Discussion Papers. A Dynamic Incentive Mechanism for Transmission Expansion in Eletricity Networks: Theory, Modeling, and Application

Transcription

1 Deutsches Insttut für Wrtschaftsforschung Dscusson Papers 1019 Juan Rosellón Hannes Wegt A Dynamc Incentve Mechansm for Transmsson Expanson n Eletrcty Networks: Theory, Modelng, and Applcaton Berln, June 2010

2 Opnons expressed n ths paper are those of the author(s) and do not necessarly reflect vews of the nsttute. IMPRESSUM DIW Berln, 2010 DIW Berln German Insttute for Economc Research Mohrenstr Berln Tel. +49 (30) Fax +49 (30) ISSN prnt edton ISSN electronc edton Avalable for free downloadng from the DIW Berln webste. Dscusson Papers of DIW Berln are ndexed n RePEc and SSRN. Papers can be downloaded free of charge from the followng webstes:

3 A Dynamc Incentve Mechansm for Transmsson Expanson n Electrcty Networks: Theory, Modelng, and Applcaton Juan Rosellón* and Hannes Wegt* Abstract We propose a prce-cap mechansm for electrcty-transmsson expanson based on redefnng transmsson output n terms of fnancal transmsson rghts. Our mechansm apples the ncentve-regulaton logc of rebalancng a two-part tarff. Frst, we test ths mechansm n a three-node network. We show that the mechansm ntertemporally promotes an nvestment pattern that releves congeston, ncreases welfare, augments the Transco s profts, and nduces convergence of prces to margnal costs. We then apply the mechansm to a grd of northwestern Europe and show a gradual convergence toward a common-prce benchmark, an ncrease n total capacty, and convergence toward the welfare optmum. Keywords: JEL: Electrcty transmsson expanson, ncentve regulaton L51, L91, L94, Q40 Afflaton: * Centro de Investgacón y Docenca Económcas (CIDE) and German Insttute for Economc Research (DIW-Berln). Contact detals: Mexco: Dvsón de Economía, Carretera Méxco-Toluca 3655, Mexco, D.F., 01210, Mexco; Tel.: , ext. 2711, fax: ; emal: juan.rosellon@cde.edu. * Dresden Unversty of Technology, Faculty of Busness and Economcs, Char of Energy Economcs and Publc-Sector Management, Dresden, Germany. Earler versons of ths artcle were presented at the Infraday n Berln n 2007; the IDEI/Bruegel conference "Regulaton, Competton and Investment n Network Industres" n Brussels n 2007; and the Transatlantc Infraday n Washngton, DC n We wsh to thank Walter Buhr, Andreas Ehrenmann, Chrstan von Hrschhausen, Wllam Hogan, Floran Leuthold, Ann Stewart, Ingo Vogelsang, Bert Wllems, and three anonymous referees for ther helpful comments. Juan Rosellón acknowledges support of the Programa Internsttuconal de Estudos sobre la Regón de Amérca del Norte (PIERAN) at El Colego de Méxco, the Alexander von Humboldt Foundaton and Conacyt (p ).

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5 I INTRODUCTION Effcent transmsson expanson s a major concern n electrcty markets around the world. In the Unted States, network congeston s managed accordng to a varety of regulatory systems. Regons where restructurng has not occurred use transmsson-loadng relef, a procedure that physcally manages constrants by ratonng access to portons of the transmsson network. Elsewhere, areas admnstered by Independent System Operators (ISOs) or Regonal Transmsson Organzatons (RTOs) generally employ market-based methods to manage congeston and promote network expanson. 1 Yet congeston costs have ncreased consderably, partcularly n the Mdwest (Dyer 2003) and n the markets n the PJM, Southern Connectcut, and New England. In 2006 the U.S. Department of Energy dentfed two areas of crtcal congeston, the Atlantc coastal area and Southern Calforna, as well as four congeston areas of concern (U.S. DOE 2006). In addton, transmsson nvestment n the Unted States has declned n recent years (Joskow 2005). The U.S. regulator, the Federal Energy Regulatory Commsson (FERC), proposed polces based on usng merchant mechansms, but lttle effort has been made to apply regulatory mechansms based on performance. Implementaton of regulatory measures s further complcated by the dualty of attrbutons at the federal and state levels. 2 In Europe, n contrast, the lberalzaton processes ntated n the late 1990s have led to natonal electrcty networks wth lmted cross-border capactes that now form the backbone of the emergng nternal market encompassng all of Europe. The grd, however, remans segmented nto several regonal and natonal subnetworks, resultng n lttle or no competton among countres. Dvergng polcy approaches have complcated the development of a functonng market, and a central focus for the European Unon n recent years has been to harmonze congeston management (EC 2007). Europe s expected ncrease n renewable energy (prmarly offshore and onshore wnd) wll requre sgnfcant nvestment n transmsson. Several studes have proposed ambtous schedules to extend the exstng grd (e.g., DENA 2005), but an economcal techncal approach has not yet been desgned that can cope wth the need for expanson whle accountng for effects on socal welfare. Ths artcle presents a new mechansm that seeks to acheve effcent long-term expanson of transmsson networks by means of regulaton and engneerng. Our prce-cap mechansm s 1 RTOs schedule and dspatch generators on regonal networks, allocate scarce transmsson capacty, montor generators, coordnate network mantenance, plan new transmsson lnks, and operate real tme and dverse tme-ahead markets for energy and ancllary servces. 2 In each state, Publc Utlty Commssons are responsble for revewng applcatons for major new transmsson facltes, grantng permts, and regulatng the bundled charges for transmsson servce made by vertcally ntegrated frms to retal customers. FERC s responsble for regulatng the prces for transmsson servce, but t generally lacks authorty over transmsson plannng. 1

