Sysem Dynamcs models for generaon expanson plannng n a compeve framework: olgopoly and marke power represenaon J.J. Sánchez, J. Barquín, E. Ceneno, A. López-Peña Insuo de Invesgacón Tecnológca Unversdad Ponfca Comllas c/ Sana Cruz de Marcenado 26 28015 Madrd (Span) Phone: +34915422800 Fax: +34915406289 JuanJose.Sanchez@.upcomllas.es Absrac: Ths paper proposes several alernave mehods o mprove sysem dynamcs models used n he leraure for generaon expanson plannng n lberalsed elecrcy markes. Concreely, hese mehods provde a beer represenaon of olgopoly srucures and marke power. These mprovemens focus on marke prce and producons calculaons, fuure markes modellng and companes dfferenaon when decdng new nvesmens. The mehods presened n he paper are based on equlbrum approaches and cred rsk heory. Keywords: sysem dynamcs, generaon expanson plannng, elecrcy markes, fuure markes, nvesmens. 1 Inroducon Snce elecrcy sysems began o be lberalsed, new plannng problems appeared for boh elecrcy companes and regulaory auhores. One of hese plannng problems s he so-called generaon expanson plannng whch deals wh nvesmens n new elecrcy generaon capacy lookng no he long-erm. To help he companes and he regulaors o carry ou hs plannng, an nensve research acvy has provded new mehods and models. Among hese new models, sysem dynamcs has succeed n represenng long-erm behavour of elecrcy markes, and concreely has helped o gan nsghs no he way he new generaon capacy eners n he marke n a lberalsed framework. However, here are some aspecs of hese markes ha have no been represened accuraely ye and whch could help o gan new nsghs and o make beer decsons by companes and regulaors. Parcularly, he olgopoly srucure of mos of hese markes and he man consequence of hs srucure, marke power, has no been aken no accoun or has been grealy smplfed. Ths paper proposes mprovemens for he sysem dynamcs models n he leraure, n order o oban a beer modellng of olgopoly srucure and marke power. Concreely, new approaches for marke prce and oupus calculaons, fuure markes modellng and agens dfferenaon regardng nvesmen decsons are provded.
In he nex secon, he problem of plannng he expanson of elecrcy generaon s explaned n deal. Alernave echnques o sysem dynamcs, used o help o solve hs problem, are presened here. Secon 3, compares sysem dynamcs wh he alernave echnques and presens he sae of he ar regardng sysem dynamcs for generaon expanson plannng. The mprovemens proposed by hs paper, are explaned n secon 4. Then, a case sudy based on he Spansh elecrcy marke s shown n secon 5. Fnally, secon 6 concludes he paper. 2 Elecrcy generaon expanson plannng 2.1 The problem of plannng he expanson of elecrcy generaon Elecrc sysems, as well as oher ules, requre a careful plannng of producon resources. Ths plannng, for he case of elecrcy has some peculares ha complcae hs knd of decsons. Manly, s an absoluely essenal good whch leads o sgnfcan regulaor vglance. Moreover, generaon plans requre bg nvesmens ha spread over long me perods. Fnally, demand randomly flucuaes and mus be nsanly and exacly suppled by generaon whle elecrcy can no be easly sored. Dependng on he sysem, buldng new plans s n charge of eher a regulaory auhory or he generaon companes. In he one hand, n cenralzed sysems, responsbly for generaon expanson decsons devolves upon a regulaory auhory ha makes decsons based on cos, relably and fulfllng producon consrans (for example echncal, sraegc or envronmenal ones). Ths framework commonly corresponds o radonal elecrc sysems. In he oher hand, n lberalzed sysems, companes ndependenly underake he seup on new power plans a her own rsk, whle he regulaory auhory plays a supervsng role by means of regulaory acons. Ths schema s nowadays followed n mos of developed counres all over he world. Plannng elecrcy generaon can be suded boh from generaon companes or regulaory auhory pon of vew. The major am of companes s obanng he maxmum prof, bu hey also may follow sraegc objecves (.e. marke share or generaon echnologc mx) and have also o respec some producon lms. Regulaory auhory manly pursues sysem relably,.e. requred energy s avalable wh a reasonable reserve margn, and addonally oher sraegc goals always orened o maxmze socal welfare. From he prevous, elecrc generaon expanson plannng n lberalzed sysems can be defned as he funcon o be performed by generaon companes o properly evaluae her decsons of buldng, closng down, buyng, sellng or repowerng power plans, whereas n he case of regulaory auhory, he acons o assess are regulaory acons orened o gude companes decsons. Ths funcon consders a se of objecves, dependng on he sandpon, s analyzed wh a long-erm perspecve and normally consders as man condonng elemens: demand growh, dfferen avalable generaon echnologes, fuels cos and avalably, sysem relably crera, envronmenal consrans and esablshed dversfcaon polcy. 2.2 Alernave echnques Elecrc generaon expanson plannng s a complex decson problem ha has been addressed usng sysem dynamcs among oher dfferen analyss echnques. Those
