Electricity Market Theory Based on Continuous Time Commodity Model

Transcription

1 Electrcty Market Theory Based o Cotuous Tme Commodty Model Haoyog Che 1, La Ha 1 Isttute of Power Ecoomcs ad Electrcty Markets, South Cha Uversty of Techology, Guagzhou , Cha. Emal: eehyche@scut.edu.c Mathematcal & Physcal Scece School, North Cha Electrc Power Uversty, Beg 1006, Cha. Emal: hlmath@cepu.edu.c Abstract The recet research report of U.S. Departmet of Eergy prompts us to re-exame the prcg theores appled electrcty market desg. The theory of spot prcg s the bass of electrcty market desg may coutres, but t has two maor drawbacks: oe s that t s stll based o the tradtoal hourly schedulg/dspatch model, gores the crucal tme cotuty electrc power producto ad cosumpto ad does ot treat the tertemporal costrats serously; the secod s that t assumes that the electrcty products are homogeeous the same dspatch perod ad caot dstgush the base, termedate ad peak power wth obvously dfferet techcal ad ecoomc characterstcs. To overcome the shortcomgs, ths paper presets a cotuous tme commodty model of electrcty, cludg spot prcg model ad load durato model. The market optmzato models uder the two prcg mechasms are establshed wth the Rema ad Lebesgue tegrals respectvely ad the fuctoal optmzato problem are solved by the Euler-Lagrage equato to obta the market equlbra. The feasblty of prcg accordg to load durato s proved by strct mathematcal dervato. Smulato results show that load durato prcg ca correctly detfy ad value dfferet attrbutes of geerators, reduce the total electrcty purchasg cost, ad dstrbute profts amog the power plats more equtably. The theory ad methods proposed ths paper wll provde ew deas ad theoretcal foudato for the developmet of electrc power markets. Keyword: Electrcty markets, Spot prcg, Load durato, Fuctoal optmzato 1. Itroducto I the letter of Uted States Secretary of Eergy Rck Perry (Perry [017]) to Federal Eergy Regulatory Commsso (FERC) o September 8, 017, t s addressed that short-ru markets may ot provde adequate prce sgals to esure log-term vestmets approprately cofgured capacty. Also, resource valuatos ted ot to corporate superordate etwork ad/or socal values such as ehacg reslece to resource or wres to vestmet decso makg. The creased mportat of system reslece to overall grd relablty may requre adustmets to market mechasms that eable better valuato. Ths cocluso s quoted from Quadreal Eergy Revew (QER [017]). Furthermore, the curret wholesale market prce formato rules are also doubted. Rck Perry urges FERC to take mmedate acto to esure that the relablty ad reslecy attrbutes of geerato wth o-ste fuel supples are fully valued ad develop ew market rules that wll acheve ths urget