6 based on redefnng the output of transmsson n terms of pont-to-pont transactons or fnancal transmsson rghts (FTRs) and rebalancng the varable and fxed portons of twopart tarffs. We wll show how ths theoretcal framework can be extended and appled n practce, ndependently of the techncal nature of the network. Ths mechansm thus proves sutable for actual projects of transmsson expanson n real-world grds. Our approach s novel n two ways. Frst, t combnes the regulatory approach (an ncentve mechansm) wth the electrcal-engneerng approach; and second, we provde applcatons to show that the mechansm can work n realty. The exstng lterature on transmsson nvestment has taken ether a merchant approach (long-term fnancal transmsson rghts or LTFTRs) or a regulatory approach (usng the ncentve-regulaton hypothess). The merchant approach s based on LTFTR auctons by ISOs. Ths approach s also known as a merchant mechansm because partcpaton by economc agents n auctons s voluntary. Loop-flow externaltes are addressed because the ISO retans some unallocated transmsson rghts (or proxy FTRs) durng the LTFTR aucton to protect FTR holders from the negatve externaltes of transmsson-expanson projects (see Krstansen and Rosellón 2006). Ths s equvalent to havng the agents responsble for externaltes pay them back (Bushnell and Stoft 1997) so that when FTR contracts exactly match dspatch, socal welfare cannot be reduced through gamng. The regulatory approach nvolves a commercal transmsson company (Transco) that s regulated through prce regulaton to provde long-term nvestment ncentves whle avodng congeston. Some mechansms suggest comparng the Transco s performance wth a measure of loss of welfare. The Transco would then be penalzed for ncreasng congeston costs n the network. Léauter (2000) analyzes a Bayesan approach under the congeston-management scheme n England and Wales. In ths setup, the regulator offers the frm a menu of contracts that (accordng to the revelaton prncple) nduces the frm to operate and buld transmsson lnes effcently whle stll allowng the frm to recover ts costs. Joskow and Trole (2002) propose a smple surplus-based mechansm to provde the Transco wth enough ncentves to expand the transmsson network. The dea s to reward the Transco accordng to the redspatch costs avoded by the expanson so that the Transco faces the entre socal cost of congeston. A regulatory varaton s the two-part tarff cap proposed by Vogelsang (2001), n whch ncentves for nvestment n expandng the grd derve from rebalancng the fxed and varable portons of the tarff. Vogelsang postulates transmsson cost and demand functons wth farly general propertes and then adapts regulatory adjustment processes to the transmsson 2

7 problem. For example, under well-behaved cost and demand functons, approprate weghts (such as Laspeyres weghts) grant convergence to equlbrum condtons. 3 One crtcsm made of ths approach, however, s that the propertes of transmsson cost and demand functons are lttle known but are suspected to dffer from conventonal functonal forms. Hence Vogelsang s assumed cost and demand propertes actually may not be vald n a real network context wth loop-flows. Moreover, a conventonal lnear defnton of transmsson output s actually dffcult to mantan because the physcal flow through a meshed transmsson network s complex and hghly nterdependent among transactons (see Bushnell and Stoft 1997, Hogan 2002a, 2002b). An extenson of Vogelsang s (2001) mechansm was recently proposed by Hogan, Rosellón, and Vogelsang (2007) (HRV). It combnes the merchant and regulatory approaches n an envronment of prce-takng generators and loads. Whle Vogelsang s orgnal mechansm (2001) s nherently relevant only to radal networks, the extended model s desgned to account for expansons wthn meshed networks. The new mechansm s desgned for Transcos but (lke the Vogelsang 2001 model) could also be appled wthn an ISO settng. Transmsson output s redefned n terms of ncremental LTFTRs n order to apply the ncentve mechansm to a meshed network. 4 The Transco maxmzes profts ntertemporally subject to a prce-cap constrant on ts two-part tarff, and the choce varables are the fxed and the varable fees. The fxed part of the tarff can be consdered a complementary charge. The varable part of the tarff s the prce of the FTR output, whch s then based on nodal prces. Fxed costs are allocated so that the varable charges can reflect nodal prces. Thus varatons n fxed charges over tme partally counteract the varablty of nodal prces, gvng some prce nsurance to the market partcpants. In the present artcle, we buld on Vogelsang (2001) and HRV to propose a regulatory logc to ncentvze a Transco to expand the network. As these authors, we focus on the Transco s monopoly prce regulaton and abstract from a possble substtuton between transmsson and generaton expanson. 5 Our model s a mathematcal program wth equlbrum constrants (MPEC), whch wll be presented n the second secton of ths artcle. It s decomposed nto upper- and lower-level programs that are solved smultaneously. The upper-level problem s 3 See Rosellón (2007) for an applcaton of the Vogelsang (2001) model to an electrcty network wth no loop flows. 4 The detaled propertes of cost functons when the transmsson output s redefned n terms of pont-to-pont transactons (or FTR-oblgatons) are stll an open feld for research. However, both n ths document and n other studes, we have found evdence that our approach s pece-wse dfferentable and thus applcable to real networks. In a related recent paper, we explore the propertes of FTR-based cost functons for dstnct network topologes and fnd evdence that cost functons defned as FTR outputs are pecewse dfferentable but that they contan sectons wth negatve margnal costs. Our smulatons, however, llustrate that such unusual propertes do not stand n the way of applyng prce-cap ncentve mechansms to real-world transmsson expanson (see Rosellón, Vogelsang, and Wegt, 2009). 3

