A Real-Time Pricing Model for Electricity Consumption

Transcription

1 A Real-Time Pricing Model Elecriciy Consumpion Ranjan Pal Universiy o Souhern Calinia rpal@usc.edu Absrac The Calinia elecric company, i.e., PG&E (Paciic Gas and Elecric Co.,), has recenly announced is inenions o charge small businesses in he sae wih dynamic prices elecriciy consumpion. In his regard, we sudy a real-ime elecriciy pricing model in he paper and compare i wih wo saic pricing models. We show ha real-ime pricing ouplays saic pricing when i comes o joinly maximizing provider and consumer welare. Keywds: elecriciy, pricing, proi, load-shedding. Inroducion The Paciic Gas and Elecric Co., (PG&E) o Calinia has recenly advocaed changes in he way small companies pay power. In a newspaper aricle in 0, PG&E announced dynamic pricing as he way o price small companies heir elecriciy consumpion. The sae regulas have already approved o his idea, and we migh see PG&E charge is approximaely 50,000 small business cusomers ime-o-use prices rom November 0. The main reasons he regulas o allow dynamic pricing are woold: (i) dynamic pricing enables he uiliy company o price is cusomers (consumers) accding o he supply available a a given ime so ha he laer can keep heir demands wihin supply limis. (ii) dynamic pricing helps reduce peak demand rom consumers. Maching ne supply wih aggregae consumer demand reduces he chances o load-shedding, and reducing peak consumer demands prevens wasage o energy - leading o a greener environmen. To provide an example he laer saemen, consider a very ho aernoon in summer Los Angeles. There is a very high demand elecriciy rom several consumer appliances like air-condiioners, ans, coolers, ec., a such a ime. Even hough a uiliy company like PG&E plans in advance o saisy cusomer demands such an aernoon, i would wan o price elecriciy high ha period so as o curail consumer demands such ha he aggregae peak demand would be wihin he supply limis. Anoher reason high pricing would be o accoun supply shage due o grid ailures a imes. In his paper, we invesigae he problem o dynamic elecriciy pricing and propose a real-ime pricing (a special ype o dynamic pricing) model. F he model, we analyze en- The aricle was eniled PG&E, small-business group seek delay o new raes, and appeared in he San Francisco Chronicle on 0h February, 0. Alhough real-ime pricing is a special ype o dynamic pricing, i is general enough o be represenable as oher ms o dynamic pricing models such as he Criical Peak Pricing (CPP) and Time-o -Use Pricing (TOU) models. ergy provider (ex., PG&E) prois, consumer welare, and loadshedding prospecs. In addiion, we compare our real-ime pricing model wih wo saic pricing models. We show ha realime pricing ouplays saic pricing when i comes o joinly maximizing provider and consumer welare.. Problem Seup We consider a single proi-maximizing energy provider like PG&E having marke power (e.g., monopoly energy provider) and servicing n small businesses wihin is service localiy. We model he elecriciy disribued by he energy provider o various consumers (businesses) as lows, where each low perains o a paricular consumer. In he res o he paper, we rea he erms low and consumer inerchangeably. In his regard, we model all lows in he elecriciy grid o be conained in a se F o cardinaliy n. In der o model he regular periods, we consider a ime period T divided ino discree ime slos o he m [, ]. T is generally a single day wih ime slos based on a hal-hourly a hourly basis. We assume ha he amoun o energy (elecriciy) consumed in he inerval [, ] by a low ɛ F o be e. Thus, wihin a ime period T, he oal energy consumpion by a low ɛ F is T e. We also assume ha in each ime slo, he energy provider can provide (supply) a maximum o L energy unis. We represen he laer ac by he ollowing inequaion: ɛ F e L, () Each consumer (low) is associaed wih an uiliy uncion U ( ), which is a uncion o e, he amoun o energy consumed by during a given ime inerval [, ]. We assume he consumer uiliy uncion o be o he ollowing m a given low : U (e ) (e ) 0 <. () Preprin submied o SIAM Conerence on Financial Mahemaics and Engineering May, 0

