Comparison of congestion management techniques: Nodal, zonal and discriminatory pricing

Transcription

1 Comparso of cogesto maagemet techues: odal, zoal ad dscrmatory prcg Pär Holmberg * ad Ewa Lazarczy ** Wholesale electrcty marets use dfferet maret desgs to hadle cogesto the trasmsso etwor. We compare odal, zoal ad dscrmatory prcg geeral etwors wth trasmsso costrats ad loop flows. We coclude that large games wth may producers ad certa formato, the three maret desgs result the same effcet dspatch. However, zoal prcg wth couter-tradg results addtoal paymets to producers export-costraed odes, whch leads to effcet vestmets the log-ru. Keywords: Cogesto maagemet, wholesale electrcty maret, trasmsso etwor, odal prcg, zoal prcg, couter-tradg, dscrmatory prcg, large game * Correspodg author. Research Isttute of Idustral Ecoomcs (IF). P.O. Box 55665, SE Stocholm, Swede, phoe , fax E-mal: [email protected]. Assocate Researcher of Electrcty Polcy Research Group (EPRG), Uversty of Cambrdge. ** Research Isttute of Idustral Ecoomcs (IF). P.O. Box 55665, SE Stocholm, Swede, fax , Stocholm School of Ecoomcs, Departmet of Ecoomcs, Sveaväge 65, Stocholm, Swede. E- mal: [email protected]. We are grateful for very helpful commets from Rchard Frberg, Sve-Olof Frdolfsso, Jey Frdström, Håa Phl, Thomas Tagerås, Bert Wllems, aoymous referees, semar partcpats at Stocholm School of Ecoomcs, Research Isttute of Idustral Ecoomcs (IF) ad the Swedsh Mstry of Eterprse, Eergy ad Commucatos, ad coferece partcpats at IAEE 2011 Stocholm, EWGCFM 2012 Lodo ad EPRG Ole Symposum o Electrcty Trasmsso Prcg ad Cogesto Maagemet. We also wat to tha Er Lud for research assstace, ad Chrsta Löblad ad Da ema for proof-readg our paper. The wor has bee facally supported by the Research Program The Ecoomcs of Electrcty Marets ad the Torste Söderberg foudato. 1

2 1. ITRODUCTIO Storage possbltes are eglgble most electrc power etwors, so demad ad supply must be statly balaced. Oe coseuece s that trasmsso costrats ad the way they are maaged ca have a large fluece o maret prces. The Europea Uo s regulato 1228/2003 (ameded 2006) sets out gudeles for how cogesto should be maaged Europe. System operators should coordate ther decsos ad choose desgs that are secure, effcet, trasparet ad maret based. I ths paper, we compare the effcecy ad welfare dstrbuto of three maret desgs that are operato real-tme electrcty marets: odal, zoal ad dscrmatory prcg. Characterstcs of the three desgs are summarzed Table 1. The zoal maret s specal that t has two stages: a zoal clearg ad a redspatch. We show that compettve marets wthout ucertates the three desgs result the same effcet dspatch. However, zoal prcg wth a maret based redspatch (couter-tradg) results addtoal paymets to producers exportcostraed odes, as they ca mae a arbtrage proft from prce dffereces betwee the zoal maret ad the redspatch stage. Ths strategy s ofte referred to as the crease-decrease (cdec) game. Ths s the frst paper that proves these results for geeral etwors wth geeral producto costs. Dj ad Wllems (2011) are closest to our study. However, ther aalyss s lmted to two-ode etwors ad lear producto costs. The parallel study by Ruderer ad Zöttl (2012) s also aalyzg smlar ssues, but the redspatch of the zoal maret that they cosder s ot maret based, thus ther model does ot capture the crease-decrease game. Table 1: Summary of the three cogesto maagemet techues. Cogesto maagemet Cosdered trasmsso Aucto format Uform-prce Pay-as-bd techue costrats odal All X Dscrmatory All X Zoal stage 1 Iter-zoal X Redspatch Itra-zoal stage 2 X 2

3 1.1 Cogesto maagemet techues Producers submt offers to real-tme marets just before electrcty s gog to be produced ad delvered to cosumers. Durg the delvery perod, the system operator accepts offers order to clear the real-tme maret, tag trasmsso costrats to accout. The aucto desg decdes upo accepted offers ad ther paymets. odal prcg or locatoal margal prcg (LMP) acowledges that locato s a mportat aspect of electrcty whch should be reflected ts prce, so all accepted offers are pad a local uform-prce assocated wth each ode of the electrcty etwor (Schweppe et al., 1988; Hoga, 1992; Chao ad Pec, 1996; Hsu, 1997). Ths desg s used Argeta, Chle, ew Zealad, Russa, Sgapore ad several U.S. states, e.g. Southwest Power Pool (SPP), Calfora, ew Eglad, ew Yor, PJM 1 ad Texas. odal prcg s ot yet used sde the Europea Uo. However, Polad has serous dscussos about mplemetg ths desg. Uder dscrmatory prcg, where accepted offers are pad as bd, there s o uform maret prce. Stll, the system operator cosders all trasmsso costrats whe acceptg offers, so there s locatoal prcg the sese that producto mport-costraed odes ca bd hgher tha producto export costraed odes ad stll be accepted. Dscrmatory prcg s used Ira, the Brtsh real-tme maret, ad Italy has decded to mplemet t as well. A coseuece of the pay-as-bd format s that accepted producto s pad ts stated producto cost. Thus oe (somewhat aïve) motvato for ths aucto format s that f producers would bd ther true cost, the ths format would crease cosumers ad/or the auctoeer s welfare at producers expese. The thrd type of cogesto maagemet s zoal prcg. Marets whch use ths desg cosder ter-zoal cogesto, but have a uform maret prce sde each rego, typcally a coutry (cotetal Europe) or a state (Australa), regardless of trasmsso cogesto sde the rego. Demar, orway ad Swede 2 are also dvded to several zoes, but ths dvso s motvated by propertes of the etwor rather tha by borders of admstratve regos. 3 Brta s oe zoe ts day-ahead maret, but uses dscrmatory prcg the real-tme maret. Itally the zoal desg was thought to mmze the complexty of the prcg 1 PJM s the largest deregulated wholesale electrcty maret, coverg all or parts of 13 U.S. states ad the Dstrct of Columba. 2 The Swedsh govermet troduced four zoes Swede from ovember 2011, as a result of a attrust settlemet betwee the Europea commsso ad the Swedsh etwor operator (Sadowsa ad Wllems, 2012). 3 The optmal defto of zoes for a gve etwor s studed by e.g. Stoft (1997), Bjørdal ad Jörste (2001) ad Ehrema ad Smeers (2005). 3

4 settlemet ad poltcally t s sometmes more acceptable to have just oe prce a coutry/state. 4 Orgally, zoal prcg was also used the deregulated electrcty marets of the U.S., but they have ow swtched to odal prcg, at least for geerato. Oe reaso for ths chage the U.S. s that zoal prcg s, cotrary to ts purpose, actually ute complex ad the prcg system s ot very trasparet uder the hood. The ma problem wth the zoal desg s that after the zoes of the real-tme maret have bee cleared the system operator eeds to order redspatches f trasmsso les sde a zoe would otherwse be overloaded. Such a redspatch creases accepted supply mport costraed odes ad reduces t export costraed odes order to relax tra-zoal cogesto. There are alteratve ways of compesatg producers for ther costs assocated wth these adjustmets. The compesato schemes have o drect fluece o the cleared zoal prces, but drectly the detals of the desg may fluece how producers mae ther offers. The smplest redspatch s exercsed as a commad ad cotrol scheme: the system operator orders adjustmets wthout referrg to the maret ad all agets are compesated for the estmated cost assocated wth ther adjustmets (Krause, 2005). I ths paper we stead cosder a maret oreted redspatch, also called couter-tradg. Ths zoal desg s used Brta, the ordc coutres ad t was used the old Texas desg. 5 I these marets a producer s adjustmets are compesated accordace wth hs stated costs as uder dscrmatory prcg. Thus the maret has a zoal prce the frst stage ad pay-as-bd prcg the secod stage. We cosder two cases: a sgle shot game where the same bd curve s used both the frst ad secod stage, ad a dyamc game where frms are allowed to submt ew bd curves the secod stage. The dyamc model s approprate f, for example, the frst stage represets the dayahead maret ad the secod stage represets the real-tme maret. 1.2 Comparso of the three maret desgs Our aalyss cosders a geeral electrcty etwor, whch could be meshed, where odes are coected by capacty costraed trasmsso les. We study a dealzed maret where 4 Polcy maers ad the dustres crtue of the odal prcg desg s summarzed, for example, by Alaywa et al. (2004), de Vres et al. (2009), Leuthold et al. (2008), Oggo ad Smeers (2012) ad Stoft (1997). 5 ote that Brta s dfferet that t has pay-as-bd prcg for all accepted bds the real-tme maret. The ordc real-tme marets oly use dscrmatory prcg for redspatches; all other accepted bds are pad a zoal real-tme prce. 4

