Market-Clearing Electricity Prices and Energy Uplift

Transcription

1 Marke-Clearng Elecrcy Prces and Energy Uplf Paul R. Grbk, Wllam W. Hogan, and Susan L. Pope December 31, 2007 Elecrcy marke models requre energy prces for balancng, spo and shor-erm forward ransacons. For he smples verson of he core economc dspach problem, he formulaon produces a well-defned soluon o he energy prcng problem n he usual form of he nersecon of he supply margnal cos curve and he demand bds. In he more general economc un commmen and dspach models, here may be no analogous energy prce vecor ha s conssen wh and suppors he quanes n he economc dspach soluon. Uplf or make-whole paymens arse n hs condon. Comparson of hree alernave prcng models llusraes dfferen ways o defne and calculae unform energy prces and he assocaed mpacs on he energy uplf requred o suppor he leas cos un commmen and dspach. Inroducon Elecrcy markes requre energy prces for balancng, spo and shor-erm forward ransacons. In he core framework of bd-based-secury-consraned-economcdspach, locaonal energy prces ha are conssen wh marke equlbrum appear as he locaonal sysem margnal coss assocaed wh he economc dspach soluon. These prces are charged o loads and pad o supplers. The prces suppor he equlbrum soluon n he sense ha a hese prces compeve supplers and loads would have no ncenve o change her bds and would have an ncenve o follow he dspach. However, he core model does no ncorporae all he feaures of he elecrcy marke, such as he dscree naure of un commmen. Gong beyond he smple core model, here are mporan cases where here are no exac prces ha suppor he quanes deermned n he economc elecrcy dspach,.e., here are no exac prces ha suppor he economc equlbrum. In pracce, wo general ypes of problems arse n choosng and usng elecrcy prces. Frs, here may be producs and servces ha are no ncluded n he formal model, so ha he model does no produce marke prces for hese producs n he form of margnal coss. For example, reacve power may no be formally represened n he model so ha here are no reacve power prces. 1 Hence, reacve power and smlar ancllary servces mus be addressed ousde he formal model. Second, even when producs are ncluded n he formal model, here may be no se of prces for hese producs ha suppor he dspach soluon. For nsance, he core model assumes all he decson varables are connuous and, because of hs, s relavely easy o denfy he 1 Includng reacve power explcly n he model of dspach s possble n prncple, bu s no ye common pracce. W. Hogan, "Markes n Real Elecrc Neworks Requre Reacve Prces," Energy Journal, Vol.14, No.3,

2 opmal soluon n erms of boh he opmal dspach quanes and he assocaed prces. Bu for ceran crcal choces n he un commmen sage, he relevan decsons are dscree and no connuously varable. The ndvdual plan s eher off or s commed, a zero-one choce. In addon, he acual dspach may be nearly bu no exacly opmal because of sofware lmaons or operaor nervenon o address specfc consrans no accouned for n he formal model. Boh dscree decson varables and less han fully opmal soluons can produce crcumsances where here are no exac prces ha suppor he elecrcy dspach. The dscusson and analyss here address he second of hese wo ypes of problems, n whch exac prces do no exs o suppor he dspach soluon because he underlyng problem conans dscree decsons, or because he soluon s nearly bu no exacly opmal. When here s no se of energy prces ha suppors he soluon, hs requres some accommodaon n selecng a workable rule for prcng elecrc energy and reang he mplcaons for any devaon from he equlbrum soluon. The frs half of he paper develops he general nerpreaon wh accompanyng graphcal llusraons. The second half of he paper summarzes a formal model. Marke-Clearng Prces Before consderng he deals of a represenave elecrcy commmen and dspach model, a more general saemen of he ssues n erms of he fundamenals of consraned opmzaon hghlghs he crcal deas and ssues relaed o prcng. We sae he basc ssue n erms of he fundamenals of consraned opmzaon, and laer exend he conceps o he elecrcy marke model as a specal case. Consder a generc opmzaon problem for fxed y, x X ( ) ( ) = Mn f x s.. g x y. The decson varables n he vecor x mus mee wo ypes of consrans. Frs, he decson varables mus be n he se X, whch may ncorporae many dfferen consrans or specal characerscs. For example, par of he specfcaon of X may be ha some or all he varables ake on dscree values. The vecor y represens some exernal requremen ha mus be me by he choce of he decson varables, x. For nsance, n he economc dspach problem he exernal requremen mgh be o mee a ceran level of ne demand. The consran g x map he commmen and dspach decson varables no he exernal funcons ( ) requremen o mee load y. The separaon of he g( x ) consrans from hose n he se X mplcly recognzes ha here are nsances n whch he consrans g( x ) are complcang and he opmzaon problem would be easy o solve f hese consrans could be removed. For example, n he elecrcy un commmen and economc g x descrbng neracons across dspach problem we mgh have he consrans n ( ) many separae un decsons, whle he se X decomposes no many separable and ndvdually smple problems. 2

3 In he broad conex of he opmzaon leraure, he general objecve funcon f ( x ) could be some form of cos-benef funcon. The elecrcy marke nerpreaon mgh be ha hs s he cos of commmen and dspach o mee load. A more general formulaon would see he objecve funcon as some cos-benef aggregaon. For he presen purposes s smpler o explan he conceps usng he nerpreaon of mnmzng coss o mee load. As he load vares, he value of he leas-cos soluon changes accordngly. Defne he value funcon (a.k.a., mnmum cos funcon, perurbaon funcon, auxlary funcon) as { } ( ) ( ) ( ) v y = Inf f x g x = y. x X The value funcon plays a cenral role n he defnon and dervaon of equlbrum prces. The slope of he value funcon represens he margnal cos of meeng an addonal un of load. Along wh he more general dervaon when boh load and generaon are opmzed, hs margnal cos defnes he marke-clearng prce. 2 The requremens for hs nerpreaon of he margnal cos as he approprae prce are me by he core elecrcy dspach model wh connuous dspach varables and well-behaved cos funcons. For example, consder hree plans as shown n he able. The frs llusraon ulzes jus he frs wo generang uns, each wh wo levels of varable cos for up o a 100 MW each for a oal capacy of 400 MW. Plans q (MW) A B C Fxed Cos ($) Var cos1 ($/MWh) Var cos2 ($/MWh) In he frs llusraon, we also gnore he fxed cos of commng he plans, and ake no accoun only he varable coss, so ha all of he dspach varables are connuous. The ndvdual and aggregae leas oal cos for each level of load.e., he value funcon for hs example would be as shown n he fgure. 2 For a furher dscusson of marke equlbrum conceps, see Wllam W. Hogan and Brendan R. Rng, On Mnmum-Uplf Prcng for Elecrcy Markes, March 19, 2003, (avalable a hp://ksghome.harvard.edu/~whogan/mnuplf_ pdf ). 3

4 Aggregae Cos Illusraon Toal Varable Cos A&B Toal Cos A B Load Here he frs level of varable cos for plan B s he cheapes, and defnes he aggregae cos curve up o 100 MW. The succeedng ncremens n he oal cos funcon follow n order of ncreasng varable cos. A more famlar way of showng he same nformaon would be n he ndvdual and aggregae margnal supply curves, whch show he margnal cos of each un of oupu, as a funcon of ncreasng oupu, nsead of showng he aggregae cos. When he varable coss are consan over ranges, as n hs example, he resulng margnal supply funcon consss of a seres of seps. The vercal seps are he pons where he aggregae cos funcon s no dfferenable, and any prce whn he vercal segmen would suppor he quany dspach soluon. These vercal segmens presen some echncal ssues o be gnored here as no mporan o he man dscusson. The aggregae supply funcon s he horzonal sum of he supply funcons of ndvdual generang uns. Ths supply funcon could be cos-based, or he varable cos of generaor operaon, or could conss of he supply offers n a coordnaed elecrcy marke. 4

5 Margnal Cos Illusraon Margnal Varable Cos 120 A A&B Margnal Cos ($/MWh) B Load For any gven level of load, he aggregae supply funcon deermnes he marke clearng prce a he level of demand or he pon of nersecon wh he demand curve. In he elecrcy marke model, hese prces are he marke clearng prces ha sasfy he no arbrage condon ha here are no remanng profable rades among he marke parcpans. 5

6 SHORT-RUN ELECTRICITY MARKET Energy Prce ( /kwh) Prce a 7-7:30 p.m. Shor-Run Margnal Cos Demand 7-7:30 p.m. Prce a 9-9:30 a.m. Prce a 2-2:30 a.m. Demand 9-9:30 a.m. Demand 2-2:30 a.m. Q1 Q2 Qmax MW Addng more plans and more seps, allowng for offers ha are no sep funcons, embeddng n a ransmsson sysem, and so on, presen no concepual dffcules. The generalzaon of he supply funcon apples, wh ncreasng load gvng rse o ncreasng prces. Wh offers equal o varable coss, he resulng prces are ncenve compable n he sense ha gven he prces no parcpan would wsh o change s offer or change he level of supply. Movng beyond hs core model, he nex sep s o nclude he fxed cos of plan B and consder ha would no be opmal o comm plan B unl he level of load was hgh enough o absorb hese fxed coss. Ths would produce a more complcaed aggregae cos pcure, as n: 6