2 obectve. I the U.S. Departmet of Eergy s Staff Report to the Secretary (DOE [017]), the problems wth the curret wholesale electrcty markets ad the relatoshp to relablty/reslece of power grds are vestgated ad several mportat fdgs are reported. It s suggested that FERC should expedte ts efforts wth states, RTO/ISOs, ad other stakeholders to mprove eergy prce formato cetrally-orgazed wholesale electrcty markets. Eergy prce formato reform s supported after several years of fact fdg ad techcal cofereces. DOE staff detfed several research topcs cludg market structure ad prcg mechasm for eablg equtable, value-based remuerato for desred grd attrbutes. The defto of electrcty commodty model, cost structure ad prce formato mechasm are the most basc problems regardless of market structure. As the physcal dfferece (electrcty produced by the power plats caot be separated oce ected to the power grd), ad the exstece of the complex physcal etwork (power system), electrcty becomes the world's most complex commodty. So the defto ad prcg theory of electrcty commodty are ot so obvous. Spot Prcg of Electrcty publshed 1988 by Prof. F. C. Schweppe of the Massachusetts Isttute of Techology s the classc lterature of electrcty prcg theory ad became the theoretcal bass of spot electrcty market desg dfferet coutres (Schweppe [1988]). Electrcty prce theory research should be dvded to two parts, amely, electrcty cost aalyss (that s, what s a reasoable prce) ad electrcty prce formato mechasm electrcty markets. I the deal electrcty market, the clearg prce should be equal to the margal cost of power plats ad margal utlty of power users. The theory of spot prcg s delcate mathematcs, but t does ot coform to the physcal characterstcs of electrcty producto ad cosumpto. I realty, the wholesale markets based o spot prcg theory has more or less problems. Besdes those reported DOE (017), spot prce ofte chages dramatcally, brgs great facal rsk to market partcpats, facal strumets ad other hedgg measures are dspesable, whch are ofte accompaed wth ufar arbtrage; most power users are capable (or utetoal) to respod to the rapdly chagg real-tme electrcty prce ad rely o retalers to covert the real-tme electrcty prce of the wholesale market to retalg packages wth smple prce structure, ad thus the theoretcal obectve of ehacg demad-supply teracto through resposve spot prcg s ot acheved; spot prces caot completely cover the vestmet cost ad result suffcet vestmet capacty, etc.. Prcple of Spot Prcg The hourly spot prce s determed by the supply/demad codtos that exst at that hour (Schweppe [1988]). I partcular, t depeds o that hour s: demad ( total ad by locato); geerato avalablty ad costs (cludg purchases from other utltes); trasmsso/dstrbuto etwork avalablty ad losses. The hourly spot prce (wthout reveue recoclato) s gve by the margal cost: () t k d () t k [Total cost of provdg eergy to all customers ow ad through the future] where () t s hourly spot prce for kth customer durg hour t ($/kwh); d () t s demad of kth k customer hour t (kwh), subect to costrats such as: eergy balace: total geerato equals total demad plus losses; k

3 geerato lmts: total demad durg hour t caot exceed the capacty of all the power plats avalable at hour t; Krchoff s laws: eergy flows ad losses o a etwork are specfed by physcal laws; le flow lmts: eergy flows over a partcular le caot exceed specfed lmts wthout causg system operatg problems. Spot prce s formed o bass of the prcple of socal welfare maxmzato classcal mcroecoomcs. I ts orgal theory, short-term ad log-term, operato ad plag are take to accout (Boh [198]). I real-world electrcty markets, spot prce s ofte calculated by Securty Costraed Ut Commtmet (SCUC), Securty Costraed Ecoomc Dspatch (SCED) ad other short-term operato optmzato model. However, the spot prce calculato stll adopts the hourly power balace model of the tradtoal ecoomc dspatch, whch dvdes the whole tradg terval to a seres of cycles, ad further dvdes each cycle to several perods, ad the the eergy balace model s calculated perod by perod. Sce the power (system demad) for each perod s assumed to be costat, the eergy balace model s equvalet to the power balace model. As show Fg. 1, the calculato, the electrcty commodty model s fact defed as follows: The area uder the etre load curve s dvded to several "slps" by tradg perod, ad the each slp s dvded to several "segmets". I each perod, each wg bdder (ut) takes a "segmet" (that s, a commodty), ad margal prce clearg mechasm, the settlemet prce for all commodty the same tradg perod s the same (that s, the hghest prce of the wg ut). Fg. 1 Electrcty commodty model spot prcg theory 3. The cotuous tme electrcty commodty model Electrcty markets based o spot prcg have the followg shortcomgs. 1) Spot prce calculato s based o tradtoal hourly dspatch (or power balace) model, ad therefore overlooks the crucal tme cotuty electrc power producto ad cosumpto, whch s of specal mportace wth large-scale tegrato of wd, solar ad other reewable eerges to power systems ad the sharp rsg requremets of flexblty; the ter-temporal costrats are ot cosdered serously (although they are metoed Schweppe [1988] but ot vestgated thoroughly). ) Formato of spot prce has a mpled hypothess that the electrcty commodty the same tradg perod are homogeeous, ad the the techcal characterstcs ad cost structure of the base