5 producers costs are commo owledge, ad demad s certa ad elastc. There s a cotuum of ftesmally small producers that choose ther offers order to maxmze ther dvdual payoffs. 6 Subject to the trasmsso costrats, the system operator accepts offers to mmze total stated producto costs,.e. t clears the maret uder the assumpto that offers reflect true costs. We characterze the ash eulbrum (E) of each maret desg ad compare prces, payoffs ad effceces for the three desgs. I the odal prcg desg, we show that producers maxmze ther payoffs by smply bddg ther margal costs. Thus, ths case, the accepted offers do fact maxmze short-ru socal welfare. We refer to these accepted eulbrum offers as the effcet dspatch ad we call the clearg prces the etwor s compettve odal prces. We compare ths outcome wth eulbra the alteratve maret desgs. For fxed offers, the system operator would crease ts proft at producers expese by swtchg from odal to dscrmatory prcg. But we show that eve f there are ftely may producers the maret, dscrmatory prcg ecourages strategc bddg amog framargal producto uts. They ca crease ther offer prces up to the margal prce ther ode ad stll be accepted. 7 I the ash eulbrum of the pay-as-bd desg, accepted producto s the same as the effcet dspatch ad all accepted offers are at the etwor s compettve odal prces. Thus, maret effcecy ad payoffs to producers ad the system operator are the same as for odal prcg. As payoffs are detcal all crcumstaces, ths also mples that the log-ru effects are the same terms of vestmet cetves. Uder our dealzed assumptos, the zoal maret wth couter-tradg has the same effcet dspatch as the two other maret desgs. We also show that producers buy ad sell at the compettve odal prce the couter-tradg stage. Stll producers payoffs are larger uder zoal prcg at cosumers ad the system operator s expese. The reaso s that the two-stage clearg gves producers the opportuty to ether sell at the zoal prce or at the dscrmatory eulbrum prce the secod stage, whchever s hgher. I addto, eve whe they are ot producg ay eergy, producto uts export-costraed odes ca mae moey by sellg at the uform zoal prce ad buyg bac the same amout at the dscrmatory prce, whch s 6 The dea to calculate ash eulbra for a cotuum of agets was frst troduced by Auma (1964). The theory was further developed by Gree (1984). 7 Related results have bee foud for theoretcal ad emprcal studes of dscrmatory auctos (Holmberg ad ewbery, 2010; Evas ad Gree, 2004). However, prevous studes of dscrmatory prcg have ot tae the etwor to accout. 5

6 lower, the secod stage. Ths crease-decrease game has bee observed durg the Calfora electrcty crss (Alaywa et al., 2004), t destroyed the tal PJM zoal desg, ad s preset the UK the form of large paymets to Scottsh geerators (euhoff, Hobbs ad ewbery, 2011). Our results show that c-dec gamg s a arbtrage strategy, whch caot be removed by mprovg competto the maret. If t s a serous problem, t s ecessary to chage the maret desg as the U.S. We show how producers profts from the c-dec game ca be calculated for geeral etwors, cludg meshed etwors. Our results for the zoal maret are the same for the statc game, where the same offer s used the two stages, ad the dyamc game, where frms are allowed to mae ew offers the couter-tradg stage. Addtoal paymets to producers the zoal maret cause log-ru effceces; producers overvest export-costraed odes (Dj ad Wllems, 2011). 8 Zoal prcg also leads to effceces the operato of flexble plats wth log ramp-rates, whch are ot allowed to trade the real-tme maret. Related ssues are aalyzed by Gree (2007). I practce odal prcg s cosdered superor to the other desgs, as t esures effcet allocato a compettve maret also for ucerta demad ad termttet wd power producto; a advatage whch s stressed by Gree (2010). The orgazato of the paper s as follows. I Secto 2 we preset a smple two ode example llustratg the eulbrum uder the odal prcg. Secto 3 dscusses our model ad Secto 4 we preset a aalyss of the three cogesto maagemet desgs. I secto 5, maret eulbra for the dscrmatory ad zoal prcg desgs are dscussed wth the meas of a smple example. The paper s cocluded secto 6, whch also brefly dscusses how more realstc assumptos would chage our results. Three techcal lemmas ad all proofs are placed the Appedx. 2. EXAMPLE ODAL PRICIG I the followg secto we descrbe a smple example of bddg uder odal prcg ad the eulbrum outcome of ths desg. We cosder a two-ode etwor wth oe costraed trasmsso-le -betwee. I both odes producers are ftesmally small ad demad s 8 Ruderer ad Zöttl (2012) show that zoal prcg addto leads to effcet vestmets trasmsso-les, at least f the zoal maret s regulated such that redspatches are compesated accordg to producers true costs. 6

7 perfectly elastc. For smplcty, we mae the followg assumptoss for each ode: the margal cost s eual to local output ad the producto capacty s 15 MW. I ode 1, demad s 5 MW; ode 2 demad s 18 MW. The trasmsso le betwee thesee odes s costraed ad ca carry oly 4 MW. Demad ode 2 exceeds ts geerato possbltes so the mssg electrcty must be mported from the other ode. Fgure 1. Eulbrum for odal prcg. ODE 1 ODE 2 prce 14 D 1 D 1 +export prce p 2 =15 =14 D 2 -mport D 2 p 1 = 9 C =o1 1 () 5 C 2 =o2 () =9 0 2 = MW MW Wth odal prcg, the eulbrum offers wll be as show Fg. 1. I the frst ode ftesmally small producers mae odal offers o( () at ther margal cost. I orderr to satsfy local demad ad export, 9 MW are gog to be dspatched. Out of these, 5 MW wll be cosumed locally ad 4 MW wll be exported; the hghest possble export level that the trasmsso le allows for. The margal cost ad odal prce s eual to 9, whch correspods to the total producto of ths ode. I the secod ode, the odal prce s 14 as there are 14 MW that have to be produced the secod ode order to satsfy demad ad the trasmsso costrat. Producto above those margal costs (9 ode 1 ad 14 ode 2) wll ot be 7