7 Aggregae Cos: Two Generaor Example Toal Commmen and Dspach Cos Toal Cos V A A&B Load As shown n he fgure, aggregae coss follow he paern of plan A unl he load level s hgh enough (a approxmaely 178 MW) o suppor commmen of plan B and swchng o he oal cos curve of he combnaon of A & B. Noe ha n hs case he rae of ncrease of oal cos drops, echncally, he margnal cos s no monooncally ncreasng. Ths change n he margnal cos of meeng an ncremen of new load s seen more readly n he companon fgure showng he mpled margnal cos. 7

8 Margnal Cos: Two Generaor Example Margnal Cos Margnal Cos ($/MWh) Load MC v Ths looks que dfferen han he well-behaved margnal cos or supply curve n he core model. Now he margnal cos ncreases and hen decreases, and hen ncreases wh ncreasng load. Furhermore, here may be no se of prces ha sasfy he marke equlbrum condons ha here s no arbrage, meanng ha supplers would no wan o change he dspach a he gven prces. Ths rases queson of how o defne he marke clearng prces and how o rea oher paymens needed o suppor he soluon. The anomales perss as we consder addonal plans and more complcaed suaons. For example, repeang he analyss wh all hree plans (A, B, & C) n he llusraon produces he followng represenaon of he leas-cos value funcon across dfferen load levels. 8

9 Aggregae Cos: Three Generaor Example Toal Commmen and Dspach Cos Toal Cos V A A&B A&B&C A&C Load The correspondng mpled margnal cos curve for hs hree plan llusraon exhbs repeaed nsances where he commmen decson changes he progresson of he mpled margnal coss. 9

10 Margnal Cos: Three Generaor Example Margnal Dspach Cos Margnal Cos ($/MWh) 120 MC v Load As s well known, he dscree commmen varables grealy complcae soluon of he economc commmen and dspach problem. Wh many plans and many levels of operaon, here are oo many possble combnaons. The same complexy arses n he analyss of he approprae prces. The furher examples ha follow emphasze hs pon for he case of wo plans, each wh wo levels of varable coss. Ths case s suffcen o llusrae he basc heorecal pons n a manner ha capures he essenal feaures bu s sll easy o check. The ask s o develop furher he concep of he marke-clearng prces and o deal wh he addonal measures needed o address he key feaure ha here may be no marke clearng prces ha suppor he leas-cos soluon of he value funcon v( y ). Energy Prces and Uplf In he core model for elecrcy markes, energy prces derved from margnal coss suppor he equlbrum soluon. Ths s rue n he lmed sense ha whn he formal srucure ncluded n he model he energy prces provde he approprae charges o loads and paymens o supplers. However, prces ha suppor he equlbrum soluon for energy do no provde he necessary paymens for producs and servces no ncluded n he core model. Ancllary serves such as reacve suppor, black sar capably and so on are no prced n he same way and mus be pad for n a maer separae from he formal srucure embedded n he core model. The parcular rules for deermnng hese paymens are ofen ad hoc and no derved from an nclusve model. Charges appled o 10

11 cusomers o cover hese coss are smlarly based on reasonable bu ad hoc rules ha ofen approxmae some pro raa allocaon across cusomers. These charges appled n addon o energy paymens go under he headng of he uplf followng a nomenclaure esablshed n he UK elecrcy marke resrucurng. Hence, an uplf paymen s an nheren par of energy markes. The oal cos of uplf paymens s usually relavely small and he effecs on marke ncenves are ofen gnored n formal analyss as beng de mnms. However, hs may no be rue as more and more charges are ncluded n he uplf. When we move o he more general energy model wh un commmen coss and dscree decsons, new opporunes or requremens arse o add o uplf charges. In he more general model, energy prces based on margnal coss wll no always be able o suppor he equlbrum soluon. Gong furher, here may be no se of energy prces ha would suppor an equlbrum soluon and somehng else s requred. One approach ha has been suggesed, bu no appled, s o develop alernave prcng models ha mgh elmnae a need for energy relaed uplf paymens. Snce he problem begns wh an exsence problem here s no se of prces ha would work hese alernave approaches nvolve some form of prcng rule ha replaces he unform or lnear marke-clearng prces of he elecrcy marke. A lnear prce s a sngle prce ha s apples o all ransacons and he oal revenue s smply he prce mes he quany, pq. A nonlnear prcng rule s anyhng else ha mgh nvolve dscrmnaon across ransacons or volumes wh a rule needed o deermne he oal revenue. 3 Such rules mgh volae usual prescrpons for non-dscrmnaon hrough unform energy prces. In addon, he rules could creae added ncenve problems ha would rase oher dffcules. Alhough hs s an area of possble research, s no pursued furher n he presen dscusson whch addresses lnear or unform energy prcng models. Wh lnear or unform marke clearng prces, he need arses for uplf paymens. Absen a nonlnear prcng scheme, he poenal for confscaon could lead generaors o whhold hemselves from he marke or o dsor he cos or consran parameers n her offers o ensure hemselves suffcenly hgh energy rens, wh he poenal o lead o an neffcen commmen. Mos ISOs overcome hs confscaon problem by payng unform hourly energy and ancllary servce prces wh supplemenal make-whole paymens, whch guaranee ha a un wll recover any 3 Marcelno Madrgal and Vcor H. Qunana, Exsence and Unqueness of Compeve Equlbrum n Uns Commmen Power Pool Aucons: Prce Seng and Schedulng Alernaves, IEEE Transacons on Power Sysems, Vol. X, No. X, Ocober 2001, pp Marcelno Madrgal, Vcor H. Qunana and Jose Aguado, Sable Exended Prcng o Deal wh Mulple Soluons n Un Commmen Power Pool Aucons, IEEE Poro Power Tech 2001 Conference Proceedngs, Sepember Alexs L. Moo and Francsco D. Galano, Equlbrum of Aucon Markes wh Un Commmen: he Need for Augmened Prcng, IEEE Transacons on Power Sysems, Vol. 17, No. 3, Augus 2002, pp Shangyou Hao and Fuln Zhuang, New Models for Inegraed Shor-Term Forward Elecrcy Markes, IEEE Transacons on Power Sysems, Vol. 18, No. 2, May 2003, pp Jeovan E. Sanago Lopez and Marcelno Madrgal, Equlbrum Prces n Transmsson Consraned Elecrcy Markes: Non-Lnear Prcng and Congeson Rens, November 1,

12 poron of s offer-based coss no covered by nframargnal energy and ancllary servce rens over he plannng horzon. 4 The make-whole paymens are par of he aggregae uplf charges. In he case of energy and un commmen coss, he need for uplf paymens can arse because he generaor has an ncenve o change he commmen or dspach. For example, gven he unform energy prce applcable o a parcular plan, he un commmen and economc dspach soluon may no produce enough energy revenue o cover he oal fxed and varable coss. The defc should be bounded by he oal fxed coss, bu wh only he energy paymens he generaor would be operang a a loss and would prefer no o be commed. The uplf paymen makes he generaor whole and adds he needed suppor for he dspach. Anoher possbly s ha a generaor s parally dspached and has remanng unused capacy. If he energy prce s above s varable cos, he prof maxmzng soluon mgh be o ncrease oupu and upse he aggregae energy balance. Ths condon canno occur n he core model, bu can arse n he more general framework. Dependng on how he energy prce s deermned he generaor sees opporuny coss n foregone profs from complyng wh he dspach. An uplf paymen for he opporuny cos makes he generaor whole and furher suppors he dspach. A more exensve form of opporuny coss arses for he case of uncommed plans ha would be profable a he unform energy prces. Agan, hs condon does no arse n he core model bu may well exs for any gven se of energy prces n he more general framework. Whle s somewha more conroversal o compensae generaors who are consraned off, hs s anoher form of opporuny cos and hese paymens have been par of he uplf charges ha suppor he economc dspach. These varous forms of energy and commmen cos uplf charges reduce o a smple general prncple and calculaon. Gven he energy prce, we can calculae he energy prof or loss ha would be earned by each generaor a he proposed equlbrum soluon. Gven he same energy prce, we can calculae he prof maxmzng poson for ha same generaor. In he core model, he wo resuls would be he same. In he more general model here could be a dfference and hs dfference reveals he makewhole paymen ha suppors he proposed commmen and dspach decson. The analyss below formalzes hs vew of he necessary uplf paymens. The focus s on he un commmen and energy dspach wh unform energy prces. The uplf s reaed as separae from he formal model wh de mnms ncenve effecs. In assumng ha he uplf s small and has small ncenve effecs, we do no consder he rules for allocaon of he uplf. However, he analyss of dfferen energy prcng rules addresses he mplcaons for assocaed uplf paymens and provdes a bass for evaluang alernave uplf magnudes. 4 Rameen Soshans, Rchard O Nell, and Shmuel S. Oren, Economc Consequences of Alernave Soluon Mehods for Cenralzed Un Commmen n Day-Ahead Elecrcy Markes, January (hp:// ). 12