dfferen o sysem dynamcs wll be brefly menoned n hs secon. Nex pon deals wh sysem dynamcs models. Regulaory auhores of cenralzed sysems face a problem wh a se of nfluencng facors ha are exogenous o he elecrc sysem ha nclude demand growh, fuel prces, hydro nflows, echnology evoluon, and macroeconomcs. The mos used echnques, n addon o sysem dynamcs, for hs knd of sysems are opmzaon (cos mnmzaon) and mulcrera decson. The laer echnque corresponds o a more sophscaed approach where an negraed resources plannng s consdered ncludng decsons as demand-sde managemen and envronmenal and socal crera, beyond cos. Some examples of boh ypes follow. (Lee e al., 1990) presens a survey ha ncludes plannng models. (EIA, 2002) presens a model developed by USA Energy Deparmen ha conans an opmzaon module for plannng. In (Mllán e al., 1998) an opmzaon model s used o analyze Cenral Amerca generaon expanson plannng. (Hobbs and Meer, 2000) summarze he use of mulcrera echnques for decsons n he area of energy. (Merrll and Schweppe, 1984) and (Connors, 1996) presen a rade-off rsk model used for example for plannng n New England. Envronmenal aspecs of generaon expanson are suded n (Schenler and Gheorge, 1998) wh a smlar model. For lberalzed sysems, analyss ges more complcaed, because of addonal uncerany sources ha are endogenous o he sysem: elecrcy prces, regulaory changes and compeors decsons. Generaon companes requre new models o manage he hgh level of rsk ha s presen n hese sysems. The man echnques used n hs framework are scenaro analyss, rsk analyss, real opons, agen-based smulaon, game heory and sysem dynamcs (Dyner and Larsen, 2001). The focusng of hese echnques s dfferen, bu all of hem conrbue o analyze plannng decsons. Scenaro analyss s a broad concep ha allows dealng wh uncerany of plannng, as n (UPME, 2000). Rsk analyss s an neresng alernave and allows copng wh he sudy of long-erm conracs ha are assocaed o plannng decsons, (Fleen e al., 1997), (Cabero e al., 2005). Real opons approach assess he nvesmen n generaon asses consderng hem as a fnancal produc and s broadly used (Frayer and Uludere, 2001), (Boerud, 2003). Agen-based smulaon s more adaped o shor- and medumerm analyss, for example for marke bddng sraegy, bu can be also used o address generaon expanson plannng problems whle explcly represens each sysem agen, s objecves and s decsons o acheve hem. An neresng example can be found n (Cosa and Olvera, 2005). Oher echnque ha has been manly used for medum- and shor-erm sudes, ncludng marke bddng elaboraon, s game heory. Neverheless, here exs some neresng works devoed o generaon expanson plannng: (Murphy and Smeers, 2001), (Venosa e al., 2002), (Ceneno e al., 2003) and (Muro, 2000). 3 Sysem dynamcs for elecrcy generaon expanson plannng 3.1 Comparson wh oher echnques The se of echnques presened n he prevous secon, can be classfed n wo groups: he frs hree -scenaro analyss, rsk analyss and real opons- ha are focused on uncerany analyss, and he oher wo, -agen-based smulaon and game heory- ha
manly deal wh sraegc analyss of compeors and sysem. These wo approaches are complemenary and a complee generaon plannng sudy should be addressed wh models from boh ses. Sysem dynamcs can be seen as a complemenary ool for any of he oher echnques. Raher han a forecas of he fuure, sysem dynamcs models are used o gan nsghs no he sysem behavour, by represenng n deal he relaonshps beween he man varables of he sysem, wh explc recognon of feedbacks and delays. Sysem dynamcs models may provde nformaon abou dynamcs of how new plans ener he sysem exendng he prevous echnques scope. Frs, scenaro analyss requres a prevous defnon of he alernave suaon ha wll be consdered as alernave soluons o he problem. As lberalzaon has been recenly nroduced n mos of he counres, here s no much experence abou long-erm evoluon of elecrcy markes and sysem dynamcs models can be of help o buld hese scenaros. The second echnque, rsk analyss, when appled o plannng usually evaluaes a deermnae nvesmen, n a parcular plan. A prevous analyss of he mos suable alernaves can be performed by means of sysem dynamcs. Real opons heory s he hrd possbly ha has been menoned. I s also cenred n profably of a deermnae new plan and addonally deermnes he bes momen o buld. Some hypohess abou sysem behavour mus be made (manly prce evoluon), and sysem dynamcs echnque allow o se hese hypohess. Wh respec o he fourh echnque, agenbased modellng s orened o suaons where agens decsons are made n a connuous way. Thus, generaon expanson plannng problem, n whch decsons are more separaed n me, s more naurally address ackled wh sysem dynamcs approaches. Fnally, game heory, he ffh alernave, provdes a soluon o dynamcs games when hey are no oo complcaed and do no exend oo much n me. An alernave o represen more complcaed problems s o use he so-called open-loop Courno games, bu here, he decsons depend jus on he me. The analyss of dynamcs of more dealed games ha represen fahfully plannng dynamcs can be advanageously performed wh sysem dynamcs paradgm. 3.2 Sae of he ar Sysem dynamc echnques have been exensvely used o analyze dfferen aspecs of elecrc energy sysems, generaon expanson plannng among hem. Two neresng surveys can be found n (Ford, 1997) and (Bunn and Larsen, 1997). Cenralzed sysems have been represened wh dfferen models as IDEAS (AES, 1993), Energy 2020 (CMPC, 1989) or RPSM (Ford and Bull, 1989). Elecrcy sysems lberalzaon has requred updang hs knd of models ncludng he new characerscs of expanson decsons made by companes and marke dynamcs. Sysem dynamcs acqures a new sgnfcance n hs suaon. Cenralzed sysems models usually represen a whole counry or a bg regon, wh a lo of deals abou he sysem, orened o assess regulaory auhory decsons. Ths makes an mporan dfference wh models ha represen lberalzed sysems, ha end o be smaller and are orened o analyze parcular problems. However, some exensons o prevous models have been suggesed (Amln and Backus, 1996), (BPA, 1994) and (Dyner and Bunn, 1996). The frs man works n he feld of lberalzed model for elecrcy generaon plannng can be classfed n hree bg sources: Andrew Ford and collaboraors, Derek W. Bunn -
