 and EU2 sell 35 and 5 kwh respectvely to the SFA. Hence, the revenues of EU1
TABLE I: Numercal example of a dscrmnate prcng scheme where an SFA requres 40 kwh of energy from two EUs and the SFA s total prce per unt of energy to pay to the EUs s 40 cents/kwh. Case 1 Case 2 Payment to EU1 (cents/kwh) 20 18 Payment to EU2 (cents/kwh) 20 22 Energy suppled by EU1 (kwh) 35 32 Energy suppled by EU2 (kwh) 5 8 Revenue of EU 1 (cents) 700 576 (-17%) Revenue of EU 2 (cents) 100 176 (+76%) Cost to the SFA (cents) 800 752 (-6%) and EU2 are 700 and 100 cents respectvely, and the total cost to the SFA s 800 cents. In case 2, the SFA uses dscrmnate prcng to motvate EU2 to sell more to the SFA. Therefore, t pays 22 cents/kwh to the EU2 and 18 cents/kwh to EU1. Now, due to ths ncrement of prce EU2 ncreases ts sellng amount to 8 kwh, and the SFA procures the remanng 32 kwh from EU1. Therefore, the revenues changes to 576 and 176 cents for EU1 and EU2 respectvely, and total cost to the SFA reduces to 752 cents. Thus, from ths partcular example t can be argued that dscrmnate prcng can be consderably benefcal to EUs wth small energy (revenue ncrement s 76%) n expense of relatvely lower revenue degradaton (e.g., 17% n the case of EU1) from EUs wth larger DERs. It also reduces the cost to the SFA by 6%. Therefore, dscrmnate prcng s advantageous for reducng SFA s cost and also for crcumstances where the SFA motvates the partcpaton of EUs wth both large and small DERs n the energy tradng. Hence, there s a need for nvestgaton as to how ths prcng scheme can be adopted n a smart grd envronment. To ths end, we take the frst step towards dscussng the propertes of a dscrmnate prcng scheme. The dea of dscrmnate prcng was frst used to desgn a consumercentrc energy management scheme n [4]. However, no nsght was provded nto the choce of dfferent prces that are pad to dfferent EUs. In ths paper, we frst propose a scheme by usng a two-stage Stackelberg game. In the proposed scheme, the EUs wth smaller energy generaton can expect hgher unt sellng prce, and the prce s adaptve to ther avalable energy for sale and ther senstvty to the nconvenence of energy exchange. At the same tme, the scheme s desgned to mnmze the total purchasng cost to the energy controller whereas each EU also receves ts best utlty based on ts avalable energy, ts senstvty to the nconvenence, and the offered prce by the SFA. We prove the exstence of a soluton to the proposed game, and use a backward nducton method to determne how the unt prce set by the energy controller s affected by an EU s varous parameters. We further derve a closed form expresson for dfferng prce generaton consderng some condtons on the energy controller s cost functon. Fnally, we present some numercal cases to show the propertes of the proposed dscrmnate prcng scheme. We stress that current grd systems do not allow such dscrmnate prcng among EUs. However, we envson t as a further addton to real-tme prcng schemes n future smart grd. Examples of such dfferentaton can also be found n standard Feed-n-Tarff (FIT) schemes [12]. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION Consder a smart communty consstng of a large number of EUs, an SFA and the man electrc grd. Each EU can be a sngle user, or group of users connected va an aggregator that acts as a sngle entty [13]. EUs are equpped wth dstrbuted energy resources (DERs) such as wnd turbnes and solar arrays. They can sell ther energy, f there s any remanng after meetng ther essental loads, to the SFA or to the man grd to make some extra revenue. Snce the grd s buyng prce s sgnfcantly low n general [14], t s reasonable to assume that each EU would be more nterested n sellng ts energy to the SFA nstead of sellng to the grd. Alternatvely, an EU can store ts energy or schedule ts equpment nstead of sellng the energy to the SFA f the return beneft s not attractve,.e., f the prce s not convenent enough for the EU to trade ts energy. We brefly explan ths phenomenon by an example n