13 Snce here s no unque defnon of he energy prce, here are alernave prce models and assocaed uplf coss. Alernave Prce Models The dscusson of he need for uplf paymens recognzes ha he exensons o he core dspach model are mporan n defnng marke equlbrum prces. In he core model, wh connuous varables and well-behaved cos funcons, he analyss pons clearly o he margnal coss as he approprae prces. In he more general case wh he dscree varables of he un commmen decsons, here s no an obvous se of prces o use. If here s o be a marke-clearng prce, uplf paymens wll be necessary o suppor he soluon. And dfferen defnons of he approprae marke-clearng prce have dfferen mplcaons for he naure of he uplf paymens. To address alernave possble prcng convenons, redefne he general problem noaon slghly o dsngush he dscree varables. Here we assume ha par of feasble se X s he requremen ha some of he varables ake on he values zero or one, hese varables may represen he on/off saus of a generang un. Resae he orgnal problem wh explc denfcaon of he neger consrans. ( ) ( xu, ) s.. X (, ) v y = Mn f x u ( ) g x = y u = 0,1. In erms of hs resaed general problem, he consrans nduce prces. We address hree alernaves o defne marke clearng energy prces: a resrced model ha comes closes o he src defnon of margnal cos; a dspachable model ha relaxes he dscree requremen and assumes ha all uns are fully dspachable and pro-raes he fxed coss; and a model ha uses he convex hull of he value funcon o fnd he bes well-behaved convex approxmaon ha emulaes he properes of he core model. Resrced Model The frs approxmaon provdes a prcng defnon and nerpreaon condoned on knowng he opmal commmen decsons. Gven he opmal O O soluon x, u, resrc ( r ) he model o mach he opmal commmens as n: ( ) ( xu, ) s.. (, ) r v y = Mn f x u X ( ) g x u = u = y O. 13

14 Ths resrced model s proposed n O Nell e al. 5 If he underlyng problem s convex for oher han he neger requremens, hen hs resrced problem s a well-behaved problem ha yelds marke clearng prces for boh energy load n y (assocaed wh he consrans n g( x )) and for capacy commmen (assocaed wh he consrans u O = u ). An aracve feaure of hs formulaon s ha wh he resrcon he model reduces o a sandard convex opmzaon problem subjec o he usual range of analyses of he prces and assocaed properes. In effec, he approach embeds he problem n a hgher dmenson ncludng he prcng of commmen varables (u) as well as energy load. Then he resrcon lms he applcaon o he soluons whch mach he opmal commmen. The soluon of he resrced model reproduces he economc dspach and provdes unform energy prces. r Over he range where v ( ) y s dfferenable, he resrced model produces mpled margnal coss exacly equal o he margnal coss descrbed above. Hence, n he case of he wo plan example, he resrced model mpled margnal coss appear as: Resrced Model Margnal Cos Example Impled Margnal Cos Margnal Cos ($/MWh) Load MC r 5 Rchard P. O Nell, Paul M. Sokewcz, Benjamn F. Hobbs, Mchael H. Rohkopf, Wllam R. Sewar, Jr., Effcen Marke-Clearng Prces n Markes wh Nonconvexes, European Journal of Operaonal Research, vol. 164, pp

15 The margnal coss may no suppor he equlbrum soluon, bu wh he approprae defnon of uplf paymens o compensae for devaons from he equlbrum, hese margnal coss could be used as he marke prces. The prces assocaed wh he resrcon consrans (.e., he requremen ha u equal he opmal commmen u O ) could be vewed as he componens of an uplf calculaon. In he dervaon of he sandard resuls from he opmzaon model, he prces for he commmen decsons can be boh posve and negave. In he fully lnear model, he effec of he commmen prces s o capure all he scarcy rens and leave he shor-run prof for each generaor exacly zero. The approach suggesed n O Nell e al. s o calculae all he commmen prces bu o apply only hose ha are posve. In effec, hs would leave he scarcy rens wh generaors who earn hem a he marke clearng prce, and pay generaors he uplf needed when he marke prces do no cover all her coss. The analyss below addresses hese uplf paymens n he comparson of he mplcaons of he alernave prcng models. Dspachable Model The second model ofen dscussed follows a procedure movaed by he reamen of nflexble uns by he New York Independen Sysem Operaor as par of he New York elecrcy marke desgn. 6 The basc dea s o approxmae he aggregae cos funcon wh a closely relaed well-behaved opmzaon model wh connuous varables. In essence, he dea s o replace he neger requremen wh a smple se of bounds bu whn hose bounds rea he commmen varables as connuous: ( ) ( xu, ) s.. (, ) d v y = Mn f x u X ( ) g x = y 0 u 1. In he conex of he un commmen problem, hs amouns o reang all he plans as connuously dspachable ( d ) wh modfed varable coss ha nclude a pro raa share of he fxed coss averaged across he full capacy of he plan. Ths oo yelds a well-behaved opmzaon problem and produces margnal coss and proposed marke clearng energy prces assocaed wh he consrans n g( x ). An aracve feaure of hs model s s smplcy. I would be easy o mplemen usng he sandard economc 6 Real-me prces are se by he deal dspach pass, n whch nflexble (.e., hey mus operae a zero or her maxmum oupu) gas urbnes are dspached economcally over her enre operang range, even f hey are no acually capable of runnng a anyhng oher han zero or her maxmum oupu. Federal Energy Regulaory Commsson Order, FERC Docke No. ER , July 19, New York Independen Sysem Operaor, Inc. FERC Elecrc Tarff, hp:// Block Uns, Impor offers, Expor Bds, vrual supply and demand Bds and commed non-fxed Block Uns are dspached o mee Bd Load wh Fxed Block Uns reaed as dspachable on a flexble bass. LBMPs are calculaed from hs dspach. Thrd Revsed Shee No See also hp:// for a descrpon of he uplf paymens. 15

16 dspach models by smply reang he uns wh fxed coss as beng commed and dspachable wh a modfed varable cos. Soluon of he model would no replcae he economc dspach, bu would produce an mpled unform energy prce. Usng he assumpons of he wo plan example, we can llusrae he aggregae cos approxmaon of he oal commmen and dspach coss n he dspachable model. By consrucon, he dspachable value funcon always les a or below he aggregae d v y v y ). cos funcon ( ( ) ( ) Dspachable Model Aggregae Cos Example Impled Toal Cos Toal Cos V d V Load The mpled aggregae cos of he dspachable model follows a form smlar o he resul of he core model. Ths yelds a paern of margnal coss ha are ncreasng n load and look lke a sandard supply curve. The mpled margnal coss are boh below and above he resrced model margnal coss. 16

17 Dspachable Model Margnal Cos Example Impled Margnal Cos Margnal Cos ($/MWh) MC r MC d Load Alhough he margnal cos curve s more lke a convenonal supply curve, he mpled prces by hemselves may no suppor he marke equlbrum soluon. There are mes when hese dspachable prces yeld oucomes where generaors would prefer o produce more or less han he equlbrum soluon, and uplf paymens wll sll be requred. Convex Hull Model One way o frame he hrd alernave prcng model s as an alernave wellbehaved convex approxmaon of he aggregae cos funcon. Any convex funcon wll produce a supply funcon ha has margnal coss ncreasng n load. The dspachable model s an example of a convex funcon ha provdes a lower bound for he aggregae cos funcon. The convex hull of a funcon s he convex funcon ha s he closes o h approxmang he funcon from below. In oher words, he convex hull v ( y) of v( y ) h s he greaes convex funcon ha s also everywhere such ha v ( y) v( y). Se asde for he momen how o oban he convex hull n general. We can examne he mplcaons of hs funcon for approxmaon of he aggregae oal coss and dervaon of he assocaed margnal cos curve. Usng he wo plan example, he convex hull yelds oal cos as n: 17

18 Convex Hull Model Aggregae Cos Example Impled Toal Cos Toal Cos V h V Load By consrucon, he oal cos of he convex hull s as close as possble o he oal aggregae cos n v( y) whle preservng he well-behaved properes of he core model. The convex hull approxmaon wll no reproduce he economc dspach, bu wll provde ncreasng unform energy prces. The resulng mpled margnal cos or supply curve appears as: 18

