Interruptible Physical Transmission Contracts for Congestion Management

Transcription

1 Energy Laboraory MIT EL 0-00 WP Massachses Inse of Technology Inerrpble Physcal Transmsson Conracs for Congeson Managemen Febrary 00

2 Inerrpble Physcal Transmsson Conracs for Congeson Managemen anosh Raar and Mara Ilć Energy Laboraory Pblcaon # MIT EL 0-00WP Energy Laboraory Massachses Inse of Technology Cambrdge Massachses Febrary 00

3 Inerrpble Physcal Transmsson Conracs for Congeson Managemen anosh Raar Mara Ilć Energy Laboraory Massachses Inse of Technology Cambrdge MA 039 Absrac Ths paper presens a novel congeson managemen proocol applcable for boh preblaeral and hybrd mare srcre. The mechansm s based on an Inerrpble Physcal Transmsson Conrac whch garanees physcal access o he ransmsson newor sers and provdes fnancal ncenves for he blaeral conrac holders o forfe he physcal access o he ransmsson newor. The conrac specfes a fnancal rembrsemen payable o he nsred f he sysem operaor dspach reqres a cralmen of a blaeral conrac nvolvng an necon no and whdrawal of power a a specfed se of nodes. The nsrer s compensaed n he form of an nsrance premm for provdng he servce. The conrac s srcred sch ha he rembrsemen payable o he nsred pary eqals he acal loss ncrred so ha he nsred pary s made whole wh he nsrance paymen. mlarly when he nsrer s he ysem Operaor self res o dspach he generaors sch ha he aggregae nsrance rembrsemens payable o he nsred pares s mnmzed. In hs way he ransmsson conrac mechansm ensres ha a near opmal cralmen polcy concdes wh he effcen dspach n he sysem..0 Inrodcon I s generally agreed pon ha an effecve ransmsson congeson managemen sysem shold provde adeqae economc sgnals for an effcen se of he ransmsson grd n a smple manner [5]. In addon he congeson managemen sysem shold be able o accommodae long-erm frm and non-frm blaeral conracs along wh he real-me spo mare. Dfferen ransmsson congeson managemen proocols have been sggesed so far based on hese gdelnes. The mos noable among hese nclde he Fnancal Transmsson Rghs or Transmsson Congeson Conracs TCC [7] Flowgae Rghs [4] and Usage based Physcal Transmsson Rghs [ 6]. In he TCC based approach congeson managemen s performed hrogh a bd-based cenralzed opmal power dspach and he ransmsson rens are calclaed ex-pos as he nodal spo prce dfferences. Ths approach yelds he mos opmal shor-erm dspach solon. Combnaon of Locaon Based Margnal Prce LBMP for spo congeson prcng and avalably of long-erm TCC based FTRs enable near opmal dspach and bndled ransmsson and generaon mplemenaon of shor-erm relably. However only he holders of TCCs are rs neral wh respec o any nexpeced eqpmen oages. Ths rs s aomacally born by he Transmsson Provders and/or consmers n case he sold TCCs are no smlaneosly feasble n he acal sysem operaon.