Fg. 1. In Fg. 1, we use the same example of Table I and show how senstve an EU s to the nconvenence of tradng energy caused by the change of prce per unt of energy. As can be seen from the fgure, EU1 has consderably lower essental load than EU2, and thus has a larger avalable energy to supply to the SFA. As a result, n case 1, EU1 supples 35 kwh of energy to the SFA whereas EU2 supples 5 kwh for the same per unt prce of 20 cents/kwh after usng the energy for ther other flexble loads. However, n case 2, the SFA adopts a dscrmnate prcng scheme and changes the per unt prce to 18 cents/kwh and 22 cents/kwh to pay to EU1 and EU2 respectvely. Due to the change of prce, the expected return for EU2 becomes larger from tradng ts energy at the expense of the revenue degradaton from EU1. Consequently, energy tradng becomes more nconvenent for EU1 where as at the same tme t becomes more appealng for EU2. As shown n Fg. 1, due to ther senstvtes to the nconvenence caused by the change of prce, EU1 reduces ts amount of energy for sellng to 32 kwh (.e., by ncreasng ts use of the remanng avalable energy for other purposes such as storage) whereas EU2 ncreases ts amount of energy for sellng to 8 kwh n case 2. In ths paper, we quantfy ths Essental load.! EU 1! EU 2! 35 kwh! 32 kwh! Case 1! Case 2! 50 kwh! 10 kwh! Energy stored or used for other purpose (Flexble load).! Energy that the EU sells to the SFA.! 5 kwh! 8 kwh! Fg. 1: Example of how EUs are senstve to the nconvenence caused by a change of prce, whch thus affects ther amount of energy to trade wth the SFA.
senstvty of each EU to the relatve nconvenence through an nconvenence parameter 1, as we wll see shortly, and analyze ts effects on the total cost to the SFA. The SFA refers to an energy controller 2 that controls the electrcty consumed by the equpment and machnes that are shared and used by EUs on daly bass. The SFA does not have any energy generaton capacty, and therefore depends on EUs and the man grd for ts requred energy. The SFA s connected to the man grd and all EUs va power and communcaton lnes [13]. To ths end, let us assume that N EUs n a set N are takng part n energy tradng wth the SFA. At a partcular tme of the day, the SFA s energy requrement s E r, and each EU N has an avalable energy of E after meetng ts essental load from whch t can sell e to the SFA. The man objectve of each EU s to make some extra revenue by sellng e to the SFA at a prce c per unt of energy. However, the choce of e s reasonably affected by the nconvenence parameter α, whch s a measure of senstvty of EU to the nconvenence t faces to trade ts energy. In ths regard, we defne a utlty functon U for each EU that captures the effect of ths nconvenence, and s assumed to possess the followng propertes: ) The utlty functon s an ncreasng functon of e and c, and a decreasng functon of nconvenence parameter α. That s δu δe, δu δc > 0, and δu δα < 0. α captures the fact that the utlty wll decrease for an EU f ts senstvty to the nconvenence of tradng energy ncreases. ) The utlty functon s a concave functon of e,.e., δ 2 U δe 2 < 0. Therefore, the utlty can become saturated or even decrease wth an excessve e. Ths can be nterpreted by the fact that snce EUs wth DERs are equpped wth a battery wth lmted capacty n general, excessve supply of energy once exceedng a certan lmt would rsk the depleton of battery due to the agng effect upon the battery, and consequently decrease the EU s utlty. Formally, we defne U as U = e c + (E α e )e. (1) In (1), e c s the drect ncome that the EU receves from sellng ts energy to the SFA at a prce c per unt of energy. (E α e )e refers to the possble loss for the EU s nconvenence senstvty α > 0. Dfferent values of α reflect dfferent negatve mpacts of energy supply on an EU s utlty, and an EU can set hgher α f t prefers to sell less. For example, the effects of c and α on an EU s utlty from ts energy tradng s shown n Fg. 2. Now, wth the goal of maxmzng utlty, the objectve of each EU can be expressed as max e [e c + (E α e )e ]. (2) 1 Where a hgher and lower value of ths parameter refers to the hgher and lower senstvty of an EU respectvely to the nconvenence caused by energy tradng. 