8 dspatched. All accepted producto wll be pad the odal prce of the ode. The dspatch leads to a socally effcet outcome. We use the superscrpt to desgate ths outcome. We call odal producto ad odal prces of compettve ad socally effcet outcomes, the etwor s effcet dspatch ad the etwor s compettve odal prces, respectvely. As our aalyss wll show, the offers Fgure 1 caot costtute E the other two desgs. For dscrmatory prcg t wll be proftable for framargal offers to crease ther prce up to the margal offer of the ode. For zoal prcg, the average demad the two zoes would be 11.5 MW, so 11.5 MW would be accepted each ode at the zoal prce 11.5 for the offers Fgure 1. Producto would be adjusted the redspatch stage. However, as t apples dscrmatory prcg, t would ot fluece the payoff of producers that bd ther true margal cost. Thus producers the export-costraed ode 1 would fd t proftable to chage ther offers dowwards. They would le to sell as much as possble at the zoal prce ad the buy t bac at a lower prce the redspatch stage. Producers the mport costraed ode 2 would shft ther offers upwards as the pay-as-bd desg, so that all producto that s dspatched the redspatch stage s accepted at the margal offer of the mport costraed ode. 3. MODEL The model descrbed ths secto s used to evaluate ad compare three maret oreted cogesto maagemet techues: odal prcg, pay-as-bd ad zoal prcg wth coutertradg. We study a geeral electrcty etwor (possbly meshed) wth odes that are coected by capacty costraed trasmsso les. Demad a ode 1,, s gve by D, whch s certa ad elastc up to a reservato prce p. C ( ) s the margal cost of producg uts of electrcty ode. We assume that the margal cost s commo owledge, cotuous ad strctly creasg up to (ad beyod) the reservato prce. 9 We let >0 be the relevat total producto capacty ode, whch has a margal cost at the reservato prce or lower. Thus we have by costructo that p ' reservato prce wll ot submt ay offers. C. Capacty wth a margal cost above the 9 ote that t s possble for a producer to geerate beyod the rated power of a producto ut. However, t heats up the ut ad shortes ts lfespa. Thus the margal cost creases cotuously beyod the rated power towards a very hgh umber (above the reservato prce) where the ut s certa to be permaetly destroyed durg the delvery perod. Ed (2007) uses a smlar margal cost curve wth a smlar motvato. 8

9 I each ode there s a cotuum of ftesmally small producers. Each producer the cotuum of ode s dexed by the varable g 0,1. For smplcty, we assume that each producer s oly actve oe ode. Wthout loss of geeralty, we also assume that producers are sorted wth respect to ther margal cost each ode, such that a producer wth a hgher g value tha aother producer the same ode also has a hgher margal cost. The relevat total producto capacty a ode s dvded betwee the cotuum of producers, such that frm g ode has the margal cost C' g. Smlarly, we let g ô represet the offer prce of frm g ode. The system operator s clearg of the real-tme maret must be such that local et-supply euals local et-exports each ode ad such that the physcal costrats of the trasmsso etwor are ot volated. Ay set 1 of odal producto that satsfes these feasblty costrats s referred to as a feasble dspatch. We say that a dspatch s locally effcet f t,.e. producto mmzes the local producto cost each ode for gve odal outputs 1 uts ode are rug f ad oly f they have a margal cost at or below C'. We cosder a set of demad outcomes D 1, such that there s at least oe feasble dspatch. I prcple the etwor could be a o-lear AC system wth resstve losses. But to esure a uue cost effcet dspatch we restrct the aalyss to cases where the feasble set of dspatches s covex. Hece, f two dspatches are possble, the ay weghted combato of the two dspatches s also feasble. The feasble set of dspatches s for example covex uder the DC load flow approxmato of geeral etwors wth alteratg curret (Chao ad Pec, 1996). 10 The system operator sorts offers ascedg order case a odal offer curve ô would be locally decreasg. We deote the sorted odal offer curve by o. The system operator the chooses a feasble dspatch order to mmze the stated producto cost or euvaletly to maxmze 10 Alteratg currets (AC) result a o-lear model of the etwor. Hece, ecoomc studes ths model s ofte smplfed by a lear approxmato called the drect curret (DC) load flow approxmato. I addto to Chao ad Pec (1996), t s used, for example, by Schweppe et al. (1988), Hoga (1992), Bjørdal ad Jörste (2001, 2005, 2007), Glachat ad Pgo (2005), Gree (2007) ad Adler et al. (2008). 9

10 W 1 y o dy, (1) 0 Stated cost whch maxmzes socal welfare f offers would reflect the true costs. Thus, we say that the system operator acts order to maxmze the stated socal welfare subject to the feasblty costrats. I a maret wth odal prcg the system operator frst chooses the optmal dspatch as explaed above. All accepted offers the same ode are pad the same odal prce. The odal prce s determed by the ode s margal prce,.e. the hghest accepted offer prce the ode. We say that margal prces or odal prces are locally compettve f the dspatch s locally effcet ad the margal prce each ode euals the hghest margal cost for uts that are rug the ode. A offer at the margal prce of ts ode s referred to as a margal offer. I the dscrmatory prcg desg all accepted offers are pad accordg to ther offer prce. Ths gves producers cetves to chage ther offers ad thereby state ther costs dfferetly. Stll, the dspatch s determed the same way; by mmzg stated producto cost. I the zoal prcg desg wth couter-tradg, the maret s cleared two stages. Frst the system operator clears the maret dsregardg the tra-zoal trasmsso costrats (costrats sde zoes). ext, case tra-zoal trasmsso les are overloaded after the frst clearg, there s a redspatch where the system operator creases accepted producto mport costraed odes ad reduces t export costraed odes. Secto 4.3 explas our zoal prcg model greater detal. 4. AALYSIS We start our game-theoretcal aalyss of the three maret desgs by meas of three techcal results that we wll use the proofs that follow. Lemma 1. Assume that offers are shfted upwards (more expesve) some odes ad shfted dowwards (cheaper) others, the the dspatched producto s wealy lower at least oe ode wth more expesve offers or wealy hgher at least oe ode wth cheaper supply. Oe mmedate mplcato of ths lemma s that: 10

11 Corollary 1 (o-creasg resdual demad) If oe producer ulaterally creases/decreases ts offer prce, the accepted sales ts ode caot crease/decrease. The system operator accepts offers order to mmze stated producto costs. Thus for a gve acceptace volume a ode, a frm caot crease ts chaces of beg dspatched by creasg ts offer prce. Thus Corollary 1 mples that a producer s resdual demad s ocreasg. The ext lemma outles ecessary propertes of a ash eulbrum. Lemma 2. Cosder a maret where a accepted offer s ever pad more tha the margal prce of ts ode ad ever less tha ts ow bd prce. I ash eulbrum, the dspatch must be locally effcet ad margal prces of the odes are locally compettve. 4.1 odal prcg Below we prove that the odal prcg desg has at least oe E ad that all E results the same compettve outcome. 11 It s oly offers above ad below the margal prces of odes that ca dffer betwee eulbra. Proposto 1 A maret wth odal prcg has oe E where producers offer at ther margal cost. All E result the same locally effcet dspatch ad the same compettve odal prces p C. 1 As the system operator clears the maret order to maxmze socal welfare whe offers reveal true costs, we ote that the eulbrum dspatch must be effcet. We use the superscrpt to desgate ths socally effcet outcome. We refer to the uue eulbrum outcome as the 11 Exstece of pure-strategy E etwors wth a fte umber of producers s less straghtforward. The reaso s that a producer a mportg ode ca fd t proftable to devate from a locally optmal proft maxmum by wthholdg producto order to cogest mports ad push up the odal prce (Boreste et al., 2000; Wllems, 2002; Dowward et al., 2010; Holmberg ad Phlpott, 2012). Such ulateral devatos are ot feasble a etwor wth ftesmally small producers, whch maes exstece of pure-strategy E more straghtforward. Escobar ad Jofré (2008) show that etwors wth a fte umber of producers ad o-exstg pure-strategy E ormally have mxed-strategy E. Exstece of E large games wth cotuous payoffs has bee aalyzed by Carmoa et al. (2009). 11

12 etwor s effcet dspatch 1 ad the etwor s compettve odal prces p 1. ote that as the dspatch s locally effcet, the uue eulbrum outcome exactly specfes whch uts are rug; producto uts ode are rug f ad oly f they have a margal cost at or below C'. Schweppe et al. (1988), Chao ad Pec (1996) ad Hsu (1997) ad others outle methods that ca be used to calculate effcet dspatches for geeral etwors. 1 Exstece of the compettve outcome also drectly establshes exstece of a Walrasa eulbrum, whch has prevously bee prove for radal (Cho, 2003) ad meshed etwors (Escobar ad Jofré, 2008). Proposto 1 proves that all of our E correspod to the Walrasa eulbrum, so ths sese our E s euvalet to the Walrasa eulbrum a maret wth odal prcg. The reaso s that the ftesmal producers that we cosder are prce taers odal marets, where all agets the same ode are pad the same maret prce. 4.2 Dscrmatory prcg Dscrmatory prcg s dfferet to odal prcg that each aget s the pad ts dvdual offer prce rather tha a uform odal prce. Thus, eve f agets are ftesmally small, framargal producers ca stll fluece how much they are pad, so they are o loger prce taers. Ths meas that the Walrasa eulbrum s ot a useful eulbrum cocept whe studyg dscrmatory prcg. Ths s the reaso why we stead cosder a large game wth a cotuum of small producers ths paper. Proposto 2. There exst ash eulbra a etwor wth dscrmatory prcg. All such E have the followg propertes: 1) The dspatched producto s detcal to the etwor s effcet dspatch each ode. 2) All producto ode wth a margal cost at or below etwor s compettve odal prce p C. C s offered at the 3) Other offers are ot accepted ad are ot uuely determed eulbrum. However, t ca, for example, be assumed that they offer at ther margal cost. 12