wh hs research group from London Busness School- and works carred ou for he Nordpool elecrcy sysem by A. Boerud and K. Vogsad among ohers. Andrew Ford s work covers dfferen aspecs relaed o generaon plannng. He has suded nheren dynamcs o buldng new plans n he wes USA marke ha forecas cyclng dynamcs (boom and bus) producng perods of overcapacy and oher of scarcy ha can be dangerous for he sysem. A consan capacy paymen s suggesed o mgae hs effec (Ford, 1999). An mproved verson of he model used n he prevous work was wdely used o sudy Calfornan marke and he causes ha led o a crcal suaon durng 2000 and 2001 (Ford, 2001a), (Ford, 2001b), (Ford, 2002). Derek Bunn s researches cener on he Englsh marke, bu s conclusons can be exended o oher counres. In (Larsen and Bunn, 1999) he usefulness of sysem dynamcs models boh for generaors and regulaors n lberalzed markes s jusfed as a powerful ool o face arsng new rsks. In (Bunn e al., 1993), complemenares wh oher alernave echnques are shown. Dynamcs lkely o appear afer elecrcy marke lberalzaon n England, consderng he dfferen szes and characerscs of he new agens ha consued ha sysem are analyzed n (Bunn and Larsen, 1992) and (Bunn and Larsen, 1994). Oher model, he one n (Gary and Larsen, 2000), ncludes neracon wh gas markes and how mpacs n plans profably and consequenly n plannng. In (Boerud e al., 2002), a plannng model o sudy Norwegan elecrcy marke s presened. Oher works for he Nordpool consder: regulaory mechansm o promoe renewable energes (Vogsad e al., 2003), dynamcs of ranson o a echnologcal generaon mx ncludng more renewable producon capacy (Vogsad e al., 2002), coordnaon beween hydraulc and wnd power (Vogsad, 2000) and effecs of massve enerng of gas plans n lberalzed markes (Vogsad, 2004). In he recen years, some new works have appeared, mprovng some modellng aspecs of prevous sudes such as marke prce represenaon, ransmsson nework modelng, dynamcs of expors and mpors or neracon beween sysem dynamcs and oher dfferen modellng approaches. Man references here are (Olsna e al., 2006), (Kadoya e al., 2005), (Ochoa, 2007), (Ford, 2006) and (Vogsad, 2006). In (Olsna e al., 2006), no acual marke s smulaed bu focuses on he formulaon of he mahemacal framework o exend he prevous modellng mehodology. They nclude several echnologes for new nvesmens decsons, a vnage model o represen progress n hermal effcences, an annual dsrbuon represenaon of marke prce and a delay concernng he opon o defer rreversble nvesmens under uncerany. A dfferen modellng approach for he annual dsrbuon of marke prce whn a sysem dynamcs model can be found n (Sánchez e al., 2005). The man conrbuons of (Kadoya e al., 2005) are he represenaon of a full merorder dspach and he calculaon of a complee NPV o assess nvesmen decsons, usng forward curves of expeced fuure values for prces, capaces and capacy facors based on hsorcal averages and a rend exrapolaed from curren condons. Ths model was calbraed successfully for PJM and ISO-NE markes. Boh conrbuons have been addressed also n (Sánchez e al., 2005) were a prce duraon curve s calculaed for each year (usng a full mer-order dspach) and exrapolaed no
he fuure (akng no accoun hsorcal prce-duraon curves) o calculae a complee NPV. In hs case, he model was based n he Spansh marke. An neresng new represenaon of he dynamcs governng mpors and expors of he Swss marke s carred ou n (Ochoa, 2007) o es he nfluence of dfferen polcy changes n hs counry concernng de-regulaon, nuclear dsmanlng and mpors dependence reducon. Fnally, boh n (Ford, 2006) and n (Vogsad, 2006) a combnaon of sysem dynamcs wh complemenary modellng approaches s made. On he one hand, n (Ford, 2006) an engneerng model whch smulaes load power flows s combned wh a sysem dynamcs model o smulae long-erm behavour n sx dfferen regons of he wes of he US. Concreely, sudes he poenal reducons n carbon emssons n he US wesern elecrcy sysem, under a cap and rade marke. On he oher hand, (Vogsad, 2006) combnes fnancal sochasc prce models wh a sysem dynamcs model descrbed n (Vogsad, 2005) n order o provde long-erm prce prognoses for nvesmen decsons, whch ake no accoun sochasc processes for gas and coal prces, hydro nflow and wnd. 4 Proposed alernaves for olgopoly srucure and marke power modellng 4.1 General srucure of a generaon expanson plannng model AVAILABLE POWER PLANTS DEMAND Buldng NEW POWER PLANTS Decson Marke ELECTRICITY POWER PRICE PRODUCTION Forecasng PRICE FORECASTING POWER PRODUCTION FORECASTING Fg. 1. Overall srucure for an elecrcy generaon plannng model. In a global vew, he feedback represenng he new generaon capacy nvesmen decsons n mos of he generaon plannng models ced n he sae-of-he-ar secon can be dvded n four man blocks as shown n Fg. 1. Sarng from demand and avalable power plans, a represenaon of he marke deermnes elecrcy prces and power oupus for every plan. The second block represens he forecasng of prces for a deermnae me horzon ha marke agens make as well as plan producon forecasng. These resuls allow makng he decson -hrd sep- of how many plans o buld, whch each generaon company makes dependng on s own characerscs and wh s profably crera. Fnally, new plans ener he sysem wh some delay ha represen perm obanng and plan buldng.