4 load, termedate load ad peak load caot be dstgushed because the tme dyamc characterstcs of the dfferet types of geerator output are ot cosdered. 3) Although the orgal spot prcg model covers resource optmzato from operato to plag over a log horzo (Boh [198]), such a large-scale optmzato problem caot be appled practce. The actual market ofte uses Securty Costraed Ut Commtmet (SCUC) or Securty Costraed Ecoomc Dspatch (SCED) models to calculate the clearg prces, whch caot gve prce sgals reflectg log-term capacty vestmet, ad the caot guaratee the adequacy of geerato capacty. 4) The producto ad cosumpto of electrcty has the dstgushg feature of tme cotuty, ad the commodty for sale or purchase s a "eergy block" wth certa durato for both the geerator ad the cosumer. Eergy ( MWh) s the ma cocer for power producers ad users, ad power ( MW) balace s maly used as the physcal costrat of power system operato (maly power flow equato). I the spot prcg theory, because tme s ot cluded the commodty model, power balace ad eergy balace are equvalet, the physcal balace costrats are drectly used as the market equlbrum codtos, ad ths s ot approprate ecoomcs. 5) Eergy type electrcty commodty has characterstcs dfferet from power type electrcty commodty. Eergy type commodty s more smlar to ordary commodty. I log term trasactos eergy type commodty actually ca be stored, maly form of coal (or other fuel) ad reservor storage, whch s cotrast to power type commodty whch caot be stored. 6) Electrcty as a kd of the basc socal product ad meas of producto, the most crtcal cosderato s to esure sustaed ad stable supply. Socety places value o attrbutes of electrcty provso beyod those compesated by the curret desg of the wholesale market, such as obs, commuty ecoomc developmet, low emssos, local tax paymets, etc. The hghest market effcecy (especally the short-term spot market effcecy) s ot so mportat. I order to emphasze the role of tme electrcty commodty, ths paper redefes the cotuous tme commodty model of electrcty P, tt t t by (power, tme) par, ad the area 1 uder the power curve s eergy, as show Fg.. Whe t t1 1 ad P cost t t t, t s 1 degraded to hourly electrcty commodty model spot prcg theory. Sce power ca be vewed as a fucto of tme, the cotuous tme electrcty commodty model ca be wrtte as Pt, tt1 t t defed the tme terval t t, ad the umercal power value the spot prce defto becomes a fucto 1,. Mathematcal theores, such as fuctoal aalyss, varatoal method, etc. are eeded for aalyss. After the troducto of the cotuous tme electrcty commodty model, the problem of socal welfare maxmzato chages from the mult-stage statc optmzato to the cotuous tme fuctoal optmzato. The soluto method s also chaged from Lagrage method to soluto of Euler-Lagrage equato.

5 P t 1 t Fg. Cotuous tme electrcty commodty model t 4. Modellg of electrcty market based o cotuous tme commodty The mcroecoomc model of socal welfare maxmzato Caramas (198) s stll used ths paper to defe the electrcty market for referece. Some varable otatos ad expressos Caramas (198) are stll used. Because the purpose of ths paper s maly to expla the basc prcples, the market model Caramas (198) s smplfed. Market partcpat cost/beeft Supposg B ( ) to be the varable cost (beeft) of market partcpat a market cycle, the cost (beeft) fucto ca be wrtte as:, 0 T B B P t t where s the set of all partcpats the market; B s a fuctoal of power curve P t for the market partcpat, B 0 deotes the geerato cost whe s a power plat ad B 0 deotes the cosumpto utlty whe s a power user; T s the durato of the market cycle. Socal welfare ad market obectve The obectve fucto s the stadard socal welfare maxmzato of the market mcroecoomcs, amely, to maxmze socal welfare = the value of electrcty cosumpto - the cost of electrcty producto The obectve fucto ca be wrtte as Costrats max W( P1, P P ) B ( P ( t)) dt (1) The key costrat s the power balace costrat, amely, whch ca be wrtte as 0 = geerato losses - cosumpto 0 e t P t L t, 0 t T, L t 0 () Note that tme t () s a cotuous varable ad o loger dscretzed as tradtoal ecoomc dspatch. P t 0 whe s a power plat, ad otherwse 0 P t whe s a power user. Lt s the loss of actve power.