13 Thus, the dscrmatory aucto s detcal to odal prcg terms of payoffs, effcecy, socal welfare ad the dspatch. As payoffs are detcal for all crcumstaces, ths also mples that the log-ru effects are the same terms of vestmet cetves etc. ote that t s ot ecessary that rejected offers bd at margal cost to esure a eulbrum. As producers are ftesmally small, t s eough to have a small fte amout of rejected bds at or just above the margal offer each ode to avod devatos. Fally we aalyze how cotracts fluece the eulbrum outcome. We cosder forward cotracts wth physcal delvery a specfc ode at a predetermed prce. For smplcty, we cosder cases where each ftesmally small producer ether has o forward sales at all or sells all of ts capacty the forward maret for physcal delvery ts ow ode to cosumers. I the real-tme maret, cosumers aouce how much more power they wat to buy each ode, addto to what they have already bought wth cotracts, ad producers mae offers for chages relatve to ther cotractual oblgatos. The system operator accepts chages producto order to acheve a feasble dspatch at the lowest possble et-crease the stated producto costs. Proposto 3. I a real-tme maret wth odal or dscrmatory prcg, the eulbrum dspatch s detcal to the etwor s effcet dspatch ad margal prces of the odes are compettve, for ay set of forward cotracts that producers have sold wth physcal delvery ther ow ode. We wll use ths result our aalyss of the zoal prcg desg, where the frst-stage clearg of the zoal maret ca be regarded as physcal forward sales. 4.3 Zoal prcg wth couter-tradg otato ad assumptos Zoal prcg wth couter-tradg s more complcated tha the other two desgs ad we eed to troduce some addtoal otato before we start to aalyze t. The etwor s dvded to zoes, such that each ode belogs to some zoe. We let Z be a set wth all odes belogg to zoe. To smplfy our euatos, we umber the odes a specal order. We start wth all odes zoe 1, ad the proceed wth all odes zoe 2 etc. Thus, for each zoe, 13

14 odes are gve umbers some rage to. Moreover, sde each zoe, odes are sorted wth respect to the etwor s compettve odal prces p, whch ca be calculated for the odal prcg desg, as dscussed Secto 4.1. Thus, the cheapest ode zoe s assged the umber ad the most expesve ode zoe s assged the umber. Couter-tradg the secod-stage oly chages tra-zoal flows. Thus t s mportat for a beevolet system operator to esure that the ter-zoal flows are as effcet as possble already after the frst clearg. I the ordc mult-zoal maret, system operators acheve ths by aoucg a arrow rage of ter-zoal flows before the day-ahead maret opes. I partcular, flows the wrog drecto, from zoes wth hgh prces to zoes wth low prces, due to loop flows, are predetermed by the system operator. We smplfy the zoal clearg further by lettg the well-formed system operator set all ter-zoal flows before offers are submtted. Total etmports to zoe are deoted by I aalyss shows that t leads to a effcet outcome:. We mae the followg assumpto for these flows, as our Assumpto 1: The system operator sets ter-zoal flows eual to the ter-zoal flows that would occur for the etwor s effcet dspatch. These ter-zoal flows are aouced by the system operator before offers are submtted. 1 Assumpto 1 sets all ter-zoal flows. Thus offers to each zoal maret ca be cleared separately at a prce where zoal et-supply euals zoal et-exports. We assume that the hghest potetal clearg prce s chose case there are multple prces where zoal et-supply euals zoal et-exports. 12 The clearg prce Π zoe s pad to all producto the zoe that s accepted the zoal clearg. I case tra-zoal trasmsso-les are overloaded after the frst clearg, there s a redspatch where the system operator creases accepted producto mport costraed odes ad reduces t export costraed odes. We cosder a maret oreted redspatch (couter-tradg), so all devatos from the frst-clearg are settled o a pay-as-bd bass. I the couter-tradg stage, the system operator maes chages relatve to the zoal 12 ormally ths choce does ot matter for our eulbra. However, t esures exstece of eulbra for degeerate cases whe exogeous zoal demad ad exogeous et-exports happe to cocde wth producto capactes oe or several odes for some zoe. 14

15 clearg order to acheve a feasble dspatch at the lowest possble et-crease stated producto costs. We cosder two versos of the zoal desg: a oe shot game where the same offers are used the two clearg stages of the maret ad a dyamc game where agets are allowed to mae ew offers the couter-tradg stage. The frst model correspods to the old pool Eglad ad Wales, whle the latter model could for example be represetatve of the reformed Brtsh maret, where producers ca frst sell power at a uform zoal prce the day-ahead maret ad the submt a ew bd to the real-tme maret wth dscrmatory prcg Aalyss The eulbrum a zoal maret wth couter-tradg has some smlartes wth the dscrmatory aucto. But the zoal case s more complcated, as the two clearg stages mply that eulbrum some producers ca arbtrage betwee ther zoal ad dvdual (dscrmatory) couter-tradg prces. Thus producers odes wth low margal prces wll play the c-dec game,.e. sell all ther capacty at the hgher zoal prce ad the buy bac the capacty at a lower prce the couter-tradg stage or produce f the margal cost s eve lower. We cosder physcal marets. Ths prevets producers from buyg power or sellg more tha ther producto capacty the zoal maret. Thus a producer a ode wth a margal prce above ts zoal prce caot mae a arbtrage proft. To maxmze ther proft the redspatch stage, bds of dspatched producto such mport costraed odes are shfted upwards to the ode s compettve odal prce, smlar to the case wth dscrmatory prcg. Frst we cosder a statc game where producers caot mae ew offers to the coutertradg stage; the same offers are used the two stages of the zoal maret. 13 The dyamc model could also represet cogesto maagemet the ordc maret, where the system operator does ot accept offers the zoal clearg of the real-tme maret f these offers wll cause tra-zoal cogesto that eeds to be coutertraded the secod-stage. Ths s to avod uecessary costs for the system operator ad uecessary paymets to producers. I our model where there s o ucertaty, the zoal day-ahead maret the taes the role of the frst-stage of the real-tme maret. The zoal real-tme maret becomes obsolete as wthout ucertaty, the day-ahead maret has already cleared the zoes. I ths case offers to the real-tme maret, whch are allowed to dffer from day-ahead offers, are oly used the dscrmatory couter-tradg stage. Proposto 5 shows that uder our dealzed assumptos swtchg to the ordc verso of zoal cogesto maagemet s va, producers stll get the same payoffs ad the system operator s couter-tradg costs are uchaged. 15

16 Proposto 4. Uder Assumpto 1 there exsts ash eulbra a zoal maret wth coutertradg ad the same offers the zoal ad coutertradg stages. All of them have the followg propertes: * 1) The zoal prce zoe s gve by Π pm, where:,, m : I 1 f D I D I f D I 2) As the odal prcg ad pay-as-bd desgs, the dspatched producto each ode s gve by the etwor s effcet dspatch,. 3) I strctly export-costraed odes Z, such that costs at or above p < (2) * Π, producto wth margal p are offered at the etwor s compettve odal prce p C strctly mport-costraed odes zoe where at or below C s offered at p C.. For p > Π *, all producto wth a margal cost 4) Other offers are ot uuely determed eulbrum. However, t ca be assumed that they offer at ther margal cost. Euato (2) defes a margal ode, where the compettve odal prce euals the zoal prce. ext we show that the eulbrum outcome does ot chage the dyamc game, where agets are allowed to up-date ther offers the couter-tradg stage. Proposto 5. Uder Assumpto 1, t does ot matter for payoffs or the eulbrum outcome of the zoal maret whether producers are allowed to up-date ther offers the couter-tradg stage. We ca ow coclude that the dspatch for zoal prcg wth couter-tradg s the same as for odal prcg ad dscrmatory prcg. Thus, the short ru, the desgs effceces are euvalet. Ths also cofrms that the system operator should set ter-zoal flows eual to the correspodg flows the compettve odal maret, as assumed Assumpto 1, f t wats to maxmze socal welfare. However, t drectly follows from Euato (2) ad Propostos 4 ad 5 16

