Demand Sde Managemen n Smar Grds usng a Repeaed Game Framework Lnq Song, Yuanzhang Xao and Mhaela van der Schaar, Fellow, IEEE Absrac Demand sde managemen (DSM) s a key soluon for reducng he peak-me power consumpon n smar grds. To provde ncenves for consumers o shf her consumpon o off-peak mes, he uly company charges consumers dfferenal prcng for usng power a dfferen mes of he day. Consumers ake no accoun hese dfferenal prces when decdng when and how much power o consume daly. Imporanly, whle consumers enjoy lower bllng coss when shfng her power usage o off-peak mes, hey also ncur dscomfor coss due o he alerng of her power consumpon paerns. Exsng works propose saonary sraeges for he myopc consumers o mnmze her shor-erm bllng and dscomfor coss. In conras, we model he neracon emergng among self-neresed, foresghed consumers as a repeaed energy schedulng game and prove ha he saonary sraeges are subopmal n erms of long-erm oal bllng and dscomfor coss. Subsequenly, we propose a novel framework for deermnng opmal nonsaonary DSM sraeges, n whch consumers can choose dfferen daly power consumpon paerns dependng on her preferences, rounes, and needs. As a drec consequence of he nonsaonary DSM polcy, dfferen subses of consumers are allowed o use power n peak mes a a low prce. The subse of consumers ha are seleced daly o have her jon dscomfor and bllng coss mnmzed s deermned based on he consumers power consumpon preferences as well as on he pas hsory of whch consumers have shfed her usage prevously. Imporanly, we show ha he proposed sraeges are ncenve-compable. Smulaons confrm ha, gven he same peak-o-average rao, he proposed sraegy can reduce he oal cos (bllng and dscomfor coss) by up o 50% compared o exsng DSM sraeges. Keywords Smar Grds; Demand Sde Managemen; Crcal Peak Prcng; Consumer Dscomfor; Repeaed Games; Incenve Desgn. I. ITRODUCTIO Smar grds am o provde a more relable, envronmenally frendly and economcally effcen power sysem [1][2]. The uly company sells elecrcy o consumers, who are equpped wh smar meers. Smar meers exchange nformaon beween consumers and he uly company, and schedule he household energy consumpon for consumers. The nformaon gahered hrough smar meers can be used by he uly company o adjus he elecrcy prces. Demand Sde Managemen (DSM) s a key mechansm o make smar grds more effcen and cos-effecve [1][2]. DSM refers o he programs adoped by uly companes o drecly or L. Song, Y. Xao and M. van der Schaar are wh Deparmen of Elecrcal Engneerng, UCLA. Emal: songlnq@ucla.edu, {yxao, mhaela}@ee.ucla.edu.
2 ndrecly nfluence he consumers power consumpon behavor n order o reduce he Peak-o- Average Rao (PAR) of he oal load n he smar grd sysem. A hgher PAR resuls n much hgher operaon coss and possbly ouages of he sysem. DSM ams o ncenvze consumers o shf her peak-me power consumpon o off-peak mes, hereby resulng n sgnfcan PAR reducons n he power sysem. A. Relaed Work Drec Load Conrol (DLC) and Smar Prcng (SP) are wo popular exsng approaches for mplemenng DSM. DLC refers o he program n whch he uly company can remoely manage a fracon of consumers applances o shf her peak-me power usage o off-peak mes [3]. Alernavely, SP [6]-[18] provdes an economc ncenve for consumers o volunarly manage her power usage. Examples are Real-Tme Prcng (RTP) [6], Tme-Of-Use Prcng (TOU) [14], Crcal Peak Prcng (CPP) [16]-[18], ec. However, he above works [3][6]-[8][16]- [18] do no consder he consumers dscomfor coss whch s nduced by alerng her power consumpon paerns. Some recen works consdered consumers dscomfor coss [4][5][9]-[15][20][23] and amed o jonly mnmze he consumers bllng and dscomfor coss (referred o subsequenly as he oal cos). These works can be classfed no wo caegores, dependng on he deployed consumer model. The frs caegory assumed ha he consumers are prce-akng (.e., hey do no consder how her consumpon wll affec he prces). By assumng ha consumers are prceakers, he decson makng of a sngle foresghed consumer s formulaed as a sochasc conrol problem amng o mnmze s long-erm oal cos n [9]-[11]. Alernavely, n [12][13], mulple myopc consumers am o mnmze her curren oal coss and her decsons are formulaed as sac opmzaon problems among cooperave users for whch dsrbued algorhms are proposed o fnd he opmal prces. The second caegory assumed ha he consumers are myopc and prce-ancpang (.e., hey ake no accoun how her consumpon wll affec he prces). In hs case, each consumer s power usage affecs he oher consumers bllng coss. These works [6]-[8][15] modeled he neracons emergng among myopc consumers as one-sho games and suded he ash
3 DLC or SP Prce-akng or prce-ancpang consumers TABLE I. COMPARISO WITH EXISTIG WORKS Myopc or foresghed consumers Sngle consumer or mulple consumers Model Consumer Dscomfor [3] DLC - Myopc Mulple Opmzaon o [4][5] DLC - Myopc Mulple Opmzaon Yes [6]-[8] SP Prce-ancpang Myopc Mulple One-sho game o [9]-[11] SP Prce-akng Foresghed Sngle Sochasc conrol Yes [12][13] SP Prce-akng Myopc Mulple Opmzaon Yes [14] SP Prce-akng Myopc Sngle One-sho game Yes [15] SP Prce-ancpang Myopc Mulple One-sho game Yes [16] SP Prce-akng Myopc Mulple Opmzaon o Our Work SP Prce-ancpang Foresghed Mulple Repeaed game Yes equlbrum (E) of he emergng game. In hs paper, we also model he consumers as prceancpang. However, n our model, he consumers nerac wh each oher repeaedly and are foresghed, hereby engagng n a repeaed game. I s well-known ha n one-sho games wh myopc prce-ancpang consumers, he sysem performance (.e., he oal cos) a he equlbrum can be much worse han he opmal performance acheved by myopc prce-akng consumers [25]. One way o acheve he same performance for myopc prce-ancpang consumers as n he case of myopc prce-akng consumers s o model he sysem as a repeaed game, and o ncenvze he prce-ancpang consumers o choose (n equlbrum) he opmal consumpon adoped by myopc prce-akng consumers [24]. Imporanly, n hs paper we go one sep furher, and adop a novel repeaed game framework whch can sgnfcanly ouperform he opmal performance obaned by he myopc prce-akng consumers [12][13]. All he exsng works consderng mulple consumers [3]-[8][12]-[16] assumed ha he consumers are myopc and mnmze her curren coss. The opmal DSM sraeges n hese works are saonary,.e., all consumers adop fxed daly/weekly power consumpon paerns as long as he sysem parameers (e.g., he consumers desred power consumpon paerns) do no change. However, as we wll show laer n he paper, he saonary DSM sraeges are subopmal n erms of he long-erm oal cos. To mnmze he oal cos, some consumers are requred o shf her peak-me power usage o he off-peak mes whle he remanng consumers can use energy when desred. By deployng hs opmal sraegy, he consumers who shf her peak-me consumpon ncur dscomfor coss, bu hs leads o a reducon of he peak-me prce and of he bllng cos of all he consumers. Imporanly, our proposed nonsaonary DSM