6 Partcpat behavor Wth the cotuous tme electrcty commodty model, the proft maxmzato model of market partcpats wll take the form of tegral (or fuctoal of the power curve). The proft maxmzato model uder spot prcg correspods to the Rema (Rema) tegral mathematcs, ad the proft maxmzato model uder load durato (called "measure" mathematcs) prcg s troduced ths paper, whch correspods to the Lebesgue tegral. Note that the basc dea of load durato prcg has bee descrbed Elmaghraby (1999). The bascs of Rema ad Lebesgue tegral are troduced the appedx. 1) The partcpat proft maxmzato model uder spot prcg (Rema tegral form) Assumg that partcpat obtas come (for a power plat) or pays purchasg cost (for a power user) at spot prce 0 T t t chagg over tme, the partcpat's obectve s to maxmze the et proft uder producto (cosumpto) capacty costrats, amely T R t 0 max N P ( t) [ B P t t P t ] dt max s.. t P t P t P t m (3) m max For a partcular partcpat,, P t P t are ofte costats (ot chagg over tme). ) The partcpat proft maxmzato model uder load durato prcg (Lebesgue tegral form) Assumg that partcpat obtas come (for a power plat) or pays purchasg cost (for a m max power user) at the prce yp y P determed by load durato (the load curve s assumed to be mootocally creasg ths paper, so the prce fucto ca also be wrtte as a fucto of the load), the partcpat's obectve s to maxmze the et proft uder producto (cosumpto) capacty costrats, amely max P m L max N m ( y) [ B y m ( y) m ( y)] dy P max s.. t P t P t P t m where m( y) mt : P( t) y s the measure of load fucto at value y; the sg before m( y) takes + sg whe s a power plat ad takes - sg whe s a power user. Commodty model By the two dfferet prcg methods, the models of electrcty commodty are also dfferet. Uder spot prcg, the power curve P t, t t t 1 (4) of partcpat wth a tme horzo s regarded as oe tem of commodty, ad the total prce s P t t t t 1 ca further defe the per ut electrc eergy prce of the commodty: t t1 d. The we

7 t t1 t t1 P d P d (5) Uder load durato prcg, the "eergy block" of the partcpat wth a certa power rage 1 P t P t P s cosdered to be oe tem of commodty,.e. : : P t P1 P t P P t, t t : Pt P1 P P t P, Its total prce s P P1 y m ( y) dy, ad the per ut electrc eergy prce of the commodty s: Market mechasm P P1 P P1 y m m The obectve of the overall market optmzato s to maxmze the socal welfare (1) subect to the costrats (). It s obvous that the problem s a varatoal problem, ad accordg to ts y dy y dy optmalty codto, the dual varable (shadow prce) () t ca be obtaed. I the market mechasm based o spot prcg, () t s drectly used as the market prce () t for each partcpat, ad that s, t t. () t s the market equlbrum prce, ad at ths prce the dvdual's optmal of (3) s also the soluto of the overall market optmzato problem (1). I the market mechasm based o load durato, t s ecessary to fd the market prce m( y) for partcpat, whch satsfes m( y) = m( y). my ( ) (6) s the market equlbrum prce, ad at ths prce the dvdual's optmal of (4) s also the soluto of the overall market optmzato problem (1). A specfc market model ad case study s used to expoud the detals of the ew market modellg theory. 5. Soluto of electrcty market model based o cotuous tme commodty For coveece of soluto ad aalyss, ths paper cosders the ulateral competto model of geerato sde electrcty market. Assumg that the cost fucto of power plats takes the form of quadratc fucto ad gorg etwork loss, the problem of socal welfare maxmzato (1) ad () becomes geerato cost mmzato, amely,