17 that producers strctly export-costraed odes receve uecessarly hgh paymets a zoal prcg desg: Corollary 2. I comparso to odal prcg, the total extra payoff from the system operator to producers zoe euals: m 1 p m p uder Assumpto 1. Eve f zoal prcg s as effcet as odal prcg the short ru, the extra payoffs wll cause welfare losses the log ru. Producto vestmets wll be too hgh strctly exportcostraed odes where p < Π. I addto, flexble producto that caot tae part the real-tme maret are pad the zoal prce the day-ahead maret. Thus, the accepted flexble supply ths maret s gog to be too hgh strctly export-costraed odes ad too low strctly mport-costraed odes. 5. EXAMPLE DISCRIMIATORY AD ZOAL PRICIG I the followg secto, we llustrate the eulbra for the dscrmatory ad zoal prcg desgs. The example that we use has a detcal structure as the odal prcg case that we descrbed secto 2. Aga, we cosder a two-ode etwor wth oe costraed trasmsso-le -betwee. I both odes producers are ftesmally small ad demad s perfectly elastc. I each ode the margal cost s eual to local output ad the producto capacty s 15 MW. I ode 1, demad s 5 MW; ode 2 demad s 18 MW. The trasmsso le betwee these odes s costraed ad ca carry oly 4 MW. Demad ode 2 exceeds ts geerato possbltes so the mssg electrcty must be mported from the other ode. The dscrmatory desg wll result the eulbrum offers preseted Fg. 2. I ths desg, geerators are pad accordg to ther bd. Kowg ths ad havg perfect formato, producers who wat to be dspatched wll bd the compettve odal prce of ther ode, to esure that they wll be dspatched at the hghest possble prce. Thus, ode 1, they wll bd 9 ad ode 2 they wll bd 14. Producers who do ot wat to be dspatched may, for example, bd ther margal costs, whch are hgher tha the odal prces of the respectve odes. The dspatch wll be the same as uder odal prcg desg. Although producers wll have dfferet bddg 17

18 strateges both desgs, the overall result wll be the same. Accepted producto wlll be pad 9 ode 1 ad 14 ode 2. Fgure 2: Eulbrum for dscrmatory prcg. ODE 1 ODE 2 prce prce 14 9 o 1 () D 1 +export =15 14 o 2 () D 2 -mport 5 C 1 C MW MW I the zoal desg wth couter-tradg, producers wll offer as follows: 18

19 Fgure 3: Zoal offer eulbrum for zoal prcg wth couter-tradg. prce 14 o 2 () D=D 1 +D 2 9 o 1 () MW Fgure 4: odal offers eulbrum for zoal prcg wth couter-tradg. ODE 1 ODE 2 prce 14 9 D 1 +export C 1 prce 14 o 2 () D 2 2 -mport C 2 o 1 () MW MW 19

20 ode 1: Due to trasmsso costrats, producers ode 1 ow that after the two stages, the system operator ca accept a maxmum of 9 MW ther ode. Therefore, producers wth a margal cost at or below the compettve odal prce, offer at or below the compettve odal prce as they wll, ay case, be accepted ad pad the zoal prce, whch s 14. The remag 6 uts ode 1 have a margal cost above the compettve odal prce. They wll bd low order to be accepted the frst stage ad be pad the zoal prce of 14. But due to the trasmsso costrat, they wll have to buy bac ther supply at ther ow bddg prce the secod roud. As they are terested maxmzg ther proft, they wat ths prce dfferece to be as large as possble, as log as they wll ot be chose to produce. Therefore, they bd the compettve odal prce 9 so that they wll be pad ot to produce ad get 14 9 =5 (the rectagle area the fgure 4). There are o proftable devatos from these bds for producers from ode 1. I partcular, we ote that o ftesmally small producer ode 1 ca ulaterally crease the zoal prce at stage 1 above 14, as there are 6 uts ( ode 2) that offer ther producto at the prce 14 wthout beg accepted the zoal maret. ode 2: Due to the trasmsso costrat, producers ode 2 ow that the system operator eeds to dspatch at least 14 uts of electrcty ther ode after the two stages. Thus, all lowcost geerators who wat to be dspatched ow that all offers at or below 14, the compettve odal prce of ode 2, wll be accepted. 8 uts are accepted the zoal clearg ad aother 6 uts are accepted the couter-trade stage. The latter uts are pad as bd ad accordgly, they maxmze ther proft by offerg ther supply at 14, the hghest possble prce for whch they are gog to be accepted. Producers that do ot wat to be dspatched at all wll bd above 14, for example ther margal cost. I ths way, 14 uts wll be produced ode 2. There are o proftable devatos from these strateges for producers ode 2. A comparso of these two examples ad the odal prcg example Secto 2 llustrates that although the bddg strateges are dfferet, the dspatch s the same all scearos. However, the last desg zoal prcg wth couter-tradg results addtoal paymets that affect the log-term vestmet cetves. It s terestg to ote that the zoal prce our example s wealy hgher tha the odal prces both odes. Ths s always the outcome two-ode etwors where the producto 20

21 capacty the cheapest ode s ot suffcet to meet the total demad, so that t s the margal cost the most expesve ode that sets the zoal prce. The system operator wll typcally use tarffs to pass ts couter-tradg cost o to the maret partcpats, so t s actually ute plausble that swtchg to odal prcg wll lower the cost for all electrcty cosumers, cludg the oes the hgh cost ode. 6. COCLUSIOS AD DISCUSSIO We cosder a geeral electrcty etwor (possbly meshed), where odes are coected by capacty costraed trasmsso les. I our game-theoretcal model producers are ftesmally small ad demad s certa ad elastc. We fd that the three desgs, odal, zoal wth coutertradg ad dscrmatory prcg, lead to the same socally effcet dspatch. I addto, payoffs are detcal the pay-as-bd ad odal prcg desgs. However, the desg wth zoal prcg ad coutertradg, there are addtoal paymets from the system operator to producers who ca mae moey by playg the famous c-dec game. It does ot matter for our results whether we cosder a statc game where producers bds are the same the zoal ad couter-tradg stages or a dyamc game where producers are allowed to update ther offer curves the couter-tradg stage. Smlar to Dj ad Wllems (2011) two-ode model, our results for the zoal maret mply that producers overvest export-costraed odes. Whle zoal prcg s good for producers, cosumers would ga overall from a swtch from zoal to odal prcg. I two-ode marets, t s ormally the case that all cosumers (also the oes the most expesve ode) would ga from a swtch to odal prcg. I addto to the effceces mpled by our model, zoal prcg also leads to effceces the operato of flexble plats wth log ramp-rates. They are ot allowed to trade the real-tme maret, so they have to sell at the zoal prce the dayahead maret. The coseuece s that too much flexble producto s swtched o export costraed odes, where the compettve odal prce s below the zoal prce, ad too lttle mport costraed odes, where the compettve odal prce s above the zoal prce. Related ssues are aalyzed by Gree (2007). Aother result from our aalyss s that there s a sgfcat umber of frms that mae offers exactly at the margal prces of the odes the zoal ad pay-as-bd desgs, whch s ot ecessarly the case uder odal prcg. Ths supports the commo vew that the zoal desg s 21