4 sraegy can acheve he opmal oal cos whle ensurng farness among consumers by recommendng dfferen subses of consumers (referred o as he acve se) o shf her peakme consumpon each day. The acve se s deermned by he consumers preferences and he pas selecon of acve ses. A dealed comparson of our work and exsng works s hghlghed n Table I. B. Our Conrbuons In hs paper, snce he consumers say n he sysem for a long me and nerac wh each oher repeaedly, we formulae he consumers neracons as a repeaed game. The repeaed naure of he neracon provdes ncenves for prce-ancpang consumers o cooperae (as shown n [24]). Alhough he proposed framework can mprove he performance of he saonary DSM sraeges dscussed for any SP scheme, we focus on he CPP scheme, whch has been wdely used for resdenal consumers and s shown o work well n praccal scenaros [17]-[19]. CPP defnes peak days n a year or peak mes n a day, and charges hgher prces durng hese peak hours f CPP evens, such as sysem load warnng, exreme weaher condons, and sysem emergences, occur [18]. Based on he repeaed game model, we propose an opmal nonsaonary 1 DSM mechansm ha mnmzes he oal cos and ouperforms he opmal saonary DSM sraegy. In addon, he proposed sraegy s Incenve-Compable (IC), namely he self-neresed consumers wll fnd n her self-neres o follow he recommended sraegy. Each day, he DSM sraegy selecs an acve se of consumers based on her preferences of wheher or when o shf and on he pas hsory of consumpon paern shfs. These consumers sacrfce her curren dscomfor coss o mnmze he oal bllng cos. In reurn, hey wll enjoy n he fuure lower bllng coss whou ncurrng dscomfor coss when oher consumers are chosen n he acve se. In hs way, he proposed sraegy mnmzes he long-erm oal cos whle ensurng farness among he consumers. In summary, he man conrbuons of our work are as follows: 1 Recall ha n a saonary DSM mechansm, he consumers adop fxed daly/weekly power consumpon paerns as long as he sysem parameers do no change. In conras, n a nonsaonary DSM mechansm, he consumers may adop dfferen daly/weekly power consumpon paerns (e.g., a consumer may shf s peak-me consumpon oday bu no shf omorrow) even f he sysem parameers reman he same.
5 A Repeaed Game Framework: A repeaed game framework s proposed o model he neracons among foresghed prce-ancpang consumers over me. Jon Bllng and Dscomfor Coss Mnmzaon: The proposed DSM mechansm consders no only he consumers bllng coss bu also her dscomfor coss. Opmal onsaonary DSM mechansm: We analycally prove ha n he energy schedulng game wh dscomfor coss, nonsaonary DSM mechansms can ouperform saonary mechansms, and propose an opmal nonsaonary DSM mechansm ha can be easly mplemened. IC sraeges: The DSM mechansm s IC, meanng ha he self-neresed consumers have no ncenve o devae from he recommended sraegy. Consumer heerogeney: Our framework can model dfferen ypes of consumers wh dfferen dscomfor coss. Moreover, dfferen consumers may have dfferen preferences on how and when her consumpon paerns are shfed. We sudy he mpac of dfferen ypes of consumers on he performance of he proposed mechansm. The res of hs paper s organzed as follows. Secon II models he repeaed energy schedulng game and formulaes he DSM mechansm desgn problem. Secon III formally nroduces he proposed algorhm for consrucng he DSM sraegy and dscusses how o mplemen he proposed algorhm n he smar grd sysem. Secon IV provdes smulaon resuls o valdae he performance of he proposed algorhm. Secon V concludes he paper. II. SYSTEM MODEL A. Energy Schedulng Game A smar grd sysem consss of a uly company and mulple consumers, as shown n Fg.1. The DSM s mplemened hrough smar meers on he consumer sde and a DSM cener n he uly company [2]. We denoe he se of consumers by {1,2,..., }. Tme s dvded no perods 0,1,2, L. We assume ha each perod s dvded no H me slos wh equal lengh and denoe he se of me slos by H {1,2,..., H}. oe ha we use perod here o denoe
6 Informaon Flow Power Flow Smar Meer da (% ), a da (% ), a 1 1 a1 p l p,, 1 Uly Company a Smar Meer Consumer 1... Fg. 1. Smar Grd Sysem Model. Consumer each sage of he neracon among consumers and use me slo o denoe he dscree me o schedule power usage whn a perod. In hs paper, we consder a perod o be one day as n [6]- [9][12]-[16], and each slo can be one or mulple hours (e.g., H 24, H 12 ). For each consumer n perod, s acon s he power consumpon paern n ha perod, whch s he vecor of power consumpon a each me slo and denoed by a ( a,, a ), wh a A beng he power consumpon and A beng he se of power,,1 H, h consumpon a each me slo. The power consumpon a h, a me slo h consss of non-shfable load and shfable load. The non-shfable load, such as lghng, cookng, wachng TV, s no conrollable by smar meers, whle he shfable load, such as dsh and clohes washng, heang and coolng sysems, can be conrolled by he smar meers [6]-[8]. We denoe he non-shfable and shfable loads a me slo h H by sasfes a h,, h, h b h, 0 and b s and a s, 0, respecvely. Thus, consumer s power consumpon h b h, h,. oe ha here he shfable load s no only lmed o he deferrable load, bu also ncludes smar hermosas amed a conrollng heang/coolng sysems, whch can shf he energy usage ahead of me and have been shown [19] o sgnfcanly reduce he peak load n pracce, ec. We denoe by H a A he daly oal power consumpon for resdenal consumer, h1 h, where A s eher a consan or slowly varyng as n [6]-[8][13]-[16]. We denoe by a ( a, a, La 1 2 ) A he power consumpon profle of all consumers, where 1 A A. H
