Congestion management in electricity networks: Nodal, zonal and discriminatory pricing

Transcription

1 Congeston management n electrcty networks: odal, zonal and dscrmnatory prcng Pär Holmberg and Ewa Lazarczyk Aprl 2012 CWPE 1219 & EPRG 1209

2 Congeston management n electrcty networks: odal, zonal and dscrmnatory prcng EPRG Workng Paper 1209 CWPE Workng Paper 1219 Pär Holmberg and Ewa Lazarczyk EPRG WORKIG PAPER Abstract Keywords JEL Classfcaton Wholesale electrcty markets use dfferent market desgns to handle congeston n the transmsson network. We compare nodal, zonal and dscrmnatory prcng n general networks wth transmsson constrants and loop flows. We conclude that n large games wth many producers who are allowed to partcpate n the real-tme market the three market desgns result n the same effcent dspatch. However, zonal prcng wth counter-tradng results n addtonal payments to producers n exportconstraned nodes. Congeston management, wholesale electrcty market, transmsson network, nodal prcng, zonal prcng wth countertradng, dscrmnatory prcng, large game C72, D44, D61, L13, L94 Contact par.holmberg@fn.se Publcaton Aprl 2012 Fnancal Support Jan Wallander s and Tom Hedelus Research Foundaton and the Research Program The Economcs of Electrcty Markets

3 Congeston management n electrcty networks: odal, zonal and dscrmnatory prcng 1 Pär Holmberg 2 and Ewa Lazarczyk 3 March Abstract Wholesale electrcty markets use dfferent market desgns to handle congeston n the transmsson network. We compare nodal, zonal and dscrmnatory prcng n general networks wth transmsson constrants and loop flows. We conclude that n large games wth many producers who are allowed to partcpate n the real tme market the three market desgns result n the same effcent dspatch. However, zonal prcng wth counter tradng results n addtonal payments to producers n export constraned nodes. Keywords: Congeston management, wholesale electrcty market, transmsson network, nodal prcng, zonal prcng wth countertradng, dscrmnatory prcng, large game JEL codes: C72, D44, D61, L13, L94 1 We are grateful for very helpful comments from Rchard Frberg, Sven Olof Frdolfsson, Jenny Frdström, Thomas Tangerås, an anonymous referee, semnar partcpants at Stockholm School of Economcs and Research Insttute of Industral Economcs (IF), and conference partcpants at IAEE 2011 n Stockholm. We also want to thank Erk Lundn for research assstance and Chrstna Lönnblad for proof readng our paper. The work has been fnancally supported by Jan Wallander s and Tom Hedelus Research Foundaton and the Research Program The Economcs of Electrcty Markets. 2 Research Insttute of Industral Economcs (IF). P.O. Box 55665, SE Stockholm, Sweden, phone , fax E mal: Par.Holmberg@fn.se. Assocate Researcher of Electrcty Polcy Research Group (EPRG), Unversty of Cambrdge. 3 Research Insttute of Industral Economcs (IF). P.O. Box 55665, SE Stockholm, Sweden, fax , Stockholm School of Economcs, Department of Economcs, Sveavägen 65, Stockholm, Sweden. E mal ewa.lazarczyk@hhs.se 1

4 1. Introducton Storage possbltes are neglgble n most electrc power networks, so demand and supply must be nstantly balanced. The consequence s that transmsson constrants and how they are managed often have a large nfluence on market prces. The European Unon s regulaton 1228/2003 (amended n 2006) sets out gudelnes for how congeston should be managed n Europe. System operators should coordnate ther decsons and choose desgns that are secure, effcent, transparent and market based. Regulatory authortes shall regularly evaluate the used congeston management methods, wth respect to complance wth the prncples and rules establshed n the regulaton and gudelnes. In ths paper, we compare the effcency of three market desgns that are n operaton n real tme electrcty markets: nodal, zonal and dscrmnatory prcng. Real tme markets are open for offers from producers just before electrcty s gong to be produced and delvered. Durng the delvery perod, the system operator accepts offers n order to clear the real market, takng transmsson constrants nto account. The aucton desgn nfluences whch offers are accepted and ther payments. odal prcng or locatonal margnal prcng (LMP) s used n Argentna, Chle, Ireland, ew Zealand, Russa, Sngapore, n several US states (e.g. Calforna, ew England, ew York, PJM and Texas), and Poland s on the way to mplement t as well. Ths desgn acknowledges that locaton s an mportant aspect of electrcty whch should be reflected n ts prce, so all accepted offers are pad a local unform prce assocated wth each node of the electrcty network. There s no unform market prce under dscrmnatory prcng, where accepted offers are pad as bd. Stll the system operator consders all transmsson constrants when acceptng offers, so smlar to nodal prcng, dscrmnatory prcng allows producton n mportconstraned nodes to be accepted at a hgher prce than producton n export constraned nodes. Dscrmnatory prcng s used n Brtan and Iran, and Italy has decded to mplement t as well. One (somewhat naïve) motvaton for ths aucton format has been that low cost producton s supposed to bd low and accordngly to be pad a low prce, whch would ncrease consumer s and the auctoneer s welfare at producers expense. 2