4 In he flowgae based approach he ysem Operaor defnes and allocaes a lmed nmber of physcal ransmsson rghs ha reflec he maxmm power flow capacy across he ransmsson lnes or grops of ransmsson lnes [4]. The mare parcpans who are neresed n delverng elecrcy from one pon n he elecrc power sysem o anoher are reqred o acqre a porfolo of sch flowgae rghs o bac her energy ransacon. An effcen secondary mare s hen developed for hese flow-based rghs whch cold help he ysem Operaor acheve opmal allocaon of he ransmsson capacy. The approach has several advanages or feares. For example he flowgae rghs can be assgned ndependen of he power flows and only he congesed lns reqre fnancal selemen. Moreover he vale of a flowgae rgh s never negave. However alhogh he decenralzed nare of he flowgae rghs based congeson mechansm s aracve he approach sffers from several drawbacs. Frs he nmber of flowgae rghs ha ms be defned for allocaon o neresed pares may be sgnfcanly hgh [8]. econd he ransacon and nformaon cos o raders may be very hgh [5]. Thrd mares for hese rghs may be hnly raded and herefore effcen prce dscovery may be dffcl. Fnally he raders may fnd dffcl o mae nformed decsons regardng her energy ransacons snce he flowgae rghs are defned and allocaed on a ln bass whereas he raders are neresed n pon-o-pon ransacons only. In he earler approach presened n [ 0] proposed a more general mehod for congeson managemen n whch he Transmsson Provder TP and sysem sers are free o deermne he erms of ransmsson provson and prcng eravely over me. The ransmsson conracs presened n hs paper cold be easly mplemened sng hs framewor..0 Inerrpble Physcal Transmsson Conrac Ths paper proposes an alernave congeson managemen mechansm applcable for boh pre-blaeral and hybrd mare srcre where hor-ahead or day-ahead spo mare coexss wh he physcal blaeral conracs wh a varyng degree of frmness. The mechansm s based on a novel Inerrpble Physcal Transmsson Conrac whch garanees physcal access for he mare players o he ransmsson newor and provdes fnancal ncenves for he blaeral conrac holders o forfe he physcal access o he ransmsson newor. The conrac specfes a fnancal rembrsemen payable o he nsred f he sysem operaor dspach resls n a cralmen of he blaeral conrac nvolvng an necon no and whdrawal of power a a specfed se of nodes. The nsrer sally he O and/or TP s compensaed n he form of an nsrance premm for provdng he servce. The conrac s srcred sch ha he rembrsemen payable o he nsred pary eqals he acal loss ncrred so ha he nsred pary s made whole wh he nsrance paymen. mlarly when he nsrer s he O self res o dspach he generaors sch ha he aggregae nsrance rembrsemens payable o he nsred pares s mnmzed. In hs way he ransmsson conrac srcre ensres ha he opmal cralmen polcy concdes wh he effcen dspach n he sysem. pecfcally a he onse of any ransacon he blaeral ransacon holder pary generaor company and a load servng eny naes an nerrpble physcal ransmsson

5 conrac wh a ransmsson provder whch specfes ha whenever he blaeral ransacon s craled he ransmsson provder wold pay he dfference beween he prce prevalng n he load s zone and he generaor s self seleced sre prce as compensaon. The wo pares namely ransmsson provder and blaeral ransacon holder agree on he maxmm nmber of cralmens o be made drng he lfe of he conrac he dedcble applcable owards he nsrance rembrsemen and he conrac prce 3. The hors drng whch he blaeral ransacon cold be craled s no specfed a he orgnaon of he conrac. However he nmber of mes he ransacon cold be craled s specfed n he conrac. Moreover he conrac specfes ha he load always receve replacemen power a a prce prevalen n s zone. Ths a he me of he orgnaon of he conrac he wo pares blaeral ransacon holder and Transmsson Provder agree o he followng: Nodes a whch generaor and load wold nec and whdraw power: Tme perod drng whch he conrac wll be n force: T Maxmm nmber of mes he blaeral ransacon cold be craled: X Dedcble amon: C Mnmm and maxmm amon of power o be craled a a me: q mn q max Toal maxmm qany of power o be craled drng he me perod T: Q max Prce o be pad by he blaeral ransacon holders o he TP for he ransmsson conrac: P. I can be shown ha he ransmsson provson conrac s descrbed s srcred as a Callable Forward conrac descrbed n reference [6]. The blaeral ransacon holder generaor and load n a par s long one forward conrac and shor one call-le opon wh a self-seleced sre prce. The call-le opon s srcred as a swng opon de o he varos flexbles regardng he exercse rghs nclded whn he opon. The dscon ha he blaeral ransacon holder ges on he forward prce s he swng opon prce a he me of conracng. The generaor s self-seleced sre prce for he conrac reveals he nrnsc vale of he blaeral conrac and he TP lzes hs nformaon for effcen and economc allocaon of he scarce ransmsson capacy n he real me. By sbscrbng o an nerrpble conrac wh a self-seleced sre prce or dedcble of C he blaeral ransacon holder reveals hs reservaon prce for he blaeral conrac. In he real me he Transmsson Provder lzes hs nformaon for effcen and economc allocaon of he scarce ransmsson capacy. The mechansm hs ensres ha he opmal cralmen polcy for he ransmsson provder accomplshes he economc dspach of generaors. Alhogh he ransmsson conracs are explaned wh reference o a zonal elecrcy mare srcre he mehod s eqally applcable for a pre spo prcng based srcre. Here he generaor s self-seleced sre prce s smlar o he dedcble applcable n oher nsrance schemes. 3 For he problem formlaon s assmed ha he ysem Operaor O s a for-prof eny and s revene s reglaed sng he performance-based reglaon. The specfc form of he organzaon s however rrelevan for he dscsson. 3