8 the Transco s proft maxmzaton beng subject to a prce-cap regulatory constrant; the lower-level problem conssts of an ISO that fnds optmal loads and nodal prces through a power-flow model that maxmzes welfare wthn a wholesale electrcty market. Unlke HRV where the choce varables of the Transco s constraned problem are the ncremental FTR prce (varable part) and the fxed part of ts two-part tarff, whle the choce varables n our new mechansm are lne capacty and fxed fee. 6 Ths approach s equvalent, snce the network capacty s a functon of the FTR prces. Under our new prce-cap ncentve scheme, the Transco s behavor s smlar to that found n Vogelsang (2001) and HRV. Neglectng uncertanty n demand and generaton, prces and quanttes durng each perod are subject to a cap (adjusted by nflaton and effcency factors) as defned by the regulator. The cap s weghted wth FTR outputs (typcally Laspeyres). Generally speakng, network expanson occurs n ways that reduces congeston, thus mplyng a decrease n the Tranco s proft due to reduced congeston rents. Our mechansm permts the Transco to overcome the congeston-revenue decrease ntertemporally by rebalancng ts varable and fxed fees. Essentally, the dea s to make the nvestng frm the resdual clamant to the surplus created through nvestments by allowng the reducton n economc rent acheved due to congeston to be compensated by an ncrement n the lump-sum component of the two-part tarff. Ths process then expands the grd to the pont where the expected margnal revenues from congeston equal the margnal cost of addng new transmsson capacty, the typcal equlbrum condton for optmal expanson of transportaton networks (see Crew, Fernando, and Klendorfer 1995). 7 The looped-flow nature of power n meshed networks prevents us from dervng an analytcal soluton for general settngs (see HRV, pp ). We therefore apply numercal methods to our new mechansm. In the thrd secton of ths artcle, we apply the ncentve mechansm to smple transmsson structures based on a three-node network to determne the mpact of loop flows on transmsson nvestment decsons. Through the use of chaned-laspeyres weghts, our regulatory mechansm ntertemporally promotes an nvestment pattern that releves congeston, ncreases consumer surplus, augments the Transco s profts, and nduces convergence of nodal prces towards margnal costs. The results also approach those of a pure 5 We also abstract from access-prcng problems for new generators that mght be wllng to nterconnect to the Transco s network, as well as from potental strategc behavor from generators that mght wthhold capacty before or after the transmsson capacty change or by colludng wth the Transco. 6 The equvalency between usng ether the capacty or the varable-prce as choce varable s later dscussed n footnote In the non-bayesan scheme of Loeb and Magat (1979), the company receves the whole consumer surplus as subsdy, and therefore maxmzes the regulatory objectve drectly (provded there are no dstrbutonal weghts), thus mplyng mmedate convergence. In the dynamc ncremental-surplus-subsdy scheme (ISS) of Sappngton and Sbley (1988), the company receves a subsdy (or pays a tax) n each perod equal to the dfference between the ncrease n consumer surplus n the current perod and proft of the prevous perod. Ths mechansm converges to welfare-optmal prces (assumng equal weghts) n a sngle perod. 4

9 welfare-maxmzng model. They are robust to changes n lne-extenson costs and ntal lne capactes and reactances as well as to changes n demand elastctes. However, when generaton costs at a certan node are ncreased (relatve to other nodes), the transmsson network wll be expanded to allow mports of cheap generaton from other nodes. Ths wll then also mply larger profts for the Transco. In contrast, assumpton of a capacty constrant on cheap generaton mples a decrease n expanson of the grd and a consequent reducton n the Transco s profts. Smlarly, hgher dscount rates mply hgher nvestment rates by the Transco because future earnngs are weghted less n ts proft stream. In the fourth secton, we apply our new ncentve mechansm to a more realstc representaton of an exstng network wth a dversfed generaton park to test whether the theoretcal conclusons obtaned are approprate. We use a smplfed grd wth ffteen nodes and twentyeght lnes n northwest Europe that connects Germany, the Benelux countres, and France. Eght types of generaton technologes are ncluded, and t s assumed that ntal congeston exsts between Belgum and France and between Germany and the Netherlands. The results show that when startng from dvergent prces, a gradual convergence occurs to a common prce benchmark. At the same tme, the total transmsson capacty s sgnfcantly ncreased, the Transco s profts are augmented, and a sgnfcant convergence occurs toward the welfare optmum. Our contrbuton to the lterature conssts of mprovng the Vogelsang (2001) mechansm, both theoretcally and numercally, to acheve the expanson of meshed loop-flowed transmsson grds. We accomplsh ths upgrade by reformulatng Vogelsang s orgnal model nto a two-stage problem that synchronzes the regulatory logc on expanson wth the optmal power-flow determnaton by an ISO. Ths approach also mples a new way to analyze the economc nature of the cost and demand functons of transmttng electrcty and to promote ther effcent expanson. Overall, our model proves to be workable from both theoretcal and emprcal standponts. It provdes consderable ncreases n socal welfare, congeston relef, and profts for frms. The mechansm can be appled relatvely easly and at low cost because the regulator needs only mnmal nformaton that s provded by market prces. We have observed that most of the larger electrcty markets n the Unted States already have the three nsttutons necessary to mplement our approach: an ISO managng the wholesale market, locatonal prcng, and FTR auctons. 5

10 II MODEL FORMULATION Our model bulds on the concept presented n HRV that redefnes the output of a Transco n terms of fnancal transmsson rghts (FTRs). Accordngly, the appled mechansm has the followng sequence of actons: 1. Gven an exstng grd wth nformaton on hstorc market prces, the regulator sets up the two-part prcng constrant. 2. Based on the market nformaton avalable about demand, generaton, network topology, and smlar factors, the Transco dentfes whch lnes to expand. 3. The Transco auctons off the avalable transmsson capacty as pont-to-pont FTRs to market partcpants. 4. The ISO manages actual dspatch. Accordng to locatonal margnal prces, the ISO collects the payoffs from loads and pays the generators. The dfference between the two values represents the congeston rent of the system that s redstrbuted to the FTR holders. 5. The Transco can also set the fxed fee accordng to the regulatory prce cap. The mathematcal model to capture ths sequence s dvded nto upper- and lower-level problems that are solved smultaneously as an MPEC. The upper-level problem conssts of the proft maxmzaton of the Transco beng subject to the regulatory constrant. The lower-level problem s the market-clearng of a wholesale electrcty market that s network-constraned. The objectve functon of the (non-myopc) Transco s ts proft over all perods and ts choce varables are the lne capactes k and the fxed fee F: (1) T t t t t t t t max π = τ ( k ) q ( k ) + F N c k j k, F j j t j, j ( ) Transco proft functon The frst term of ths functon represents the collected congeston rent defned as pont-topont FTRs q j between two nodes and j multpled by the FTR aucton prce τ j. 8 The second term represents the transmsson revenue from fxed fee F collected by the Transco from the N consumers. The Transco s free to set the fxed fee as long as the regulatory prce constrant s met (see equaton 2 below). 9 The thrd term represents the costs the Transco bears when extendng the capacty k j between two nodes accordng to the extenson cost functon c(.). We 8 Followng Vogelsang (2001), who employs total transmsson output, we use the total avalable FTRs n each perod t, HRV use ncremental FTRs followng the practce n actual markets that hold FTR auctons (such as PJM). Hence, ther output Δq j refers to the addtonal or ncremental FTRs between and j sold between perods t and t-1. 9 An nterestng extenson would consder more general nonlnear tarffs wth dstnct prvate-preference parameters for each consumer. Ths undertakng would extend the model to an asymmetrcal nformaton envronment. 6