7 The oal load a me slo h, denoed by l a h, 1 h, s he sum of all consumers power consumpon. We assume ha he desred power consumpon paern n each perod for consumer s a [ a, a, La ] A, whch refers o s preferred daly power consumpon paern [22]. H,1,2, H The correspondng desred power consumpon profle of all consumers and oal load are denoed by a [ a, a, La ] A and 1 2 l a h 1 h,. Defne h argmax lh hh as he peak me of he day, he lengh of whch changes accordng o how H s se. Emprcal sudes show ha, compared wh ndusral and commercal consumers, resdenal consumers have very smlar peak-me shfable loads [19], mplyng haa b s, for each consumer. h, h, h Remark: Frs, dfferen consumers may have dfferen desred power consumpon paerns a. For example, mos consumers may have her hghes power consumpon n he evenng, whle some consumers may have hghes power consumpon n he mornng. Second, each consumer can have s ndvdual peak consumpon n hours oher han h ; h s deermned based on he aggregae power consumpon of all consumers, nsead of he power consumpon of ndvdual consumers. Thrd, n he model currenly presened, we le he desred daly power consumpon paern a o be he same for each day. One may argue ha he desred daly power consumpon should be dfferen n weekdays and n weekends. In hs case, we can easly exend he model such ha one perod s one week and a s he desred weekly power consumpon paern, where he desred daly power consumpon paerns are dfferen each day. The cos of a consumer consss of bllng and dscomfor coss. The bllng cos s he power consumed mulpled by he un prce and he dscomfor cos s he consumer s dscomfor caused by reschedulng s daly power consumpon from s desred power consumpon paern. We denoe c : A a R o be he cos funcon of consumer : H h1 h h, c( a ) p ( a ) a d ( a ), (1)
8 where p : h A a R denoes he prce a me slo h, a denoes he daly bllng H p ( ) a h1 h h, H cos, d : A a R denoes he dscomfor cos of consumer. Whou loss of generaly, he oal cos s he sum, nsead of he weghed sum, of bllng and dscomfor coss, because any wegh he consumer pus on he dscomfor cos can be absorbed n he expresson of he dscomfor cos funcon d. We wll dscuss laer he prce p and dscomfor cos d n deal. h To model he neracons among consumers, we frs formalze he one-sho energy schedulng game G {, { A },{ c } }, where, 1 1 { A } and 1 { } denoe he se of consumers, c 1 ses of acons and cos funcons for each consumer, respecvely. ex, we formalze he repeaed game model. In each perod, consumer deermnes s power consumpon paern a based on s hsory, whch s a collecon of all s pas power consumpon paerns and he pas prces made publc o he consumers. The hsory of consumer up o perod can be wren as: he nal hsory s defned as { a, p, a, p,..., a, p } ) 0 0 1 1 1 1 H H ( A P for 0 and 0. The correspondng publc hsory s defned as { p, p,..., p } ( P ) for 0 and he nal hsory s defned as 0 1 1 H 0. The publc sraegy of consumer s defned as a mappng from publc hsory o curren acon, denoed by H : U( P ) 0 H 0 a A, where ( P ) [24]. Due o realzaon equvalence prncple [24, Lemma 7.1.2], we lose nohng, n erms of he achevable operang pons, by resrcng o publc sraeges, compared o sraeges usng he enre hsory. Gven he sraegy profle of all consumers, denoed by (,, L ), consumer s 1 2 average long-erm cos s dscouned by a facor : C ( ) (1 ) c ( ( )) 0, (2) where c ( ( )) s he cos of consumer n perod. The dscoun facor [0,1) represens how consumers dscoun her moneary coss n he fuure, e.g. due o neres raes/nflaon. A smaller means ha he consumers dscoun her fuure coss more. Snce he neres
9 rae/nflaon s usually he same for all he consumers, we assume o be equal for all consumers as n [24][25]. The correspondng long-erm dscouned dscomfor cos s denoed by D ( ). Hence, he nfnely repeaed game model can be wren as RG H {, U ( ),{ },{ C ( )} } 1 1 0 P, where, H ( ) 0 U P, { } and 1 { C ( )} 1 denoe he se of consumers, se of publc hsores, ses of sraeges and ses of cos funcons, respecvely. B. Crcal Peak Prcng Scheme Recall ha n Crcal Peak Prcng (CPP) scheme, he uly company charges a hgher prce n he crcal peak me when CPP evens occur (such as sysem load warnng, exreme weaher condons, and sysem emergences), amng a reducng he peak-me load of he sysem [16]- [18]. The maxmum number of crcal days and of crcal hours whn a day s predefned. We model he CPP prcng scheme wh a sngle crcal peak me and only consder he CPP evens rggered by he oal load n he sysem. The me-varyng prce funcon p ( ) h a s defned as: where p H p Lo p, 0 l l p ( ) p ( l ) h h h p l l Lo h h, H h h a, (3) are he peak prce and off-peak prce of he prcng model and l h s he hreshold of he oal load. When l h l h, he hgher prce wll no be rggered and p wll be adoped. Lo When l h l, he CPP even occurs and he hgher prce h H p wll be adoped. To provde ncenves for consumers o shf her peak-me loads, he hreshold l h s se o be below he peak load and above he off-peak load, namely, l l, l l, h H \{ h}. (4) h h h h Gven peak load reducon goal, namely h l, we furher se m l l / s h h h, where m s he smalles number of consumers needed o shf her peak-me consumpon such ha he peak-me prce s low. We denoe he prces whn a day by p ( p, p, Lp ). 1 2 H
10 C. Consumer Dscomfor Cos We use a dscomfor cos funcon o model he consumers dscomfor from reschedulng her power consumpon paerns. Many papers defne he dscomfor cos funcon based on he mposed change n her consumpon paerns,.e., he dsance beween consumer s desred demand and acual consumpon [14][20]-[23]. As n [14][20], we use a lnear weghed funcon o model he dscomfor cos: where k h,, H k ( a a ), a a h h, h, h, ( ) 1 0, a a d a, (5) R are parameers of he dscomfor cos funcon. Consumer s dscomfor cos d ( a ) s an ncreasng funcon of he dsance beween he rescheduled and he desred power consumpon paerns. The above dscomfor cos funcon s able o capure several mporan consumer preferences n erms of consumpon paern shfng. Frs, capures he consumer s wllngness o shf: a consumer wh a large s less wllng o shf. Second, k h, capures he consumer s preference on how o shf. A larger k, h ndcaes ha he consumer does no wan o reduce s peak-me consumpon; a larger k h, for h h ndcaes ha he consumer does no wan o shf s peakme consumpon o me slo h. We are neresed n wo coss ha consumer can acheve: consumer s mnmum cos achevable by any power consumpon profle, denoed by c =mn c ( a )= p A, and consumer aa Lo s mnmum cos achevable by any power consumpon profle n whch consumer shfs all s peak-me shfable load, denoed by c% mn c ( a ) p A d ( a% ) a A, a Lo h, bh,, where a% argmn d ( a ). Clearly, we always have c c%, because c% s he mnmum cos H a A, a b h, h, achevable when consraned o consumer s ceran power consumpon paerns (whch cause dscomfor coss). The cos c s mporan because s he mnmum cos ha consumer could