6 . Mehodology In hs sbsecon we wll develop a mahemacal framewor o calclae he prce of he conrac. For hs prpose le s frs defne he followng varables: T Tme perod of he conrac Any nsan drng he lfe of he conrac Tme sep N Toal nmber of me seps for me perod T po prce a node a me sep po prce a node a me sep C Dedcble for he nsrance conrac X Maxmm nmber of cralmens specfed n he nsrance conrac I Insrance paymen o he blaeral ransacon holder n he even of a cralmen I C F Prce of he forward conrac componen of he nsrance conrac J Premm for he cralmen rgh I s he dscon a blaeral conrac holder may ge for sbscrbng for X nmber of cralmen drng he lfe of he conrac P Prce of he Inerrpble Physcal Transmsson Conrac Wh hese noaons f he evolon of he prces a nodes and s gven he prce of he conrac can be calclaed as follows. respecvely Prce of he conrac P Prce of he forward componen for he conrac for he access o he ransmsson newor Dscon whch he blaeral ransacon holder ges for allowng a maxmm of X nmber of cralmens. Therefore F J Where P N F Eˆ{ } To calclae J please noe ha he overall prof of he ransmsson operaor comprses of hree componens. Prce of he nsrance conrac ha he ransmsson provder receves a he begnnng of he conrac. Insrance paymen o he blaeral ransacon holder n he even of a cralmen 3. Transmsson congeson ren ha ransmsson provder collecs from he alernae ransmsson newor ser n he even of a cralmen of he blaeral ransacon holder Ths overall prof for he ransmsson provder s gven by N N T J T { Π F I } 4

7 5 N C I C T J T F } { 3 N C T J T F 3 Therefore J can be wren as N J J 4 where ˆ max w J E w J π 5 and { } N X Wa Cral w f C 0 π 6 For dervng Eqaon 6 s assmed ha he ncremenal effec of an addonal ransmsson conrac on he prce processes whn he sysem s neglgble. Bellman s prncple of opmaly for dynamc programmng saes ha an opmal polcy ncldes opmal sb-polces [3]. The sochasc opmzaon problem saed by Eqaons 4-6 solves for he opmal cralmen polcy for he Transmsson Provder. Therefore he solon of he dynamc programmng problem yelds he opmal decson crera for he Transmsson Provder for each me sep n he lfe of he ransmsson conrac. More specfcally as mpled by Eqaon 5 a each me sep n he lfe of he Transmsson Conrac he Transmsson provder s faced wh he wo choces:. Cral he ransmsson provson o he blaeral ransacon holder pary and realze he payoff gven by he dfference of spo prces exsng n he wo zones less he nsrance paymen o he blaeral ransacon pary.. Wa nl he nex perod. The decson creron for hese choces defne he ndvdal opmal cralmen polcy for a gven me sep and he aggregaon of sch ndvdal sb-polces defnes he overall cralmen polcy for he Transmsson Provder. For he problem saemen descrbed above we are consderng he case of a sngle Inerrpble Physcal Transmsson Conrac. In real lfe he Transmsson Provder wold have several sch Inerrpble Physcal Transmsson Conracs sgned wh many blaeral ransacon holder pares. Therefore he overall opmal cralmen polcy wold nvolve a combnaon of several opmal cralmen polces wh he effec of each ransmsson