11 consder a total tme framework of T perods and assume perfect nformaton; we neglect uncertanty about demand and generaton. 10 Note that FTR prces and the demand for FTRs are a functon of the avalable capacty of the network. Thus the Transco s choce varables are the lne capactes k j and the fxed fee F. 11 In each perod, the Transco s subject to a regulatory prce cap: j t t w t t j j 1 + RPI + (2) t 1 w t 1 t τ q + F N regulatory cap j j j τ ( k ) q + F N The prces and quanttes of each perod are lnked wth a weght mechansm w (such as Paasche or Laspeyres weghts), and are subject to a cap defned by the regulator and consderng nflaton RPI and effcency factors X. A grd expanson would generally be amed at reducng the congeston rent of the system and would then decrease the Transco s proft from the FTR aucton. Gven the regulatory mechansm, the Transco can counter the decrease of aucton revenues by ncreasng ts fxed fee. By ths mechansm, the Transco s ncentvzed to expand the grd even f the congeston rent decreases. As long as the rebalancng of the varable and the fxed fee compensates possble revenue losses from reduced congeston to a pont where the per-unt margnal cost of new transportaton capacty equals the expected congeston cost of not addng an addtonal unt of capacty, we can expect to observe expanson (see Crew, Fernando, and Klendorfer 1995 and Vogelsang 2001). If the demand and optmzed cost functons can be dfferentated and nflaton and effcency factors are gnored, the frst-order optmalty condtons of maxmzng (1) subject to (2) look lke: t t t * w t t t (3) q τ ( k ) c = ( q q ( k )) τ frst-order condtons j j j j j Vogelsang (2001) and HRV analyse ths key relatonshp to dscover the ncentve propertes of the regulatory prce-cap constrant. The later study ponted out the lmted nformaton that X 10 If a postve nterest rate r > 0 were to be consdered, each monetary part of the proft functon would be multpled by (1 + r) t. In ths artcle, we follow the standard approach n the lterature on regulatory economcs and concentrate on proft maxmzaton under stable cost and demand condtons. Results are ambguous when nonstable demand and cost functons are consdered. Under such condtons, a proft-maxmzng Transco subject to a chaned Laspeyres constrant mght establsh prces that dverge from the Ramsey structure (see Neu 1993, Fraser 1995, Law 1995, and Brennan 1989). 11 HRV assume that the network capacty and the FTR demand are functons of the FTR prce τ, and thus ther choce varables are the two parts of the Transco s tarff. In such a model, the mnmum cost for each possble FTR prce (and consequently for each possble FTR quantty) s obtaned n a lower-level power-flow problem, and as a consequence, the optmal grd sze k s derved. Ths mnmum cost functon s later plugged nto an upper-level problem where the Transco maxmzes ts profts subject to a regulatory constrant smlar to the constrant n Vogelsang (2001). Our approach of usng as choce varables capacty k and the fxed fee F s thus generally equvalent (both for fxed or ncreasng demand) to the HRV approach. 7

12 can be derved. Due to the looped-flow nature of power flows n meshed networks, FTRbased demand and cost functons can only be dfferentated pecewse and generally cannot be separated. Yet the local propertes may n many crcumstances be those of well-behaved functons. To test these nsghts, we wll next apply numercal methods. One problem n dervng an MPEC mathematcal formulaton for the mechansm s uncertanty as to the outcome of the FTR aucton. We here redefne FTR aucton ncome as the congeston rent collected by the ISO gven the market-clearng prces. Because the consdered set of FTRs s smultaneously feasble and the system constrants are convex, the FTRs satsfy the condton of revenue adequacy. 12 That s, equlbrum payments collected by the ISO through economc dspatch wll be greater than or equal to payments requred under the FTR oblgatons. Gven the assumpton of perfect nformaton, ths approach allows us to abstract from the explct modelng of an aucton mechansm. Therefore, the dstrbuton of FTRs to specfc market partcpants s not an output of the model. The proft objectve functon of the Transco s then rewrtten as: ( ) (4) = T t t t t t t t max π ( p d p g ) + F N c k j adjusted proft functon k, F t, j The frst part of the Transco s proft rewrtes the outcomes of the lower-level problem, that s, payoffs from loads d and payments to generators g. The prces p are the resultng marketclearng prces at each node. FTR prces and locatonal prces are related va τ j = p j - p. In choosng the lne capactes k j the Transco wll determne resultng market prces and quanttes and thus the collected congeston rent. The prce cap s rewrtten accordngly by replacng the FTR aucton revenue wth the collected congeston rent: ( p d p g ) + F N t w t w t t (5) t 1 w t 1 w t 1 t ( p d p g ) + F N 1 + RPI + X Adjusted regulatory cap The lower-level problem defnes the wholesale-market outcome by takng nto account power flows, lne restrctons, and generaton capacty lmts: Ths has been shown for lossless networks by Hogan (1992), analyzed for quadratc losses by Bushnell and Stoft (1996), and generalzed to smooth nonlnear constrants by Hogan (2000, 2002a, 2002b). 13 Our model could be generalzed to nclude securty-constraned dspatch to take care of relablty ssues and regulate the potental reducton n network relablty (and servce qualty) typcal of prce-cap mechansms. Wthn a securty-constraned dspatch and under normal contngences, the Transco wll not optmze the relablty level on ts own ntatve. Normal securty-constraned, economc dspatch as actually employed n electrcty apples the n-1 lmts under whch the actual dspatch would reman feasble n the event of any of the montored contngences. Hence, there exsts no dchotomy between the normal dspatch and the contngency-constraned dspatch. The normal congeston cost ncludes the economc cost assocated wth the contngency. Under these assumptons, all the economc costs are congeston costs covered by the FTRs. Only on rare contngences (such as one-day-n-ten-years contngences consdered by Blumsack, et al., 2007) should the regulator/ or system operator should have an addtonal major regulatory role pursung relablty. In such a context, relablty constrants could be ntroduced more formally. In practce, regulators, such as the FERC n Order 679, dstngush between congeston relef and relablty as two separate objectves. The formal modelng of a frm optmzng relablty endogenously s beyond the scope of our artcle. 8