19 Convex Hull Model Margnal Cos Example Impled Margnal Cos Margnal Cos $/MWh) MC r MC h Load The margnal cos s ncreasng n load. However, for he same reasons as for he oher approxmaons, use of hese margnal coss may produce energy prces ha would no suppor he equlbrum soluon. There would sll be a need for a deermnaon of uplf paymens o compensae for he lack of any energy prce ha by self could suppor he oucome. Uplf Paymens The defnon of uplf paymen appled here begns wh he proposed marke clearng prce, p. The revenues receved for meeng load y wll be py and he cos of meeng hese loads wll be v( y ). Hence, he prof or loss of he preferred soluon would be ( p, y) py v( y) π =. Faced wh prces p, compeve generaors would seek o offer supply z ha maxmze profs by solvng he problem ( p) Max{ pz v ( z) } π =. z Suppose ha hs unconsraned prof soluon ncludes y as an opmal soluon. Then we say ha p suppors he equlbrum soluon y. Ths s always he case for he core model of elecrcy markes, and wll be rue for many un commmen and dspach soluons whch could be suppored by a marke-clearng prce. 19

20 However, when y s no a prof-maxmzng soluon n hs sense, he prce p does no suppor he soluon. Then n order o suppor he soluon he uplf paymen would have o make he marke parcpans ndfferen beween he proposed soluon and he unconsraned prof. Hence, he defnon of he uplf used here s he dfference beween he acual energy profs a he proposed soluon and he opmal profs gven he proposed prce: (, ) π ( ) π (, ) Uplf p y = p p y. Under hs rubrc, he usual marke paymens of he core model are py and produce prof π ( p, y). The uplf paymen compensaes for losses or foregone opporuny coss o make he supplers whole when hey accep he proposed soluon, as long as hey receve he uplf paymen n addon o he drec paymens n he energy marke a prces p. Applyng hs defnon o he hree models above we calculae he uplf paymens per megawa for he wo plan example and compare he hree cases n he fgure: Comparson of Example Uplf Coss Impled Uplf U r Uplf ($/MWh) U h U d Load Below 100 MW he approxmang funcons are dencal o he aggregae cos funcon and here are no uplf paymens. In he nermedae range he uplf paymens for he resrced model and s volale margnal cos curve produce hgh uplf paymens, somemes much greaer han he uplf paymens of he dspachable model. Above

21 MW, he resrced model and he convex hull model are dencal o he aggregae cos funcon and here are no uplf paymens. However, above 300 MW here connue o be uplf paymens n he dspachable model because he mpled energy prces are above he hghes varable cos segmen. Imporanly he uplf paymens for he prces assocaed wh he convex hull approxmaon are always less han he uplf paymens for he oher wo alernaves. Alhough he uplf paymens for he oher wo prcng models are somemes hgher and somemes lower, as shown below he convex hull model always has he lowes uplf paymens. Therefore, margnal cos prces assocaed wh he convex hull are he mnmum uplf prces proposed by Rng. 7 Ths relaonshp s no a concdence and does no depend on he parcular assumpons of he example. The convex hull approxmaon always resuls n energy prces ha are ncreasng n load. As dscussed furher below, he prces from he convex hull approxmaon also produce he mnmum possble uplf of any se of unform energy prces. Furhermore, he convex hull prces are he same as he dual soluon for a naural formulaon of he dual problem. Ths connecon o dualy heory s mporan boh for concepual reasons and n gudng us owards feasble compuaonal approaches for obanng he approprae prces. Prce Comparsons For well-behaved problems ha are oherwse convex excep for he neger consrans, hese prce model approxmaons are convex n y and we can solve for he correspondng prce suppor. For hese alernave prce models, we can have values and suppors such ha: d h r v ( y) < v ( y) < v ( y) = v( y), d h r p p p. Comparng he approxmae coss for he llusrave wo plan example, we have: 7 Brendan J. Rng, Dspach Based Prcng n Decenralzed Power Sysems, Ph.D. hess, Deparmen of Managemen, Unversy of Canerbury, Chrschurch, New Zealand, (see he HEPG web page a hp://ksgwww.harvard.edu/hepg/). Rng emphaszed he lnear case and focused on he uplf paymens, wh he dea of mnmzng hese paymens as a bes compromse o provde workable prcng mehods n he presence of devaons from he dealzed model. 21

22 Comparson of Example Aggregae Coss Impled Toal Cos Toal Cos V h V V d Load The resrced prcng problem s easy o solve, bu produces a volale prce and uplf combnaon. 8 The relaxed problem s also easy o solve, bu produces a prce ha may resul n a large uplf. The convex hull problem requres a mehod of characerzaon and soluon bu assures he mnmum uplf. The dspachable and convex hull models produce an ncreasng energy supply curve. 8 Wllam W. Hogan and Brendan R. Rng, On Mnmum-Uplf Prcng for Elecrcy Markes, March 19, 2003, (avalable a hp://ksghome.harvard.edu/~whogan/mnuplf_ pdf ). 22

23 Comparson of Example Margnal Coss Impled Margnal Cos Margnal Cos ($/MWh) MC r MC d MC h Load The combnaon of energy paymens a he unform energy prce and he uplf paymen produces he oal paymens by load and he correspondng revenues o generaors. 23

24 Comparson of Example Toal Revenues Load Paymen or Generaor Revenue Revenue ($/MWh) U+P r U+P d U+P h Load Alhough he energy supply curves are ncreasng n load for he dspachable and convex hull models, he uplf s no monoonc and he oal revenues ncrease and decrease wh oal load. The convex hull model mnmzes hs mpac because mnmzes he uplf. An examnaon of he connecon wh dualy heory provdes addonal clarfcaon and suggess compuaonal approaches for solvng he convex-hull, mnmum-uplf, prcng problem. Dualy and Mnmum Uplf Wh hs movaon, we formulae he dual of he opmzaon problem wh respec o he complcang consrans and draw he connecons o he convex hull approxmaon, marke-clearng prces and uplf paymens. Inroduce he vecor of prces (a.k.a., Lagrange mulplers) p and he Lagrangan funcon: (,, ) ( ) ( ( )) L y x p = f x + p y g x. The Lagrangan prces ou he complcang consrans and, gven p, produces a problem ha s easer o solve. For gven prces, defne he opmzed Lagrangan value as: 24

25 { ( )} ( ) ( ) ( ) Lˆ y, p = Inf f x + p y g x. x X Noe ha hs defnon makes no assumpons abou he feasble se X, whch may nclude lmaons o dscree choces. The assocaed dual problem s defned as D choosng he prces p o maxmze he opmzed Lagrangan o oban { { ( )}} ( ) ( ) ( ) ( ) L y = Sup Lˆ y, p = Sup Inf f x + p y g x. p p x X In he case of a well-behaved convex opmzaon problem, where he decson varables are connuous and he consrans ses are all convex, he opmal dual soluon produces a vecor of prces ha suppors he opmal soluon. In parcular, usng hese prces, he correspondng soluon for x embedded n v( y ) also solves he problem n Lˆ ( y, p ). Furhermore, under hese condons, we have ( ) ( ) L y v y =. In he more general suaon whou he convenen convexy assumpons, here may be no equlbrum prces ha suppor he soluon a y and we have a dualy gap, where ( ) ( ) L y v y <. To make he connecon wh he mnmum uplf, consder hs formulaon of he dual problem. For convenence here, he neger consrans are enforced as par of he consrans mplc n he se X and are no represened separaely. By defnon, for he dual soluon we have: So L ( y) v( y). { { ( )}} ( ) = ( ) + ( ) p x X Sup{ Inf { f ( x) p ( y g ( x) ) g ( x ) y} } p x X { ( ) ( ) } L y Sup Inf f x p y g x + = { } ( ) ( ) p x X x X { } ( ) = Sup Inf f x g x = y = Inf f x g x = y = v y The dfference v( y) L ( y) as when v( y ) s convex and ceran regulary condons hold, here s no dualy gap. 9 s known as he dualy gap. When equaly holds, Grbk presens an alernave represenaon of he argumen: Mokhar S. Bazaraa, Hanf D. Sheral, and C.M. Shey, Nonlnear Programmng: Theory and Algorhms, John Wley & Sons, 2nd. Edon, 1993, pp Paul Grbk, Noes (mmeo), July See also, James E. Falk, Lagrange Mulplers and Nonconvex Programs, SIAM Journal on Conrol, Vol. 7, No. 4, November 1969; Dmr P. Beseks, 25