8 conracs aen no accon. Noneheless he basc prncple for he decson crera for he Transmsson Provder wold reman he same. 3.0 Algorhm For he Valaon of he Transmsson Conracs In hs sbsecon we wll develop a generalzed algorhm whch wll delneae he procedre o prce he ransmsson rgh nder he nflence of dfferen nceranes for he scheme proposed n he paper. In general he prcng of he aforemenoned Callable Forward conrac employng he swng opon cold be done sng he evolon of he prces a he generaor and load nodes. pecfcally he forward conrac prce cold be obaned by calclang he expeced prce of sochasc nodal prce dfferenal exsng beween he nodes and. The call-ype swng opon cold be prced sng he sochasc prce process of a node he spply node as descrbed n he earler secon. Unfornaely he evolon of he prce process s complcaed for several reasons. Frs elecrcy can be sored and he load demand needs o be mached wh he generaon spply n real me on an nsananeos bass. econd he power flow on a ransmsson newor has o follow Krchoff s laws. In he presence of ransmsson congeson he reslng loop flows may be dffcl o esmae de o varos nceranes ha exs n a real-lfe power sysem namely load varaon generaon bd ncerany and eqpmen oage ncerany. One of he ways o crcmven hese nceranes and proec he prce processes s o se he Probablsc Opmal Power Flow POPF as descrbed n references [5 7]. pecfcally assmng an approprae load process one cold rn POPF o esmae he evolon of he prces a dfferen nodes. The reslng prce processes cold hen be sed o prce he forward and swng opon conracs. 3. Mean Reverng Prce Process The las sbsecon descrbed a generalzed algorhm for prcng an nerrpble physcal ransmsson conrac nder he nflence of dfferen prce nceranes. ch an elaborae algorhm may be approprae for he Transmsson Provder. Oher mare players may rely on he exogenos prce processes o calclae he far vale of he conracs. ch prce processes may range from he smple lognormal prce process o more sophscaed spply-load dynamcs based prce process [3]. Ths secon wll lze a one-facor mean reverng prce process o llsrae he valaon of an nerrpble physcal ransmsson conrac. The swng opons can be valed sng a dynamc programmng approach based on a dscree me approxmaon. For hs prpose he radonal bnomal ree approach sed for he lognormal prce processes cold be exended o rnomal rees n order o accon for he mean reverng nare of elecrcy prces. De o he mlplcy of he exercse rghs we need o consrc a ml-layered rnomal model also called as a rnomal fores []. pecfcally whereas he radonal Amercan opons can be exercsed a any me drng he lfe of he conrac hey can be exercsed only once. In conras he swng opons have mlple exercses. Therefore he opmal exercse polcy ncldes several exercse oppornes whch cold be ncorporaed sng a ml-layer ree srcre. Ths sbsecon 6

9 descrbes he mehod for a sngle facor mean reverng prce process. mlar procedre cold be adoped for a wo-facor model. The maor dffcly o vale any opon for a mean reverng process s o bld a rnomal ree. Therefore we wll frs concenrae on reebldng procedre for a sngle-facor prce process sng he Hll-Whe approach [9]. The one-facor mean reverng prce process assmes ha he commody prce follows he sochasc process gven by Eqaon 7. d µ dz 7 In hs eqaon po prce µ Long-erm eqlbrm vale of Mean reverson rae σ Volaly dz Wener process Defnng X ln and applyng Io s Lemma we can descrbe he commody prce behavor by he Oornsen-Uhlenbec sochasc process [] as shown by Eqaons 8-9. µ 8 dx ln dz σ µ µ 9 The frs sage n bldng a ree for hs process s o consrc a ree for a varable * ha s nally zero and follows he process * * d d σdz 0 Ths ree s symmerc abo * 0. The varable * - * s normally dsrbed. Or obecve s o bld a ree smlar o he rnomal ree shown n Fgre. To do hs we wll need o denfy whch of he ree branch paern shown n Fgre wold apply for he ree drawn n Fgre. I can be shown ha f he branchng paern from any node s as drawn n Fgre a hen he probably assocaed wh he p branch mddle branch and down branch are gven as p p m p d respecvely whch are gven by he followng eqaons. p p p m d mlarly for he branch paern shown n Fgre b he probables are gven as 7