8 m C( P, P P ) C ( P ( t)) dt = 1 s. t. P t P t 0 t T d T 0 T a P t b P t c dt (7) where d P t s the system load at tme t. Frst, the varatoal problem wth oe costrat (7) s solved, ad ts Euler-Lagrage equato s C1'( P1( t)) ( t) 0 C'( P( t)) ( t) 0 C'( P( t)) ( t) 0 Pt Pd t 0 1 (8) From the above equato we ca express P ( t) 1,, by () t. The by substtutg these expresso of Pt () 1 ad P () t to the power balace equato (7), () t ca be solved out, ad the all P ( ),, ( ) 1 t P t ca be got. The, the varatoal problem (3) the sese of Rema tegral s solved as follows. By gorg the capacty costrats, (3) s regarded as a ucostraed varatoal problem, ad the Euler-Lagrage equato s By comparg (8) ad (9), we ca get Namely () t s the market equlbrum prce. C ' P t t 0 (9) () t t (10) Whe the geerato cost fucto takes the form of quadratc fucto, we have P 1 d () t a ( t) C ' P t a P t b 1 1 a a whch shows the cosstece of spot prce wth load curve. 1 1 Fally, the varatoal problem (4) the Lebesgue tegral sese s solved. Uder the assumpto of mootocally creasg load curve ad ulateral competto of geerators, (4) ca be wrtte as follows: where m( y) mt : P( t) y Pmax 1 1 m N P [(T P y ) C y T P ( y) m ( y)] dy (1) L Pm s the measure of m ( y ) at value y. b (11) Pmax ( )} max{ P t, P t[0,t] t[0,t] m m{ P( t)}

9 where s the geerato cost. hc ( y) Pmax C ( max ) 1 P 1 1 m P m C ( Pm ) [T P y ] C y dy C ( P ) T [T P C h ] dh C ( P ) T T t 0 C P t dt Whe the cost fucto takes the quadratc form, t ca be see from (11) that P () t s a strctly mootoc creasg fucto, ad there s m ( P( t)) T t. We ca use varable substtuto P t y, the (1) turs to the followg varatoal problem: The Euler-Lagrage s T max N P [(T t) C P t t m ( P t )] P ' t dt L 0 0 T = [(T t)( a P t b ) t (T t)] P ' t dt m (13) d a P ' t( T t) [(T t)( a P t b ) t T t ] 0 (14) dt By solvg the frst order lear ordary dfferetal equato (14), we ca get the prce fucto. t O the other had, from (11) there s P t P () t a a a 1 1 ( d b ) a It s easy to fd that the coeffcet ad olear term of (14) P d () t a P t, a P t b ( t) 1 a 1 are rrelevat to, ad the the soluto of (14) s relevat to, whch meas that the prces for all power plats are the same at the optmum, that s, Namely t s the market equlbrum prce. b t t (15) 6. Dstrbuto of commodty values uder two dfferet prcg mechasm Same qualty same prce s the basc prcple for commodty prcg. Uder spot prcg mechasm, the uderlyg assumpto s that all electrcty commodtes at the same tradg perod are homogeeous ad therefore have the same margal prce (or value); ad the electrcty commodtes at dfferet tradg perods are heterogeeous ad have dfferet prces (values), as show Fg. 3. For the same power plat (.e., the cross bar cosstg of multple small segmets of the same color Fg.