5 Real tme markets wth zonal prcng consder nter zonal congeston, but have a unform market prce nsde each regon, typcally a country or a state, regardless of transmsson congeston nsde the regon. Orgnally ths desgn was thought to mnmze the complexty of the prcng settlement and poltcally t s sometmes more acceptable wth one prce n a country/state. Ths s why zonal prcng was adopted by Australa and by most European countres. Orgnally, zonal prcng was also used n most deregulated electrcty markets n the US, but they have now swtched to nodal prcng, at least for generaton. One reason for ths change n US s that zonal prcng s, contrary to ts purpose, actually qute complex and prcng s not very transparent behnd the hood. The problem s that the system operator needs to order redspatches after the zones of the real tme market have been cleared f transmsson lnes nsde a zone would otherwse be overloaded. Such a redspatch ncreases supply n mport constraned nodes and reduces t n export constraned nodes n order to relax nter zonal transmsson congeston. There are alternatve ways of compensatng producers for ther costs assocated wth these adjustments. The compensaton schemes have no drect nfluence on the cleared zonal prces, but ndrectly the detals of the desgn may nfluence how producers make offers nto the real tme market. We consder the market orented redspatch, whch s called countertradng, and where countertraded offers are pad as bd. Ths desgn s used n the real tme market of the ordc countres (ord Pool) and was used n the old Texas desgn. The three market desgns that we consder are summarzed n the table below. Table 1: A summary of the three desgns of real tme markets. Congeston Consdered Aucton format technque transmsson Unform prce Pay as bd constrants odal All x Dscrmnatory All x Zonal stage 1 Inter zonal x Zonal stage 2 Intra zonal x 3

6 We compare the three market desgns by means of a game theoretcal model. Our analyss consders a general electrcty network where nodes are connected by capacty constraned transmsson lnes. Producers costs are assumed to be common knowledge, and demand s certan and nelastc. There s a contnuum of nfntesmally small producers n the market that choose ther offers n order to maxmze ther ndvdual payoffs. Subject to the transmsson constrants, the system operator accepts offers to mnmze total stated producton costs,.e. t clears the market under the assumpton that offers reflect true costs. We consder one shot games and characterze the ash equlbrum of each market desgn. We compare prces, payoffs and effcences for the three desgns. As far as we know there s only one prevous game theoretcal comparson of congeston management technques. But ths study by Wllems and Djk (2011) s lmted to two node networks wth constant margnal costs, and they do not consder the dscrmnatory prcng desgn. 4 In the nodal prcng desgn, producers maxmze ther payoffs by smply bddng ther margnal costs. Thus, n ths case, the accepted offers do maxmze short run socal welfare. We refer to these accepted equlbrum offers as the effcent dspatch and we call the clearng prces the network s compettve nodal prces. We compare ths outcome wth equlbra n the alternatve market desgns. For fxed offers, the system operator would ncrease ts proft at producers expense by swtchng from nodal to dscrmnatory prcng. But we show that dscrmnatory prcng encourages strategc bddng even f there are nfntely many producers n the market and ths wll exactly offset the system operator s ncreased payoff. Hence, n the ash equlbrum of the pay as bd desgn, all accepted offers are at the network s compettve nodal prces. Moreover, accepted producton s the same as n the effcent dspatch. Thus, market effcency and payoffs to producers and the system operator are the same as for nodal prcng. As payoffs are dentcal n all crcumstances, ths also mples that the long run effects are the same n terms of nvestment ncentves. 4 Djk and Wllems (2011) also extend ther model to consder cases wth a fnte number of producers and market power. 4

7 Equlbrum offers are also smlar n the zonal real tme market wth counter tradng, and the dspatch s the same as for the two other market desgns. But the producers payoffs are larger under zonal prcng at the system operator s expense. The reason s that the two stage clearng of the real tme market gves producers the opportunty to ether sell at the unform zonal prce or at the dscrmnatory equlbrum prce n stage 2 (whch equals the compettve nodal prce of the producer s node), whchever s hgher. In addton, even when they are not producng any energy, producton unts n export constraned nodes can make money by sellng at the unform zonal prce and then to buy back the same amount at the dscrmnatory prce, whch s lower, n the countertradng stage. Ths type of strategc behavor has also been observed n practce, and the unnecessarly hgh payoffs for strategc producers at the system operators expense s another reason why US markets have chosen to swtch from zonal to nodal prcng. Our results are the same regardless of whether producers are allowed to make new offers n the counter tradng stage or whether the rules are such that the same offers are used n the two stages. As n Djk and Wllems (2011) two node model, the addtonal payments to producers n the zonal market cause long run neffcences; producers overnvest n export constraned nodes. 5 We attrbute ths defcency to a suboptmal combnaton of the two dfferent aucton formats, unform and pay as bd prcng. If one wants to get the nvestment ncentves rght, t s better to stck to a pure format, such as nodal prcng or a pure pay as bd aucton. In secton 2 we dscuss congeston management and the lterature on ths topc n more detal. Secton 3 presents our model and secton 4 an analyss of dfferent congeston management desgns. In secton 5, market equlbra are dscussed for a smple two node example. The paper s concluded n secton 6 and a bref dscusson of the results s presented n secton 7. 5 In a model wth nfntesmally small frms, ths smply follows from excessve payments to generators n export constraned nodes. Djk and Wllems (2011) use a two stage game wth entry and Cournot competton to prove ths for ther settng wth market power. 5

8 2. Lterature revew 2.1 odal prcng The early lterature on ssues of electrcty prcng n networks concentrates on the calculaton of nodal prces. Schweppe et al. (1988) and Hsu (1997) present a model where the spot prce s derved from a socal welfare maxmzaton problem subject to a number of constrants and where the dfference n spot prces between any two locatons corresponds to a prce of transmsson costs between those two nodes. Electrc power networks normally have alternatng currents (AC), whch results n a nonlnear model of the network. Hence, ths model s often smplfed by a lnear approxmaton called the DC load flow approxmaton. It s, for example, used by Hogan (1992) and Chao and Peck (1996). They present a verson of nodal prcng that ncorporates the technologcal externaltes assocated wth transmsson congeston and transmsson losses, adopt tradable transmsson capacty rghts and ntroduce a tradng rule that specfes the transmsson loss compensaton requred for power transfers. They demonstrate that a compettve equlbrum wth property rghts and ther tradng rule can replcate a socal optmum Equlbrum Analyss of Electrcty etworks The early lterature on locatonal margnal prcng (LMP) or nodal market desgn focuses on the development of computatonal methods to determne the compettve market prces n electrcty networks. Models analyzed n ths strand of lterature are based on detaled systems of equatons stpulatng the mechancs of electrcal power flow and the physcal constrants assocated wth transmsson networks, and not much space s left to analyze whether producers and consumers have ncentves to bd strategcally. However, there are exceptons; Escobar and Jofre (2008) prove the exstence of ash equlbra (E) n electrcty auctons for a network wth one generator per node and a central agent. Wlson (2008) characterzes the supply functon equlbrum (SFE) n a mult unt aucton constraned by lmted 6