10 p p p d m and for he one shown n Fgre c hey are p p p d m 3 Hll and Whe [9] have shown ha he probables gven by Eqaons -3 are always posve f mn and max are se sch ha max s he smalles posve neger greaer han 0.84/ and mn - max where mn and max are he nodes a whch he branchng paern changes from he one gven by Fgre b o Fgre c and Fgre a respecvely. In he ree shown n Fgre he spacng beween he spo rae s gven as 3 σ 4 a b c Fgre : Branch paerns for a ree for a mean reverng prce process Fgre : Trnomal Tree for * gven by Eqaon 0

11 Once he approxmae ree for * s consrced he nex sage s o dsplace he nodes on he * ree o so ha he nal forward erm srcre s exacly mached. The branches shf n he new ree by an amon a for each me sep b he probables are ep he same. In order o faclae compaons le s defne a new varable Q as he presen vale of a secry ha pays $.00 f node s reached and 0 oherwse. The varables a s and Q s are calclaed sng forward ndcon. For nsance we assme ha he Q s are calclaed for me seps m. The nex sep s o calclae a m sch ha he ree correcly prces he forward conrac marng a m. The spo rae of elecrcy a node m s a m so ha he prce of he forward conrac marng a m s gven by olvng Eqaon 5 we ge a m P m n n m Q exp a m m n m m ln Qm exp ln Pm nm Fnally Q for he nex me sep can be calclaed as Q Q q exp a m 3. wng Opon Prcng m In order o vale he swng opon for he mean reverng process gven by Eqaon we frs consrc he dscree-me rnomal ree for he spo prces sng he procedre descrbed n he earler secon. Frs we wll consder a rhless verson of he swng opon wheren he qany o be receved a any exercse prvlege s fxed []. Le s assme ha he opon provdes wo exercse prvleges. In order o vale he opon we can envsage 3 levels of rnomal rees rnomal fores one each for: no exercses lef one exercse lef wo exercses lef and hree exercses lef.. A he boom level here are no exercses lef herefore he opon prce s zero.. A any oher level a any node here are wo opons: Exercse he opon and ae delvery realzng he dfferenal beween he spo prce and he sre prce as he payoff. Wa nl he nex perod. Accordngly he opon prce a any node n any level s deermned sng he followng eqaon: r { ˆ r max ˆ J } C e E J e E J 8 where C re Prce r Rs-free neres rae m

12 The above algorhm cold be easly exended o prce a more flexble swng opon whch allows any qany o be receved a any exercse whn he mnmm and maxmm qany lms of corse n a dscree seps for example 0 exercses wh he flexbly o receve any amon beween 0 and 00 MW n dscree seps of 0 MW []. In hs case he opon prcng formla gven for rhless swng opons s modfed sch ha now here are mlple opons o consder: Exercse he opon and ae delvery of n realzng he dfferenal beween he spo prce and he sre prce as he payoff. Exercse he opon and ae delvery of ns 3 ns 4 ns and so on p o he maxmm possble qany specfed n he conrac. Wa nl he nex perod. Accordngly he vale of he opon conrac a any node n any level s deermned by he followng eqaon: r r J max{ ˆ ˆ } C e E J e E J 9 where.n- for any level. 3.3 Prcng of he Forward Componen For he mean-reverng prce process descrbed by Eqaon 8 he forward conrac prce marng years from oday can be gven by [0] ln F e T ln 0 e T σ µ e 4 0 nce he vale of any forward conrac marng years from oday s gven by T F Eˆ Eqaon 0 can be sed n conncon wh Eqaon o vale he forward componen of he ransmsson conrac. 4.0 Example In hs secon we wll llsrae he valaon of he Inerrpble Physcal Transmsson Conrac for he Calforna Power Exchange. We calbraed he hsorcal zonal prces n he for he mean-reverng prce process gven by Eqaon 8. The parameers are gven n he Table. Table : Prce Process Parameers for he Elecrcy Prces n Calforna Zone Mean Reverson Rae Annal Volaly P % NP % AZ % 0