26 { } x X { ( ) ( ) ( ) } ( ) = ( ) + ( ( )) Lˆ y, p Inf f x p y g x = Inf f x + p y z g x = z x X, z x X, z z z z { ( ) ( ) } { { ( ) x X ( ) }} { ( )} = py + Inf f x pz g x = z = py + Inf pz + Inf f x g x = z = py + Inf pz + v z { ( )} = py Sup pz v z The Fenchel convex conjugae of v s by defnon: c Noe ha v ( ) ( ) ( ). { } c v p Sup pz v z z =. p s he supremum over a se of convex (affne) funcons of p and s herefore a convex funcon of p. Hence, ˆ c L y, p = py v p. Now ( ) ( ) { } c ( ) ( ) ( ) L y = Sup Lˆ y, p = Sup py v p. Therefore, applyng he conjugae defnon agan, we have p p ( ) ( ) cc L y v y =. cc The resulng funcon v ( y ) s a closed convex funcon of y and we have 11 Consrned Opmzaon and Lagrange Mulpler Mehods, Ahena Scenfc, Belmon, MA, 1996, pp For a dscusson of he connecon beween he convex hull and he dual problem, wh an applcaon o he specal case of separable problems, see D. L, J. Wang, and X.L. Sun, Compung Exac Soluon o Nonlnear Ineger Programmng: Convergen Lagrangan and Objecve Cu Mehod, Journal of Global Opmzaon, Vol. 39, 2007, pp C. Lemarchechal and A. Renaud, A Geomerc Sudy of Dualy Gaps, wh Applcaons, Mahemacal Programmng, Seres A 90, 2001, pp Davd E. Bell and Jeremy F. Shapro, A Convergen Dualy Theory for Ineger Programmng, Operaons Research, Vol. 25, No. 3, May-June 1977, pp Harvey J. Greenberg, Boundng Nonconvex Programs by Conjugaes, Operaons Research,, Vol. 21, No. 1, 1973, pp Fred Glover, Surrogae Consran Dualy n Mahemacal Programmng, Operaons Research, Vol. 23, No. 3, May- June 1975, pp Regardng use of he perurbaon funcon and a penaly funcon, see D. L and X.L. Sun, Towards Srong Dualy n Ineger Programmng,, Journal of Global Opmzaon, Vol. 36, 2006, pp

27 cc v ( y) = L ( y) v( y). Suppose ha v( y ) s he closed convex hull of v( y ). Then 12 cc v ( y) = v( y) v( y). cc In oher words, v ( y ) equals he convex hull of v( y ) over y. Furher, under one of he regulary condons wh D p as a soluon o he dual problem, cc hyperplane (a.k.a., margnal cos) for v ( y ) a y: ( ) λ ( λ) Therefore, D p defnes a supporng { } ( ) ( ) ( ) ( ). v cc z = Sup z v c p D z v c p D = p D z+ v cc y p D y = v cc y + p D z y λ D p s a subgraden of h v ( y) = v ( y), ( ) D h p v y. In general, he prce suppors defned by he subgradens are no unque, bu all elemens h v y, suppor he convex hull. of he se characerzed by he subdfferenal, ( ) The connecon o he uplf depends on a ceran economc nerpreaon of he dualy gap. From above Hence, Therefore, he dualy gap s { { }} ( ) ( ) ( ) Lˆ y, p = py + Inf pz + Inf f x g x = z. z x X ( ) = ( ) p Sup py Inf { pz Inf { f ( x) g ( x) z} } p z x X Sup py Inf { pz v( z) }. L y Sup Lˆ y, p = + + = p { } = + + z { } p z { { }} p z { { ( )} ( ) }. ( ) ( ) = ( ) + { + ( )} v y L y v y Sup py Inf pz v z ( ) ( ) = v y + Inf py + Sup pz v z = Inf Sup pz v z py v y Gven any oupu z, he economc prof s p z 12 R.T. Rockafellar, Convex Analyss, Prnceon Unversy Press, Prnceon, NJ, 1970, p

28 ( p, z) pz v( z) π =. Ths s he dfference beween he revenues for z a prces p and he mnmum cos of meeng he requremen n z. We gve an economc nerpreaon where he frs erm n he dualy gap s he prof maxmzng oucome gven prces p: π { } π (, ) ( ) ( ) p = Sup pz v z = Sup p z. The acual economc prof whou furher uplf paymens s z ( p, y) py v( y) π =. If we have o make up he dfference n order o compensae drec losses or for foregone opporunes, hen he oal paymen s he dfference: (, ) π ( ) π (, ) Uplf p y = p p y. In oher words, he dual problem seeks a p D ha mnmzes he uplf. 13 Wh hs nerpreaon, he dualy gap equals he mnmum uplf across all possble prces p: ( ) ( ) = { ( )} ( ) p z { } v y L y Inf Sup pz v z py v y { π ( ) π ( )} ( ) = Inf p p, y = Inf Uplf p, y. p z p If D p s a dual soluon and D arg max{ ( )} y p z v z, z hen here s no dualy gap and no uplf. In hs sense, he prces n equlbrum soluon f here s no dualy gap. However, f D y arg max{ p z v( z) }, z D p suppor he hen D p does no suppor y, and here s a dualy gap equal o he mnmum uplf. Apparenly he argumen apples o an arbrary feasble soluon wh a a x X, g x = y. Then ( ) 13 Brendan J. Rng, Dspach Based Prcng n Decenralzed Power Sysems, Ph.D. hess, Deparmen of Managemen, Unversy of Canerbury, Chrschurch, New Zealand, (see he HEPG web page a hp://ksgwww.harvard.edu/hepg/) proposed choosng prces o mnmzng he uplf. Marcelno Madrgal, Opmzaon Models and echnques for Implemenaon and Prcng of Elecrcy Markes, Ph. D. hess, Unversy of Waerloo, Canada, 2000, p. 47, descrbes he connecon wh dualy heory and esablshed an upper bound on he uplf. 28

29 { } ( a) ( ) = { ( )} ( a) p z Inf { π ( p) py a f ( x ) } Inf Uplf ( p, y). f x L y Inf Sup pz v z py f x = = p Here we nerpre he uplf as he dfference beween he opmal prof and he acual prof n he arbrary dspach. The argumen can be exended o nclude condons where g( x ) defnes mxures of equaly and nequaly consrans. Noe ha under eher defnon he uplf s nonnegave. Rng proposes choosng energy prces o mnmze he uplf n he case of approxmae soluons o dspach problems. Rng s analyss ncludes he case of he dscree un commmen problems. Usng examples wh each plan havng a sngle varable cos, Rng showed ha he uplf mnmzng soluon was he same as he soluon n he dspachable prce model. 14 In he more general case wh mulple segmens havng dfferen varable coss for he same plan, as shown n he examples above, he dspachable soluon and he mnmum-uplf, convex-hull, dual-soluon prces and assocaed uplfs can be dfferen. Hogan and Rng compare he mnmum uplf prces wh hose of he resrced model. 15 Dspach-Based Prcng Approxmaons The elecrcy marke model ulzes he formulaon of a secury-consraned economc dspach. Ths formulaon ncludes many consrans o represen ransmsson operaons and relably requremens. Compuaonal approaches for solvng hese models nvolve a grea deal of ar and echnque n evaluang and managng he soluon procedures. Gven he soluon, he prcng model can explo he resuls of he process o grealy smplfy he problem and reduce he dmensonaly of he model. Transmsson Consrans Suppose ha he ransmsson consrans defnng he feasble se are ( ) KMax K x. Then we have he value funcon of he economc dspach as: (, ) ( ) ( ), ( ) Max x X p { Max} v y K = Inf f x g x = y K x K. There are many elemens o accoun for all he possble conngency consrans. We could defne he se of bndng consrans gven a soluon x as 14 Brendan J. Rng, Dspach Based Prcng n Decenralzed Power Sysems, Ph. D. hess, Deparmen of Managemen, Unversy of Canerbury, Chrschurch, New Zealand, 1995, p Wllam W. Hogan and Brendan R. Rng, On Mnmum-Uplf Prcng for Elecrcy Markes, March 19, 2003, (avalable a hp://ksghome.harvard.edu/~whogan/mnuplf_ pdf ). 29

30 { j j jmax} ( ) ( ) ( ) K x = K x K x = K. In a convex case of he core model, we can drop he non-bndng consrans and no change he soluon. 16 Ths s no rue n he more general formulaon wh he nonconvex un commmen consrans and varables. A slghly more general soluon would be o fnd a small se of lmng consrans such ha he soluon does no change. Ths would be any subse K ( x) such ha (, Max ) = { ( ) ( ) =, ( ) Max} v y K Inf f x g x y K x K. x X a Gven he nformaon n he acual dspach, x, we seek a small subse ha drops mos of he elemens of K( x ). If no consrans can be dropped, hen hs s he full consran se. Gven hese consrans we modfy he approxmaon furher by lnearzng he K x. consrans n ( ) Wh he approprae lnearzaon and dualzaon wh respec o he consrans n K ( x) or any oher couplng consrans, he model smplfes o be separable across many componens (generaors) ha make up he cos funcon. Relably Commmen In organzed markes wh organzed un commmen and dspach, a relably concern arses n a poenal conflc beween equlbrum load soluons and operaor load forecass. In he bd-based day-ahead models, load bds could n prncple defne he oal load o be me and he soluon could be obaned conssen wh ha load. However, sysem operaors also regularly forecas load over he shor horzon and seek o manan relable condons o mee ha forecas load. If he wo load levels dffer, he queson arses as wheher o solve he un commmen problem o mee he bd-n load or he operaor-forecased load. Even economss would pause a relyng solely on he perfecon of he marke o address hs relably queson. Sysem operaors argue srongly ha deference mus be gven o preservng relably under he forecas load. The resoluon of hs ssue has been o adop a heursc mehod ha follows some varan of a hree sep procedure. The frs sep would be o solve for he economc commmen and dspach usng he bd-n load. Then n a second sep wh he economcally commed uns forced on, solve a relaed commmen and dspach problem wh he forecas load. In he second sep, he relaed problem has a relably focus and uses only he fxed coss of commmen bu reas all he varable dspach coss as zero. The nen s o mnmze he ncremenal coss mposed by he relably 16 Rng, See also Wllam W. Hogan, E. Gran Read and Brendan J. Rng, Usng Mahemacal Programmng for Elecrcy Spo Prcng, Inernaonal Transacons n Operaonal Research, IFORS/Elsever, Vol. 3, No. 3/4 1996, pp