11 possbly acheve. The cos c% s also mporan because represens he mnmum cos ha consumer could possbly acheve when shfs all of s peak-me consumpon (.e. when s seleced o be n he acve se). Based on he relaonshp beween bllng and dscomfor coss, we can classfy he consumers no hree classes. Consumers wh low dscomfor coss: Ths class ncludes consumers whose dscomfor coss are low compared wh ph p Lo. In hs case, he ncrease n he prce maers much more o consumers han he dscomfor coss. For hese consumers, he dscomfor coss can be reasonably gnored and only he bllng coss need o be consdered. Consumers wh medum dscomfor coss: Ths class ncludes consumers who have medum dscomfor coss wh respec o ph p Lo. In hs case, boh bllng and dscomfor coss need o be consdered. We say a consumer has medum dscomfor cos f ( p p ) l H Lo h m d ( a% ) and [( p p )] a. (6) H Lo h, The frs nequaly mples ha he dscomfor cos s no oo large, such ha he consumers are wllng o shf her peak-me consumpon f by dong so her bllng coss can be grealy reduced. The second nequaly mples ha he dscomfor cos s no oo small, such ha each consumer does care abou s own dscomfor cos and s no wllng o shf s peak-me consumpon every day. Consumers wh hgh dscomfor coss: Ths class ncludes consumers who have hgh dscomfor coss parameers compared wh ph p Lo. In hs case, dscomfor coss are hgh when alerng her power consumpon paerns and hence hey should be recommended no o change her power consumpon paerns. In a praccal sysem, he above hree classes of consumers coexs. However, snce he recommendaons o consumers wh low and hgh dscomfor coss can be deermned based on her cos characerscs (namely, consumers wh low dscomfor coss always shf and choose a%, whle consumers wh hgh dscomfor coss canno shf her power consumpon paern and
12 choose a ), we only need o consder he DSM sraegy for consumers wh medum dscomfor coss. Hence, n Secon III, we focus on dervng he opmal DSM sraegy for consumers wh medum dscomfor coss. The opmal DSM sraegy derved for consumers wh medum dscomfor coss, along wh he predeermned recommended sraeges for consumers wh hgh and low dscomfor coss, consues he DSM sraegy used n he praccal sysem wh hree classes of consumers. D. Problem Formulaon In hs subsecon, we consder he sysem wh self-neresed consumers and formulae he opmal IC DSM mechansm desgn problem. The desgner s he benevolen uly company ha ams a mnmzng he oal cos (maxmzng he socal welfare) n he smar grd sysem. However, excep for he oal cos, mananng farness among all he consumers s essenal [19]. Hence, he mechansm wll ensure ha he average dscomfor cos of consumer s no greaer han a maxmal value Therefore, he DSM mechansm Desgn Problem (DDP) can be formulaed as π h, h, H (DDP): mnmze C ( ) subjec o a b, T A a, T h1 h, D( ) D, s IC,max. D,max. The uly company wll solve hs problem, hen recommend he consumers wh he opmal soluon. The abovemenoned energy schedulng game model can be exended or revsed n dfferen ways o accommodae varous sysems. Frs, he CPP scheme may be only appled n he summer and wner seasons, when he sysem load s hgh. In hs case, he DSM mechansm can updae he sysem parameers and begn o solve he DDP problem agan, a he begnnng of a new season. Second, he daly power consumpon A for a specfc consumer can vary for dfferen days durng a week, due o her dfferen weekly rounes, e.g. hey do he laundry on Frdays ec.
13 In hs case, we can se he perod n he model o be a week, nsead of a day, and se he daly oal power consumpon and desred power consumpon paern as and H mod7 a A h1 h, a a mod7, where { } 6 A and 6 { a } 0 0 are he daly power consumpon and desred power consumpon paerns of a week and are dfferen for dfferen days of a week. III. OPTIMAL STRATEGIES In hs secon, we solve he DDP problem defned n he prevous secon. We frs dscuss a benchmark case - he performance of he one-sho energy schedulng game. Subsequenly, we characerze he Pareo effcen operang pons of he repeaed energy schedulng game and propose our nonsaonary algorhm ha acheves he opmal soluon of he DDP problem. Fnally, we descrbe he mplemenaon of he proposed DSM n he smar grd sysem. A. Benchmark The One-sho Energy Schedulng Game In he unque E of he one-sho energy schedulng game, he consumer chooses s desred power consumpon paern. We sae hs formally n he followng heorem. Theorem 1 (ash Equlbrum of he One-sho Game): The one-sho energy schedulng game has a unque E, n whch each consumer chooses s desred power usage as a a,. (7) Proof: The dea of he proof s o show ha a s he domnan sraegy for consumer. The complee proof s gven n Appendx A. The resul of Theorem 1 s he well-known ragedy of commons. Each consumer ams o mnmze s ndvdual bllng and dscomfor coss and hus, wll myopcally fnd n s selfneres o mnmze s ndvdual cos by sckng o s desred power consumpon paern. Ths resuls n a hgh prce and low socal welfare. Accordng o Theorem 1, he oal cos a he unque E s C 1 E,, where C c E ( a ). In he followng secon we wll quanfy he neffcency, of he E and propose a novel DSM mechansm ha can acheve he opmal socal welfare.
14 B. Pareo-opmal Regon of he Repeaed Energy Schedulng Game We formally characerze he achevable operang pons of he repeaed energy schedulng game. I s well known ha as long as s suffcenly close o 1, he achevable regon of he repeaed energy schedulng game s he convex hull of he one-sho energy schedulng game [24, Lemma 3.7.1]. Hence, we can wre he achevable regon of repeaed energy schedulng game as H C conv{ ca ( ) a A, a A, a b }, where conv{ X } s he convex hull of X and c c L c 1 2 h1 h, h, h, ca ( ) ( ( a), ( a), ( a )) s he cos profle of he consumers. We wll prove ha he Pareoopmal regon of he repeaed energy schedulng game (.e., he Pareo boundary of he se of achevable cos profles), denoed by B, s a face of C ha s a polyope wh dmenson 1. Ths means ha we can analycally express he Pareo-opmal regon. Theorem 2 formally characerzes he Pareo-opmal regon B analycally. Theorem 2: The Pareo-opmal regon of he repeaed energy schedulng game s B { C ( C, C, L, C ) ( C c %)/( c c %) mc, c %}. (8) 1 2 1 In addon, he saonary DSM sraeges can only acheve he exreme pons 2 of B. Proof: See Appendx B. Theorem 2 does no only characerze he Pareo-opmal regon (.e., par of a hyperplane) of he repeaed energy schedulng game, bu also proves ha by usng saonary DSM sraeges, we canno acheve any pons on he Pareo-opmal regon oher han he exreme pons. oe, however, ha, he exreme pons can only be acheved by he acon profles n whch m consumers shf her peak consumpon and ncur cos c, whle he oher consumers do no shf and ncur cos c%. I s clear ha he exreme pons are no desrable operang pons, because an exreme pon can be acheved only when a fxed se of m consumers shf her peak-me consumpon all he me, whch s unfar for hese m consumers because hey ncur hgh dscomfor coss all he me. Hence, he desred operang pons le n he neror of he Pareoopmal regon, whch can be acheved by nonsaonary sraeges n he repeaed energy schedulng game accordng o Theorem 2. 2 An exreme pon of a convex se s he pon ha s no he convex combnaon of any oher pons n hs se. In our case, snce B s par of a hyperplane, he exreme pons wll be he verces of B.