Usng hs srucure, he dfferen models commened above have succeed n represenng some parcular lberalzed elecrcy markes, permng he dfferen acors o gan nsghs no her long-erm behavour, and concreely no he way he new generaon capacy ener n he marke. However, hese models have no been so accurae represenng some oher mporan aspecs of he curren lberalzed elecrcy markes, as marke srucure and marke power. Ths paper proposes some mprovemens n he frs ( Marke ) and hrd ( Decsons ) blocks shown n Fg.1, o cope wh hese drawbacks. The nfluence of an olgopoly srucure n he second block ( Forecasng ) s less mporan as dfferen companes could use he same forecasng mehods. However, some dfferenaon beween companes could be represened by consderng dfferen avalable nformaon as s done n (Ford, 2001a). The fourh block ( Buldng ) s he smples one and makes no a sgnfcan dfference beween sysems wh dfferen marke srucures. A slghly modfcaon could be consderng dfferen delays beween companes (domnan companes could have greaer advanages o oban consrucon perms). Mos of hese models assume perfec compeon when calculang prces and plans oupus, whereas mos of curren lberalzed markes are domnaed by an olgopoly. Alhough some of he models have represened hs fac wh some smplfcaons as assumng some groups bddng above margnal coss, hs has been done mos of he mes usng parameers ndcang he quany of he over-bd and adjusng hese o oban credble prces. In hs paper, we presen an alernave mehod o represen olgopoly behavour n he marke, based on equlbrum approaches whch have been proved successfully n he medum-erm. Long-erm conrac markes are anoher mporan aspec of curren lberalzed elecrcy markes whch have no been represened accuraely. Nowadays, mos of lberalzed sysems, are nroducng hs knd of markes or smlar ools, n order o reduce marke power. Ineracons beween spo markes and long-erm conrac markes are a ho research opc oday, and sysem dynamcs models could provde new nsghs no. Ths paper proposes dfferen alernaves o model hese neracons. Fnally, companes dfferenaon has been avoded n mos of hese models, by consderng nvesmens n a global way. Oher models, have made smplfcaons dvdng he companes n bg groups lke leaders-followers or ncumbens-new enrans. A more dealed represenaon of hs fac could make sysem dynamcs models o provde beer undersandng on he possble marke srucure evoluon and s effecs on nvesmens and prces. In hs paper, a new dea o dfferenae generaon companes when makng new nvesmens s shown. Nex, mprovemens proposed n each block are explaned. Prevously, how hese blocks have been represened n he prevous models s commened and hen enhancemens are dealed. 4.2 Marke Marke represenaon n he above-menoned models range from hose whch do no requre explc prce compuaon o hose whch explcly nclude a compeve marke represenaon. In (Bunn and Larsen, 1992), (Bunn e al., 1993) and (Bunn and Larsen, 1994) profably s deermned, whou compung prce, from capacy paymen ha s esmaed from sysem power reserve margn. Ths s an neresng smplfcaon when deals abou prce are no requred. Oher models represen a perfec compeon marke where he agens bd her margnal coss (Ford, 1999),
(Boerud e al., 2002), (Vogsad, 2005), (Olsna e al., 06). In hs represenaon, obaned prce s he same as margnal cos n a cenralzed sysem and may represen a lberalzed marke under ceran assumpons. A smplfed compeve marke s represened n (Gary and Larsen, 2000), where prce depends on reserve margn, and wh more deal n (Ford, 2006) and (Kadoya e al., 2005), dvdng groups n hose ha bd her margnal cos and hose ha make a sraegc bd over margnal cos usng some a-pror parameers (acually, n (Kadoya e al., 2005) s no explaned how he bddng sraegy s modelled). Regardng long-erm conrac markes or fuure markes, (Vogsad, 2006) sugges he combnaon of sysem dynamcs wh fnancal sochasc prce models n order o oban forward prces expecaons a each me n he smulaon. These prces, whch could be he prces of forward conracs, are used by he companes as he man sgnal for nvesmen. However, he possble neracon beween long-erm conrac markes and spo markes and s effecs on marke power evoluon s no represened here. Ths paper suggess a dealed marke represenaon ha ncludes: a conjecured-prceresponse based marke equlbrum o calculae prces and producons under olgopoly srucures, a dynamc compuaon of conjecure-prce-responses and an explc represenaon of fuure markes. As wll be explaned nex, we based he new represenaons on some equlbrum deas. I may be argued ha one of he common assumpons of sysem dynamcs s bounded raonaly, and so, combnng hese wo echnques may no be coheren. However, when companes are lookng a dfferen me horzons a he same me can be argued ha companes may behave n erms of equlbrum n he shor-erm (when acons are more repeve, uncerany s lower and companes have more nformaon) and consderng bounded raonaly n he long-erm (nvesmen decsons). 4.2.1 Marke equlbrum Some elecrcy markes can no be assmlaed o perfec markes and olgopoly effec mus be explcly and accuraely represened. Besdes, profably of nvesmen n generaon asses s heavly condoned by he frs year s prces, and hus specal care mus be pad o represen. For hese requremens, a wdely acceped approach s marke equlbrum n he sense ha was defned by (Nash, 1950). Marke equlbrum s he se of oupus of every generaor such ha any generaon company can no mprove s benef by unlaerally modfyng s producon. Le us suppose ha each company receves as revenue, s spo marke producon q - f, a marke prce p, and besdes he prevously conraced quany f a a prce p. Prof can be compued as revenues mnus cos: ( q f ) + p f c π = p ' (1) Equlbrum condons can be obaned by maxmzng he prof wh respec o producon for each generaon company: π p = 0 = p + q q ( q f ) c q The dervave of prce wh respec o producon s he so called conjecured prce response (Ceneno e al., 2007) and wll be consdered as known for each company. Then he prevous expresson reduces o: (2)