13 (6) d* t t max W t = ( )d d, g p d d mc g t, 0 t, ISO objectve functon s.t. t t, max (7) g g,t generaton constrant at node t t (8) pfj kj j lne flow constrant between and j t t t (9) g + q = d, t energy balance constrant at node We assume that the wholesale market s managed by an ISO that s maxmzng socal welfare W n a perfectly compettve envronment. 14 Demand d and generaton g are located at specfc nodes wthn a meshed network. Welfare s derved by obtanng the gross consumer surplus and subtractng the total generaton costs. 15 We assume a lnear demand behavor p(.) and constant margnal generaton costs mc. The maxmzaton of welfare s subject to techncal constrants. Frst, actual generaton g cannot exceed the avalable generaton capactes g max (equaton 7). Second, the power flow pf j on a lne connectng and j cannot exceed the avalable transmsson capacty k j defned by the Transco (equaton 8). Thrd, the energy-balance constrant ensures that demand at each node s satsfed by ether local generaton or net njectons q (equaton 9). The choce varable of the Transco s the actual lne capacty k j, whch mpacts the allowed power flow on that lne (equaton 8) and smultaneously affects the whole flow pattern n the meshed network. The lnkage of lne capacty and power-flow dstrbuton s derved usng a DC load-flow approxmaton (see Appendx A). The dervaton of the Karush Kuhn Tucker condtons of the lower level problem s reformulated as a mxed complementarty problem. These frst-order condtons are n turn ncluded as constrants n the Transco s maxmzaton n addton to the prce-cap constrant (equaton 5). The problem s ncorporated as MPEC nto GAMS. 16 The Transco s choce of avalable capacty k j therefore drectly mpacts the resultng market prces and thus ts varable-ncome component. The locatonal prces p are derved as duals of the energy-balance constrant. 14 The lower-level problem s an approxmaton of the ISO dspatch problem. We assume that the regulator s ndependent from the ISO, and that the regulator establshes the regulatory prce-cap constrant based on the ISO nformaton. Incluson of the lower-level welfare-maxmzng ISO power-flow problem mples that (regulated) prces are always set optmally, gven the capactes determned by the nvestment process n the upper level. In the orgnal Vogelsang (2001) formulaton, the Transco would set proft-maxmzng prces accordng to a prce- cap determned by the regulator accordng to prces and quanttes n the prevous perod. But n such a model, t s possble that prces do not necessarly concde wth a welfare - maxmzng crteron. 15 Ths approach allows a more straghtforward mathematcal expresson of the three components of welfare n electrcty markets: consumer rent, generators rent, and congeston rent. 9

14 III THREE-NODE CASE We now test the propertes of our ncentve mechansm by applyng t to a three-node example, whch allows us to analyze the mpact of loop-flowed lnes on the Transco s nvestment decson. 17 We next descrbe the basc model and present the results. Because our analyss s based on several smplfyng assumptons, we also conduct a robustness test by gradually relaxng the constrants. Smlarly, the welfare propertes of our regulatory approach are evaluated. III. Basc model outcome We assume a three-node-networassume that suffcent generaton exsts at nodes 1 and 2 and that no generaton capacty exsts topology that s gven and fxed (Fgure 1). 18 We further at node 3. For smplcty, the margnal costs of generaton are fxed at zero. Each node has a lnear demand functon p(d ). The locatonal prce at nodes 1 and 2 wll be at margnal cost level and the prce at node 3 wll nclude possble congeston charges because no local generaton capactes are avalable. The ntal lne characterstcs represent a congested system. The avalable transmsson capacty s such that node 3 faces a hgher prce than nodes 1 and 2. The system s assumed to be symmetrcal wth respect to capactes and lne reactances. Accordng to ts objectve functon (equaton 4), the Transco s free to choose the lne capactes k j.. We specfy extenson costs va a lnear cost functon c(.) usng a constant extenson cost factor ecf: t t t 1 (10) = ecf ( k k ) c extenson cost functon j j j We assume that the Transco extends the capacty; thus t j t 1 j k k. 19 Table I summarzes the ntal network characterstcs. As noted, we later relax these assumptons to analyze the mpact on the Transco s extenson behavor. 16 The General Algebrac Modelng System (GAMS) s desgned for solvng lnear, nonlnear, and mxed nteger - optmzaton problems (Brooke et al., 2008). The MPEC formulaton s solved usng the NLEPC solver. 17 Three-node networks serve as the spannng structure for more generally meshed grds. Blumsack has analyzed large networks under the Wheatstone brdge (roughly defned as the frst lnk whose addton causes a loop flow n an electrcty network). See Blumsack (2006, sec. 5, also Ekelöf 2001). 18 Propertes of FTR-based transmsson cost functons under changes n ptdfs and topology reman an open queston n the lterature (see HRV, secton 4). 19 The possblty of reducng capacty n the gven MPEC formulaton presents two problems. Frst, reducton of capacty would also be costly; thus at 0 the functon would have a knk. Second, followng equaton 12, the lne s reactance ncreases wth each reducton convergng to nfnty f the lne s completely taken out (capacty kb jb set to 0). Ths may result n nfeasble solutons due to numercal lmtatons of the modelng software. The HRV general formulaton however allows at the optmum the choce of transmsson capacty reductons that are welfare mprovng. Hogan (2002a), and Krstansen and Rosellon (2006) defne expanson protocols that apply to both ncreases and decreases n grd capacty. Transmsson nvestment s based on the ISO (or the Transco) retanng some FTRs (and capacty) n order to deal wth possble negatve externaltes caused by expansons (or reductons) n the network that harm other FTR holders. FTR auctons --and the respectve capacty expansons (or reductons) -- can then be desgned n such a way so that welfare does not decrease, even for those (negatvely) affected (but hedged) nvestors. Fnally, s t possble for a transmsson company to add capacty n a congested network n a way that would not change locatonal prces or mprove delverablty to customers. Expandng a lne downstream of a sgnfcant constrant mght be such an example (see also Bushnell and Stoft, 1997). 10