31 requremen. 17 A hrd sep can hen be ncluded o solve for he economc dspach keepng he combned commmen decsons from he second sep. One way o vew hs problem would be o see he forecas load as addng a (very large) se of consrans, doublng he sze of he commmen and dspach problem. The sequenal hree-sep procedure s an ad hoc mehod o mee he relably requremen whle avodng hs currenly prohbve compuaonal ask. In he conex of prce and uplf deermnaon, he mplcaon of he sequenal mehod s o fx he commmen bu no oherwse represen he added relably consrans n he prcng model. In deermnng prces and uplf, hs s equvalen o pckng a feasble bu no necessarly opmal soluon n he smplfed prcng model, as dscussed above. The prce from he frs sage dual problem whou he relably consran sll provdes he mnmum uplf resul. Compuaonal Mehods Solvng he un commmen and economc dspach problem nvolves exensve compuaons ha presen serous challenges. Mehods for solvng hese mxed-neger programs have advanced o he pon where hey are a regular producon ool. 18 In some cases, applcaon of a Lagrangan relaxaon mehod mgh be used n he search for a soluon o he un commmen problem. 19 A concern wh hese dual mehods s ha hey may no produce a prmal feasble soluon. However, hese mehods would produce a dual prce as a by-produc. In oher cases where a dual prce s no avalable, separae calculaons may be requred o oban he approprae energy prces. One aracon of boh he resrced and he dspachable models s ha each offers a sraghforward compuaonal model for obanng he mpled energy prces. The convex-hull, mnmum-uplf model presens a less obvous soluon mehod. I would always be possble o smply apply he Lagrangan echnques drecly, bu a more focused approach for dspached-based prcng would be preferred ha explos nformaon n he proposed commmen and dspach. In he case ha a dual prce vecor s no avalable as a by-produc of solvng he un commmen problem, a furher characerzaon of he soluon suggess an algorhm for obanng: ( ) D h h p p v y =. 17 Mchael D. Cadwalader, Sco M. Harvey, Wllam Hogan, and Susan L. Pope, Relably, Schedulng Markes, and Elecrcy Prcng, May 1998, avalable a ( ). 18 D. Sreffer, R. Phlbrck, and A. O, A Mxed Ineger Programmng Soluon for Marke Clearng and Relably Analyss, n Power Engneerng Socey General Meeng, 2005, IEEE, San Francsco, CA, Marshall L. Fsher, The Lagrangan Relaxaon Mehod for Solvng Ineger Programmng Problems, Managemen Scence, Vol. 27, No. 1, January 1981, pp

32 In he un commmen problem, here s a naural separably by un, assumng ha we dualze or prce ou he jon consrans. Suppose ( ) = ( ) f x f x ( ) = ( ) g x g x X = X v( y) = Inf f x g x = y { x X} ( ) ( ). Then v( y) = Inf f( x) g( x) = y { x X} = Inf Inf { f( x) g( x) = z} z = y { z} x X = Inf v( z) z = y. { z} Followng a smlar argumen, n he separable case we can wre he dual problem as { { }} ( ) = ( ) L y Sup py Sup pz v z p z = Sup py Sup{ pz v( z) }. p z Usng he Fenchel conjugae agan, herefore, we have c ( ) = ( ) L y Sup py v p p cc h We know ha v ( z) = v ( z) s he convex hull of ( ) hc ( ) ( ) c v p v p hull. Apparenly, =. In oher words, he conjugae of hc ( ) = ( ) L y Sup py v p p. v z. Furhermore, v s also he conjugae of s convex Bu hs s he same as he dual problem obaned by subsung he convex hulls of he componens as n:. 32

33 have ( ) h ( ) v y = Inf v z z = y { z} Hence, he convex hull of v s also. L. Snce v s self a convex funcon, we h ( ) = ( ) = ( ) ( ) L y v y v y v y Ths provdes an alernave characerzaon of he convex hull of v( y ). 20 many cases, s easy o descrbe he convex hull of he ndvdual componens. Gven hese componens we could solve he convex opmzaon problem drecly o oban a dual soluon (wh no dualy gap) ha would be he dual prces for he orgnal nonconvex problem. Ths would no necessarly reproduce he economc dspach bu would provde a prce for he dual soluon. Even when we canno wre down he full convex hull of each componen n a convenen way, he formulaon of v offers an alernave way o generae suppors or cus n a cung plane approach o solvng he dual problem. In he un commmen case, f here s a sarup cos, hen ( ) v 0 = 0 s a pon of dsconnuy. Assumng he res of he cos funcon s convex away from hs pon of dsconnuy, he convex hull s of a smple form connecng he orgn and a pon on a convex funcon. Gven a good soluon o he prmal problem for he vecor of dspach O decsons across segmens and perods, x, hen: O O O 1. For each non-convex z = g x. If z = 0, pck an arbrary pon near 1 O zero. Fnd a pon z on he ray hrough he orgn and z where here s a prce suppor for v wh v ( 1 ) 1 1 z = pz. Even for a mul-perod problem, hs s a one dmensonal search and s easy o do for he smple dspach problem of a 1 separable un. By consrucon, p s also a suppor for he convex hull, v. Use 1 p and z 1 o consruc a convex approxmaon of v, say v. For he convex cos funcons v = v. v, le ( ). In 2. Solve for he dual prce n ( ) k=1. k 3. For each un, solve Max p z v ( z ) z v y = Inf v ( z) z = y. Le hs be { z} { } for k 1 z +. Le k=k+1. 1 p. Le 20 See he same resul for un commmen problems wh lnear consrans n C. Lemarchechal and A. Renaud, A Geomerc Sudy of Dualy Gaps, wh Applcaons, Mahemacal Programmng, Seres A 90, 2001, pp James E. Falk and Rchard M. Soland, An Algorhm for Separable Nonconvex Programmng Problems, Managemen Scence, Vol. 15, No. 9, Theory Seres, May, 1969, pp

34 k k 4. Solve Max py σ σ pz v( z ), k for p sop or reurn o sep 3. k p. Check for convergence o Ths s a heursc mehod for geng a good nal soluon and suppor. Sarng wh sep 3 he mehod becomes a sandard ouer approxmaon mehod. 21 If he underlyng problem s pecewse lnear, he soluon should be obaned n a fne number of seps. A smple example llusraes. Consder anoher example wh wo plans, bu smplfed o have only one varable cos segmen. Boh have fxed coss. Plan 1 has a posve varable cos, and plan 2 has zero varable cos. Boh plans have lmed capacy. ( ) v y = Mn Fu + c x + Fu 1 2 x1, x2, u1, u2 s.. 0 x1 K1u1 0 x K u u1 = 0,1 u2 = 0,1 x + x = y The convex hulls for he ndvdual plans ulze he average coss a full dspach: cˆ F = c + 1 K F 1 1 cˆ 2 2 =. K2 In oher words, snce here s a sngle sep n he varable cos, he problem reduces o he same formulaon as he dspachable approxmaon. Then he convex hull of he oal mnmum cos funcon can be found as: v y = Mn cˆ x + cˆ x ( ) 1 2 x1, x2 s.. 0 x1 K1 0 x2 K2 x + x = y Arhur M. Geoffron, Elemens of Large-Scale Mahemacal Programmng, Pars I and II, Managemen Scence, Vol. 16, No. 11, July 1970, pp

35 Convex Hull Illusraon ( ) v y = Mn Fu + c x + Fu s.. 0 x 0 x x1, x2, u1, u2 K u u = 0,1 u = 0,1 K u x + x = y Cos F1+F2 v(y) ( ) v y = Mn c1x1+ c2x2 x1, x2 s.. 0 x 0 x 1 2 K 1 1 K 2 2 x + x = y. ˆ c1 ˆ cˆ cˆ F = c + 1 K F = 2 K 2 1 F2 ĉ 1 L(y)=v(y) F1 c1 ĉ 2 K1 K2 K1+K2 Load (y) Noe ha he mplc commmen and dspach wh v s no he same as he commmen and dspach wh v. Bu he convex hull equals he dual soluon objecve funcon and he dual prces are everywhere equvalen o he slopes of he convex hull. Applyng he above algorhm and approxmaons s rval n hs case and solves he problem n one pass because he lnear suppor defned here by he average cos s he complee convex hull for each funcon. In general, here may be peces n he componen convex hulls, and more han one pass would be requred. Elecrcy Marke Model The argumens above specalze o he elecrcy un commmen and economc dspach problem. For noaonal smplcy, he formulaon here assumes ha aggregae demand s fxed and he focus s on he economc commmen and dspach of generaon over a shor horzon of a few nerconneced perods. Operang reserve requremens are no represened. Furher, generaors are reaed as a sngle represenave generaor a each node n he grd, havng he same ndex as he node. Represenng mulple generaors, demand bds, operang reserves and smulaneous deermnaon of energy and reserve prces rases no fundamenal ssues bu would complcae he noaon. Inroducon of mulple perods addresses he dynamcs over he commmen perod. Such neremporal models always presen quesons abou nal and endng condons. For example, may be ha some uns are on lne and sll operang based 35