15 By addng IC consrans and he consrans on he maxmum dscomfor coss, he feasble Pareo-opmal regon can be wren as B C { C ( C, C, L, C ) ( C c% )/( c c% ) mc, c%, C C }, (9) 1 2 1 mn{, }, C c% D.,max,max where C C C,max E, C. onsaonary DSM Mechansm Gven he Pareo-opmal regon, we can hen reformulae he DDP problem as a lnear programmng problem: mnmze subjec o C C B. (10) The soluon of (10) s an exreme pon of B, denoed by C C C C (,, L ). C 1 2 Theorem 3: The feasble Pareo-opmal regon B s achevable f he dscoun facor C sasfes C Proof: See Appendx C. 1 1. (11) m 1 Gven a desred operang pon n B C, we can use he onsaonary DSM (-DSM) algorhm, descrbed n Table II, o consruc he DSM sraegy. In perod, he -DSM algorhm chooses he acve se I () I conssng of m ou of consumers o reschedule her power consumpon paerns, where I s he se of all possble ndex combnaon ha conanng m consumers ou of. The choce of whch m consumers are seleced depends on how far hey are from her arge cos and hs s measured by ndex g () and he m consumers who have he larges g () wll aler her power consumpon paerns. Theorem 4: When he dscoun facor sasfes (11), he -DSM algorhm s IC and can acheve he opmal operang pon C C C C (,, L ). Proof: See Appendx D. 1 2
16 Theorem 3 and 4 sae ha when he dscoun facor sasfes (11), he opmal nonsaonary DSM mechansm can be consruced by he -DSM algorhm. In pracce, he daly neres rae s low enough o make condon (11) be sasfed and he proposed DSM mechansm can accommodae a large number of consumers. D. DSM Implemenaon TABLE II. OSTATIOARY DSM (-DSM) ALGORITHM Inpu: Targe average cos vecorc ( C1, C2, LC), 0. Oupu: Opmal sraegy 1: Se g () ( C c% ) ( c c% ). j j j j j 2: Repea 3: If p ( ) p,, hen h H 4: Recommend acon a o all consumer. 5: Else 6: Fnd he acve se I () {,, L } of m consumers who have he m larges ndces gj (). 1 2 m 7: Recommend acon a% o I (), and a o I (). 8: Observe consumers acon a. 9: If all consumers follow he recommendaon, hen 10: Updae g( 1) [ g() ( 1) 1 { I()} ]/ for all. 11: Broadcas p () p for all h. 12: Else h 13: Broadcas p () ph and p () p, h h. 14: End f 15: End f 16: 1 17: End Repea h Lo h In hs subsecon, we descrbe how o mplemen he DSM mechansm n he smar grd sysem based on he proposed -DSM algorhm. The publc nformaon, such as prce nformaon and oal load, s known o every consumer and he prvae nformaon of consumer, such as dscomfor cos, power consumpon, s only known o s own. There are wo phases n mplemenng he DSM mechansm: he nalzaon phase and he run me phase. As n [2], he uly company recommends he sraegy one day n advance. 3 Consumers can schedule her power usage and exchange nformaon wh uly company hrough smar meers for he upcomng day so as o mnmze her bllng and dscomfor coss based on he recommendaon. Lo 3 The DSM can also be mplemened n real-me or T-day-ahead, as long as here s enough me for he consumers o respond o he recommendaons.
17 TABLE III. IFORMATIO EXCHAGE OF IMPLEMETATIO Informaon exchange Publc nformaon (known o all consumers) Consumer s prvae nformaon Uly company o consumer Consumer o uly company Inalzaon pp,,, ( ), H plo lh m l h d ( a ), a 1: p, p, l, l, C H Lo h h Run me 4:, 7: p Inalzaon 2: d ( a% )( or c, c% ), a Run me 56: a Fg.2. The Implemenaon of DSM. The deals of publc and prvae nformaon, he wo phases and he correspondng nformaon exchange of he wo phases are shown n Fg. 2 and Table III. Inalzaon: 1 The uly company broadcass he CPP parameers: p Lo, p and l H h. 2 Consumer calculaes s cos parameers c% and c, and sends hem o he uly company. 3 The uly company calculaes he arge cos vecor C ( C, C, L C ) by solvng DDP 1 2 problem, and hen sends he arge cos vecor o all consumers. Run me: In each perod: 4 The DSM cener calculaes he acve se I() and hen makes he recommendaon () o each consumer. 5 Each consumer chooses s energy schedulng sraegy. oe ha he consumers can devae from he recommended sraegy. However, snce he sraegy s IC, s n her self-neres o comply wh he recommendaon. 6 The smar meer repors he power consumpon paerns a o DSM cener. 7 The uly company calculaes he prce based on he oal load and hen broadcass he prces. Go o nex perod and repea 4-7. The consumers dscomfor cos funcons may vary over me. For example, hey may have dfferen desred power consumpon paerns n dfferen seasons/monhs, or hey may prefer o shf he peak-me consumpon o dfferen off-peak mes (.e., dfferen k h, ) due o her change of schedules (e.g., due o a new job), or hey smply change her wllngness (.e., dfferen ) o shf. To accommodae for such occasonal changes (e.g., once a monh), he uly company can
18 run he nalzaon phase agan n order o calculae he new opmal arge cos vecor, and hen run he -DSM algorhm based on he updaed arge cos vecor. IV. UMERICAL RESULTS In hs secon, we compare he performance of our proposed DSM mechansm wh hose obaned usng he one-sho energy schedulng games wh myopc prce-ancpang consumers [6]-[8][15], jon opmzaon wh myopc prce-akng consumers [12][13], as well as sochasc conrol mehods wh a sngle foresghed consumer [9]-[11]. Then we sudy he mpac of dfferen consumer preferences. In addon, we sudy he sysem performance when he percenage of shfable load and he lengh of he peak me vary. Throughou hs secon, we use he followng sysem parameers by defaul unless we change some of hem explcly. We consder he scenaro ha H 24 me slos and se he dscoun facor of he consumers o be 0.995, whch ensures ha (11) s sasfed. The prcng scheme ses he peak prce and off-peak prce o be p 0.8 $/kwh and p 0.1 $/kwh 4. Accordng H Lo o he uly company s PAR goal (namely he percenage of reducon n peak-me load, wren as l l / l h h h ), he hreshold l h n (3) wll be se o an approprae value o conrol he parameer m. We smulae boh he scenaro wh heerogeneous consumers wh parameers shown n Table IV and Fg. 3 and he scenaro wh homogeneous consumers wh he same parameers as Type 1 consumers descrbed n Table IV and Fg. 3. In hs expermen, he shfable load of each consumer s se o be 40% of he consumer s oal load. A. Comparson wh Exsng Mechansms In hs subsecon we compare our proposed -DSM algorhm wh he exsng ones [6][7]. TABLE V. COMPARISOS OF DIFFERET MECHAISMS TABLE IV. PARAMETERS OF THREE TYPES OF COSUMERS A (kwh) kh, ( h 1 o 14) / ($) D,max ($) kh, ( h 15 o 24) ($/kwh) Type 1 10 0.2/0.1 0.7 0.71 Type 2 8 0.1/0.05 1.5 0.91 Type 3 11 0.15/0.1 1.2 0.95 Works Algorhm Sraeges [6]-[8] H OG-DSM mn{ ( ), ( )} [15] p 1 h a a h h d a a [12][13] JO-DSM mn [ ph( a ) ah, d( a)] a H n1 h1 [9]-[11] SC-DSM mn{ pa, ( 1) d ( a )} Our work a H h1 h h -DSM mn{( 1) c ( ( ))} 0 4 Accordng o [17][19], he peak prce s ofen a leas 6 mes hgher han he off-peak prce.