10 Hetero Homo 3), the value of the electrcty commodtes produced over tme s dfferet. Ths does ot coform to the actual operato of power systems. At the same perod (vertcal bar cossted of segmets wth dfferet colors), all electrcty commodtes are homogeeous. The the obvously dfferece techcal characterstcs ad cost composto amog the base load, termedate load ad peak load caot be dstgushed. Ths s oe of the defceces of spot prcg, whch caot correctly value dfferet attrbutes provded by dfferet geerators. Hetero Fg.3 Dstrbuto of commodty values uder spot prcg Uder load durato prcg mechasm preseted ths paper, all electrcty commodtes wth the same system load durato (amely o a horzotal le represetg a specfc load level Fg. 4) are homogeeous ad therefore have the same margal prce (or value). As system load durato chages, the commodty prce chages accordgly. The shorter the load durato, the hgher the prce (ts ecoomc meag s that the peak load prce s hgher tha the base load prce). As show Fg. 4, for the same power plat (.e., the cross bar wth the same color), the value of the electrcty commodtes produced over tme s the same. O the other had, at the same perod (vertcal bar cossted of segmets wth dfferet colors), all electrcty commodtes are heterogeeous, ad the attrbutes ad prces of power produced by the base load, termedate load ad peak load uts are dfferet. Homo Fg.4 Dstrbuto of commodty values uder load durato prcg

11 7. Case study Assumg the load curve s mootocally creasg ad as follows P ( t) 350 (1 t) t [0, 1] d Assumg further that there are 3 power plats the market, ad the geerato cost s dfferet by tmes successvely, represetg the low, medum ad hgh cost power plat respectvely. The cost fucto s as follows (where a 0.001, b 0.07, c 0. ) a C1 P1 t P1 t bp1 t c C P t ap t bp t c C P t ap t 4bP t 4c From (8), the spot prce ad geerato output of each power plat are solved as follows, 4 5b ( t) C1 ' P1 t ap1 t b, P1 t Pd ( t) t, 7 7a b ( t) C ' P t ap t b, P t Pd ( t) t, 7 7a 1 4b ( t) C3 ' P3 t 4aP3 t 4 b, P3 t Pd ( t) t, 7 7a The dspatch schemes of the 3 power plats ad system load are show Fg. 5. The spot prce s show Fg. 6 (a). Obvously t s lear as the system load. The geerato costs of 3 power plats ca be obtaed: 1a C1 1 0 P t bp t c dt 1a t b50 400t c dt b c C ap t bp t c dt 1 a 90 00t b 90 00t c dt b c C ap t 4bP t 4c dt 1 a t 4b t 4c dt b 4c ) Aalyss of reveue ad proft uder spot prcg The reveues of 3 power plats ca be calculated as 1 1 R t0 t0 N P t P t dt [ t]( t) dt R t0 t0 N P t P t dt [ t](90 00 t) dt 105.5

12 1 1 R t0 t0 N P t P t dt [ t]( t) dt The profts of 3 power plats are respectvely , 39.06, 8.04, whch are show Fg. 7(a). The proft rates ( proft / cost 100% ) are respectvely 77%, 59%, 30%. The total geerato cost ad total proft of 3 power plats s 3.53 ad respectvely. The total electrcty purchasg cost ad proft rate of the market are ad 67% ( total proft / total cost 100% ) respectvely. ) Aalyss of market clearg prce, reveue ad proft uder load durato prcg Because m ( P t) 1 t, a P' t =0. 1,,3, by substtutg the example data to the correspodg Euler-Lagrage equato (14), we ca get: d a (1 t) P ' t [(1 t)( a P t b ) t(1 t)] 0 dt It ca be arraged to ordary dfferetal equato: t(1 t) t 1.t ,,3 The soluto s t 1 t 1 t t C e e (1. ) e dt 1 t 1 1 = C(1 t) (1 t) [1.(1 t) 0.7] dt t [0,1), 1,,3 1 1 = C(1 t) (1 t) (0.48t 0.6 t ) Let 0 = (0),the C 0.3, ad we have 1 1 t t=0.3(1 t) (1 t) (0.48t 0.6 t ) t [0,1), =1,,3 Because the measuremet m1 t, the 1 1, ( m) 1 t =0.3 m m (0.48(1 m) 0.6(1 m) ) m(0,1] 1,,3 By substtutg the example data to (1), we ca get the correspodg measuremet fuctos: 650 y m1 ( y), y [50, 650] y m ( y), y [90, 90] y m3 ( y), y [10, 110] 100 Thus the reveues of 3 power plats ca be calculated as: 650 L N P ( m ( y)) m ( y) dy [0.3 (0.48(1 m 50 1( y)) 0.6(1 m1( y)) ) ] dy [ m 50 1( y) 0.6 m1( y) ] dy 80 4