13 Table descrbes he prce of a ransmsson conrac for delvery of power beween zones AZ and P5 for dfferen nmber of cralmens. mlarly Table 3 gves he prces for he ransmsson conracs for dfferen nerzonal blaeral conracs. Table : Inerrpble Physcal Transmsson Conrac prce for AZ-P5 Power Delvery Conrac Example Daa: Generaor Zone: AZ Load Zone: P5 Tme perod of he conrac: year Power delvery qany: MW Cralmens Forward Prce Opon Prce Conrac Prce 5 $ $90.33 $ $ $76.67 $ $ $ $ $ $49.00 $ $ $ $ Table 3: Inerrpble Physcal Transmsson Conracs for Dfferen Iner-zonal Blaeral Conracs Generaor Zone Load Zone Cralmens Conrac Prce/MW AZ P5 0 $ AZ NP5 30 $ NP5 P5 40 $ Alhogh he forward and swng opon componens of he ransmsson conrac are valed ndependenly for he nmercal example descrbed above he nderlyng mahemacal framewor s dencal o he one descrbed n econ so far as he opmal decson crera for cralmen for he TP s consdered. In oher words he solon o he sochasc dynamc programmng problem yelds boh he prce for he ransmsson conrac and he opmal ransmsson cralmen polcy for he TP. For llsraon prposes we wll consder he valaon of a swng opon for he ransmsson conrac for he load resdng n zone NP5. The example daa and he procedre for valaon of he conrac are explaned n he Appendx. Fgre 3 shows wo layers of he rnomal ree for he conrac for fve me seps. The rees n he fgre depc he lely spo prce pah for elecrcy n zone NP5 n Calforna. The pper boxes n he ree show he spo prce whereas he lower boxes show he correspondng opon prce a he gven nsan. The flled boxes for he opon prces ndcae he resl of early exercses of he opon conrac. A each of hese possble early exercses he TP s lely o cral he ransmsson conrac and le oher ransmsson sers aval he ransmsson servce snce he spo prce of

14 elecrcy s relavely hgher and wold no sfy wang. nce here are only lmed nmber of ransmsson cralmen prvleges wo n hs case he TP wold need o wegh he odds of exercsng he cralmen rgh a he gven nsan and wang nl he nex possble exercse opporny. Ths a each of he exercse opporny he TP wold need o compare he opon vale of wang wh he mmedae payoff realzable wh he exercse of one of he rghs. To llsrae a he hrd me sep when only one cralmen rgh s lef and he spo prce of elecrcy happens o be $79.57/MW he opon vale of wang s $.44/MW for he swng opon whereas he mmedae exercse of he opon wold fech $9.57/MW. Therefore he TP s lely o cral he ransacon a hs parclar nsan. As descrbed earler aggregaon of sch decson crera defnes he opmal cralmen polcy over he lfe of he conrac for he Transmsson Provder. A each node: Upper vale Underlyng Asse PrceElecrcy Prces n NP5 zone of Calforna Lower vale Opon Prce Vales n shaded boxes are a resl of early exercse P P Pm Pd 0.09 Pm P Pm Pd P P Pm Pm Pd P 0.67 P P 0.8 Pm Pm Pm Pd 0.4 Pd P Pm Pd P P Pm Pd 0.09 Pm P Pm Pd P P Pm Pm Pd P 0.67 P P 0.8 Pm Pm Pm Pd 0.4 Pd P Pm Pd Conclson Fgre 3: Trnomal Fores for swng opon prcng A sond ransmsson congeson managemen proocol s essenal for compeve elecrcy mare. The nerrpble physcal ransmsson rghs acheve mare based congeson managemen for an effcen elecrcy mare. Once mplemened sccessflly he nerrpble ransmsson rgh based scheme yelds long erm opmal economc solon whle faclang opmal allocaon of he scarce ransmsson capacy n real me hereby relevng newor congeson.