19 Toal cos Performance gan Fg.3. The Desred Power Consumpon Paerns of Type 1, 2, 3 Consumers. TABLE VI. COMPARISO OF TOTAL COSTS ACHIEVED BY DIFFERET ALGORITHMS umber of consumers (Homogeneous, PAR<2.280) umber of consumers (Heerogeneous, PAR<2.359 ) 30 50 80 100 200 30 50 80 100 200 OG-DSM 49.95 83.25 133.20 166.50 333.00 50.95 84.95 135.90 169.80 339.70 JO-DSM 42.18 70.08 111.95 139.86 279.40 42.37 70.41 112.46 140.48 280.66 SC-DSM 46.63 77.71 124.34 155.42 310.84 47.73 79.59 127.32 159.07 318.26 -DSM 30.78 50.78 80.78 100.78 200.78 26.23 43.25 68.35 84.50 168.73 Over OG-DSM 38% 39% 39% 39% 40% 49% 49% 50% 50% 50% Over JO-DSM 27% 28% 28% 28% 28% 38% 39% 39% 40% 40% Over SC-DSM 34% 35% 35% 35% 35% 45% 46% 46% 47% 47% We dvde he exsng algorhms no hree caegores as shown n Table V. The One-sho Game based saonary DSM (OG-DSM) algorhms wh myopc prceancpang consumers [6]-[8][15] calculae he E and operae a E of he one-sho energy schedulng game, whch s characerzed n Theorem 1. The Jon Opmzaon (JO-DSM) algorhms wh myopc prce-akng consumers [12][13] assume ha he obeden consumers jonly mnmze he oal cos of he sysem. In hs case, he opmal performance of saonary DSM mechansm can be acheved by approprae prcng schemes. The Sngle-consumer Sochasc Conrol (SC-DSM) mehods [9]-[11] ry o use sochasc conrol mehods o mnmze he oal cos of a sngle consumer. In hs case, he uly company ses he prce as ph p H and p p, h h, and he consumer buys energy n advance accordng o s scheduled power h Lo consumpon paern a. We assume ha renewable energy s avalable wh probably 5 0.8, n whch case he consumer can reschedule s power consumpon paern o he desred paern whou sufferng he dscomfor cos snce he energy supply s abundan. The renewable energy s no avalable wh probably 1 0.2, n whch case he consumer mus comply wh s scheduled power consumpon paern and wll ncur dscomfor cos d ( a ). 5 Ths probably comes from he uncerany of renewable energy generaon (wheher s wndy n wnd energy generaon, wheher s shny n solar energy generaon, ec.).
20 Gven he same PAR goal, he comparson of oal coss usng hese four algorhms s shown n Table VI. We can see ha when he number of consumers ncreases, he -DSM algorhm sgnfcanly ouperforms oher hree algorhms. The cos reducons compared o OG-DSM, JO- DSM and SC-DSM are 40%, 28% and 35% n homogeneous case and 50%, 40% and 47% n heerogeneous case, respecvely. oe ha our algorhm, whch s IC, can sgnfcanly ouperform he JO-DSM algorhm, even hough s no IC. B. Impac of Dscomfor Cos TABLE VII. COMPARISO WITH BILLIG COST MIIMIZATIO ALGORITHM Toal Cos/ umber of consumers (Homogeneous) umber of consumers (Heerogeneous) Performance gan 30 50 80 100 200 30 50 80 100 200 -DSM 36.21 60.09 95.52 119.40 238.80 35.21 58.39 92.82 116.10 232.10 Bllng cos mnmzaon 56.13 93.55 149.68 187.10 374.20 67.24 112.46 179.77 223.88 448.87 Performance gan 35% 36% 36% 36% 36% 48% 48% 48% 48% 48% Some works [6]-[8][16] consder mnmzng he bllng cos only, whou akng no accoun he dscomfor cos; whle we consder he problem of jonly mnmzng bllng and dscomfor coss as n [9]-[15]. We compare he resuls of he bllng cos mnmzaon algorhm wh our proposed algorhm n Table VII. In he smulaon, we assume ha he PAR reducon goal s 10%. For he bllng cos mnmzaon algorhm, snce here are numerous opmal soluons whch acheve he mnmal bllng cos, we choose a far soluon where all consumers shf he same amoun of peak-me consumpon o off-peak mes, and hen calculae he bllng and dscomfor coss. By comparng he resuls n Table VII, we can see ha he performance of our proposed algorhm sgnfcanly ouperforms he bllng cos mnmzaon algorhm, wh around 36% cos reducon for he homogeneous scenaro and 48% cos reducon for he heerogeneous scenaro, respecvely. In fac, he bllng cos mnmzaon algorhm does no consder he mpac of consumers behavor on dscomfor coss, resulng n a hgher dscomfor cos han our proposed -DSM algorhm. The -DSM algorhm nduces he consumers o cooperae wh each oher o reduce her long-erm dscomfor coss and he consumers wh hgher dscomfor coss benef more hrough cooperaon n he heerogeneous scenaro. Thus, he performance gan of he -DSM algorhm over he bllng cos mnmzaon polcy n he heerogeneous scenaro s hgher han ha n he homogeneous scenaro.
21 C. Impac of Consumer Preferences Recall from Secon III ha n order o acheve he socal opmum,.e., he soluon o he DDP problem, he sysem runs he -DSM algorhm whch requres only a subse of he consumers o reduce her peak-me power usage. However, some consumers may have ceran preferences of when o shf her consumpon. Hence we smulae he scenaro of 3, m 1, o show he mpac of consumer preferences. The consumers ypes are Type 1, 2 and 3. We compare four cases: Case 1: all consumers can aler her power consumpon paerns. Case 2: Type 1 consumer canno aler s power consumpon paern on Mondays; Type 2 consumer canno aler s power consumpon paern on Tuesdays; Type 3 consumer canno aler s power consumpon paern on Wednesdays. Case 3: Type 1 and 3 consumers canno aler her power consumpon paerns on Mondays; Type 2 consumer canno aler s power consumpon paern on Tuesdays. Case 4: Type 1, 2 and 3 consumers canno aler her power consumpon paerns on Mondays. oe ha he four cases represen dfferen levels of heerogeney n consumer preferences, whch resul n dfferen levels of flexbly n he power consumpon schedulng. In Case 1, no consumer has specfc preferences on how o shf he power consumpon paern. Hence, he flexbly of schedulng s he hghes. In Case 2, each consumer has dsnc preferences on how o shf power consumpon. Snce her preferences are dfferen, he flexbly of schedulng s sll hgh. In Case 3, Type 1 and 3 consumers have smlar preferences (.e., boh canno shf on Fg.4. Comparson wh Dfferen Consumer Preferences.