p c ( q f ) θ = p = + θ (3) q q If demand s consdered as a funcon of prce, mus be equal o he oal producer s oupu. Then, he prevous equlbrum condons can be joned wh demand funcon o consue a se of +1 equaons wh +1 unknown values, producons and demand: c p = + θ q = q d( p) ( q f ) Dependng on he applcaon, hs marke formulaon can be mplemened wh dfferen deal levels: Producon can be consdered n a sngle producon block, n wo producon blocks peak and base-load for example or more blocks, up o hourly or smaller blocks. The res of parameers mus be also dsaggregaed block by block so he sze of he problem ncreases. For example, n (Sánchez e al., 2005), a load duraon curve for each monh, dvded n blocks of 10 hours was consdered. However, n (Sánchez e al., 2005) here were no olgopoly bu perfec compeon, whch s equvalen o use a conjecured prce response equal o 0. The dervave of cos wh respec o producon margnal cos can be consdered as a consan, as a lnear funcon, or a sepwse funcon f margnal cos s consdered as consan for each power plan. For example, n (Sánchez e al., 2005), he sepwse funcon s consdered, as each group was represened ndvdually, wh s own margnal coss. Conjecured prce response can be consdered as a consan value or can be acualzed as wll be explaned laer. The quany ha s conraced can also be consdered as a known value, bu can be also compued from marke condons, as wll be shown. The funcon ha esablshes he relaonshp beween prce and demand can be a consan (nelasc demand), a lnear funcon, a quadrac funcon or a sepwse lnear funcon. (4) So, a each smulaon sep, havng he avalable generaon capacy of each company wh s coss, he demand, he quanes conraced prevously ha call for delver a hs sep and he conjecured prce response, he prce and he power produced by each group and by each company can be obaned. In (Balle and Barquín, 2005), a dealed descrpon of a medum-erm model whch calculaes prces and oupus usng hs heory s explaned. 4.2.2 Conjecured-prce-responses esmaon In he prevous secon, conjecured prce response has been consdered as a known value for each generaor. Compuaon of hese values s complex and requres sophscaed echnques o analyze hsorcal daa, see chaper 2 of (Bunn, 2003). When no hsorcal daa are avalable or when a long-erm represenaon s requred, as n our
case, alernave approaches mus be chosen. If a generc supply funcon S(p) s supposed for each company, he prevously conraced quany for each company s consdered as proporonal o demand f =α.d and supply funcon of he companes s acceped as proporonal o company sze (homogeney hypohess) S (p)=β.s(p), hen can be proven ha marke equlbrum condons lead o he followng dfferenal equaon: 1 S ( p) c S ( p) p = β α p p β (5) ( S ( p) ) 0 Ths s he Rudkevch equaon, a well-known expresson n he sudy of elecrcy marke prce usng he so-called supply funcon equlbra (Rudkevch e al., 1998). Solvng hs equaon, an analycal expresson for prce wh respec o he producon of he company s obaned, ha can be dfferenaed o oban he conjecural prce response. Ths response wll be an analycal formula dependng on he margnal cos funcon of he company, among oher varables. Some dfferen alernaves are possble a hs pon: Cos funcons can range from lnear o sepwse. Demand could be elasc; however requres reformulang he prevous expresson. Wh hs, a each me sep n he model and prevously o he marke prce calculaon, a conjecured prce response for each company can be obaned from cos funcons, quanes prevously conraced and companes szes. 4.2.3 Fuure markes So far, forwarded conraced quanes f have been consdered as known. These quanes are perodcally decded by generaon companes and depend on marke and sysem condons. There s an open dscusson n he leraure quesonng wheher forward conracng ncreases or reduces marke power, sarng wh he semnal paper (Allaz and Vla, 1993). Wha s obvous s ha he presence of hs knd of markes modfes generaor s behavour and prce dynamcs and as a consequence condon plannng decsons. Man concluson n (Allaz and Vla, 1993) s ha, even n he absence of rsk-averson, he nroducon of a fuure marke prevous o a spo marke, ncrease compeon. Some oher auhors have reached he same concluson n wha has been called he procompeve rend. Bu n he las years, a dfferen rend has appeared whch concludes he oppose. The man dfferen assumpon beween hese wo rends s ha n (Allaz and Vla, 1993), a wo-perod game s consdered (fuure marke followed by a spo marke) whle n he oppose rend, a mul-perod game s consdered. When consderng mul-perod, some parcular effecs appear reducng he compeve effec of he wo-perod case. Some of hese effecs are colluson -see for example (Lsk and Monero, 2005)- and he nfluence of he curren spo prce n he prce of he nex fuure markes -see (Amaya e al., 2006)-. (Allaz and Vla, 1993) compue an equlbrum model n wo sages n order o oban opmum quanes conraced n he fuure marke. Man assumpons for hs model are Courno compeon, symmerc duopoly, consan margnal coss for each company and a fuure marke where conracs raded call for delvery durng he nex spo marke. Wh hs, a formula for he quany o be conraced by each company n he fuure