2 For the rest of ths paper, we wll use SFA to ndcate an energy controller as dscussed n Secton I. Acheved utlty 6000 4000 2000 0 2000 4000 c = 20, α = 1 c = 40, α = 1 c = 40, α = 2 Coordnates of maxmum utlty 6000 0 10 20 30 40 50 60 70 80 90 100 Energy sold by the EU Fg. 2: Effect of parameters such as α and c on the acheved utlty of an EU are shown n ths fgure. As can be seen n the fgure, for the same α a hgher c encourages an EU to sell more to the SFA and thus the maxmum utlty shfts towards a hgher value on the rght. By contrast, a hgher α causes more nconvenence to the EU whch can lead the utlty even to a negatve value (.e., cost) for greater energy tradng. However, as a buyer of energy, the SFA wants to mnmze ts total cost J of energy purchase from EUs and the grd. In ths paper, we consder the followng cost functon to capture the total cost to the SFC for buyng ts requred energy from EUs and the grd: ( N ( J = e c k ) + a c + b + cg E r ) e, (3) =1 such that c C, c mn c c max, (4) where C s the total unt energy prce [4], and c mn and c max are the lower and upper lmts of unt prce that the SFA can pay to any EU [4]. In (3), e c k corresponds to the drect cost e c that s weghted by c k 1 to generate dscrmnate prces for EUs wth dfferent α, and the term (a c + b ), a, b > 0 accounts for other costs such as transmsson cost and store of purchased energy cost [4]. c g (E r e ) s the cost of purchasng energy from the grd. C scales a set of normalzed prces to generate the unt prce c. It s fxed for a partcular tme and can be determned by the SFA usng any real-tme prce estmator, e.g., the estmator proposed n [15]. Now, the SFA s objectve s to set a prce c per unt of energy for each EU that not only mnmzes ts total cost n (3) but also pays a prce to each EU accordng to ther nconvenence parameters, and thus encourages them to take part n energy tradng wth the SFA. Therefore, the objectve of the SFA can be defned as [ N ( ( mn e c k ) + a c + b + cg E r )] e, (5) c =1
such that (4) s satsfed. We stress that (2) and (5) are related va e and c, and can be solved n a centralzed fashon. However, consderng that the nodes n the system are dstrbuted, t s more advantageous to defne a soluton approach that can be mplemented dstrbutedly accordng to the parameter settng wthn the system [16]. In ths regard, we propose to use a game theoretc formulaton. In [4], the effect of changng C on the cost to a seller was nvestgated, and a dstrbuted algorthm was proposed to desgn a consumer-centrc smart grd va capturng ths effect. In ths paper, we focus on explorng the nfluence of dfferent EUs behavor on the choce of prce by the SFA, and the resultant cost ncurred to t. To that end, we propose a two-stage Stackelberg game n the next secton. III. TWO-STAGE STACKELBERG GAME To determne energy tradng parameters e and c, on the one hand, each EU needs to decde on the amount of energy e that t wants to sell to the SFA accordng to ts nconvenence senstvty and the offered prce. On the other hand, based on the amount of energy offered by each EU and ts nconvenence parameter, the SFA agrees on the prce vector c = [c 1, c 2,..., c N ] that t wants to pay to each EU such that the cost J to the SFA s mnmzed. Thereupon, ths sequental nteracton can be modeled as a two-stage Stackelberg game [17], whch s formally defned as Ω = {(N {SFA}), {E } N, {U } N, J, c}. (6) In (6), (N {SFA}) s the set of total players n the game where each EU N s a follower, and {SFA} s the leader. E s the strategy vector of each follower and U s the utlty that the follower receves from choosng ts strategy e E. J s the cost ncurred to the SFA for choosng the strategy vector c. As the leader of Ω, the SFA chooses ts strategy vector c n the frst stage of the game such that ts cost functon n (3) s mnmzed, and the constrants