15 Table I: Intal Network Characterstcs n1 n2 n3 Demand p(d ) = 10 - d Generaton costs mc = 0 n.a. lne1 lne2 lne3 Extenson cost ecf = 1 Intal capacty 2 MW Intal reactance 1 Fgure 1: Three-node Network n1 lne2 lne1 n3 lne3 n2 We consder a tme framework of ten perods. The Transco s maxmzng ts expected returns over all perods. Regardng the prce cap, we assume a classc chaned Laspeyres approach, weghng the current perod s prces wth the prevous perod s quanttes. We gnore nflaton or effcency targets: (11) ( p t t 1 ( p d d t t 1 1 p p t g t 1 t g 1 t ) + F 1 t ) + F t 1 1 Laspeyres cap The number of consumers N s consdered to be constant over all perods and normalzed to 1, and the ntal fxed fee s equal to 0. 11

16 Gven these condtons, the approach shows an nvestment pattern that reduces the prce at node 3 over tme close to the margnal costs of generaton (Fgure 2). The fnal prce pattern obtaned represents the pont where the per-unt margnal cost of new transportaton capacty equals the expected congeston cost of not addng an addtonal unt of capacty. Gven the ntal grd condtons, provdng an addtonal MW energy at node 3 decreases the nodal congeston prce dfference by 1 due to the assumed lnear demand functon wth a slope of - 1. Thus after the capacty expanson, an ncreased demand wll be satsfed (one MW more) but at a reduced prce (one 1 less). From ths new ncome, the Transco stll subtracts the extenson costs (of 1 per MW lne capacty). The new proft, however, s postvely counterbalanced further by the Transco va the fxed fee F. Ths combned behavor s precsely the one that promotes successve grd extensons over tme based on the rebalancng of the merchant and the regulatory mechansms represented, respectvely, by the FTR revenue and the fxed-fee revenues. Compared to the ntal congested grd condtons, consumer rent ncreases by about 35% n perod 10 due to the reduced prces and ncreased demand. Lkewse, the Transco s proft ncreases due to the balancng of congeston revenues and the fxed fee. If the Transco does not expand the grd, ts profts would be 30 % lower (see Appendx B). Due to the symmetry of the network characterstcs, the extenson pattern s not a fully unque soluton. The loop-flowed lne 1 s never subject to expansons, 20 and the Transco ncreases the capactes of lnes 2 and 3 each perod by decreasng amounts. Although the total amount of expanded capacty n each perod s unque, ts dstrbuton to ether lne 2 or 3 s not unque and can vary wth each model run. Ths outcome occurs because we assume smlar cost factors and ntal lne parameters for both lnes and have symmetrc generaton structures at nodes 1 and Rebalancng the net njectons at nodes 1 and 2 produces counter-flows that keep the power flow on lne 1 stable, and thus there s no need to expand. 12

17 Fgure 2: Prce and Fxed-fee Development for the Base Case prce node 3 fxed fee Prce [ per MWh] Fxed Fee [ ] 0 0 t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 Perod III. Tests for Robustness Testng for robustness wll depend on the assumptons made. 21 Intal lne capactes: Our base case assumed a congested system wth 2 MW of capacty on each lne. If we decrease or ncrease the ntal capacty by the same amount on each lne, the general results do not change (although absolute levels dffer). If we relax the symmetry assumpton and ncrease the ntal capacty of ether lne 2 or 3 compared wth the other lnes, then the absolute values are lower as the ntal congeston s reduced. The largest expanson occurs on the lne wth the lower ntal capacty. Intal lne reactance: Reactances defne how the power flow splts on each lne. Increasng the reactance of one lne compared wth the remander of the grd reduces the share of power flow on that lne and ncreases the flow on the others. Changng sngle lne reactances n our model, however, does not alter the basc results when compared wth the base case wth respect to prces. Lne extenson costs: Our base case assumed that all lnes have smlar expanson cost factors (ecf). Increasng the ecf for ether lne 2 or 3 does not alter the prce pattern when compared wth the base case because the entre grd expanson s focused on the cheaper lne. Increasng the reactance of the more expensve lne (e.g., to resemble an ncreased length) has no mpact on the outcome. 21 An overvew about the obtaned numercal results s presented n Table II and Appendx B. 13