36 on a prevous commmen ha canno be changed. There may be an oblgaon o make uplf paymens for hese uns, bu he decsons are fxed, he coss are sunk and he commmen decson s no a choce n he prospecve un commmen problem formulaon. 22 Hence, hese uns are no par of he uplf as reaed here. The sylzed verson of he un commmen and dspach problem s formulaed as: nf SarCos sar + NoLoad on + GenCos g g,d,on,sar ( ( )) subjec o m on g M on, ramp g g, 1 ramp, sar on sar + on, 1, sar = 0 or 1, on = 0 or 1, T e g d LossFn d g = 0 ( ) ( ) max ( g d ) Flowk Fk k, d = y. Indces: nodes (and un a node) me perods ransmsson consrans k. Varables: sar 0 f un s no sared n perod = 1 f un s sared n perod 0 f un s off n perod on = 1 f un s on n perod g = oupu of un n perod d = vecor of nodal demands n perod. Consans: 22 Sco Harvey, prvae communcaon. 36

37 y m M = vecor of nodal loads n perod ramp max k = mnmum oupu from un n perod f un s on = maxmum oupu from un n perod f un s on SarCos NoLoad F = maxmum ramp from un beween perod -1 and perod = Cos o sar un n perod = No load cos for un n perod f un s on = Maxmum flow on ransmsson consran k n perod. Funcons: GenCos () = Producon cos above No Load Cos o produce energy from un n perod LossFn () = Losses n perod as a funcon of ne nodal whdrawals Flow = Flow on consran k n perod as a funcon of ne nodal njecons. k () In he noaon of he value funcon descrpon above, f ( x) s he objecve funcon, x s all he varables, y s he vecor or nodal loads for each perod { y }, he consran g( x) = y reduces o T { e ( g d) LossFn( d g) } max { Flowk ( g d )} { Fk } { d } = { y }. = 0 The remanng consrans defne he se X, and he opmal soluon value of he objecve s v( y ). The dual varables assocaed wh he fnal consrans defne he prces of neres. Dspach Based Approxmaon A sandard pracce n he dspach models s o lnearze he ransmsson flow funcons abou gven generaon and load vecors: g, d 37

38 T k ( g d) k( g d ) + ( k( g d ) ) ( g d ( g d ) ) T max k ( g d ) + ( k( g d ) ) ( g d ( g d ) ) k T max T ( Flowk( g d ) ) ( g d) Fk Flowk( g d) + ( Flowk( g d) ) ( g d) max max T Fk = Fk Flowk( g d ) + ( Flowk( g d ) ) ( g d ). T max ( Flowk ( g d )) ( g d ) Fk Flow Flow Flow Flow Flow F hen. Defnng Then. Assumng ha we are operang n a range where he volage angle dfferences are small, we wll have ha T max max ( g d ) ( Flow ( g d )) ( g d ) Fk Fk Flow + k k 0 or. Gven ha he economc dspach denfes a proposed soluon, lnearze he loss funcon abou gven generaon and load vecors: g, d T ( d g) ( d g ) + ( ( d g ) ) ( d g ( d g ) ) T T ( LossFn( d g ) ) ( d g) ( LossFn( d g) ) ( d g) LossFn( d g) LossFn LossFn LossFn Defnng LossSen = LossFn d g, ( ) = ( ) T ( d g ) ( d g ) LossFn ( d g ) (( LossFn ) ) and OffSe = T ( d g ) LossSen ( d g ) hen LossFn OffSe. We can re-wre he approxmaon of he economc un commmen and dspach wh he lnearzed funcons v max ( 0, { Fk },{ y} ) nf ( SarCos sar + NoLoad on + GenCos ( g )) g,d,on,sar subjec o m on g M on, ramp g g, 1 ramp, sar on sar + on, 1, sar = 0 or 1, on = 0 or 1, T e g d T T + LossSen g LossSen d + OffSe = 0 ( ) T max ( g d ) Flowk Fk k, d = y, 38

39 ( ) In he above, we suppressed g, d when wrng Flow g d and wre Flow k k If he value funcon v were convex as n he core model, he assocaed dual varables for he las hree complcang consrans correspond o he prces and have an nerpreaon as he sysem margnal cos of energy, he margnal cos of ransmsson congeson and he locaonal margnal cos of energy o nclude he effec of congeson and ransmsson. Unforunaely, he un commmen problem value s no a convex funcon n general, as llusraed above. Resrced Prcng Model The correspondng verson of he resrced prce model akes he proposed opmal commmen decsons { sar, on} as gven and resrcs he soluon o mach hs commmen. v r max ( 0, { Fk },{ y} ) ( SarCos sar + NoLoad on + GenCos ( g )) nf g,d,on,sar subjec o m on g M on, ramp g g, 1 ramp, sar on sar + on,, 1 = on T T T ( ) OffSe T max k ( g d ) k sar = sar, on, e g d + LossSen g LossSen d + = 0 Flow F k, d = y. Ths s a specal case of a dspach problem ha can be solved usng normal economc dspach sofware. Dspachable Prcng Model The dspachable prcng model relaxes he neger requremens for dscree zero-one represenaon of he un commmen decsons. The relaxaon model reas hese as connuous varables beween zero and one. 39

40 v d max ( 0, { Fk },{ y} ) ( SarCos sar + NoLoad on + GenCos ( g )) nf g,d,on,sar subjec o m on g M on, ramp g g, 1 ramp, sar on sar + on, 1, 0 sar 1, 0 on 1, T T T e ( g d) + LossSen g LossSend + OffSe = 0 T max Flowk ( g d ) Fk k, d = y. Ths s a dfferen ype of specal case of a dspach problem ha can be solved usng normal economc dspach sofware. Convex Hull, Mnmum Uplf, Dual Prcng Model The convex hull prcng model ha mnmzes he uplf corresponds o he dual soluon. In he core model hs s he same as he value funcon. In he general case he convex hull soluon can be dfferen. We form a dual opmzaon problem by dualzng or prcng he power balance equaon and flow consrans no he objecve funcon. 40

41 v h max ( 0, { Fk },{ y} ) max λoffse μ F + T k k py k nf g,d,on,sar T T T λ LossSen ( g d ) + μ Flow ( g d ) p d k subjec o sup m on g M on, p, λμ, + ramp g g, 1 ramp, sar on sar + on, 1, sar = 0 or 1, on = 0 or 1, subjec o μ k 0. T ( SarCos sar + NoLoad on + GenCos ( g )) λe ( g d ) k k By nspecon, we see ha he nner problem s unbounded unless he prces sasfy he relaon p = λe + λ LossSen μ Flow k k k. Therefore, an equvalen resaemen of he dual problem ha s more ransparen would be 41

42 v h max ( 0, { Fk },{ y} ) max λoffse + μ F T k k py k T sup pg + + g,on,sar subjec o nf m on g M on, p, λμ, + ramp g g, 1 ramp, sar on sar + on, 1, sar = 0 or 1, on = 0 or 1, subjec o μk 0 p = λe+ λ LossSen μ Flow. ( SarCos sar NoLoad on GenCos ( g )) k k k Ths equvalen formulaon would be useful for desgnng a compuaonal procedure. However, for he connecon o uplf, anoher equvalen formulaon presens he mnmum uplf verson of he dual problem or convex hull model: max h max = ( 0, { k },{ y} ) ( 0, { k },{ y} ) = max max ( 0, { },{ y }) + λ + μ MnUplf v F v F T v Fk OffSe kfk p y k T sup pg + + g,on,sar subjec o nf m on g M on, p, λμ, + ramp g g, 1 ramp, sar on sar + on, 1, sar = 0 or 1, on = 0 or 1, subjec o μk 0 p = λe+ λ LossSen μ Flow. k k k ( SarCos sar NoLoad on GenCos ( g )) Gven he prces, he neror problem separaes no ndvdual plan commmen and dspach problems. Ths s he Lagrangan relaxaon nerpreaon for he dual problem. The value funcon s convex n s argumens. The dual objecve s concave n 42