9 transport capactes and restrcted nput/output capactes of partcpants. There are also several Cournot E studes of networks, see e.g. Stoft (1998), Borensten et al. (2000), euhoff et al. (2005) and Downward et al. (2010). Cho (2003) derves a Cournot E of a radal electrcty network wth one generator per node. Glbert et al. (2004) show how the allocaton of transmsson contracts can decrease the market power n transmsson constraned wholesale markets. Adler et al. (2008) compute a two settlement equlbrum n a transmsson constraned olgopolstc electrcty market. Hobbs et al. (2004) use a conjectural varatons approach to evaluate a range of transmsson prcng methods n networks wth olgopoly producers. 2.2 Zonal prcng Although provdng correct short term ncentves, the nodal prces desgn s often crtczed to be too complex and n need of a large degree of coordnaton. 7 Ths fault s often dealt wth by bundlng partcular nodes together nto areas wth one prce whch s referred to as zonal prcng or market splttng. In many European countres, the whole country consttutes a zone, so there s one prce n the whole country (ths s called unform prcng). In Australa each state consttutes a zone. Denmark and orway are also dvded nto several zones. The frst desgn of zonal prcng n orway was based on flexble boundares but ths created some uneasness among market partcpants, snce the procedure was not suffcently transparent. Thus, the regme was changed n 2000 nto a system where zone boundares are predetermned once or twce a year. The Swedsh government has ntroduced four zones n Sweden from ovember Zonal prcng models wthout nternal congeston Advsors of zonal prcng argue that ths approach does not only lmt the complexty of nodal prcng by breakng up the system nto just a few zones, but some also clam that the system n queston dvdes naturally nto a few unformly prced areas. In some cases, zonal dvson 7 Advantages wth the zonal desgn are for example dscussed by De Vres et al. (2009), Leuthold et al. (2008) and Stoft (1997). 7

10 can be qute straghtforward, corresponds well wth nodal prcng and can thus be consdered as an effectve smplfcaton of the LMP approach. However, the grd s normally more complex, and as shown by Stoft (1997) n hs examples, the zonal approach dstorts the nodal prces and assgns unform prces to nodes that n realty bear dfferent costs and should thus be prced dfferently. Whle zonal prcng s supposed to be a smplfcaton of nodal prcng, t trades smplcty for effcency. Björndal and Jörnsten (B&J) (2001) show that the zonal approach s just a secondbest soluton. They dscuss some problems wth the system and pont out dffcultes n defnng the zones and redstrbuton effects. They show that a zone allocaton system based on the absolute values of optmal nodal prce dfferences does not necessarly lead to a zonal system wth maxmal socal surplus and they present an example of a small network where they dentfy a multtude of possble zone constructons. Ther concerns about zone composton are confrmed by Ehrenmann and Smeers (2005). Usng a sx node model n whch two lnes have capacty restrctons, they show that n a meshed network, the number of possble zone decompostons can be large and thus, the selecton of the nodes nto rght areas s not trval. Moreover, freezng of zones s not always a good soluton, as the characterstcs of a good parttonng may change wth tme. Zonal prcng also appears to be nferor to nodal prcng when t comes to market power. Hogan (1999), Harvey and Hogan (2000) and Green (2007) present a set of examples where they show that LMP s better suted to prevent market power when compared to the zonal approach (although t does not elmnate t and addtonal measures are necessary n order to mtgate market power) Zonal prcng wth counter tradng Zones are supposed to be chosen such that ntra zonal congeston s lmted. However, the frst clearng stage, where the zonal prce s calculated, only consders nter area/crossborder congeston. Intra zonal congeston s dealt wth by means of a redspatch n the 8

11 second stage. The smplest redspatch s exercsed as a command and control scheme ( Krause, 2005); the system operator orders adjustments wthout referrng to the market and all agents are compensated for the estmated cost assocated wth ther adjustments. When the redspatch s market based as n the ordc countres and the old zonal market n Texas, t s called countertradng. In ths case, all changes after the frst clearng are compensated as n a pay as bd aucton, so that all agents are pad the stated costs assocated wth ther adjustments. Although nternal congeston s qute mportant, t s normally omtted from the analyss of zonal markets. Djk and Wllems (2011) s the only excepton that we know of. They argue that counter tradng leads to strategc bddng by generators who have an ncentve to bd a very low prce n the energy market n order to create congeston and be pad for not producng. They show that under perfect competton, entry s effcent wth nodal prcng, but neffcent wth counter tradng and they derve an E of counter tradng bds for the market wth an ncumbent and a fnte number of entrants. They conclude that compared to nodal prcng, counter tradng can be seen as a subsdy to entrants n the export constraned area whenever there s congeston. Countertradng s costly for the system operator. To avod ths cost, a network operator sometmes has ncentves to resolve ntra zonal congeston by mposng fctve nter zonal constrants. Björndal et al. (2003) llustrate ths by numercal examples. The authors also conclude that zonal prcng s not just a mere smplfcaton of nodal prcng and the partcular aggregaton of nodes nto a zone can change the allocaton of the socal surplus. Smlar dstortons of the congeston sgnal resultng n perverse ncentves to the system operators (SO) were dentfed by Glachant and Pgnon (2005) who analyze congeston technques and ther mpact on grd users on an example of a stylzed orwegan and Swedsh nterconnected grd. 9