15 Appendx wng Opons wng opons have been n se n he energy mares for ol and naral gas. Wh he dereglaon of elecrcy mares hey are also beng sed o hedge he rs n he elecrcy mares. The swng opon gves he holder of he conrac a rgh o repeaedly exercse an opon p o a gven nmber of mes o receve any amon of energy whn a specfed lm. The exercse of he opons has an mplc assmpon for dependence on me. Ths swng opons gves he holder he flexbly o receve any amon of energy a any me whn he consrans specfed n he conrac. These consrans relae o he qany o be receved a each exercse and he oal amon o be receved over he perod of he conrac. Followng parameers and consrans defne any swng opon. po prce of he commody a any me K re prce r Rs-free neres rae T Perod of he conrac q Qany o be receved a any exercse Mn T Max T Mnmm and maxmm qany o be receved drng perod T Mn Max Mnmm and maxmm qany o be receved a me N Nmber of exercse rghs R Refracon me perod beween consecve exercses Based on he above noaons swng opons cold be sbeced o he followng consrans: Mn Mn T N q Max q Max The frs consran defnes he consran relaed o he oal qany o be receved over he me perod T. The second consran specfes he consran regardng he amon ha cold be receved a any exercse. Fnally he hrd consran saes ha a mnmm me perod nown as he refracon me necessary o be elapsed beween wo consecve exercses. Dfferen varaons of swng opons cold be consrced based on he opon descrbed above. The examples nclde: A pre mng opon n whch he qany o be receved a each exercse s fxed. However he opon cold be exercsed mlple mes a any me drng he lfe of opon. An opon wh a predeermned sre prce K where he payoff a any exercse s deermned by he dfference beween he spo prce prevalen a me and he prespecfed sre prce K. R T 3

16 An opon where he sre prce K s se eqal o he spo or forward prce observable a some fre dae. Appendx Illsraon Of wng Opon Prcng In hs appendx we wll llsrae prcng of a rhless swng opon for elecrcy prces n he NP5 zone of Calforna. Example Daa: Load Zone: NP5 Volaly σ Mean reverson rae Inal spo prce 0 $50.00/MW re prce K $50.00/MW Lfe of he conrac T.0 year Nmber of rghs X Nmber of me seps N 5 The rnomal fores for he opon s as shown n Fgre 3. In hs Fgre a each node he vales n he pper box represen he nderlyng elecrcy prce and he vales n he lower box represen he opon prce. The flled lower boxes represen he resls of early exercses. For example he second elemen n he hrd colmn of he boom ree represens an early exercse wh a payoff of $9.57 $79.57-$ The boom level of he fores represens he ree when one of he wo rghs has been exercsed. mlarly he pper level represens he rnomal ree when here are wo rghs lef o be exercsed. To llsrae he procedre he elemen 9.57 n boom level ree nd row 3 rd Colmn s obaned as 9.57 max{ P.06P m 3.637P d } where P 0.4 P m and P d 0.8. mlarly he elemen s row 4 h colmn n he op ree s obaned as max{ p.8p m 0.0P d 6.006P.8P m 0.0P d } where P P m and P d 0.09 As he nmber of me seps N ncreases he opon prce converges o s exac vale. Acnowledgmens The ahors wold le o acnowledge he fnancal sppor from he consorm New Conceps and ofware for Compeve Power ysems a MIT Energy Laboraory. Also he frs ahor wold le o han Mr. Peer anze for he helpfl dscssons regardng he sochasc prce processes and he algorhm for swng opon prcng. 4