2 When, we se U (e ) log(e ). Our consumer uiliy uncion is similar o he widely used Cobb-Douglas uiliy uncion used in classical demand hey []. We assume here ha a consumer s uiliy level is ime-varian bu shapeinvarian. The laer assumpion is realisic because a consumer may be happier han nmal i i can ge a cerain amoun o energy a a ime when i needs i mos. In his regard, le U (e ) be he uiliy o low a ime insan, where is he ime sensiiviy ac o s uiliy level. The energy provider can iner his parameer in he long-run by analyzing consumer consumpion daa over ime and aking a saisical esimae. A he beginning o each ime slo, each low, he energy provider ses a price ha is operaive o ha paricular slo 3. In response o he price se, he consumers decide o consume a cerain amoun o energy in ha slo so as o maximize is own ne uiliy. We assume here ha he energy provider knows he uiliy uncions o he consumers in he localiy in addiion o heir ime sensiiviy acs. Having his knowledge enables he provider o se is opimal prices each ime slo assuming ha he consumers would choose a level o energy usage ha maximizes heir uiliies he cresponding slo. Le p be he price charged by he energy provider o low he ime inerval [, ]. Thus, he ne uiliy opimizaion problem consumer he cresponding ime slo is argmax e U (e ) p (e ), (3) The opimal amoun o energy demanded by a consumer in every ime slo is he soluion o he above opimizaion problem. We assume ha he price p charged by he energy provider o a low a each given ime slo is o he ollowing linear m: p (e ) a + b e, (4) where a is he la componen o p irrespecive o consumer consumpion in he ime slo, and b is he usage componen o p denoing he price per uni o consumer demand. We adop he linear pricing m p because a monopolisic energy provider having marke power, he linear price combinaion ransers a consumer s enire ne-surplus ino he energy provider s proi []. Thus, a low, he reined version o opimizaion problem (3) is argmax e U (e ) (a + b e ), (5) Since we have an unconsrained opimizaion problem in (5), he soluion o i is obained by considering he ac ha he opimal consumer ne-uiliy should no be less han zero, in which case he consumer decides o have a demand o zero unis. Given ha consumer ne uiliy is greaer han equal o zero, we obain is opimal demand in he cresponding ime slo by evaluaing he irs derivaive o he objecive uncion in (5) and equaing i o zero [3]. In doing so we obain he opimal consumer demand as (U ) ( b ),. a 3 In pracice, he slo prices are inmed o consumers a day wo in advance. 3. Real-Time Pricing Model In his secion, we propose our real-ime pricing model. We consider he scenario ha he energy provider has energy provision limis and will no be able o service consumer demands when hey exceed supply limis, hus leading o load shedding being enced on he consumers. We also assume ha he energy provider can charge dieren prices dieren consumers in a given ime slo. We have he ollowing proi opimizaion problem (OPT) he energy provider. argmax e,a (a,b + b e ) s.. e ( b ),,, e L,, U (e ) (a + b e ) 0,,. The irs consrain saes ha every ime slo he energy provider always provisions an amoun low which is less han he opimal demand in he cresponding ime slo. The RHS o he consrain is he opimal consumer demand, (U ) ( b ) o on being charged p a, and is obained by aking he irs derivaive o (5) and equaing i o zero. The second consrain saes ha in each ime slo he energy provider can provision a mos L unis o energy across all lows. The hird and inal consrain saes ha he ne consumer uiliy each should be greaer han equal o zero each ime slo. I is eviden rom he problem mulaion above ha when he consumer has an opimal demand greaer han wha he energy provider could guaranee, he laer ences load shedding on he consumer. However, noe ha in every ime slo he energy provider akes he irs sep in seing he prices ollowed by consumers choosing heir energy consumpion. Given ha he energy provider has access o he uiliy uncions o all consumers as well as heir ime sensiiviy acs, i can se usage prices high enough such ha no consumer exceeds demand greaer han wha he mer can guaranee, and a he same ime guaranee ha all consumers have non-negaive ne uiliy. In his case, we have OPT, a slighly modiied version o OPT. argmax e,a (a,b + b e ) s.. e ( b ),,, e L,, U (e ) (a + b e ) 0,,. The change rom OPT is in he irs consrain and he hird consrain. The irs consrain in OPT saes ha energy