13 90 L 90 N P ( m ( y)) m ( y) dy [ m 90 ( y) 0.8 m( y) ] dy L N P ( m ( y)) m ( y) dy Here 110 [ m 10 3( y) 0.8 m3( y) ] dy y m ( y) dy dy m y 1( ) dy y m ( y) dy dy m y dy ( ) y m ( y) dy ( ) dy m y dy 3( ) The load durato prce s show Fg. 6(b), whch s olear sce t s related to the durato of system load. The profts of 3 power plats are respectvely 84.37, 34.38, 1.7, whch are show Fg. 7(b). The proft rates are respectvely 60%, 5%, 48%. The total geerato cost ad total proft of 3 power plats s 3.53 ad respectvely. The total electrcty purchasg cost ad proft rate of the market are 364 ad 57% respectvely. From Fg. 7 we ca see that because spot prcg mechasm caot dstgush base load, termedate load ad peak load, the profts of 3 power plats dffer greatly. Power plat 1 s allocated too much proft. Load durato prcg ca reduce the total electrcty purchasg cost, ad the proft allocato amog the power plats s more equtable. Compared wth spot prcg mechasm, the proft of power plat 1 s reduced ad that of power plat 3 s creased. As the two prcg mechasms have ther ow pros ad cos, the future research wll cosder combato of the two prcg mechasms. Fg.5 Dstrbuto of commodty values uder load durato prcg

14 (a) spot prce (b) load durato prce Fg.6 Prces uder two dfferet prcg mechasm (a) Profts uder spot prcg (b) Profts uder load durato prcg Fg.7 Profts uder two dfferet prcg mechasm 8. Cocluso I the letter of Secretary of Eergy Rck Perry to FERC, t s hghlghted that dstorted prce sgals the Commsso-approved orgazed markets have resulted uder-valuato of grd relablty ad reslecy beefts provded by tradtoal baseload resources. The Commsso has recogzed that there are defceces the way the regulated wholesale power markets prce power ad that these defceces are udermg relablty ad reslecy. So t s the Commsso s mmedate resposblty to take acto to esure that the relablty ad reslecy attrbutes of geerato wth o-ste fuel supples are fully valued ad partcular to exercse ts authorty to develop ew market rules that wll acheve ths urget obectve. I partcular, the value of o-ste fuel storage capablty must be accouted for. To acheve these goals, the basc theores of electrcty commodty ad market mechasm are urgetly eeded for further study. Begg wth the aalyss of wholesale electrcty markets based o spot prcg theory, ths paper presets the cotuous tme electrcty commodty models, cludg commodty models uder spot prcg ad load durato prcg, based o whch the market models of socal welfare maxmzato are establshed. The market optmzato models uder spot prcg ad load durato prcg correspod to Rema tegral ad Lebesgue tegral mathematcs respectvely. The market equlbrum soluto s obtaed by solvg the respectve Euler-Lagrage equato. I partcular, the feasblty of load durato prcg s proved by strct mathematcal dervato. Fally, a example s gve to verfy the correctess of the theores ad methods proposed. Ths paper s expected to provde ovel deas ad theoretcal bass for electrcty market desg.