17 References. Zad Alaywan and Jac Allen Calforna Elecrc Resrcrng: A Broad Descrpon of he Developmen of he Calforna IO IEEE Transacons on Power ysems vol. 3 No. 4 Nov Erc Allen Mara Ilc and Zad Yones Provdng for Transmsson n Tmes of carcy: an IO Canno Do All Elecrcal Power and Energy ysems pp Dmr Berseas Dynamc Programmng and Opmal Conrol- Volme I Ahena cenfc Hng-po Chao ephen Pec hmel Oren and Rober Wlson Flow-based Transmsson Rghs and Congeson Managemen Unversy of Calforna Energy Inse Worng Paper ep he Deng and hmel Oren Prory Newor Access Prcng for Elecrc Power Unversy of Calforna Energy Inse Worng Paper Febrary Thomas Gedra and Pravn Varaya Mares and Prcng for Inerrpble Elecrc Power IEEE Transacons on Power ysems Vol. 8 No. Febrary co Harvey Wllam Hogan and san Pope Transmsson Capacy Reservaons and Transmsson Congeson Conracs March Wllam Hogan Flowgae Rghs and Wrongs Harvard Elecrcy Polcy Grop Worng Paper Ags John Hll and Allen Whe Nmercal Procedres for Implemenng Term rcre Models I: ngle-facor Models Jornal of Dervaves Fall 994 pp Mara Ilc and Fran Galana Power ysems Resrcrng: Engneerng and Economcs Chaper pp 5-08 Klwer Academc Pblshers Parc Jalle Ehd Ronn and ahs Tompads Modelng Energy Prces and Prcng and Hedgng Energy Dervaves Preprn Edardo chwarz The ochasc Behavor of Commody Prces: Implcaons for Valaon and Hedgng Jornal of Fnance Vol. 5 Isse 3 Jly 997 pp Peer anze Andre Gbna and Mara Ilc Bd-based ochasc Model for Elecrcy Prces: The Impac of Fndamenal Drvers on Mare Dynamcs MIT Energy Laboraory Worng Paper November Yong Yoon anosh Raar and Mara Ilc Congeson Managemen for Large Elecrc Power ysems Proceedngs of he 6 h Inernaonal Conference on Probably Mehods Appled o Power ysems. 5. Yong Yoon Elecrc Power Newor Economcs: Desgnng Prncples for For-prof Independen Transmsson Company and Underlyng Archecre for Relably Ph.D. Thess Febrary Rober Wlson Implemenaon of Prory Insrance n Power Exchange Mares The Energy Jornal 8 pp Chen-Nng Y Herarchcal Congeson Managemen for a Dereglaed Power Indsry Ph. D. Thess epember

18 Bographes anosh Raar s a gradae sden n he Technology and Polcy Program a MIT. He holds Bachelor of cence and Maser of cence degrees n Elecrcal Engneerng from Unversy of Mmba and Arzona ae Unversy respecvely. Pror o onng MIT he wored for GE-Harrs Energy Conrol ysems a Melborne FL. Hs research neress nclde Power ysem Economcs and Operaon. Mara Ilć s a enor Research cens n he Deparmen of Elecrcal Engneerng and Comper cence a MIT where she eaches several gradae corses n he area Elecrc Power ysems and heads he research n he same area. he has over weny years of experence n eachng and research n he feld of Power ysem. Pror o onng MIT n 987 she wored as an Asssan Professor a Cornell Unversy and an Assocae Professor a Unversy of Illnos- Urbana Champagne. Dr. Ilc receved her M.c. and D.c. degrees n ysem cence and Mahemacs from Washngon Unversy. Ls. Her man neress nclde Power ysem Economcs Operaon and Conrol. 6