marke s obaned as a funcon of coss, and demand elascy. In (Amaya e al., 2006) he same problem s solved under he same assumpons bu consderng he nfluence of he curren spo prce n he nex fuure marke. The funcon obaned depends now also on he esmaed quany o be conraced n he nex fuure marke and on a dscoun facor. For he symmerc duopoly case hese formulas can be expressed as: f 1 = c λ δ ( ) 1 2 a + f21 + f22 q1 p1 5 a d q = + (7) 1 In hese formulas, f j s he quany conraced by company n he fuure marke a nsan j. When calculang quany for fuure marke 1, quanes for forward marke 2 are esmaons. Parameer δ s he dscoun facor. The dervave of he nex fuure marke prce λ 2 wh respec o he curren spo prce p 1 s equal o 0 n (Allaz and Vla, 1993) and equal o 1 n (Amaya e al., 2006). Compung hese formulas whn he sysem dynamcs model for each smulaon sep allows calculang quanes conraced n long-erm conrac markes ha call for delvery n he nex spo markes. The above formulas can be easly exended o nclude several asymmerc companes. Moreover, oher assumpons can be nroduced n hese formulaons lke dfferen compeon models (Berrand, conjecured prce responses), nelasc demand, quadrac funcons for he coss of each company or even rsk averson. Whou rsk averson, prce for he long-erm conracs s equal o he expeced spo prce. When consderng rsk averson, a rsk premum s added o he expeced spo prce. For example, a rsk-averse demand can be consdered usng a smple uly quadrac funcon lke hs: µ U = E[ pr pelec] var[ pr p (8) elec] 2 elec ( ) p = f λ + q f p (9) In hese equaons, p elec s he prce o be pad by he demand for he elecrcy, p r s a reservor prce for he demand, E[x] s he expeced value of x, var[x] s he varance of x and µ s a rsk-averson parameer. Maxmzng hs uly funcon, a funcon whch relaes he rsk premum wh he quany conraced s obaned. [ p] [ p] λ E f = 1 (10) a var So, fuure markes represenaon whn a sysem dynamcs model can be done usng formulaons as he above explaned, wh dfferen complexy levels dependng on he assumpons consdered. The funcons for he quanes conraced and he prce of he conracs depend on varables ha can be obaned easly, eher exogenously or endogenously, wh he sysem dynamcs model for each smulaon sep. These varables are cos funcons, demand, conjecured prce responses and rsk-averson parameers. Is mporan o noe, ha whaever approach and assumpons are chosen, hey mus be coheren wh hose chosen n he spo marke equlbrum and n he (6)
conjecure prce varaon esmaon mehod. For example, f demand s consdered elasc n he fuure markes equaons, should be elasc also n he conjecured-prceresponse esmaon. Alernaves a hs pon are: exendng formulaon o consder more ypes of conracs (peak and off-peak conracs or longer conracs, for example) consder effecs of curren spo prces for he nex year only, or for more years 4.3 Invesmen decsons All he models consder expeced profably as he man decson crera for buldng new plans. In some cases addonal crera are ncluded. Some models consder new plans globally whou assgnng hem an owner, as (Ford, 1999), (Boerud e al., 2002), (Vogsad, 2005), (Olsna e al., 2006), (Ochoa, 2007) and (Kadoya e al., 2005). The res of models dsaggregae he agens consderng her decsons separaely. In (Ford, 2001a) decsons are dfferen because of dfferen prces forecasng dependng on agen s nformaon. The agens are dvded n belevers, pre-couners and followers. The model n (Gary and Larsen, 2000) dsngushes decson crera for bg generaon companes ha consue a duopoly, ha decde usng profably and an objecve marke share; and IPPs (ndependen power producers) ha subsue marke share by an opmsm facor. Profably s also consdered n (Bunn and Larsen, 1992), (Bunn e al., 1993) and (Bunn and Larsen, 1994) as decson crera, and s compued comparng capacy paymens wh a reference value based on relably compuaons. The agens use dfferen dscoun raes, o nroduce her marke share objecves. Ths paper suggess an alernave mehod o dfferenae he agens when hey decde her nvesmens based on cred rsk heory deas. In many real suaons, agen s decsons are based on ne presen value (NPV) ha s compued for each of hem usng a dfferen dscoun rae. Dfferences n hs dscoun rae are relaed o cred rsk. Dscoun rae uses o be consued by rsk-free rae r, ha s commonly known, and a rsk premum w, ha s relaed o cred rsk. If no dvdends are pad o shareholders (γ = 0) he value v of a quany V ha s len n me o be reurned n me T s affeced by hs rsk: ( ) wt vt (, ) = e V Ths cred rsk can be advanageously modelled usng he Black-Scholes-Meron deb prcng model (See chaper 5 of (Duffe and Sngleon, 2003). In hs model a deb s faled by a company f s asses value A T goes below s deb V. Company asses value s s equy value plus s deb. Consequenly, a me of deb ssue and evaluang shares value as a call opon C over asses wh V srke: v(, T ) = A C( A, V, r, γ, T, σ ) (12) ( 1 2 γ ( T ) r( T ) (13) C A, V, r, γ, T, σ ) = A e N( v ) V e N( v ) v log A logv + = σ 2 ( r σ / 2) γ ( T ) 1 (14) T (11)
v = v σ T (15) 2 1 N(x) s probably for a sandard normal o be below x and σ s a parameer ha can be esmaed from company value A evoluon volaly. From he prevous expressons w can be obaned. The use of hs schema n he model requres separaely compung agens dscoun rae r+w for each smulaon sep o compue NPV of a possble nvesmen. Asses value A and deb D mus be also recompued a each sep. A ncludes company lqud asses L and nfrasrucures I. Lqud asses are updaed as: ' L = 1 L + M rd + D + NI ϕ (16) M s operaonal profs, rd deb neres, D new deb, NI new nvesmen (assumng D = NI ) and φ deb redempon. I can be updaed makng I equal o s nomnal value B. Le ab be nfrasrucures deprecaon, hen: B = B + ab + NI (17) 1 Oher alernave s o compue I from marke value. If nfrasrucures profably s assumed as consan wh a value: ' M r = (18) B hen nfrasrucures, wh a esmaed lfe spam T, have a value: ' T ' r ' = ' = 0 I e r B Deb s also updaed, usng he followng expresson. (19) D = 1 D + N + ϕ (20) By compung he above equaons for each smulaon sep, and applyng Black- Scholes-Meron deb prcng model, a value for he dscoun rae of each company s obaned based on s fnancal and economc srucure. Ths dscoun rae allows compung a dfferen NPV for each company whch leads o dfferen