22 more lud. Although, the stadard motvato for ths s that the zoal desg has less maret prces ad thus fewer products to trade, ad hece ludty ca be cocetrated o these. Stll t s ow from PJM that t s also possble to have a lud maret wth odal prcg (euhoff ad Boyd, 2011). However creased ludty ca have more drawbacs tha advatages. As llustrated by Aderso et al. (2009), the elastc offers, especally the pay-as-bd desg but also the zoal desg, mea that gettg ts offer slghtly wrog ca have a huge effect o a frm s dspatch. Ths creases the chaces of gettg effcet dspatches whe demad or compettors output s ucerta, whle the effcecy of the odal prcg desg s more robust to these ucertates. Smlarly, Gree (2010) stresses the mportace of havg desgs that ca accommodate ucertates from termttet power. There are other drawbacs wth the zoal desg. We cosder a beevolet system operator that uses couter-tradg to fd the socally optmal dspatch. However, eve f coutertradg s socally effcet, t s costly for the system operator tself. Thus strategc system operators have cetves to fd the feasble dspatch that mmzes couter-tradg costs. I practce, couter-tradg s therefore lely to be mmalstc ad less effcet tha our framewor. Moreover, Bjørdal et al. (2003) ad Glachat ad Pgo (2005) show that etwor operators have cetves to mapulate ter-zoal flows order to lower the couter-tradg cost (ad maret effcecy) further. I our aalyss we assume that the system-operator has full cotrol of the system ad that t ca set ter-zoal flow as effcetly as uder odal prcg, but practce maret ucertaty, coordato problems ad mperfect regulatos lead to sgfcatly less effcet cross-border flows (Leuthold, 2008; euhoff, et al., 2011; Ogo ad Smeers 2012). Studes by Hoga (1999), Harvey ad Hoga (2000), ad Gree (2007) dcate that odal prcg s also better suted to prevet maret power. REFERECES Adler, I., S. Ore, J. Yao (2008). Modelg ad Computg Two-Settlemet Olgopolstc Eulbrum a Cogested Electrcty etwor. Operatos Research 56(1):

23 Alaywa, Z., T. Wu, ad A.D. Papalexopoulos (2004). Trastog the Calfora Maret from a Zoal to a odal Framewor: A Operatoal Perspectve. Power Systems ad Exposto 2: Aderso, E.J., P. Holmberg, ad A.B. Phlpott, (2013). Mxed Strateges Dscrmatory Dvsble good Auctos. Rad Joural of Ecoomcs 44(1): Auma, R.J., (1964). Marets wth a Cotuum of Traders. Ecoometrca 32 (1/2): Berard, J. T., ad C. Guert (2002). odal Prcg ad Trasmsso Losses: A Applcato to a Hydroelectrc Power System. Dscusso Paper 02-34, Resources for the Future, Washgto DC. Bjørdal, M., ad K. Jörste (2001). Zoal Prcg a Deregulated Electrcty Maret. The Eergy Joural 22 (1): Bjørdal, M., K. Jörste, ad V. Pgo (2003). Cogesto maagemet the ordc power maret: couter purchases. Joural of etwor Idustres 4: Bjørdal, M., ad K. Jörste (2005). The Deregulated Electrcty Maret Vewed as a Blevel Programmg Problem. Joural of Global Optmzato 33: Bjørdal, M. ad K. Jörste (2007). Beefts from coordatg cogesto maagemet The ordc power maret. Eergy Polcy 35: Boreste, S., J. Bushell, J. ad S. Stoft (2000). The compettve effects of trasmsso capacty a deregulated electrcty dustry. RAD Joural of Ecoomcs 31(2): Bruereeft, G., K. euhoff, ad D. ewbery (2005). Electrcty trasmsso: A overvew of the curret debate. Utltes Polcy 13: Carmoa, G., ad K. Podczec (2009). O the Exstece of Pure Strategy ash Eulbra Large Games. Joural of Ecoomc Theory 144: Chao, H-P ad S. Pec (1996). A Maret Mechasm For Electrc Power Trasmsso. Joural of Regulatory Ecoomcs 10: Cho, I Koo (2003). Compettve eulbrum a radal etwor. The RAD Joural of Ecoomcs 34(3): de Vres, L.J., J. de Joode, ad R. Havoort (2009). The regulato of electrcty trasmsso etwors ad ts mpact o goverace. Europea Revew of Eergy Marets 3(3): Dj, J., ad B. Wllems (2011). The effect of couter-tradg o competto the Dutch electrcty maret. Eergy Polcy 39(3):

24 Dowward, A., G. Zaer, ad A.B. Phlpott (2010). O Courot Eulbra Electrcty Trasmsso etwors, Operatos Research 58(4): Ehrema, A., ad Y. Smeers (2005). Ieffceces Europea cogesto maagemet proposals. Utltes Polcy 13: Ed, K.A. (2007). Hydro-Thermal Aucto Marets. Mmeo, Tetum, Stocholm. Escobar, J.F., ad A. Jofré (2008). Eulbrum Aalyss of Electrcty Auctos., Departmet of Ecoomcs, Staford Uversty. Evas, J.E., ad R.J. Gree (2005). Why Dd Brtsh Electrcty Prces Fall After 1998? Worg paper 05-13, Departmet of Ecoomcs, Uversty of Brmgham. Glachat, J. M., ad V. Pgo (2005). ordc cogesto s arragemet as a model for Europe? Physcal costrats vs. ecoomc cetves. Utltes Polcy 13: Gravelle, H. ad R. Rees (1992). Mcroecoomcs. Lodo:Logma. Gree, E.J. (1984). Cotuum ad Fte-Player ocooperatve Models of Competto. Ecoometrca 52 (4): Gree, R. (2007). odal prcg of electrcty: how much does t cost to get t wrog? Joural of Regulatory Ecoomcs 31: Gree, R. (2010). Are the Brtsh tradg ad trasmsso arragemets future-proof? Utltes Polcy 18: Harvey, S. M., W.W Hoga (2000). odal ad Zoal Cogesto Maagemet ad the Exercse of Maret Power, Joh F. Keedy School of Govermet, Harvard Uversty. Hoga, W. W. (1999). Trasmsso Cogesto: The odal-zoal Debate Revsted. Joh F. Keedy School of Govermet, Harvard Uversty. Hoga, W. W. (1992). Cotract etwors for Electrc Power Trasmsso. Joural of Regulatory Ecoomcs 4: Holmberg, P., ad D. ewbery (2010). The supply fucto eulbrum ad ts polcy mplcatos for wholesale electrcty auctos. Utltes Polcy 18(4): Holmberg, P., ad A. Phlpott (2012). Supply Fucto Eulbra etwors wth Trasport Costrats., IF Worg Paper 945, Research Isttute of Idustral Ecoomcs, Stocholm. Hsu, M. (1997). A troducto to the prcg of electrc power trasmsso. Utltes Polcy 6(3):

25 Krause, T. (2005). Cogesto Maagemet Lberalzed Electrcty Marets Theoretcal Cocepts ad Iteratoal Applcato, EEH Power Systems Laboratory, Edgeösssche Techsche Hochschule Zürch. Leuthold, F., H. Weght, ad C. vo Hrschhause (2008). Effcet prcg for Europea electrcty etwors The theory of odal prcg appled to feedg- wd Germay. Utltes Polcy 16: euhoff, K., ad R. Boyd (2011). Iteratoal Expereces of odal Prcg Implemetato, Worg documet, Clmate Polcy Itatve, Berl. euhoff, K., R. Boyd, T. Grau, J. Baru, F. Echabarre, J. Bale, C. Det, C. vo Hrschhause, B. F. Hobbs, F. Kuz, H. Wegt, C. abe, G. Papaefthymou ad C. Weber (2011). Reewable Electrc Eergy Itegrato: Quatfyg the Value of Desg of Marets for Iteratoal Trasmsso Capacty. Worg documet, Clmate Polcy Itatve, Berl. euhoff, K., B. F. Hobbs, ad D. ewbery (2011). Cogesto Maagemet Europea Power etwors, Crtera to Asses the Avalable Optos. Dscusso Papers 1161, DIW Berl. Oggo, G. ad Y. Smeers (2012). Degrees of Coordato Maret Couplg ad Couter- Tradg. The Eergy Joural 33 (3): Ruderer, D., ad G. Zöttl (2012). The Impact of Trasmsso Prcg etwor Idustres. EPRG Worg Paper 1214, Uversty of Cambrdge, UK. Sadowsa, M., ad B. Wllems (2012). Power Marets Shaped by Attrust. TILEC Dscusso Paper , Uversty of Tlburg, etherlads. Schweppe, F. C., M.C. Caramas, R.D. Tabors, ad R.E. Boh (1988). Spot Prcg of Electrcty. Lodo: Kluwer academc publshers. Stoft, S. (1997). Trasmsso prcg zoes: smple or complex? The Electrcty Joural 10 (1): Wllems, B. (2002). Modelg Courot Competto a Electrcty Maret wth Trasmsso Costrats. The Eergy Joural 23(3):