22 Toal Cos JO-DSM SC-DSM -DSM TABLE VIII. IMPACT OF PERCETAGE OF SHIFTABLE LOAD PAR Percenage of shfable load (Homogeneous) Percenage of shfable load (Heerogeneous) Reducon goal 20% 30% 40% 50% 60% 20% 30% 40% 50% 60% 10.0% 166.80 155.40 149.30 145.80 143.20 186.80 168.80 159.10 153.60 149.50 8.0% 160.10 151.00 146.10 143.30 141.20 175.70 161.40 153.80 149.40 146.10 6.7% 155.60 148.00 144.00 141.60 139.90 168.80 156.80 150.40 146.80 144.00 10.0% -- 161.70 155.40 149.20 142.90 -- 161.60 159.10 154.20 147.80 8.0% -- 161.70 155.40 149.20 142.90 -- 161.60 159.10 154.20 147.80 6.7% -- 161.70 155.40 149.20 142.90 -- 161.60 159.10 154.20 147.80 10.0% 135.80 124.90 119.10 115.80 113.30 144.80 123.90 115.80 112.40 109.90 8.0% 128.60 120.00 115.30 112.60 110.60 130.20 116.70 112.00 109.30 107.30 6.7% 123.80 116.60 112.80 110.50 108.90 120.40 113.30 109.40 107.20 105.50 Mondays), whch makes he schedulng less flexble (.e. we can only aler Type 2 consumer s consumpon on Mondays). In Case 4, all consumers have he same preference (.e., all consumers canno shf on Mondays). Hence, he schedulng s he leas flexble. We can see from Fg. 4 ha he average dscouned daly cos ncreases from Case 1 o Case 4, as expeced. Ths mples ha he sysem performance depends on he heerogeney of consumer preferences. Regardng he mpac of he dscoun facor on he convergence rae, we can see ha here s no sgnfcan performance loss beween he case when 0.90 and he case when 0.75. D. Impac of Percenage of Shfable Load In order o evaluae he sysem performance wh dfferen percenages of he shfable load, we show he oal cos wh dfferen PAR reducon goals, as shown n Table VIII. The OG-DSM canno mee he PAR reducon goals, so we compare he oher hree algorhms. We se he number of consumers o be 100. Smulaon resuls show ha gven he consran mposed by he PAR reducon goal, our algorhm can acheve around 8% and 9% reducon n he oal cos when he percenage of shfable load vares from 30% o 60%, n he homogeneous and heerogeneous scenaros, respecvely; whle hese reducons are 7% and 10% wh JO-DSM algorhm and 12% and 9% wh SC-DSM algorhm. Ths mples ha he performance of our algorhm s no sgnfcanly affeced by consumers beng homogeneous or heerogeneous, whle he performances of he oher wo algorhms sgnfcanly depend on he consumer heerogeney. A rade-off beween he opmal oal cos and he PAR can be observed: when a hgher hreshold l h (.e. a smaller m ) s chosen, fewer consumers are requred o change her power
23 TABLE IX. IMPACT OF THE LEGTH OF THE PEAK TIME Toal Cos umber of consumers (Homogeneous) umber of consumers (Heerogeneous) 30 50 80 100 200 30 50 80 100 200 H=24 44.80 74.66 119.46 149.32 298.65 47.70 79.60 127.30 158.91 318.11 JO-DSM H=12 50.83 84.71 135.54 169.43 338.85 53.68 89.56 143.23 178.83 357.94 H=6 62.89 104.81 167.70 209.63 419.25 65.61 109.44 175.05 218.61 437.50 H=24 46.63 77.71 124.34 155.42 310.84 47.73 79.59 127.32 159.07 318.26 SC-DSM H=12 54.47 90.79 145.26 181.58 363.16 55.95 93.31 149.26 186.47 373.08 H=6 70.17 116.95 187.12 233.90 467.80 71.55 119.29 190.84 238.45 477.03 H=24 36.21 60.09 95.52 119.40 238.80 35.60 59.31 93.78 116.62 234.31 -DSM H=12 36.59 60.71 96.48 120.60 241.20 35.59 59.01 93.78 117.30 234.50 H=6 37.36 61.96 98.40 123.00 246.00 36.36 60.26 95.70 119.70 239.30 consumpon paerns, resulng n a hgher PAR bu a lower dscomfor cos. As a resul, hs rade-off should be consdered when choosng he desgn parameer l h (or m ). E. Impac of he Lengh of he Peak me In hs subsecon, we evaluae he sysem performance n erms of he oal cos n Table IX, when he lengh of he peak me vares. We se he lengh of he peak me o be 1 hour ( H 24 ), 2 hours (H 12 ) and 4 hours (H 6 ). We need o change he desred power consumpon paern accordng o he lengh of he peak me (for example, he desred power consumpon level n me slo 1 when H 12 s he sum of he desred power consumpon levels n me slos 1 and 2 when H 24 ). We fx he PAR reducon goal a 10%. The OG- DSM canno mee he PAR reducon goal. We compare he oher hree algorhms. Frs, we can see ha our algorhm acheves a lower oal cos han he oher wo under all lenghs of he peak me wh boh homogeneous and heerogeneous consumers. Second, we can observe ha he oal cos rses when he lengh of he peak me ncreases. The ncrease n he oal cos comes from he ncreased dscomfor cos, because when he lengh of he peak me ncreases, he consumers need o shf more energy o off-peak mes n order o oban a low peak-me prce. Hence, s mporan o see how he performances of he algorhms vary wh he lengh of he peak me. Under our algorhm, here s a slgh ncrease n he oal cos,.e., 3% (homogeneous) and 2% (heerogeneous), respecvely, when he lengh of he peak me vares from 1 hour o 4 hours. In conras, he ncreases n he oal cos are sgnfcan usng he oher wo algorhms, namely 40% (homogeneous) and 38% (heerogeneous) for JO-DSM, and 50% (homogeneous) and 50% (heerogeneous) for SC-DSM. Clearly, our algorhm s less sensve o he lengh of he peak
24 me, yeldng hgh performance for dfferen lenghs of he peak me as compared o he oher wo algorhms, whose performances degrade a lo when he lengh of he peak me ncreases. V. COCLUSIOS In hs paper, we proposed a nonsaonary DSM mechansm, whch explos he repeaed neracons of he consumers over me. We rgorously prove ha he proposed DSM mechansm can acheve he socal opmum n erms of he oal cos, and ouperform exsng saonary DSM sraeges. Moreover, he proposed mechansm s IC, meanng ha each self-neresed consumer volunarly follows he power consumpon paerns recommended by he opmal DSM mechansm. Smulaon resuls valdae our analycal resuls on he DSM mechansm desgn and demonsrae up o 50% performance gans compared wh exsng mechansms, especally when here are a large number of heerogeneous consumers n he sysems. In addon, compared o he exsng mechansms, he performance of he proposed mechansm s much less sensve and hus much more robus o sysem parameers such as he consumer heerogeney and he lengh of he peak me. APPEDIX A. Proof of Theorem 1 Gven he oher consumers acon a, we calculae he bes response of consumer. We denoe l h a., j, j jh, (a) If l a l, he bes response of consumer s obvously a h, h, h, snce p A d ( a ) p A. Lo Lo (b) If l a l, hen h, h, h a A, a a, we have H p A ( p p ) a d ( a ) p A [ k ( a a ) ] Lo H Lo h, Lo h1 h, h, h, H h1 pa d ( a) h h, where he frs nequaly s due o (6). Therefore, he unque E s a a. H, (12)