12 2.3 Quanttatve comparson of market desgns Another strand of lterature compares dfferent prcng strateges for real markets; often the system n place wth an optmal electrcty dspatch based on LMP. Bernard and Guertn (2002) smulate a three node model of Hydro Quebec s electrc network. The smulated nodal prces dffer by 18% between Montréal s (load) and the hydro generator s node, whch s sgnfcantly larger than the 5.2% flat rate for transmsson losses that was mposed at the tme of the study. The large dfference ndcates that nvestors are provded wth erroneous prce sgnals when choosng the ste for new generatons. Leuthodl et al. (2008). analyze the mpact of ncreased wnd power producton on the German power grd. They show that changng from a unform natonal prce to nodal prcng n Germany has a neglgble nfluence on welfare, but the socal welfare rses by 0.8% f seven of ts neghbours also ntroduce nodal prcng, whch would make cross border flows more effcent. Green (2007) analyzed three dfferent transmsson prcng schemes: unform wth one natonal prce for both generaton and demand; nodal prce for generators and unform prce for demand; and nodal sngle prce for each node for both generaton and demand. All three prcng schemes were set up for a 13 node network n England and Wales. Hs results show that LMP rases welfare by 1.5% n comparson wth a unform approach (where demand cannot actvely bd nto the market). He calculates the welfare for dfferent elastctes and shows that t ncreases wth a larger elastcty value. Moreover, a system wth nodal prces also provdes correct nvestment sgnals. In a later artcle (Green, 2010), he once agan stresses the mportance and usefulness of a market desgn that accommodates spatal prces. He argues that the current unform prcng of electrcty s not well suted for accommodatng ntermttent wnd generaton that s often located far from demand. 10

13 2.4 Pay as bd auctons The debate between proponents of unform prce and proponents of pay as bd auctons has a long hstory, manly n the lterature on treasury auctons. For unchanged offers, dscrmnatory prcng would lead to lower electrcty prces. But n theory and practce, agents compensate for ths by adjustng ther offers upwards and, n the end, the desgn s nfluence on the spot prce s small. Electrcty prces dd go down when Brtan changed from unform to pay as bd prcng. But as shown by Evans and Green (2004), ths can manly be explaned by a smultaneous change n the ownershp structure. Pay as bd auctons have been studed both theoretcally and emprcally; see, for example, the survey by Holmberg and ewbery (2010). But these studes have not taken the network nto account Model The model descrbed n ths secton s used to evaluate and compare three market orented congeston management technques: pay as bd, nodal prcng and zonal prcng wth counter tradng. For the three desgns we compare the ash equlbrum of a one shot game wth a contnuum of nfntesmally small and perfectly nformed producers. 8 We study an electrcty network (possbly meshed) wth n nodes that are connected by capacty constraned transmsson lnes. In each node there s a contnuum of nfntesmal producers. Each producer n the contnuum of node can be ndexed by the varable q. C (q) s the margnal cost of the producer q and the producer submts an offer prce o(q) for ts producton to the real tme market. Wthout loss of generalty, we assume that producers are sorted wth respect to ther margnal cost and that the ndex q s scaled such that f producer q s the margnal producer n node, then the local output s q and the total producton cost n node s gven by C (q ) (provded offers follow the mert order). We assume that the margnal cost s contnuous and strctly ncreasng n q. Maxmum total producton n node s denoted by q >0. Demand n each node s gven by D, whch s certan and nelastc. It s 8 The dea to calculate ash equlbra for a contnuum of agents was frst ntroduced by Aumann (1964). 11

14 assumed that costs are common knowledge among producers; they have perfect nformaton. For smplcty, we assume that each producer s only actve n one node of the network. The system operator s clearng of the real tme market must be such that local net supply equals local net exports n each node and such that the physcal constrants of the transmsson network are not volated. Any set { q} n = 1 of nodal producton that satsfes these feasblty constrants s referred to as a feasble dspatch. The system operator chooses a dspatch among the feasble dspatches n order to mnmze the stated producton cost or equvalently to maxmze W = n = 1 q ( y) o dy, (1) Stated cost whch maxmzes socal welfare f offers reflect the true costs. Thus, we say that the system operator acts n order to maxmze the stated socal welfare subject to the feasblty constrants. In a market wth nodal prcng, all accepted offers n the same node are pad the same nodal prce. The nodal prce s determned by the node s margnal offer. In a pay as bd aucton, all accepted offers are pad accordng to ther offer prce. In the zonal prcng desgn wth counter tradng, the real tme aucton s cleared n two stages. Frst the system operator clears the market wthout regard for the ntra zonal transmsson constrants (constrants nsde zones). The zonal clearng prces are chosen such that welfare s maxmzed (costs are mnmzed) and total net supply n the zone equals total net exports from the zone. Smlar to a unform prce aucton, the zonal prce Π k s pad to all accepted producton n zone k. In case ntra zonal transmsson lnes are overloaded after the frst clearng, there s a second clearng stage where the system operator ncreases accepted producton n mport constraned nodes and reduces t n export constraned nodes. Ths s called counter tradng. All devatons from the frst clearng are settled on a pay as bd bass. We consder two verson of the zonal desgn: one where the same offers are used n the two stages of the realtme market and one verson where agents are allowed to make new offers n the countertradng stage. 12