26 APPEDIX A: TECHICAL LEMMAS Lemma 3. m() s uuely defed by Euato (2). Proof: We frst ote that the etwor s effcet dspatch s feasble as the ter-zoal flows are effcet,.e. D I. Thus I D. We have m() = f D I. Otherwse we have 1 I D I. Moreover, 1 I s strctly creasg, because 0. Thus Euato (2) always has a uue soluto. The followg two techcal lemmas are used to prove that all ash eulbra must result the same dspatch. Lemma 4. If there s a set of odal offer fuctos * oˆ 1 results a locally effcet dspatch wth the odal output margal prces, the ay set of strctly creasg odal offer fuctos * * * oˆ oˆ 1,,, wll result the same dspatch. (ot ecessarly creasg) that * 1 ad locally compettve oˆ 1, such that Proof: Frst, cosder the case whe offers * oˆ 1 are also strctly creasg output. I ths case, the objectve fucto (stated welfare) s strctly cocave the supply,. Moreover, the set of feasble dspatches s by assumpto covex our model. Thus, t follows that the objectve fucto has a uue local extremum, whch s a global maxmum (Gravelle ad Rees, 1992). Thus the system operator s dspatch ca be uuely determed. It follows from the ecessary Lagrage codto that the uue optmum s ot flueced by chages ode s offers below ad above the uatty *, as log as offers are strctly creasg output. Thus the 26

27 dspatch must be the same for ay set of strctly creasg odal offer fuctos * * * that ˆ oˆ o 1,,. Wth perfectly elastc segmets the offer curves * whch o 0 * o 1 oˆ 1, such there are output levels, for some ode. Ths meas that the objectve fucto s o loger strctly cocave the supply. However, oe ca always costruct strctly creasg curves that are arbtrarly close to curves wth perfectly elastc segmets. Moreover, the system operator s objectve fucto s cotuous offers. Thus, we ca use the same argumet as above wth the dfferece that the system operator may sometmes have multple optmal dspatches, addto * oˆ 1 to the dspatch above, for a gve set of offer curves.14 However, the same dspatch as above s ped dow by the addtoal codtos that the dspatch s locally effcet ad margal prces locally compettve. * oˆ 1 Fally, we realze that there could be cases wth o-mootoc offers. However, the dspatch s locally effcet ad margal prces locally compettve, so such offers would * * * have to satsfy the followg propertes ˆ oˆ 1,, o for * * * ad ˆ oˆ * o for *. Thus as the system operator sorts offers to ascedg order, we ca go through the argumets above for sorted offers ad coclude that the statemet must hold for such cases as well. Lemma 5. If two sets of odal offer fuctos both result a locally effcet dspatch wth locally compettve margal prces, the the two resultg dspatches must be detcal. Proof: Mae the cotradctory assumpto that there are two pars of offer fuctos wth a correspodg dspatch, * propertes, except that 1 * * o ˆ, ad ˆ o, 1 1, that satsfy the stated. Lemma 4 states how these offers ca be adjusted to strctly creasg offer curves wthout chagg the dspatch. We mae such adjustmets to get 14 Multple optmal dspatches for example occur f several uts a ode have the same stated margal cost ad some but ot all of these uts are accepted a dspatch that mmzes stated producto costs. 27

28 two sets of adjusted odal offer fuctos, o 1 * ˆ ad ˆ the same dspatch ad o-crossg the other odes. o 1 * * * By assumpto we have oˆ C ad ô C curve s strctly creasg output. Thus adjusted offers expesve) compared to adjusted offers offers where * oˆ 1 oˆ 1 all odes where that are detcal odes wth. The ode s margal cost * oˆ 1 * must be below (cheaper) compared to adjusted offers must be above (more. Smlarly, adjusted oˆ 1 * *. However, ths would volate Lemma 1. Thus, the dspatches 1 must be detcal. all odes ad 1 APPEDIX B: OTHER PROOFS Proof of Lemma 1 We let the old dspatch refer to the feasble dspatch old 1 at old offers whe supply ode s gve by o. Let o curve, so that o o that maxmzed stated socal welfare deote the shft of the supply s the ew supply curve ode. The ew dspatch refers to the feasble dspatch that maxmzes stated socal welfare for ew offers. Thus for ew offers, o o tha the old dspatch ew 1 ew, the ew dspatch 1 old,.e. 1 ew 1 0 x o xdx o x o x dx. old 1 0 should result a wealy hgher socal welfare o (3) ow, mae the cotradctory assumpto that comparso to the old dspatch, the ew dspatch has strctly more producto all odes where offers have bee shfted upwards (more expesve) ad strctly less producto all odes where offers have bee shfted dowwards ew (cheaper). Thus > old ew whe o 0 wth strct eualty for some 0, old whe 0 wth strct eualty for some 0, so that old o,, ad < ew 28

29 ew 1 0 o x dx old 1 0 o x But summg Euato (3) ad Euato (4) yelds ew 1 0 o x dx old 1 0 o x whch s a cotradcto sce, by defto, the old dspatch stated welfare at old offers. dx. (4) dx, (5) old 1 s supposed to maxmze Proof of Lemma 2 The statemet follows from that: 1) offers caot be dspatched at a prce below ther margal cost eulbrum, ad that 2) all offers from producto uts wth a margal cost at or below the margal prce of a ode must be accepted eulbrum. If 1) dd ot hold for some frm the t would be a proftable devato for the frm to crease ts offer prce to ts margal cost. 2) follows from that there would otherwse exst some ftesmally small producer the ode wth a margal cost below the margal prce, but whose offer s ot dspatched. Thus, t would be a proftable devato for such a producer to slghtly udercut the margal prce ad we ow from Corollary 1 that such a devato wll ot decrease the dspatched producto ts ode, so the revsed offer wll be accepted. Proof of Proposto 1 We ote that the objectve fucto (stated welfare) Euato (1) s cotuous the odal output whe offers are at the margal cost. Moreover, the feasble set (the set of possble dspatches) s closed, bouded (because of capacty costrats) ad o-empty. Thus, t follows from Weerstrass theorem that there always exsts a optmal feasble dspatch whe offers reflect true costs (Gravelle ad Rees, 1992). ext, we ote that o producer has a proftable devato from the compettve outcome. Margal costs are cotuous ad strctly creasg. Hece, t follows from Corollary 1 that o producer wth a accepted offer ca crease ts offer prce above the margal prce of the ode ad stll be accepted, as ts offer prce would the be above oe of the prevously rejected offers 29