3 provider always provisions he opimal consumer demand in each ime slo. The hird consrain in OPT saes ha in each ime slo he consumer exracs consumer ne-uiliy. The laer consrain is valid as we consider a monopoly energy provider who charges consumers accding o a linear uncion []. In his case, he provider services opimal consumer demand, which is releced in he irs consrain. Applying Karush-Kuhn-Tucker (KKT) condiions [3] o problem OPT, we ge he ollowing opimal parameers as a soluion each ime slo. (a )op U ((e )op ) λ (e )op, provider can charge dieren prices dieren households in every ime slo. In our second saic pricing model condiion (ii) holds, bu we assume ha he provider does no ence load shedding on consumers. 4.. Saic Pricing - Model We have he ollowing proi opimizaion problem (OPT3) he energy provider in Model. argmax e,a,b (a + b e ) (b )op λ, (e )op (U ) ( λ ), and (e )op L. s.. e ( b ),,, e L,, U (e ) (a + b e ) 0,. Soluion Observaions and Implicaions: λ is he Lagrange muliplier he opimizaion problem in each ime slo. We observe ha under opimaliy condiions, he usage componen o (p )op is consumer independen and is he same all lows in a ime slo. On he oher hand, he la componen o (p )op is consumer dependen and allows he energy provider o ully exrac ne-consumer uiliy, i.e., when energy provider proi is maximized, each consumer has zero ne uiliy. The inuiion behind such observaions is ha he la componen is independen o usage and so he energy provider could discriminae amongs lows by uning his parameer as i knows ha all consumers would have o pay he la componen irrespecive o he amoun o energy usage. On he oher hand, since usage is a personal preerence each consumer in each ime slo, he energy provider inds i opimal o charge he same per uni o energy price o all consumers in a ime slo, and in urn provides equal reedom o all in erms o he willingness o consume energy. Since in every ime slo, consumer opimal demands are always me, and ha provider is able o exrac consumer ne-uiliy, he consumer welare as well as provider prois are joinly maximized. 4. Saic Pricing Models In his secion we propose wo saic pricing models elecriciy consumpion and compare he various resuls beween our saic pricing and real-ime pricing models. In a saic pricing model, he energy provider ses a single price each low a he beginning o he enire period o operaion, and ha price says ixed across all imes. In he irs saic pricing model, similar o he real-ime pricing model, we consider (i) he scenario ha he energy provider has energy provision limis in every ime slo and will no be able o service consumer demands when hey exceed supply limis, hus leading o load shedding being enced on he consumers and (ii) he energy 3 The irs consrain saes ha every ime slo he energy provider always provisions an amoun low which is less han he opimal demand in he cresponding ime slo. The RHS o he consrain is he opimal consumer demand, (U ) ( b ) o on being charged p a, and is obained by aking he irs derivaive o (5) and equaing i o zero. The second consrain saes ha in each ime slo he energy provider can provision a mos L unis o energy across all lows. The hird and inal consrain saes ha he ne consumer uiliy each should be greaer han equal o zero aer period T. Remark on he Opimal Soluion: Opimizaion problem OPT3 is ineresing he ollowing reason: Since he energy provider canno charge a every ime slo, i canno manipulae he consumer demands o remain wihin wha i wans o provide. Thus, we expec ha he provider would no be able o exrac ne consumer uiliy. However, we observe somehing dieren. F each low, le he la componen o he opimal ime invarian price ha he energy provider would charge be given as a op U ((e )op ) b op (e )op, (6) where b op is he usage componen o he opimal ime invarian price and equals min{(b )op }, and e (e )op. The raionale choosing he usage componen o be as low as possible is o encourage consumers push in as much demand as possible in every ime slo. Le (b )op and (e )op be assumed rom he opimal soluion o OPT. Under he se o parameers described in his paragraph, he energy provider s opimal revenue (proi) is U ((e )op ), which is equal o he opimal revenue when a energy provider is able o charge a he beginning o each ime slo. However, in doing so he second consrain in OPT3 does no always sricly bind a opimum, unlike in he case o OPT, when i does.