15 We consder networks where a ash equlbrum exsts for a nodal prcng desgn wth a contnuum of nfntesmal producers, and where the market outcome s unque n the sense that any ash equlbrum has the same nodal prces and dspatches. 9 Our analyss apples to general networks wth possble loop flows. In prncple t could be a lossy AC network. But to ensure a unque outcome we restrct the analyss to cases where the feasble set of dspatches s convex,.e. f two dspatches are possble, then any weghted combnaton of the two dspatches s also feasble. Ths s, for example, the case under the DC load flow approxmaton of general networks wth alternatng current (AC) (Chao and Peck, 1996). Ths s a smplfed, lnear form of modelng an AC system, whch s normally used n economc studes of complcated networks (Green, 2007; Björndal and Jörnsten, 2001, 2005, 2007; Adler et al., 2008; Schweppe et al., 1988; Glanchant and Pgnon, 2005) Analyss We start our game theoretcal analyss of the three market desgns by means of a techncal result that we wll use n the proofs that follow. Lemma 1. Assume that offers are shfted upwards (more expensve) n some nodes and shfted downwards (cheaper) n others, then the accepted supply s weakly lower n at least one node wth more expensve offers or weakly hgher n at least one node wth cheaper supply. Proof: We let the old dspatch refer to the feasble dspatch { that maxmzed stated old q } n = 1 socal welfare at old offers when supply n node s gven by o ( q ). Let Δo ( q ) shft of the supply curve, so that o ( q ) Δo ( q ) denote the + s the new supply curve n node. The new dspatch refers to the feasble dspatch { that maxmzes stated socal welfare for new offers. Thus we have new q } n = 1 9 Generally, t s only margnal offers that are unquely determned n an equlbrum of an aucton wth nodal or unform prcng and no uncertantes. Techncally, offers above and below the margn, whch are not prcesettng, may dffer from the margnal cost. So techncally these non margnal offers are not unquely determned n equlbrum, but on the other hand they do not nfluence the market outcome. 13

16 n new q = 1 0 ( ( x) + Δo ( x) ) dx ( o ( x) + Δo ( x) ) dx. n old q o (2) = 1 0 ow, make the contradctory assumpton that n comparson to the old dspatch, the new dspatch has a strctly hgher accepted supply n all nodes where offers have been shfted upwards (more expensve) and strctly lower accepted supply n all nodes where offers have been shfted downwards (cheaper). Thus Δ o ( q ) <0, so that n new q ( x) dx > Δo ( x) Δo = = 1 0 n old q 1 0 But summng (2) and (3) yelds n new q = 1 0 o ( x) dx > o ( x) n old q = 1 0 dx. (3) dx, (4) q > old new q when Δ ( ) whch s a contradcton snce, by defnton, the old dspatch { stated welfare at old offers. o q old q } n = 1 >0 and q < q old when new s supposed to maxmze One mmedate mplcaton of ths lemma s that: Corollary 2. If one producer unlaterally ncreases/decreases ts offer prce, then accepted sales n ts node cannot ncrease/decrease. 2.5 odal prcng ext, we characterze the equlbrum n the nodal prcng desgn. The lemma below proves that the nodal prcng desgn has an E where frms offer at ther margnal cost. We get ths compettve market outcome as an nfntesmally small producer has too lttle capacty to nfluence the market prce, so they are prce takers. The correspondng equlbrum has been used n prevous studes by for example Hogan (1992), Joskow and Trole (2000) and Green (2007). 14

17 Lemma 3. In a market wth nodal prcng, the large game wth a contnuum of producers has at least one E where producers offer at ther margnal cost. Proof: We note that the objectve functon (stated welfare) n (1) s contnuous n the supply q for the consdered offers. Moreover, the feasble set (the set of possble dspatches) s closed, bounded (because of capacty constrants) and non empty. Thus, t follows that there always exsts an optmal dspatch when offers reflect true costs (Gravelle and Rees, 1992). ext, we note that no producer has a proftable devaton from the compettve outcome. Margnal costs are contnuous and strctly ncreasng and from Corollary 2, we know that f a frm unlaterally devates and ncreases ts offer n node, the accepted supply n that node wll not ncrease. Hence, no producer wth an accepted offer can ncrease ts offer prce above the margnal offer of the node and stll be accepted, as ts offer prce would then be above one of the prevously rejected offers n the same node. o producer wth a rejected offer would gan by undercuttng the margnal offer, as the changed offer would then be accepted at a prce below ts margnal cost. Thus, there must exst an E where all frms offer to produce at ther margnal cost. As the system operator clears the market n order to maxmze socal welfare when offers reveal true costs, we note that ths equlbrum dspatch must be effcent. Offers above and below the margn are not unquely determned n the ash equlbrum of a market wth nodal prcng, but t can be shown that the dspatches and nodal prces are the same n all ash equlbra. Proposton 4. The E of a nodal prcng market has the followng propertes: 1) Margnal offers are at the margnal cost. q = 1 2) Dspatches { } n and nodal prces p C ( q ) = are the same n every E. Proof: We frst realze that offers cannot be accepted below ther margnal cost n equlbrum. Moreover, margnal offers must be at the margnal cost n each node and all offers from producton unts wth a margnal cost below the margnal offer n the same node must be accepted. Otherwse, there would exst some nfntesmally small producer n the node wth a margnal cost below the margnal offer, but whose offer s not accepted. Thus, t 15