30 the same ode. 15 o producer wth a rejected offer would ga by udercuttg the margal prce, as the chaged offer would the be accepted at a prce below ts margal cost. Thus, there must exst a E where all frms offer to produce at ther margal cost. Offers above ad below the margal prce of a ode ca dffer betwee eulbra. But t follows from Lemma 2 ad Lemma 5 Appedx A that all E must have the same locally effcet dspatch ad the same locally compettve margal prces, so odal prces, whch are set by margal prces, must also be the same. Proof of Proposto 2 Proposto 1 esures exstece of the etwor s effcet dspatch ad compettve odal prces. Both odal ad dscrmatory prcg are marets where a accepted offer s ever pad more tha ts ode s margal prce ad ever less tha ts ow bd prce, so both cases the eulbrum dspatch must be locally effcet ad margal prces of the odes are compettve eulbrum, because of Lemma 2. Thus statemet 1) follows from Lemma 5 Appedx A. I a dscrmatory maret t s proftable for a producer to crease the prce of a accepted offer utl t reaches the margal prce of ts ode, whch gves statemet 2). Fally, we realse that there are o proftable devatos from the stated eulbrum f rejected offers are at ther margal cost. Proof of Proposto 3 We ote that the stated producto cost of cotracted sales s a costat. Thus we ca add t to the objectve fucto of the system operator s optmzato problem wthout fluecg the optmal dspatch. The set of feasble dspatches s ot flueced by producers forward sales. Thus to solve for the optmal dspatch we ca add producers forward sales to ther offered uattes, so that offers clude cotracted uattes stead of beg et of cotracts, ad the solve for the feasble dspatch that mmzes the total stated producto costs as defed by Euato (1). Rewrtg the dspatch problem ths way, mples that Lemma 1, Corollary 1, Lemma 4 ad Lemma 5 Appedx A also apply to stuatos wth cotracts. Thus the stated result would follow f we ca prove that the dspatch must be locally effcet ad margal prces of the odes 15 Also ote that the last ut a ode caot crease ts offer above ts margal cost due to the reservato prce p C'. 30

31 are compettve eulbrum, also for cotracts. Smlar to the proof of Lemma 2, ths follows from that: 1) a producto ut caot be dspatched at a real-tme prce below ts margal cost eulbrum, ad that 2) all producto uts wth a margal cost at or below the margal realtme prce of ts ode must be dspatched eulbrum. The proof of Lemma 2 explas why 1) must hold for ucotracted frms. If 1) would ot hold for a cotracted frm, the t would be a proftable devato for the frm to crease ts offer prce (to buy bac the cotract ad avod beg dspatched) to a prce above the margal real-tme prce ad below ts margal cost. It follows from Corollary 1 that such a ulateral devato caot crease the odal producto the cotracted frm s ode. Thus ts offer to buy bac the cotract s accepted at a prce below ts margal cost, whch s cheaper tha to follow the cotracted oblgato ad produce at margal cost. 2) follows from that there would otherwse exst some ftesmally small producer the ode wth a margal cost below the margal prce, but whose offer s ot dspatched. We already ow from the proof of Lemma 2 that such a producer would fd a proftable devato f t was ucotracted. We also realze that a producer that has sold ts producto forward ad that has a margal cost below the margal prce would lose from bddg above the margal prce (to buy bac the cotract), so that ts ut s ot dspatched. It would be a proftable devato for such a producer to lower ts bd to ts margal cost. It follows from Corollary 1 that such a chage would ot decrease accepted producto. Thus t creases ts payoff by at least the dfferece betwee ts odal margal prce ad ts margal cost. Proof of Proposto 4 Exstece of a compettve eulbrum the odal desg follows from Proposto 1. Assumpto 1 restrcts ter-zoal flows to be effcet. However, we realze from the proof of Proposto 3 that ths extra costrat does ot chage the statemet Proposto 3. A producer s accepted offer the zoal maret s euvalet to a forward posto wth physcal delvery ts ode. Thus t follows from Proposto 3 that, depedet of the zoal clearg, the eulbrum dspatch s detcal to the etwor s effcet dspatch ad margal prces of the odes are compettve the couter-tradg stage. Ths gves the uue dspatch 1 as stated 2). The couter-tradg stage uses dscrmatory prcg, but all agets wat to trade at 31

32 the best prce possble, so all accepted offers the couter-tradg stage are margal offers at the etwor s compettve odal prces. Cosder a zoe wth ts assocated odes sde zoe where the etwor s compettve odal prce or euvaletly,, Z. A ode p s strctly below the zoal prce Π s referred to as a strctly export costraed ode. Prce-tag producers such odes wat to sell as much producto as they ca at the zoal prce, ad the buy bac producto the dscrmatory couter-tradg stage at the lower prce p or produce at a eve lower margal cost. Thus all capacty a strctly export costraed ode s offered at or below p < Π. As the real-tme maret s physcal, producers strctly mport-costraed odes of zoe (where the etwor s compettve odal prce p s strctly above the zoal prce Π ) are ot allowed to frst buy power at a low prce the zoal maret ad the sell power at p the coutertradg stage. Thus they ether buy or sell ay power the zoal maret, so they mae offers above Π. We ca coclude from the above reasog that a margal offer at the zoal prce caot come from a producto ut that s located a ode that s strctly export or mport costraed. I eulbrum there must be at least oe margal ode m wth p Π. Recall that odes have bee sorted wth respect to compettve odal prces ad that that the hghest clearg prce s chose case there are multple prces where zoal et-supply euals zoal et-exports. Thus we ca defe oe margal ode by Euato (2). 16 It follows from Lemma 3 that ths euato uuely sets the zoal prce Π. p m Offers strctly mport costraed odes, whch are above the zoal prce, are ever accepted the frst stage of the zoal maret. For these odes, t s the rules of the coutertradg stage that determe optmal offer strateges. Thus, the aucto wors as a dscrmatory aucto, ad we ca use the same argumets as Proposto 2 ad Proposto 3 to prove the secod part of statemet 3). m 16 It s possble that odes wth umbers adjacet to m() have the same compettve odal prces as ode m(), but t wll ot chage the aalyss. It s eough to fd oe margal ode to determe the zoal prce. As a example, t follows from Proposto 1 ad our cost assumptos that the specal case whe zoal demad euals the zoal producto capacty plus effcet mports, the the compettve odal prce euals the prce cap all odes. Thus ay ode could be chose to be the margal ode, but s the most atural exteso of the frst part of Euato (2). 32

33 Producto uts a strctly export-costraed ode that have a hgher margal cost tha ther compettve odal prce ca sell ther power the zoal maret at the zoal prce ad the buy t bac at a lower offer prce the couter-trade stage. Thus, to maxmze profts ths power s offered at the lowest possble prce, for whch offers are ot dspatched,.e. at the margal prce of the ode. Ths gves the frst part of statemet 3). o-dspatched producto uts would ot ga by udercuttg the margal prce. Offers that are dspatched strctly exportcostraed odes are pad the zoal prce. It s ot possble for oe of these uts to crease ts offer prce above p < Π ad stll be dspatched, as o-dspatched uts such odes offer at p. Moreover, t s wealy cheaper for dspatched uts to produce stead of buyg bac power at p. Thus, they do ot have ay proftable devatos. Accordgly, the stated offers must costtute a ash eulbrum. Proof of Proposto 5 We solve the two-stage game by bacward ducto. Thus we start by aalysg the coutertradg stage. A producer s accepted offer the zoal maret s euvalet to a forward posto wth physcal delvery ts ode. Thus t follows from Proposto 3 that, depedet of the zoal clearg, the eulbrum dspatch s detcal to the etwor s effcet dspatch ad margal prces of the odes are compettve the couter-tradg stage. The couter-tradg stage uses dscrmatory prcg, but all agets wat to trade at the best prce possble, so all accepted offers the couter-tradg stage are margal offers at the etwor s compettve odal prces. We calculate a subgame perfect ash eulbrum of the game, so ratoal agets realse what the outcome of the secod-stage s gog to be, ad mae offers to the zoal maret order to maxmze profts. Thus, smlar to the oe-stage game, all producto capacty strctly export-costraed odes Z, such that p < Π, s sold at the zoal prce. As before, producto capacty strctly mport costraed odes maxmze ther payoff by sellg o power the zoal maret; all producto that s dspatched strctly mport costraed odes s accepted the couter-tradg stage. As the oe-stage game, the zoal prce zoe must be set by the margal prce of some margal ode m as defed Euato (2). Otherwse there 33

34 must be some offer to the zoal maret from a producto ut a strctly export costraed ode (wth p < Π ) that s rejected, ad whch would fd t proftable to slghtly udercut the zoal prce. All producto uts that are dspatched margal odes are sold at the zoal prce. As the oe-stage game, there are always rejected offers from uts margal odes that ca be placed at or just above the zoal prce. Ths rules out that proftable devatos for producto uts margal odes. Thus all agets get the same payoffs as the game Proposto 4, where the same offers were used the zoal ad coutertradg stages. 34