15 9. Appedx The bascs of Rema tegral (Zadma [1999]) ad Lebesgue tegral (Bear [1988]) are troduced as follows. 1) Rema tegral Gvg a bouded fucto f x o a, b ad a dssecto of b a x x x x x b The the terval terval s x x 1 a, b s devded to small tervalsx, x. A Rema sum for x 1 1 x f s a expresso x f, a,, ad the legth of each where are arbtrarly chose umbers x x, 1,,, 1. We say that f x s Rema-tegrable o b a, f there exsts a real umber I wth the followg property: for 0, ( ) 0, such that 1 f x x I for all parttos wth x 1 ad for ay choce of x x umber I s the Rema-tegral of 1max f x over b b a a,, we deote I f x dx,,. 1 The above foud The dea of Rema tegral s makg ay parttos a, b ad costructg Rema sum, the take lmtato over for the Rema sum s the tegral b f x dx. a The Rema tegral permts a precse defto of the geometrcal cocept of "area" uder a curve. So Rema tegral s a mportat tool computg "area". ) Lebesgue tegral The ma dfferece betwee the Rema ad Lebesgue tegrals s that the former uses tervals ad ther legths whle the latter uses more geeral pot sets ad ther measures. Thus t s ot surprsg that Lebesgue tegrals s more geeral tha the Rema tegral. Frst, we troduce the defto of Lebesgue measure. We deote I as the ope set 1 x, x,, x a x b, 1,,, R ad I as the volume of I. For ay set E R,we defe the measure of set E to be the mmum of the sums of the volumes of famles of ope sets whch cover E. To make ths precse, we defe the Lebesgue outer measure m * E as followg

16 * m E f I, I s coutable such that E I Smlarly, we ca defe the Lebesgue er measure m * E as m* E sup I, I s coutable such that I E If * m E m E, we say E s Lebesgue measurable. * me, Next, we troduce the defto of Lebesgue tegral. Assume E s a measurable set, f x s a bouded fucto over E. sup ( ),, B f f ( x), x E A f x x E Suppose a dssecto of the tervala, B A y y y y y B Deote E x x y 1 f y. For ay, 1 1,,, y y, take summato S f D, me 0 We say that x f s Lebesgue-tegrable o E f lm S f, D 0 exts, where max y y 1. 1 We deote lm, E f x dx S f D 0 We could descrbe Lebesgue tegral usg Fg. 8. I order to obta the area bouded by the curves y f x, y 0, x a, x b, we could use the rectagles from the dssecto of axs y, Whe y y 1 b a y1 y0 x4 x1 b x5 y y1 x x b x y y b x y y max 1 0, the the summatos of the areas of the rectagular s the Lebesgue tegral of f x.

17 Fg. 8 Referece R. Perry. Re: Secretary of Eergy s Drecto that the Federal Eergy Regulatory Commsso Issue Grd Reslecy Rules Pursuat to the Secretary s Authorty Uder Secto 403 of the Departmet of Eergy Orgazato Act. Washgto DC. September 8, 017. Quadreal Eergy Revew (QER). Trasformg the Nato s Electrcty System: the Secod Istallmet of the Quadreal Eergy Revew. Jauary 6, 017. U.S. Departmet of Eergy (DOE). Staff Report to the Secretary o Electrcty Markets ad Relablty. August, 017. F. C. Schweppe, M. C. Caramas, R. D. Tabors, R. E. Boh. Spot Prcg of Electrcty. Bosto: Kluwer Academc Publshers R. E. Boh. Spot Prcg of Publc Utlty Servces. Doctoral thess, Massachusetts Isttute of Techology M. C. Caramas, R. E. Boh, F. C. Schweppe. Optmal Spot Prcg: Practce ad Theory. IEEE Trasactos o Power Apparatus ad Systems, PAS-101(9): 334~345, 198. W. Elmaghraby, S. S. Ore. The Effcecy of Mult-Ut Electrcty Auctos. The Eergy Joural, 0(4): 89~116, S. Zadma. Advaced Calculus: A Itroducto to Mathematcal Aalyss. World Scetfc Press, H.S. Bear, A Prmer of Lebesgue Itegral. Academc Press, 1988.