18 would be a proftable devaton for such a producer to slghtly undercut the margnal offer and we know from Corollary 2 that such a unlateral devaton wll not decrease the accepted supply n the node. ow, consder the case when offers are strctly ncreasng n output. In ths case, the objectve functon (stated welfare) s strctly concave n the supply, q. Moreover, the set of feasble dspatches s by assumpton convex n our model. Thus, t follows that the objectve functon has a unque local (and global) maxmum (Gravelle and Rees, 1992) and the system operator s dspatch can be unquely determned. From the above, t follows that the unque dspatch of an E must be such that margnal offers are at the margnal cost. Actually, the dspatch s not nfluenced by changes n offers below and above the margn. 10 Thus, we realze that any E wth strctly ncreasng offers must result n the optmal dspatch { q } n = 1 wth nodal prces { as n Lemma 3. p } n = 1 Fnally, we argue that perfectly elastc segments n the offer curves would not change the result above. One can always construct strctly ncreasng offers that are arbtrarly close to such offers, and the system operator s objectve functon s contnuous n offers. Thus, we can use the same argument as above wth the dfference that the system operator may sometmes have multple optmal dspatches for a gven set of offer curves, but t s only one of them, {, where the margnal offers n each node are at the margnal cost, a q } n = 1 necessary condton for an E. ext, we wll analyze the other two market desgns. In these calculatons, we wll refer to the nodal prcng equlbrum. From the analyss above, t follows that we can calculate a unque market outcome for every network wth nodal prcng that we consder and we refer to t as the network s effcent dspatch { q } n = 1 and the network s compettve nodal prces { p }n = 1. Schweppe et al. (1988), Chao and Peck (1996) and Hsu (1997) and others outlne methods that can be used to calculate these effcent dspatches for real networks. 10 To verfy ths statement, one can for example wrte down the Lagrange condton of the system operator s optmzaton problem. 16

19 2.6 Dscrmnatory prcng The contnuum of producers that we consder are prce takers n nodal markets where all agents n the same node are pad the same market prce. Thus under nodal prcng we fnd E where each agent offers ts supply at margnal cost. But even f the market outcome s the same, we see below that equlbra where producers bd ther margnal cost do not exst n the dscrmnatory desgn. The reason s that each agent s then pad ts ndvdual offer prce. Thus, even f agents are nfntesmally small, they can stll nfluence ther own offer prce, so they are no longer prce takers. Proposton 5. The ash equlbrum of a dscrmnatory market has the followng propertes: 1) The dspatched producton s dentcal to the network s effcent dspatch, node. 2) All producton n node wth a margnal cost at or below ( ) network s compettve nodal prce p C ( q ) =. q q, n each C s offered at the 3) Other offers are not accepted and are not unquely determned n equlbrum. However, t can, for example, be assumed that they offer at ther margnal cost. Proof: We frst realze that offers cannot be accepted below ther margnal cost n equlbrum. Moreover, margnal offers must be at the margnal cost n each node and all offers from producton unts wth a margnal cost below the margnal offer n the same node must be accepted. Otherwse, there would exst some nfntesmally small producer n the node wth a margnal cost below the margnal offer, but whose offer s not accepted. Thus, t would be a proftable devaton for such a producer to slghtly undercut the margnal offer and we know from Corollary 2 that such a unlateral devaton wll not decrease the accepted supply n the node. We also note that n a dscrmnatory aucton, t s proftable for a producer to ncrease the prce of any accepted offer untl t reaches the margnal offer of the node. We have assumed that margnal costs are strctly ncreasng n output and we have establshed that accepted offers must be at the margnal cost of the margnal unt n 17

20 equlbrum. Thus, n comparson to the nodal prcng dspatch, the accepted supply must be hgher and supply s shfted upwards (more expensve) n all nodes wth a margnal offer hgher than p and the accepted supply must be lower and supply s shfted downwards (cheaper) n all nodes wth a margnal offer lower than p. However, ths would volate Lemma 1. Thus, f an equlbrum exsts, the optmal dspatch must be same as for nodal prcng. From the proof of Proposton 4, we realze that ncreasng offers below the margnal offer does not change the optmal dspatch. Thus, equlbrum offers must be as stated and the dspatch must be the same as under nodal prcng. Fnally, we realze that there are no proftable devatons from the stated equlbrum f accepted offers are at the margnal cost of the margnal unt n the node, and non accepted offers are at ther margnal cost. Thus, the dscrmnatory aucton s dentcal to nodal prcng n terms of payoffs, effcency, socal welfare and the dspatch. As payoffs are dentcal for all crcumstances, ths also mples that the long run effects are the same n terms of nvestment ncentves etc. Fnally we show that physcal forward postons (such as day ahead sales) do not nfluence the equlbrum outcome. Proposton 6. The equlbrum dspatch and equlbrum prces n a market wth nodal or dscrmnatory prcng s { q } n = 1 and { p }n = 1, respectvely, for any physcal forward poston. Proof: Assume that producers q [ 0, ] n node have together sold f physcal forward f contracts. In the real tme market the system operator accepts addtonal offers on top of physcal forward tradng and t maxmzes stated socal welfare for these addtonal offers, whch s gven by: ΔW = n q n q ( y) dy = o ( y) dy + o ( y) o = 1 = = f n f dy, (5) n but o ( y) f = 1 0 dy s a constant that s not nfluenced by the system operator s dspatch. Thus t can be dsregarded from the optmal dspatch problem, so maxmzng ΔW s equvalent to 18