4 Soluion Observaions and Implicaions: We observe ha i b op min{(b )op } hen he second consrain in OPT3 always holds (no necessarily binds). Thus, in his model he energy provider may gain he same maximum proi as i did in he case when i charged consumers in every ime slo, i.e., as in OPT, bu i has o always ence consumer load-shedding. Such shedding o load was no required o consumers accding o he real-ime pricing model in Secion 3. Theree, he disadvanage ha saic pricing has on he consumers is ha hey canno be made o adjus heir demand in each ime slo so as never experience load shedding. Since in every ime slo, consumer opimal demands are no me, bu he provider is able o exrac consumer ne-uiliy, he consumer welare is no maximized bu he provider prois are. Thus, real-ime pricing ouplays his saic pricing model in erms o joinly maximizing consumer welare and provider prois. We now quaniy he amoun o load ha ges shed via he load-shedding process in our model o saic pricing. Le S be he load shed each consumer in ime slo. We represen S as S ( ) (b op ) (e )op 0,, (7) S ( ) (b op ) (e )op 0,, (8) Considering he special case when, we ge ( S ) ( min { ) ( ) } { ( ) L,, (9) } An ineresing observaion rom Equaion (9) is ha i he ime sensiiviy acs o individual consumers lucuae a lo rom ime slo o ime slo, he consumers will need o shed signiican amouns o heir energy demands in every ime slo energy providers o no run a a proi loss aer period T. 4.. Saic Pricing - Model We have he ollowing energy provider proi maximizaion problem (OPT4) Model. argmax e,a,b (a + b e ) s.. e ( b ),,, e L,, U (e ) (a + b e ) 0,. 4 Applying he KKT condiions, we have b op λ ( ) ( ). The opimal value o a op depends on he spread o he ime sensiiviy ac o uiliy levels o consumers. I he ime sensiiviy acs are well spread hen he energy provider can se a op so as o nearly exrac ne consumer uiliy aer period T(See Theem ). Soluion Observaions and Implicaions: We observe ha even hough load shedding is no enced on consumers, he energy limis in each ime slo - specially in non-peak periods o a day, are no uilized o he maximum, and as a resul he energy provider canno exrac consumer ne-uiliy. The ineiciency in uilizing ull energy capaciy in a ime slo arises due o a one-ime high pricing by he energy provider in der o preven consumer demand rom exceeding energy supply limis in any ime slo. Since in every ime slo, consumer opimal demands are me, bu he provider is no able o exrac consumer ne-uiliy, he consumer welare is maximized bu he provider prois are no. Thus, real-ime pricing ouplays his saic pricing model in erms o joinly maximizing consumer welare and provider prois Comparison Sudy In his secion we compare beween he prois ha an energy provider makes in he real-ime pricing and he saic pricing scenarios. We observe ha he opimal prois made by an energy provider rom OPT is he same as ha made rom OPT3. However, in OPT, consumers do no have o shed load in any ime slo as in OPT3, where consumers may have o. The reason his dierence is ha energy providers can charge in every ime slo in OPT and conrol consumer demand o no exceed provider provision, whereas in OPT4 energy providers have o price saically and does no have ha conrol in every ime slo. Regarding he comparison beween provider prois in OPT and OPT4 we have he ollowing resul in he m o a heem. Theem. Le and all lows,. The raio o he opimal provider prois rom OPT4 o OPT is given by ( max ) ( max ), (0) where is he opimal proi rom OPT4, is he opimal proi rom OPT, and max max{ } is he maximum ime sensiiviy ac any low. In addiion, NPop he ollowing inequaliy ( ) E() max is bounded by () We provide he proo o he heem in he Appendix secion. Theem Implicaions: We observe rom he heem ha