n (4) are satsfed. In the second stage of the game, each EU N ndependently chooses e n order to maxmze ts utlty n (1) n response to c chosen by the SFA. Consequently, Ω reaches the equlbrum soluton of the game. A. Soluton Concept A general soluton of a mult-stage Stackelberg game such as the proposed Ω s the sub-game perfect equlbrum (SPE) [17]. A common method to determne the SPE of a Stackelberg game s to adopt a backward nducton technque that captures the sequental dependences of decsons between stages of the game [17]. To that end, we frst analyze how each EU would maxmze ts beneft by playng ts best response to the prce offered by the SFA n stage two. Then, we explore how the SFA decdes on dfferent prces to pay to dfferent EUs accordng to ther offered energy and nconvenences. We note that, due to the method for game formulaton, Ω wll possess a SPE f there exsts a soluton n both stages of the decson makng process by the SFA and EUs. In fact, the exstence of a soluton n pure strateges s not always guaranteed n a game [18], and hence there s a need to nvestgate the exstence of a soluton n the proposed Ω. Theorem 1. A unque SPE exsts for the proposed two-stage Stackelberg game Ω f k = 2 n (3). Proof: Accordng to the backward nducton technque, each EU N decdes on ther energy tradng parameters e at the second stage of the game to maxmze (2). It s a strctly concave functon of e as δ2 U = 2α δc 2 and α > 0. Hence, EU s decson makng problem has a unque soluton. Furthermore, n the frst stage of the game, the SFA optmzes ts prce c to pay to each EU. Now, we note that f k = 2 n (3), whch s a general choce of quadratc cost functon for electrcty utlty companes and controllers [4], [16], the cost functon (3) s strctly convex wth respect to c. Thus, for the amount of energy offered by each EU, the choce of dfferent prce to pay to each also possesses a unque soluton, whch mnmzes (5). Hence, the game Ω possesses a unque SPE, and thus Theorem 1 s proved. B. Analyss of Energy Tradng Behavor In ths secton, we show how the energy tradng behavor of the SFA and EUs are affected by dfferent decson makng parameters such as the prce set by the SFA, and the nconvenence that s caused to each EU for tradng ts energy. Frst, we consder the second stage of the game where each EU plays ts best response to the prce c offered by the SFA. Snce the utlty functon n (1) s dfferentable, we obtan the frst order dervatve δu δe, and U attans ts maxmum when δu δe = 0. Therefore, from (1), the best response functon of EU to a gven c can be expressed as e (c ) = c + E 2α, (7) whch leads to the followng proposton: Proposton 1. For an offered prce c, the amount of energy e that an EU s wllng to sell, from ts avalable energy E, to the SFA decreases wth the ncrease of ts senstvty to nconvenence α. In other words, an EU wth nconvenence parameter α would be more wllng to sell ts energy to the SFA for a hgher prce per unt of energy. The SFA s cost, on the other hand, s determned by the prce c that t wants to pay to each EU for ts offered energy e. Therefore, n the frst stage of the game the SFA determnes the prce c havng the knowledge of the energy vector e = [e 1, e 2,..., e N ] offered by all EUs va (7). Now, the Lagrangan for the SFA s optmzaton problem n (5) s gven by Γ = ( e c k ) + a c + b + c g (E r e ) + λ(c c ), (8)
where λ s the Lagrange multpler and δγ δc = 0. (9) In (8), we only consder the case when c mn c c max,.e., the Lagrange multpler assocated wth c mn and c max are assumed to be zero 3. Now replacng the value e n (8) from (7), (9) can be expressed as k + 1 c k + ke c k 1 + a c g λ = 0. (10) 2α 2α c n Now, for the general case 4 k = 2, and consequently, 3c 2 + 2E c + 2α (a λ) c g = 0, (11) c = E + [ E 2 3 (2α (a λ) c g ) ] 1 2. (12) 3 In (12), λ and a are desgn parameters, and thus constant for a partcular system. λ needs to be chosen sgnfcantly hgher than a such that c always possesses a postve value. Note that we skp the other soluton of c n (12) for the same