43 he prces and lends self o a number of soluon procedures developed for he Lagrangan relaxaon. 23 Denoe he soluon o he prmal problem by [{ g } { sar }, { on }],. The soluon o he prmal problem wll sasfy: T e ( g d ) + LossSen ( g d ) + OffSe = 0 d = y Usng hese relaons, we can wre he mnmum uplf as: MnUplf = sup ( p g SarCos sar NoLoad on GenCos ( g )) gonsar,, subjec o m on g M on, ramp g g, 1 ramp, sar on sar + on, 1, nf sar 0 or 1 p, λμ, =, on = 0 or 1, ( p g SarCos sar NoLoad on GenCos ( g )) max T + ( μk Fk μk Flowk ( g d )) k subjec o μ 0 p k k, = λ e + λ LossSen k μ Flow k The frs erms n he above uplf are he dfferences beween he opmal profs for each generang un gven he proposed prce and he acual profs a he proposed soluon. The las erms are he dfference beween he mpled value of avalable ransmsson capacy a he proposed prces and he value of he flows on he ransmsson a he proposed soluon. Dependng on he confguraon of ransmsson rghs, hs could be he dfference he paymens ha he RTO may have o make o k 23 L. Dubos, R. Gonzalez, C. Lemarechal, A Prmal-Proxmal Heursc Appled o he French Un-Commmen Problem, Mahemacal Programmng, Seres A, Vol. 104, 2005, pp

44 holder of fnancal ransmsson rghs (FTRs) and he congeson charges ha collecs from flows n he marke. 24 T max A he proposed soluon, we have ( ) Flow k g d F k. Consequenly, he FTR paymen oblgaon could exceed he congeson charges colleced by he RTO. Ths can be vewed as anoher uplf. How such a shorfall s handled vares among he RTOs. The flow on a consran a he opmal soluon o he prmal problem can be below he consran lm whle he dual assgns a nonzero shadow prce o he consran. One way n whch hs can happen s f he sysem operaor mus comm a un o enforce a consran bu he mnmum oupu of he un would cause he flow on he consran o drop below he lm. We noe ha hs approach o seng prces can be appled o mnmze requred uplfs even f he proposed commmen and schedule s no he mnmum cos soluon. Exensons and Compuaonal Tess The smple examples above llusrae he basc properes of he alernave prce models. Prelmnary ess of smple models wh mulple locaons, nework consrans, mulple perods, rampng lms, and demand bds produce resuls wh prcng properes ha are smlar o he smple graphcal llusraons. The equvalence of mnmum uplf prces wh he prces obaned from he Lagrangan relaxaon, even n he presence of a dualy gap, provdes a rch source of experence abou he behavor of such prces. 25 The need for alernave prcng models arses because of he dualy gap. I s known ha he relave magnude of he dualy gap, and hence he mnmum uplf, decreases as he sze of he problem ncreases. In oher words, as he number of plans wh maeral fxed coss ncreases, here are more choces and dsconnues a he pon of a change n he commmen decson are less pronounced. 26 The presence of ransmsson conngency consrans rases he queson of how many of he consrans can be defned as non-lmng and dropped from he dspached-based prcng model. Furher smplfcaons and specalzaons of each model would be avalable dependng on he nformaon ha would be avalable from he economc un commmen and dspach sofware. Alhough he heory esablshes he convex hull approxmaon as he dual soluon and hs mnmzes he uplf, s no clear how large he dfferences n prces and uplf would be across he hree prce models 24 Wh fnancal ransmsson rghs, he wors case would be f all he consrans wh posve shadow prces were bndng n he allocaon of FTRs, n whch case he mpled ransmsson paymen s he congeson revenue defcency. 25 For example, see A. Borghe, G. Gross, C. A. Nucc, Aucons wh Explc Demand-Sed Bddng n Compeve Elecrcy Markes, n Benjamn J. Hobbs, Mchael H. Rohkopf, Rchard P. O Nell, Hung-po Chao (eds.), The Nex Generaon of Elecrc Power Un Commmen Models, Kluwer Academc Publshers, Boson, pp L.A.F.M. Ferrera, On he Dualy Gap for Thermal Un Commmen Problems, IEEE Inernaonal Symposum on Crcus and Sysems, Volume 4, May 3-6, 1993 Page(s): Seven Sof, Power Sysem Economcs, Wley-Inerscence, 2002, pp

45 n realsc applcaons. The relave magnude mgh affec he workably of he assumpon ha he shor-run ncenve effecs of uplf paymens are de mnmus. 27 The neracon beween day-ahead and real-me prces s anoher area o address. In he core model wh rsk neuraly here s a naural connecon wh expeced realme prces approxmaely equal o day-ahead prces. Wh he nroducon of uplf, he formal connecon would presumably be more complcaed. As Rng 28 and Sof 29 pon ou, he full long run ncenve effecs of hese prcng rules o nclude he complcaons of mul-par bds, marke power, and nvesmen are no well undersood. The magnude of he dfferences s an emprcal queson ha would be addressed hrough sensvy analyss of realsc problems wh he full array of plans, offers, loads, bds, relably commmens, and ransmsson consrans. Summary Elecrcy marke models requre energy prces for balancng, spo and shor-erm forward ransacons. For he smples verson of he core economc dspach problem, he formulaon produces a well-defned soluon o he prcng problem n he usual nersecon of he supply margnal cos and he demand bds. Ths prcng suppors he equlbrum soluon and sasfes a no arbrage condon. In he more general economc un commmen and dspach models, here may be no correspondng unform energy prce vecor ha suppors he soluon. Ths nroduces a need boh defne he approprae energy prces and deermne he assocaed uplf make-whole paymens needed o suppor he soluon. Dfferen approxmaon of he opmal value funcon yeld dfferen prce and uplf resuls. Smple examples llusrae he dfferences. Examnaon of he relave magnudes of he dfferences would requre praccal compuaonal esng. Endnoes Paul R. Grbk s he Drecor of Marke Developmen and Analyss for he Mdwes Independen Sysem Operaor (MISO). Wllam W. Hogan s he Raymond Plank Professor of Global Energy Polcy, John F. Kennedy School of Governmen, Harvard Unversy and a Drecor of LECG, LLC. Susan L. Pope s a Prncpal a LECG, LLC. Ths paper was suppored by he Mdwes Independen Sysem Operaor and draws on work 27 The ncenve o manpulae uplf paymens by changng sarup cos offers s balanced by he relave ease of monorng hese coss and Regonal Transmsson Organzaons can requre hese offers o reman consan for long perods Rng, 1995, p Sof, 2002, p

46 for he Harvard Elecrcy Polcy Group and he Harvard-Japan Projec on Energy and he Envronmen. The auhors are or has been a consulan on elecrc marke reform and ransmsson ssues for Allegheny Elecrc Global Marke, Amercan Elecrc Power, Amercan Naonal Power, Ausralan Gas Lgh Company, Avsa Energy, Barclays, Brazl Power Exchange Admnsraor (ASMAE), Brsh Naonal Grd Company, Calforna Independen Energy Producers Assocaon, Calforna Independen Sysem Operaor, Calpne Corporaon, Canadan Imperal Bank of Commerce, Cenerpon Energy, Cenral Mane Power Company, Chubu Elecrc Power Company, Cgroup, Comson Reguladora De Energa (CRE, Mexco), Commonwealh Edson Company, Conecv, Consellaon Power Source, Coral Power, Cred Frs Susse Boson, Dero Edson Company, Deusche Bank, Duquesne Lgh Company, Dynegy, Edson Elecrc Insue, Edson Msson Energy, Elecrcy Corporaon of New Zealand, Elecrc Power Supply Assocaon, El Paso Elecrc, GPU Inc. (and he Supporng Companes of PJM), Exelon, GPU PowerNe Py Ld., GWF Energy, Independen Energy Producers Assn, ISO New England, Luz del Sur, Mane Publc Advocae, Mane Publc Ules Commsson, Merrll Lynch, Mdwes ISO, Mran Corporaon, JP Morgan, Morgan Sanley Capal Group, Naonal Independen Energy Producers, New England Power Company, New York Independen Sysem Operaor, New York Power Pool, New York Ules Collaborave, Nagara Mohawk Corporaon, NRG Energy, Inc., Onaro IMO, Pepco, Pnpon Power, PJM Offce of Inerconnecon, PPL Corporaon, Publc Servce Elecrc & Gas Company, PSEG Companes, Relan Energy, Rhode Island Publc Ules Commsson, San Dego Gas & Elecrc Corporaon, Sempra Energy, SPP, Texas Genco, Texas Ules Co, Tokyo Elecrc Power Company, Torono Domnon Bank, TransÉnerge, Transpower of New Zealand, Wesbrook Power, Wesern Power Tradng Forum, Wllams Energy Group, and Wsconsn Elecrc Power Company. Sco Harvey and Brendan Rng provded helpful commens. The vews presened here are no necessarly arbuable o any of hose menoned, and any remanng errors are solely he responsbly of he auhors. (Relaed papers can be found on he web a ). 46