nvesmens. A dfferen aspec regardng nvesmen decsons s he oal quany ha a company s gong o nves dependng on he expeced profably calculaed. Ths paper does no provde a concree alernave o hs pon. Wha s obvous s ha he greaer he expeced profably, he greaer he quany nvesed. And seems obvous also ha here should be a maxmum lm. Reasons for hs maxmum lm are dscussed n (Olsna e al., 2006). For example, under a very hgh expeced profably suaon, companes area aware of he poenal danger of massve enres. Moreover, n olgopoly markes, domnan players mgh lm her own nvesmens because could decrease he profably of her own capacy n place. Fnally, here s a fnancal consran o fund smulaneously many nvesmen projecs. Increasng he deb of he company wll ncrease s cred rsk and would make new nvesmens less profable. Furhermore, nowadays, companes are very worred abou her cred rang, whch depends also on
he deb o equy rao. The mehod proposed n (Olsna e al., 2006), where he quany nvesed by each company depends on a profably ndex (nernal rae of reurn akng no accoun he opporuny cos of posponng he nvesmen- dvded by he requred rae of reurn) seems o be an accurae mehod o represen hs quany choce. Ths mehod could be combned wh he Black-Scholes-Meron deb prcng model explaned above n order o calculae endogenously he requred rae of reurn for each company. 5 Case Sudy In hs secon, a smple case sudy based on he Spansh elecrcy marke s presened. In order o es he model rgorously, more cases and sensvy and robusness analyss should be carred ou, apar from he resuls shown here. As he man am of hs paper s a mehodologcal one, we have preferred o avod hs n order o focus he paper on he explanaon of he new mehods proposed (prevous secons). So he objecve of hs case sudy secon s jus o presen a smple case n order o show one possble applcaon of a model ncludng a dealed represenaon of olgopoly srucures and marke power lke ours. The case sudy analyses he nfluence of he nroducon of a fuure marke n a sysem whch s based manly n a spo marke. Sraegc neracons beween fuure and spo markes have been a ho research opc n he las years as explaned n secon 4.2.3. These sudes focus manly on evaluang he effecveness of fuure markes as marke power mgaon ools. In our case, we analyse no only hs effecveness bu also he possble long-erm effecs of hese markes on nvesmens. The sysem under sudy s he Spansh marke whch s represened n grea dealed (each group of each company). The oal capacy and number of hermal groups by uly, arranged by margnal cos, are shown n Table I. Oher characerscs of hs sysem used n hs case sudy can be found n (Sánchez e al., 2005). TABLE I CAPACITY (MW) AND NUMBER (IN PARENTHESIS) OF THERMAL GROUPS BY UTILITY ARRANGED BY COST /MWh C1 C2 C3 C4 C5 C6 C7 0 15 4984 (7) 5047 (8) 160 (1) 2417 (6) 0 (0) 0 (0) 0 (0) 15 20 5291 (15) 160 (1) 1699 (7) 160 (1) 732 (3) 1577 (4) 0 (0) 20 30 0 (0) 2715 (6) 1927 (5) 400 (1) 0 (0) 0 (0) 1980 (3) 30-45 1846 (6) 2715 (7) 1014 (3) 0 (0) 753 (2) 0 (0) 120 (1) The model used follows he srucure explaned n 4.1 bu ncludng some of he mprovemens commened n hs paper. Concreely, o calculae marke prces and oupus, he model explaned n (Balle and Barquín, 2005) s used. Ths model was used also n (Sánchez e al., 2005) bu consderng perfec compeon. Now, he dfference s ha he equlbrum approach based on conjecured prce responses explaned above s used. The conjecured prce responses are calculaed endogenously
usng he mehod presened n hs paper, consderng lnear margnal cos funcons for each company and nelasc demand. Noe ha when calculang he marke equlbrum we used he sepwse funcon for he margnal cos bu when we esmae he conjecured prce response we make a smplfcaon consderng hem as lnear funcons. The fuure marke s modelled followng (Allaz and Vla, 1993) formulaon bu exendng hs o consder conjecured-prce-responses compeon n he spo, nelasc demand, quadrac cos funcons for each company and rsk-averse companes. Demand s consdered rsk-averse usng he mehod shown n he prevous. One-year conracs are assumed, ha call for delvery n he nex spo marke. Tha s, each year, a fuure marke s smulaed and hen a spo marke whch akes no accoun he quanes conraced n ha prevous fuure marke n he companes sraegy s calculaed. To smplfy, we consder ha jus he wo man companes n he Spansh sysem are able o conrac forward. Companes based her nvesmen decsons on a NPV calculaon. To calculae he dscoun rae used by each company n her own NPV, he Black-Scholes-Meron deb prcng model commened n hs paper s used. Then, each company nvess each year jus n he mos profable echnology and bulds a number of groups of hs echnology whch s funcon of he NPV and whch has a maxmum value expressed as a percenage of he asses value of he company. To compue he NPV, he companes make forecass of prces and producons of a new group of each echnology. To do hs, an esmaed prce-duraon curve s calculaed for a gven year n he fuure (n our case, 40 years afer he curren one) as f n ha year here s an opmal generaon porfolo n he sysem. Then, he curren prce-duraon curve s approxmaed sofly o he one esmaed n he fuure durng he followng years. Wh hs, an esmaon of he fuure prces s obaned whch akes no accoun he curren suaon and a reasonable hypohess for he long-erm (opmal porfolo). Once he forecass of prce-duraon curves have been made, he esmaed producon of a new group of each echnology s calculaed consderng s gong o be bded by s margnal coss. Regardng he buldng block of he general scheme n 4.1, we have consdered wo delays: one for obanng he consrucon perms and one for buldng he new plan (dfferen delays for each echnology). Once a perm s obaned, he nvesmen may be revaluaed, and he company may decde no o use he perm. In our case sudy we compare a suaon whou