reason. From (12), we note that for the same generaton and grd prce, a hgher prce needs to be pad to an EU wth hgher nconvenence parameter α compared to an EU j, j wth α j < α to encourage t to sell energy. Nevertheless, n cases when c > c max and c < c mn, the SFA sets c to the respectve lmts. Hence, the choce of prce by the SFA to pay to each EU at the SPE can be expressed as c mn, c < c mn c = E +[E 2 3(2α(a λ) cg)] 2 1 3, c mn c c max. (13) c max, c > c max IV. CASE STUDY To show the propertes of the proposed dscrmnate prcng scheme, we consder an example n whch a number of EUs are nterested n tradng ther energy wth the SFA n the tme slot of nterest. We assume that the avalable energy to each EU, after meetng ts essental load, s unformly dstrbuted wthn [50, 250], and the energy requred by the SFA for the consdered tme slot s 650 kwh. The value of λ s chosen to be 1000. The grd s sellng prce s set to be 50 cents/kwh, and c max and c mn are assumed to be 38 and 10 cents/kwh 5 respectvely. These two values are chosen such that the SFA can pay to each EU a prce, whch s lower than the grd s sellng prce, and at the same tme s hgher than the grd s buyng prce. Ths condton s necessary to motvate all EUs to trade ther energy only wth the SFA nstead of the grd. Nonetheless, we hghlght that all parameter values are chosen 3 The condtons for c = c mn and c = cmax are consdered at the soluton of the c n (13). 4 We wll consder k = 2 for the rest of the paper. 5 Prce c mn s margnally greater than the prce of 8.45 cents/kwh that a grd typcally pays to buy energy from DERs [12]. partcularly for ths case study only and that these values may vary between dfferent case studes. In Fg. 3, we show how the prce per unt of energy s decded by the SFA for each EU. Accordng to (13), for a partcular grd prce c g, the unt prce c that the SFA pays to each EU depends on 1) EU s nconvenence parameter α, and 2) the avalable energy E to each EU. Frst, we consder fve EUs wth the same E = 150 kwh, but wth dfferent nconvenence n sellng ther energy to the SFA. We note that the SFA tends to pay more, wthn the constrant n (4), to the EU wth hgher senstvty to nconvenence. In fact, a hgher nconvenence parameter refers to the state at whch tradng energy wth the SFA s not a convenent opton for an EU. Therefore, to encourage the EU to sell the energy the SFA needs to ncrease ts unt prce to pay. However, f c becomes more than c max, the SFA pays c max to the EU as shown n the case of the last EU wth nconvenence parameter α = 3 n Fg. 3. By contrast, for the same senstvty to nconvenence, the SFA pays a hgher prce to an EU wth lower avalable energy and vce versa. In fact, a lower avalable energy could stop an EU from sellng the energy to the SFA as t mght not brng sgnfcant beneft to the EU at a lower prce. Hence, to provde more ncentve to the EU, the SFA needs to pay a relatvely hgher prce per unt of energy. However, EUs wth larger amount of energy can stll obtan hgher utltes from tradng a consderable amount of energy wth the SFA even at a relatvely lower prce, as explaned by the example n Table I. Thus, the SFA pays comparatvely a lower prce to such EUs to mnmze the cost of energy tradng, such that the energy tradng does not effect ther utltes sgnfcantly 6. 6 We note that the lowest prce per unt of energy c mn s assumed to be hgher than the buyng prce of the grd. Therefore, any EU wth a hgher avalable energy would beneft more from tradng wth the SFA nstead of tradng wth the grd. Prce per unt of energy set by the grd 40 35 30 25 20 15 Prce for EUs wth dfferent α (E = 150 kwh) Prce for EUs wth dfferent E (α = 1.5) (1.0, 50) (1.5, 100) (2.0, 150) (2.5, 200) (3.0, 250) Dfferent values of E and α of each EU (α, E ) Fg. 3: Effect of avalable energy and nconvenence parameters on the prce per unt of energy that the SFA selects to pay to each EU. The frst term of each tuple on the horzontal axs refers to the nconvenence parameter α of an EU, and the second term ndcates the avalable energy E.