21 maxmzng W. Ths mples that we can go through the same arguments used n Lemma 3 and Propostons 4 5 to prove that these statements also apply wth contractng. Thus we can conclude that the equlbrum outcome s not nfluenced by physcal forward tradng. We wll use ths result n our analyss of zonal prcng, where the frst stage clearng of the zonal market can be regarded as physcal forward sales. 2.7 Zonal prcng wth counter tradng Zonal prcng wth counter tradng s more complcated than the other two desgns and we need to ntroduce some addtonal notaton to analyze t. The network s dvded nto zones, such that each node belongs to some zone k. We let Zk be a set wth all nodes belongng to zone k and each node s gven a number { 1, K, n}. To smplfy our equatons, we number the nodes n a specal order. We start wth all nodes n zone 1, and then proceed wth all nodes n zone 2 etc. Thus, for each zone k, nodes are gven numbers n some range n k to Moreover, nsde each zone, nodes are sorted wth respect to the network s compettve nodal prces p, whch can be calculated for the nodal prcng desgn, as shown n Secton 4.1. Thus, the cheapest node n zone k s gven the number zone k s assgned the number n k. n k and the most expensve node n n k. Power flows between zones are determned and announced by the system operator before offers are submtted to the real tme market. 11 Total net mports to zone k are denoted by flows, as our analyss shows that t leads to an effcent outcome: I k. We make the followng assumpton for these Assumpton 1: The system operator sets nter zonal flows equal to the nter zonal flows that would occur for the network s effcent dspatch {. These nter zonal flows are q } n = 1 announced by the system operator before offers are submtted to the real tme market. 11 In the ordc countres nter zonal flows are determned and announced by the system operator before offers are submtted. 19

22 The equlbrum n a zonal market wth counter tradng has some smlartes wth the dscrmnatory aucton. But the zonal case s more complcated, as the two stages mply that n equlbrum some producers can arbtrage between ther zonal prce and ther ndvdual counter tradng prce, whch we below show s equal to the network s compettve nodal prce p. Thus t s proftable for producers n export constraned nodes, where the network s compettve nodal prce p s below the zonal prce, to offer producton at even f ths prce s below the margnal cost of the unt. Such offers are accepted n the zonal market at the zonal prce, but due to ntra zonal congeston, the offers are then bought back from the system operator n the counter trade stage at the lower offer prce, p p,. Hence, the offered supply n nodes where the network s compettve nodal prce s below the zonal prce s gven by the total producton capacty n the node. As the real tme market s physcal, producers n mport constraned nodes, where the network s compettve nodal prce p s above the zonal prce, are not allowed to frst buy power at a low prce n the zonal market and then sell power at p n the counter tradng stage. Thus they nether buy or sell any power n the zonal market. Thus, the zonal prce Π k n zone k s set by the network s compettve nodal prce n node m(k), whch we defne by: m n 1 k k ( k) = n { nk,, nk} : Ik + q D and Ik + q = n k n K D. (6) = n k We wll consder two types of zonal markets. In the frst case we assume that producers cannot make new offers to the counter tradng stage; the same offers are used n the two stages of the zonal market. n = n k n = n k Proposton 7. Under Assumpton 1, the ash equlbrum of a zonal market wth countertradng and the same offers n the zonal and countertradng stages has the followng propertes: 1) The zonal prce n zone k s gven by Π k = p m ( ), where m k s defned n (6). k 20

23 2) In nodes Zk, such that costs above n zone k where p = C. p < Πk, the margnal offer and producton wth margnal p are offered at the network s compettve nodal prce p C ( q ) p > Π k, all producton wth a margnal cost at or below ( q ) ( q ) =. For nodes C s offered at 3) Other offers are rejected n both stages and not unquely determned n equlbrum. However, t can be assumed that they offer at ther margnal cost. 4) As n the nodal prcng and pay as bd desgns, the dspatched producton n each node s gven by the network s effcent dspatch, q. Proof: Offers above the zonal prce are never accepted n the frst stage of the zonal market. For these nodes, t s the rules of the counter tradng stage that determne optmal offer strateges. Thus, the aucton works as a dscrmnatory aucton, and we can use the same arguments as n Proposton 5. The remander of ths proof deals wth offers n nodes such that p < Πk. We let the margnal offer of the node be the last offer n a node that s dspatched. It follows from Proposton 6 that the zonal clearng does not nfluence the fnal dspatch, so Lemma 1 and Corollary 2 are applcable. Thus the margnal offer must be at the margnal cost and all offers n the same node wth a lower margnal cost must be dspatched. Otherwse, there would exst some nfntesmally small producer n the node wth a margnal cost below the margnal offer, but whose offer s not accepted. Thus, t would be a proftable devaton for such a producer to slghtly undercut the margnal offer and we know from Corollary 2 that such a unlateral devaton wll not decrease the accepted supply n the node. Unts wth a hgher margnal cost than the margnal offer can stll sell ther power n the zonal market at the zonal prce and then buy t back at a lower offer prce n the countertrade stage. Thus, to maxmze profts ths power s offered at the lowest possble prce, for whch offers are not dspatched,.e. at the margnal offer of the node. We have now establshed that margnal offers are at the margnal cost of the node. Thus, n comparson to the nodal prcng dspatch, dspatched supply must be hgher n nodes wth 21

24 a margnal offer hgher than margnal offer lower than p p and dspatched supply must be lower n nodes wth a. The effcent dspatch s feasble under Assumpton 1 and t follows from Proposton 6 that the zonal clearng does not nfluence the fnal dspatch. Thus any devaton from the effcent dspatch would volate Lemma 1. From the proof of Proposton 4, we realze that ncreasng offers below the margnal offer does not change the optmal dspatch. Thus, as long as offers that are margnal for nodal prcng are unchanged, the optmal dspatch wll be the same as for nodal prcng. We have already verfed that non dspatched producton unts would not gan by undercuttng the margnal offer. Offers that are dspatched n nodes wth p < Π k are pad the zonal prce. It s not possble for one of these unts to ncrease ts offer prce above p < Πk and stll be dspatched, as non dspatched unts n such nodes offer at weakly cheaper for dspatched unts to produce nstead of buyng back power at p. Moreover, t s p. Thus, they do not have any proftable devatons. Accordngly, the stated offers must consttute a ash equlbrum. The next result shows that our conclusons for markets wth zonal prcng wll stll hold even f producers are allowed to up date ther offers n the counter tradng stage. 12 Proposton 8. Under Assumpton 1, t does not matter for payoffs and the equlbrum outcome of the zonal market whether producers are allowed to up date ther offers n the counter tradng stage. 12 Ths two stage model could also represent congeston management n the ordc market, where the system operator does not accept offers n the zonal clearng of the real tme market f these offers wll cause ntrazonal congeston that needs to be countertraded n the second stage. Ths s to avod unnecessary costs for the system operator and unnecessary payments to producers. In our model where there s no uncertanty, the zonal day ahead market then takes the role of the frst stage of the real tme market. The latter becomes obsolete as wthout uncertanty, the day ahead market has already made the zonal clearng. In ths case offers to the real tme market, whch are allowed to dffer from day ahead offers, are only used n the dscrmnatory counter tradng stage. Proposton 8 shows that under our assumptons, swtchng to the ordc verson of zonal congeston management s n van, producers stll get the same payoffs and the system operator s counter tradng costs are unchanged. 22