25 B. Proof of Theorem 2 We frs prove ha -DSM can acheve he operang pons n B. Gven any subse I of m consumer, le I choose acon a% and I () choose acon a. Then s cos s C [ I] c% 1 c 1. I s easy o see ha he cos profle { I} { I} ( C [ I], C [ I],, C [ I]) 1 2 L s n B. Snce he oher cos profles n B are convex combnaons of {( C1[ I], C2[ I], L, C [ I])} I, all he cos profles n B can be acheved [24, Lemma 3.7.1]. ex, we prove B o be he Pareo-opmal regon by showng ha % % a, ( c ( a ) c )/( c c ) m and equaly holds only by choosng he acon profles 1 descrbed above, whch proves he second saemen of he heorem. To show hese, we analyze he soluon of he followng problem: ( C1, C2, L, C ) C mnmze ( C c% )/( c c% ) 1 (13) (a) Suppose p ( a ) h p H, hen he opmal acon s a and he correspondng opmal cos vecor s ca ( ) ( C, C, L, C ) due o Theorem 1. However, 1, E 2, E E, ( C c% )/( c c% ) ( p p ) a / d ( a% ) 1 E, 1 H Lo h, ( p p ) a /max{ d ( a% )} ( p p ) l /max{ d ( a% )} m H Lo 1 h, H Lo h where he las nequaly s due o (6).. (14) (b) Suppose p ( a ) p and he opmal acon s a. Obvously, h Lo 1 a h, l h, oherwse a leas one consumer can reduce s dscomfor cos by shfng peak-me power whle keepng he oal bllng cos and oher consumers dscomfor coss unchanged. ex, we show ha eher a a or a b. h, h, h, h, Suppose for and j, we have b a a and b a a. Whou loss of h, h, h, jh, jh, jh, generaly, we assume ( k mn k )/ d ( a% ) ( k mn k )/ d ( a% ). Then we compare he h, h, jh, jh, j j hh hh coss of and j usng acon a, where a a V, h, h, a argmn c ( aˆ ), h h h, a ˆA, aˆ a V h, h,,
26 a a V, jh, jh, a argmn c ( aˆ ), h h jh, a ˆA, aˆ jh, a V jh, j, wh V mn{ a b, a a } h, h, jh, jh,, and a a, n j,. Thus, we have: n n ( c ( a) c% ) ( c ( a) c% ) j j d ( a ) d ( a ) j j ( c c% ) ( c c% ) d ( a% ) d ( a% ) j j j j [ ( k mn k )( a a )]/ d ( a% ) [ ( k mn k )( a a )]/ d ( a% ) h, hh h, h, h, j jh, hh jh, jh, jh, j j.(15) [ ( k mn k )( a ( a ))]/ d ( a% ) [ ( k mn k )( a ( a ))]/ d ( a% ) h, hh h, h, h, j jh, hh jh, jh, jh, j j ( c ( a) c% ) ( c ( a) c% ) j j ( c c% ) ( c c% ) j j We noce ha eher a a% or a a,.e., eher c( a ) c or c ( a ) % c. By repeang j j j hs procedure, we can fnally ge a subse I, where acon a. Therefore, he soluon of problem (13) s n he form of I chooses acon a%, and I chooses C [ I] and he Pareo-opmal regon can be wren as (8). In addon, any oher acon profle wll resul n a % % ( c ( ) c )/( c c ) m. In oher words, he oher cos profles n he se B canno be 1 acheved by saonary sraeges. C. Proof of Theorem 3 Based on -DSM, n me perod, c %, c I () and c c, I (). Then, C (1 ) [ c 1 c 1 ] 0 { I()} { I()} where 1{} g s he ndcaor funcon. We denoe g g C c% g (,, L, g ), g 1 2 c c% % (16), g C c% c c% G { ( %)/ %), C B }. Then we noce ha and g g C c ( c c C C g (1 ) 0 { I ()} 1 and C B s equvalen o g G C C. We denoe by g () (1 ) 1 and g (0) g he connuaon cos a me { I ( )} and 0, respecvely. Then we use a backward nducon mehod o show ha connuaon cos a
27 any gven me can be decomposed of he curren cos and he connuaon cos a me 1. We call he vecor g() G C a feasble vecor and hrough he decomposon we wll show ha when he dscoun facor sasfes (11), he connuaon cos a any me 1 s also a feasble vecor, namely g( 1) show hs. C For me perod, suppose g() G C, hen G. Snce he orgnal g(0) G C, we use mahemacal nducon o Or equvalenly, We need o show g () (1 ) g ( 1) g ( 1) { I()} 1. (17) [ g () (1 )]/, I () g ()/, I () g ( 1) m and 0 g ( 1) g 1. (18). The former one s obvous due o hypohess. For he laer one, 0 g ( 1) g s obvous for I (), and for I (), we also need Ths can be smplfed as g () (1 ). (19) 0 g, I () 1 g () 1 g, I () 1 g () The frs erm s sasfed due o hypohess. The second erm requres max mn max{1 g ()} g () G I() I I() C Whou loss of generaly, we sor g () n an decreasng order, namely,. (20). (21) I s easy o calculae: g () g () L g () L g (). (22) 1 2 m max{1 g ()} 1 g (), (23) I() max{} I( ) and mnmax{1 g ()} (1 g ()) I() I I() m. (24)
28 Thus, he wors case g () n (21) s Thus we requre m1 m1 ( m g ) ( m g ) 1 1 g () ( g, L, g,, L, ). (25) 1 m1 ( m 1) ( m 1) m 1 Snce ( m g ) m ( m 1) 1 ( m ) 1 ( m 1) m1 g 1. (26), B s achevable, when (11) s sasfed. C D. Proof of Theorem 4 Obvously, C C C C (,, L ) s n B. Accordng o he proof of Theorem 3, C can be 1 2 C decomposed of curren cos plus connuaon cos: ( ) % ( %) () (1 )[ % ] ( ). (27) C c c c g c 1 c 1 C { I()} { I() } 1 When he dscoun facor sasfes (11), he opmal operang pon C can be acheved by decomposng no curren cos plus connuaon cos n each perod. We also noce ha he bes sraegy of I() n (21) s o choose I() {1,2, L m},.e., he ndexed wh he m larges g (). Therefore, he proposed -DSM can acheve opmal performance C C n he sysem. To show -DSM s IC, we need o show ha gven he operang pon ( C, C, LC 1 2 ) B C and an arbrary me perod, he connuaon cos s he mnmum cos achevable for consumer. By decomposon, we have ( ) (1 )[ % ] ( ). (28) C c1 c 1 C { I( )} { I( ) } 1 Obvously for consumer I (), has no ncenve o devae from sraegy For consumer c a (, ( )) I (), f devaes from sraegy n perod., he mnmum cos can acheve s, snce he bes response s o choose a accordng o Theorem 1. Accordng o one- sho devaon prncple [24], when consumer devaes, he oher consumers wll play E sraegy from me perod 1. So he cos from perod s C ( ) (1 ) c ( a, ( )) C ( ) C. (29) E 1 E,