Amount of energy traded wth dfferent class of EUs 500 450 400 350 300 250 200 150 100 50 0 α = 1 α = 2 α = 3 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Dfferent cases as per Table II Fg. 4: Effect of behavor of EUs on the total amount of energy that the SFA buys from each class of EUs. After showng how prces are set by the SFA to pay to dfferent EUs, we now show the effect of dfferent behavor of EUs n a group on the total cost to the SFA. Frst we note that EUs behavors are domnated by ther nconvenence parameters α. For example, f an EU wth very large avalable energy does not want to sell ts energy at the offered prce t can set ts α hgh and thus nsgnfcantly (or, not at all) take part n energy tradng. Hence, we can model dfferent EUs behavors by smply changng ther α. To ths end, we assume a network wth 10 EUs that have the same avalable energy 150 kwh but dfferent nconvenence parameters to sell ther energy to the SFA. For ths partcular case, we consder total unt energy prce C = 380 cents/kwh so that even when all EUs are pad at c max, the constrants n (4) are stll satsfed, and the unt prce for each EU remans lower than the grd s sellng prce. We compare the performance wth an equal dstrbuton scheme (EDS) such as n [19], where C s equally dvded to pay to each EU for buyng ts energy. That s n EDS each EU s pad a prce 7 C N per unt of energy, where N s the total EUs n the network. TABLE II: Dfferent behavoral cases of EUs n the network (a total of 10 EUs) where the number of EUs wth a partcular nconvenence parameter α {1, 2, 3} s specfed. Cases α = 1 α = 2 α = 3 1 6 EUs 2 EUs 2 EUs 2 4 EUs 3 EUs 3 EUs 3 2 EUs 4 EUs 4 EUs 4 2 EUs 2 EUs 6 EUs 5 1 EU 1 EU 8 EUs 6 0 0 10 EUs To that end, we categorze the behavor of EUs nto sx dfferent cases based on the number of EUs wth partcular nconvenence parameters α n the group as shown n Ta- 7 For N = 10, each EU s pad a prce 38 cents/kwh. ble II. Although we have chosen only three nteger values of α {1, 2, 3}, other fractonal values wthn ths range are equally applcable to defne dfferent levels of senstvty to nconvenence. Now frst we see from Fg. 4 that as the number of EUs wth α = 1 domnates the group, the SFA buys most of ts energy from them. For example, n case 1 and case 2, the number of EUs wth α = 1 s hgher n the system and consequently, the SFA buys sgnfcantly large amount of energy from them n these two cases compared to the other cases, as shown n Fg 4. However, as ther number reduces the SFA needs to buy more energy from the other two types of EUs, based on ther percentage of presence n the group, wth relatvely hgher payment. In the extreme case,.e., case 6, the SFA needs to buy all ts energy from EUs wth α = 3 as there are no other types of EUs n the system. Consequently, ths trend of energy tradng affects the total cost to the SFA from buyng ts energy from EUs and the grd. We show these effects separately n Table III. From Table III, frst we note that the amount of energy that the SFA buys from the grd ncreases as the categores of EUs change from case 1 to case 6 n the system. Ths s due to the fact that as the number of EUs wth hgher senstvty to nconvenence ncreases n the group, the total amount of power that the SFA can trade wth EUs becomes lower. Hence, the SFA needs to procure the remander of requred energy from the grd at a hgher prce. Secondly, the cost to the SFA to buy energy from EUs wth hgher nconvenence parameters also ncreases ts cost sgnfcantly as the SFA needs