25 Proof: We solve the two stage game by backward nducton. Thus we start by analyzng the countertradng stage. Accepted offers n the frst stage of the real tme market are equvalent to physcal forward tradng. Thus t follows from Proposton 6 that the dspatch after the countertradng stage s the same effcent dspatch as n Proposton 5. We calculate a subgame perfect ash equlbrum of the game, so ratonal agents realze what the outcome of the second stage s gong to be, and make offers to the zonal market n order to maxmze profts. Thus all producton n nodes Zk, such that p < Πk, s sold at the zonal prce, and the zonal prce n zone k s set by the node m(k) as defned n (5). Thus all agents get the same payoffs as the game n Proposton 7, where the same offers were used n the zonal and countertradng stages. We can now conclude that the dspatch for zonal prcng wth counter tradng s the same as for nodal prcng and dscrmnatory prcng. Thus, n the short run, the desgns effcences are equvalent. Ths also confrms that the system operator should set nter zonal flows equal to the correspondng flows n the compettve nodal market, as assumed n Assumpton 1, f t wants to maxmze socal welfare. However, t drectly follows from (5) and Propostons 7 and 8 that some producers get unnecessary hgh payments n a zonal prcng desgn: Corollary 9. In comparson to nodal prcng, the total extra payoff from the system operator ( ) m( k ) 1 to producers n zone k equals: p m ( k ) p q under Assumpton 1. = nk Even f zonal prcng s as effcent as nodal prcng n the short run, the extra payoffs wll cause welfare losses n the long run. Producton nvestments wll be too hgh n nodes where p < Πk. In addton, nflexble producton that cannot take part n the real tme market are pad the zonal prce n the day ahead market. Thus, the accepted nflexble supply n ths market s gong to be too hgh n nodes wth p < Π k and too low n nodes wth p > Π k. 23

26 3. Example In the followng secton, we present equlbra for the three market desgns that we have been analyzng. Fgure 1. We consder a two node network wth one constraned transmsson lne n between. In both nodes producers are nfntesmally small and demand s perfectly nelastc. For smplcty, we make the followng assumptons for each node: the margnal cost s equal to local output and the producton capacty s 15 MW. In node 1, demand s at 5 MW; n node 2 demand s at 18 MW. The transmsson lne between these nodes s constraned and can carry only 4 MW. Demand n node 2 exceeds ts generaton possbltes so the mssng electrcty must be mported from the other node. Wth nodal prcng, the equlbrum offers wll be as follows: 24

27 Fgure 2. In the frst node nfntesmally small producers make offers at ther margnal cost (as n Lemma 3). In order to satsfy local demand and export, 9 MW are gong to be dspatched. Out of these, 5 wll be consumed locally and 4 wll be exported; the hghest possble export level that the transmsson lne allows for. The margnal cost and nodal prce s equal to 9, whch corresponds to the total producton of ths node. In the second node, the nodal prce s 14 as there are 14 MW that have to be produced n the second node n order to satsfy demand and the transmsson constrant. Producton above those margnal costs (9 n node 1 and 14 n node 2) wll not be dspatched. All accepted producton wll be pad the nodal prce of the node. The pay as bd desgn wll result n the equlbrum offers presented n Fg. 3. In ths desgn, generators are pad accordng to ther bd. Knowng ths and havng perfect nformaton, producers who want to be dspatched wll bd the nodal prce of ther node, to ensure that they wll be dspatched at the hghest possble prce. Thus, n node 1, they wll bd 9 and n 25

28 node 2 they wll bd 14. Producers who do not want to be dspatched may, for example, bd ther margnal costs, whch are hgher than the nodal prces of the respectve nodes. The dspatch wll be the same as under nodal prcng desgn. Although producers wll have dfferent bddng strateges n both desgns, the overall result wll be the same. Accepted producton wll be pad 9 n node 1 and 14 n node 2. Fgure 3. 26

29 In the zonal desgn wth counter tradng, producers wll offer as follows: Fgure 4. ode 1: Producers wth a margnal cost at or below the nodal prce 9 may, for example, offer at ther margnal cost; 13 they wll be pad the zonal prce whch s 14. o nfntesmally small producer n node 1 can unlaterally ncrease the zonal prce at stage 1, as the system operator would then just accept more producton from node 2. 6 unts wth a margnal cost above the nodal prce wll bd low n order to be dspatched n the frst stage and be pad the zonal prce of 14, but as there s a transmsson constrant n the second round, they wll have to buy back ther supply at the bddng prce. As they are nterested n maxmzng ther proft, they want ths dfference n prces to be as large as possble but, at the same tme, they want to ensure that they wll not be chosen to produce. Therefore, they bd the nodal prce 9 13 They mght as well bd at the nodal prce or anythng below; t does not matter as they wll anyway be dspatched and pad the zonal prce. 27