18 These results suggest that alterng the network assumptons under the gven smplfed-grd topology has no major mpact. We observe a gradual grd expanson that reduces congeston wthn the system and thus the prce at node 3 as well as an ncrease n consumer rents and the Transco s proft. In addton to modfyng the ntal value of lne parameters, changes n the assumptons about generaton, demand, and the length of the observaton perod may mpact the model outcome: Generaton costs: Gven that most electrcty markets face a varety of generaton technologes, the Transco must consder prce structure because t greatly affects the resultng power flows. Increasng generaton costs at node 2 n our model leads to a new prce pattern (node 2 now faces a prce larger than zero). A large fracton of local generaton at node 2 wll be replaced by njectons from the grd, whch n turn ncreases the Transco s profts due to a hgher transport volume. The results confrm these assumptons n that we observe a sgnfcantly larger amount of capacty extenson and hgher profts for the Transco. Prces at nodes 2 and 3 gradually decrease, whereas prces at node 3 behave dentcally to the base case. Generaton capactes: If cheaper generaton has already reached ts maxmum capacty lmt, addtonal transmsson capacty may not ncrease the transmsson volume. We model ths scenaro by reducng the capacty of the cheap generaton at node by one to ffteen MW whle keepng the more expensve generaton at node 2 unrestrcted. Although we fnd the same ntal market outcome as n the upper-level case, the Transco should not ncrease the network capacty sgnfcantly because only a lmted amount of addtonal transmsson wll occur, thus producng a lower proft. The results confrm our assumptons: the prces at nodes 2 and 3 drop but reman hgher than n the cases wthout restrctons, and the proft s lowered as the transported energy decreases. Demand behavor: The ntal demand functon represents a relatvely elastc consumer behavor. In the short term, the demand for electrcty s nelastc, and congeston therefore can lead to sgnfcantly hgher prces wthout alterng the transported energy by large amounts. Consequently, we changed the demand functon to p (d ) = d and kept the remanng parameters the same as n the base case. The results show that although the absolute values are sgnfcantly hgher, consumers stll have an equvalent decrease n prce. Thus the ncreased congeston rent due to the more nelastc demand does not change the ncentves for the Transco to releve congeston. Number of perods: The lmted tme frame of the model as well as the mssng dscountng of future payments can bas the outcome. If we extend the tme frame to 100 perods, we observe a much slower prce decrease than n the case of ten perods. Ths result, however, s caused 14

19 by the mssng dscountng as current and future payments are weghted equally by the Transco. Introducng dscountng nto the model reduces the mpact of the number of perods consdered. The hgher the dscountng, the more rapd the nvestment becomes because future earnngs are weghted less by the Transco. III. Welfare Propertes To determne whether the results produced by our model also mply desrable ncreases n welfare, we compare t wth a model n whch a benevolent welfare-maxmzng ISO admnsters the lne capactes as choce varables. 22 In ths model, the Transco s omtted from the formulaton, and the basc market-clearng equatons are extended. Thus network extensons become part of the welfare-objectve functon: d* t t t t (12) max W = ( )d ( j ISO objectve functon d, g p d d mc g c k ) t, 0 t,, j s.t. equatons (7) to (13) There s a large number of robustness tests and we present only two n detal: asymmetrc lmted capacty and unlmted generaton capacty. The remanng results show a smlar general behavor, although wth dfferent absolute values. 23 Table II summarzes the outcomes of comparng the base case of no grd extenson, the case of capacty extenson under our regulatory approach, and the case of a benevolent ISO. The case of non-extenson represents the ntal network condtons, and thus congeston and prces are the hghest. In the case of our regulatory mechansm, we observe an ncrease n consumer surplus and n the Transco s proft as well as a decrease n congeston rents and nodal prces. The results of the regulatory approach are relatvely close to a pure welfare-maxmzng outcome and suggest a convergence over tme toward the welfare optmum wthn a relatve short tme frame (see Appendx D). Furthermore, the case wth lmted generaton capactes demonstrates a common outcome n the case of congested markets: f congeston s releved, prces converge to a common level, and thus areas wth ntally low prces may afterward face hgher prces due to ncreased export. Ths may cause partcular concern when mplementng the approach n exstng markets. 22 Ths model s bult only as a benchmark to evaluate numercally the welfare-convergence propertes of our own mechansm. Our model seeks to decentralze nvestment decsons, so that market players are ncentvzed to nvest. In realty, the ISO mght not be able to obtan the necessary nformaton to carry out expanson based on market crtera as modeled n the welfare benchmark. 23 A lst of results can be found n Appendx B. 15

20 We therefore conclude that gven the smplfed topology of a three-node network, our approach has robust welfare-enhancng propertes. Although the absolute values of prces, rents, and capactes are changed by varyng the assumptons, the Transco s ncentvzed to expand the grd n such a way that market prces converge toward the welfare optmum. Table II: Comparson of Regulatory Approach wth Welfare Maxmzaton (values refer to the last perod, t10) Asymmetrc generaton costs, lmted generaton capactes Asymmetrc generaton costs, unlmted generaton capactes No grd extenson Regulatory Approach Welfare maxmzaton No grd extenson Regulatory approach Welfare maxmzaton Consumer rent Producer rent Congeston rent Total welfare Extenson sum Capacty: lne MW 2.00 MW 2.00 MW 2.00 MW 9.75 MW 9.90 MW Capacty: lne MW 5.90 MW 6.00 MW 2.00 MW 9.53 MW 9.90 MW Capacty: lne MW 2.70 MW 2.90 MW 2.00 MW 2.00 MW 2.00 MW Prce: node Prce: node Prce: node IV APPLICATION TO A REAL-WORLD NETWORK IV. Model and Data We test the nsghts obtaned n the prevous sectons by applyng our approach to a more realstc representaton of an exstng electrcty network wth a dversfed generaton park. Fgure 3 llustrates a smplfed grd of northwestern Europe connectng Germany (D), the Benelux countres, and France (F) based on Neuhoff et al. (2005). The model conssts of ffteen nodes and twenty-eght lnes. The nodes connectng France and Germany wth ther neghbors are auxlary nodes wthout assocated demand or generaton. The lnes connectng the German and French nodes wth these auxlary nodes are assumed to have unlmted capactes. Thus ntra-country congeston n those two countres s not consdered n detal. Each country node possesses gven generaton capactes (VGE 2006) and a reference demand (UCTE 2007). For addtonal smplfcaton, we classfy generaton capactes nto eght types and assume equal margnal cost levels for each type n all countres. Wth the excepton of hydro power, renewable generaton s not consdered wthn ths model. Table III provdes an overvew of the types, nstalled capactes, and margnal generaton costs. A lnear-demand behavor at the nodes s derved from the average load level for the node, a reference prce of 30 /MWh, and an assumed prce elastcty of at that reference pont. We do not assume 16