5 is less han equal o, which indicaes he ineiciency o he energy provider o exrac ne consumer uiliy in OPT4 when compared o OPT. The ineiciency arises due o he inabiliy o he provider o charge prices in every ime slo in OPT4, hus leading o a siuaion where consumers do no demand suicien energy amouns (o avoid making negaive uiliy) o uilize he energy limis se in every ime slo. The ineiciency is higher lower values o (high price sensiiviy [4] o consumer), which is inuiive. We also observe rom he lower bound o NPop ha energy provider proi ineiciency (he inabiliy o hen energy provider o exrac ne consumer uiliy) is small i and only i here is a low spread in he ime sensiiviy ac o consumer uiliy levels, i.e., consumers having similar preerence energy consumpion. 5. Conclusion Moivaed by PG&E s recen announcemen o price small business consumers real-ime prices elecriciy consumpion, we provided a real-ime pricing model in he paper and compared i o wo saic pricing models. We observed ha he mer ouplays he laer in erms o joinly maximizing consumer welare and provider prois. 6. Appendix Proo o Theem. We begin our proo wih he opimal soluion o he usage ee componen in OPT4. The opimal soluion each ime slo is given by he ollowing se o equaions. (b )op (b )op λ δe δb δe δb λ ( ) λ η e η e () ( ), (3) where η is he elasiciy (demand sensiiviy o price lucuaions) o consumer demands and is represened as η δe b δb e,. (4) The complemenary slackness condiion in he se o KKT condiions implies ha and λ 0 i λ > 0 i ( ) < L (5) b ( ) L. (6) b 5 Le PU be he subse o ime slos when here is peak usage by consumers. Thus, during he PU ime slos, equaion (8) holds all ɛ PU. I ollows ha (b )op ɛ PU λ ( ) ( ). (7) Now consider he case ha all lows are idenical. The saic price b op b op in he peak usage imes have o be se so as o consrain he energy consumpion o be wihin energy availabiliy limis. Considering he ac ha consumers could op he maximum ime sensiiviy ac o heir uiliy levels, max, b op is given by he ollowing equaion. ( ) b op max L. (8) The cresponding opimal energy provision by he provider in each ime slo in a saic pricing environmen is given by ( ) (e ) op ( ) L max. (9) In a real-ime pricing scenario, he opimal energy provision by he provider in each ime slo is given by ( ) e op L. (0) The raio o he opimal provider prois rom OPT4 o OPT is given by ( L ) U((e ) op ) U((e ) op ) ( ) ( ) max ( ) L () ( ) ( ( ) L ) max ( max ) ( max ). () Le min min{ }. Then he ollowing relaionship holds o characerize he lower bound o he raio opimal prois. ( ) ( ) max min. (3) ( ) max max A igh lower bound o he opimal proi raio can be derived by modeling as realizaions o a random variable and applying he Jensen s inequaliy as ollows. ( max ) E( ) E() Thus Theem is proved. Reerences ( E() max ). (4) [] H.R.Varian, Inermediae Microeconomic Analysis, Non, 99. [] W. Oi, A disneyland dilemma: Two-par aris a mickey mouse monopoly, Quarerly Journal o Economics, (). [3] S.Boyd, L.Vanderberghe, Convex Opimizaion, Cambridge Universiy Press, 005. [4] H.R.Varian, Microeconomic Analysis, Non, 99.