to pay a hgher prce to them. For example, consder the dfferent cost that s ncurred to the SFA for buyng energy from dfferent types of EUs n case 1. From Fg 4, we can see that the amount of energy that the SFA buys from EUs wth α = 1 s almost fve tmes the amount t buys from EUs wth α = 2 and 3. However, the resultant cost s only three tmes more than the cost to buy from EUs wth hgher senstvty. Therefore, more EUs wth lower senstvty to nconvenence allows the SFA to procure more energy at a comparatvely lower cost. Therefore, the total cost ncurred by the SFA ncreases sgnfcantly wth an ncrease n the number of EUs wth hgher nconvenence parameters as can be seen from Table III. We also compare the total cost that s ncurred to the SFA wth the case when the SFA adopts an EDS scheme for energy tradng n Table III. In an EDS scheme, the cost to the SFA remans the same for all type of EU groups as the cost does not depend on ther categores. From Table III, the proposed scheme shows consderable beneft for the SFA n terms of reducton n total cost when there are a relatvely hgher number of EUs wth lower nconvenence parameters n the group. For example, as shown n Table III, the cost reducton for the SFA s 49.9% and 36.63% respectvely for case 1 and 2. Accordng to the current case study, the average total cost reducton for the SFA s 23.18% compared to the EDS. However, the cost ncreases wth the ncrease of number of EUs wth hgh nconvenence parameters, and becomes the same as the EDS scheme when all the EUs n the group become hghly senstve to the nconvenence of energy
TABLE III: Cost to the SFA n dollars for dfferent EUs behavors (cases stated n Table II). Dfferent Costs Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Cost for buyng from EUs wth α = 1 68.92 45.94 22.97 22.97 11.48 0 Cost for buyng from EUs wth α = 2 23.6 35.4 47.2 23.6 11.8 0 Cost for buyng from EUs wth α = 3 24.36 38.4 48.72 73.08 97.44 121.8 Cost for buyng from the grd 53.91 98.16 142.42 155.18 186.88 218.58 Total cost for proposed scheme 170.79 216.06 261.32 274.89 307.62 341 Total cost for EDS 341 341 341 341 341 341 % reducton n total cost 49.91% 36.63% 23.36% 19.38% 9.78% 0% tradng,.e., α = 3, as can be seen from Table III. V. CONCLUSION In ths paper, a dscrmnate prcng scheme has been studed to counterbalance the nconvenence experenced by energy users (EUs) wth dstrbuted energy resources (DERs) n tradng ther energy wth other enttes n smart grds. A sutable cost functon has been desgned for a shared faclty authorty (SFA) that can effectvely generate dfferent prces per unt of energy to pay to each partcpatng EU accordng to an nconvenence parameter for the EU. A two-stage Stackelberg game, whch has been shown to have a unque sub-game perfect equlbrum, has been proposed to capture the energy tradng between the SFA and dfferent EUs. The propertes of the scheme have been studed at the equlbrum by usng a backward nducton technque. A theoretcal prce functon has been derved for the SFA to decde on the prce that t wants to pay to each EU, and the propertes of the scheme are explaned va numercal case studes. By comparng wth an equal dstrbuton scheme (EDS), t has been shown that dscrmnate prcng gves consderable beneft to the SFA n terms of reducton n total cost. One nterestng future extenson of the proposed scheme would be to desgn an algorthm that can capture the decson makng process of the SFA and EUs n a dstrbuted fashon. 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