21 a demand growth rate. However, due to the lnear demand functon, ncreased network capacty and lower electrcty prces results n a hgher demand level. The model formulaton used for applcaton to the European network s smlar to the prevous three-node approach. We consder a tme frame of twenty perods and nclude a dscountng factor wth an nterest rate of 8%. 24 We do not account for nflaton or an effcency factor wthn the Transco s prce cap. The derved market results for one tme perod represent one hour; thus the Transco s revenue s multpled by 8,760 for each perod to approxmate yearly ncomes. Due to the average nature of both the load level and the generaton structure, ths approach omts the varable nature of real-world electrcty systems. Lne-expanson costs are assumed to behave lnearly. Followng Brakelmann (2004) and DENA (2005], we chose a value of 100 per km per MW. 25 The Transco can only expand lnes that already exst. The startng condtons n the market are classfed by a hgh prce level n the Netherlands (Krm, Maas, Zwol), a dvded prce structure n Belgum (Gram, Merc), modest prces n Germany, and low prces n France. Congeston thus occurs between Belgum and France as well as between Germany and the Netherlands. 24 Twenty years are assumed to represent the dscountng tme of assets n electrcty markets, and 8% trepresents an nvestment wth rather low rsk. 25 Ths value s derved from upgrade costs for addtonal lnes of the same voltage level as well as upgrades from 220 to 380 kv. The lumpy character of network nvestments s omtted. 17

22 Fgure 3: Smplfed Grd of Northwestern Europe (based on Neuhoff et al. 2005) Netherlands Belgum Germany France Table III: Plant Characterstcs 26 Plant type Installed capacty Margnal generaton cost Plant type Installed capacty Margnal generaton cost Nuclear GW 10 /MWh Steam GW 45 /MWh Lgnte GW 15 /MWh Gas turbne GW 60 /MWh Coal GW 18 /MWh Hydro GW 0 /MWh CCGT GW 35 /MWh Pumped storage GW 28 /MWh IV. Results To classfy the results obtaned, we agan compare them wth a case wthout grd extenson (the startng condtons) and a benevolent ISO. Fgure 4 shows the prce developments at the country nodes over the perods consdered. Wth an ntally hgh prce dvergence n the market, we observe a gradual convergence to a common prce level resemblng the margnal cost of the last unt runnng. But the prce convergence already occurs wthn the frst ten perods. In the second half of the consdered, the prces change only margnally, and addtonal expansons are relatvely small. Consequently also, the convergence of the regulatory approach towards the welfare optmum takes place wthn the frst ten perods, wth most of the benefts acheved wthn the frst fve (see Appendx D). After the consdered tme 26 A detaled locatonal overvew of the demand and generaton data s provded n Appendx C. 18

23 frame, we observe a close convergence of the regulatory approach toward the welfare optmum (Table IV). But even though overall welfare ncreases, consumer surplus decreases due to the ncreased prces that French consumers must pay. In the ntal grd case, congeston separates the French market from the remander of the grd. Thus the prces are at the margnal costs of local nuclear unts, resultng n the lowest prce n the market. If the grd s expanded, more of the avalable French nuclear power s utlzed to satsfy demand at other nodes, and the prce n France ncreases, thus reducng consumer surplus. Ths reducton cannot be offset by the lower prces n the Benelux countres because the demand n France s hgher than the total demand n the Benelux. Yet the decrease n consumer rent s compensated by a sgnfcant ncrease n producer rent. The largest fracton of the addtonal company proft wll beneft France because nuclear generaton s now prced above ther margnal costs. The grd expanson totals about 300 mllon euros wthn the twenty perods (Table IV), a relatvely small amount gven the market s total welfare of 10 bllon euros per year. Ths result arses manly from assumng that the grd upgrades that occur wll be less expensve than entrely new connectons. But the total amount of new nstalled ncreases sgnfcantly, doublng the grd capacty ntally avalable. The Transco s proft over all twenty perods s about 25% hgher under the regulatory approach than f the company had not extended the grd at all and just collected the ntal congeston rent durng all perods. Consequently the fxed fee ncreases as the congeston rent (represented by the prce dfferences) decreases (see Fgure 4). 19

24 Fgure 4: Prce Development n the European Model D F GRAM KRIM MAAS MERC ZWOL Fxed Fee Prce [ /MWh] t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 Perod bn per year Table IV: Comparson of Regulatory Approach wth Welfare Maxmzaton (values refer to the last perod, t20) No grd extenson Regulatory Approach Welfare Maxmzaton Consumer rent [Mo /h] Producer rent [Mo /h] Congeston rent [T /h] Total welfare [Mo /h] Total extenson sum [Mo ] Total grd capacty [GW] Average prce [ /MWh] 28 28,4 18,5 18,1 V CONCLUSION Ths artcle has presented a combnaton of the merchant-ftr approach and the regulatory approach to transmsson expanson n a compettve envronment of prce-takng generators and loads. We mplemented a regulatory mechansm as an MPEC problem wth a proftmaxmzng Transco and a compettve wholesale market based on nodal prcng. Startng wth a congested grd, the Transco s free to choose grd expansons that enhance ts own profts (the congeston rent and a fxed fee). The Transco s profts are subject to a prce cap wth Laspeyres weghts. The results show that the Transco expands the network and that prces converge toward margnal costs over the perods analyzed. 27 Excludng auxlary lnes. 28 Excludng auxlary nodes. 20