fuure markes wh one where a fuure marke s nroduced a he begnnng of each year, from he second year on. The am s o observe he nfluence of hs new fuure marke n he exercse of marke power and also n he nvesmen decsons. In Fg. 2 can be seen how, when a new fuure marke s nroduced he elecrcy prce decreases consderably. In our case, hs occurs because he wo man companes ener volunarly n fuure conracs and because of ha hey have less marke power n he spo where hey behave more compevely. As has been proved n he leraure regardng neracons beween fuure and spo markes, he effecveness of fuure markes as marke power mgaon ools depends grealy on he modellng assumpons. As we have commened before, we have consdered smlar assumpons as he ones n (Allaz and Vla, 1993), whch are he mos commonly acceped, bu we have exended hem. Even wh hese exensons, he resuls n (Allaz and Vla, 1993) do no change sgnfcanly and seems ha he fuure marke helps o reduce he
exercse of marke power. 180 Prce-Duraon Curve Year 2 160 Base Fuure Marke 140 120 /MWh 100 80 60 40 0 100 200 300 400 500 600 700 800 900 Hours/10 Fg. 2.- Prce-duraon curve of year 2 n he base case and n he case wh he fuure marke The average elecrcy prce for each year of he sudy horzon (20 years) s presened n Fg. 3. If we look jus a he 4 frs years, he concluson commened before s sll vald: ha s, fuure markes help o nroduce compeon n a sysem. However, f we look o he whole horzon, can be seen ha he prces n he fuure marke s case are somemes even hgher han n he base case. Here, our model s ponng ou a possble dangerous effec of fuure markes. Whou modfyng oher characerscs of he sysem, fuure markes mply a reducon n he expeced profably of he marke. Ths reducon leads o a suaon where companes wa oo much n order o make new nvesmens wha make he sysem o ener n dangerous zones of non-suppled energy as can be seen n Fg. 4, when he reserve margn (nsalled capacy dvded by peak demand) s below 1.
180 160 Average Prce Base Fuure Marke 140 120 100 /MWh 80 60 40 20 0 0 2 4 6 8 10 12 14 16 18 20 Years Fg. 3.- Average elecrcy prce n he base case and n he case wh he fuure marke 1.5 Reserve Margn 1.4 Base Fuure Marke 1.3 1.2 p.u. 1.1 1 0.9 0.8 0.7 0 2 4 6 8 10 12 14 16 18 20 Years Fg. 4.- Reserve margn n he base case and n he case wh he fuure marke
The quanes conraced by each company can be seen n fgure Fg. 5 (as a fracon of he maxmum oupu of each company). Changng he hypohess here, consderng for example he nfluence of he curren spo-prce n he followng fuure-conrac prce as n (Amaya e al., 2006) may change hese quanes conraced and so he conclusons. 0.7 0.6 Fracon conraced forward C1 C2 0.5 0.4 p.u. 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 20 Years Fg. 5.- Fracon of he maxmum oupu of each company (C1 and C2) whch s conraced n he fuure marke In our cases, nvesmens are leaded manly because of curren prce levels. Alhough he dscoun raes change consderably durng he horzon (n Fg. 6 can be observed he dscoun rae calculaed for he man company) hey do no seem o nfluence he level of nvesmens sgnfcanly. Because of our forecasng mehod, he curren suaon of he marke nfluences oo much he decsons of he companes and so he dscoun raes are less mporan. Regardng he values observed n fgure Fg. 6, n he wo cases of our sudy, as profably obaned by he companes s que hgh (eher because of he exercse of marke power or because of he hgh prces nduced by nonsuppled energy) he lqud asses of he companes ncrease a lo durng he horzon. Ths makes he rao beween asses and deb o become hgher and hgher durng he sudy horzon, decreasng consderably he dscoun rae.
0.046 0.044 Dscoun Rae C1 Base Fuure Marke 0.042 0.04 p.u. 0.038 0.036 0.034 0.032 0.03 0 2 4 6 8 10 12 14 16 18 20 Years Fg. 6.- Dscoun rae for he man company n he base case and n he case wh he fuure marke 6 Conclusons Ths paper presens mprovemens n he sysem dynamcs models n he leraure ha have been used for generaon expanson plannng, n order o oban a beer represenaon of olgopoly srucures and marke power. Several curren lberalzed elecrcy markes are domnaed by few companes whch use or could use her marke power o oban greaer profs beng dermenal o socal welfare. These facs should be aken no accoun when plannng generaon expanson by boh he companes and regulaory auhores. In he presen leraure of sysem dynamcs, olgopoly srucures and marke power have no been represened or have been represened grealy smplfed. Ths paper proposes mprovemens n marke prces and oupu calculaons, fuure markes modellng and agens dfferenaon when decdng new nvesmens. To calculae marke prce and producons, an equlbrum approach based on conjecured prce responses s proposed. Addonally, a new esmaon mehod for he conjecured prce responses parameers s shown. A fuure marke modellng approach based also on equlbrum deas s presened. Ths approach allows calculang quanes and prces conraced n fuure markes ha called for delvery n spo markes. Fnally, a mehod o dfferenae companes when decdng new nvesmens based on cred rsk heory s explaned. Ths mehod allows calculang dfferen dscoun raes for each company n order o oban he expeced profably of new nvesmens perceved by hem. These dscoun raes depend on economc and fnancal srucure of each company.
A smple case sudy has been presened o show one of he possble applcaons of a model ncludng he above mprovemens. Ths case sudy analyses he nfluence of he nroducon of fuure markes n olgopolsc sysems based manly n a spo marke. I has been shown how fuure markes may reduce marke power n he sysem bu also how hese fuure markes may have dangerous effecs n he long-erm, reducng he expeced profably n he sysem and so he level of nvesmens, decreasng he sysem relably.
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