In the UC problem, we went a step further in assuming we could even remove a unit at any time if that would lower cost.

Transcription

1 uel Schedulg (Chapter 6 of W&W.0 Itroducto I ecoomc dpatch we aumed the oly lmtato were o the output of the geerator: m g. h aumed that we could et ge to ay value we dered wth the rage, at ay tme, to acheve optmalty. I the UC problem, we wet a tep further aumg we could eve remove a ut at ay tme f that would lower cot. I both of thee problem, we were aumg that the fuel upply would alway exactly match our eed, that, we could tur o the fuel whe we wated t or tur t off whe we dd ot wat t, wthout regard to how much or how lttle fuel we mght be ug. h may ot alway be the cae. It may be poble that oe or more power plat are eergy-cotraed ome faho. h mea that the tegral of the plat power output over tme wll eed to be ether le tha a certa value or greater tha a certa value. h doe ot cotra power at ay partcular momet but t doe cotra the tme-tegrated power (eergy over a terval of tme. Sce eergy ca be uatfed term of amout of fuel (atural ga, coal, ol, cotrat o eergy a certa tme terval are euvalet to cotrat o fuel ue that tme terval. h why W&W-Ch 6 called Ge w/lmted Eergy Supply. It poble to have lower boud o fuel uage or upper boud o fuel uage. I hydro ytem, uch boud are dctated by water level reervor upplyg hydroelectrc power plat. Hydroelectrc faclte are made more complex, however, becaue may reervor ytem have coupled reervor uch that the eergy cotrat o oe hydro plat are coupled wth the eergy reuremet of the dowtream ad uptream hydro plat. We addre hydro chedulg uder the hydro-thermal coordato problem decrbed Chapter 7.

2 he problem of corporatg eergy cotrat for thermal ut referred to a the fuel chedulg (S problem. I the S problem, boud o fuel uage are dctated by the cotract the power plat ower mae wth the fuel uppler. Although th problem ha bee of teret for may year, referece [] artculate t the curret cotext of LM electrcty maret: I th cotext, t ow that a thermal plat that geerate oly whe pot prce are above t operatve cot ca meet t facal cotract oblgato wth a low effectve operatg cot: the plat doe ot operate bae load, beg hut dow durg the moth whe pot prce are low (ad beeft from pot maret purchae at very low cot. I other word, operatoal flexblty a very attractve charactertc for thermal plat the hydro-baed ytem. However, th operatoal flexblty, alog wth a low dverfcato of fuel maret, oppoto wth the eed of fuel producer whch have hgh fxed cot due to captal expedture developg producto ad traportato fratructure. A a coeuece, fuel upply agreemet are heavly tructured over cotract cludg tae-or-pay (o claue. hee are ut facal agreemet to reduce the volatlty of the fuel producer reveue ad (uually are ot aocated to coumpto oblgato. he o claue mpoe a atcpated purchae of a mmum amout of fuel (o a daly, mothly ad/or yearly ba, depedetly of t coumpto. Ofte, the amout of fuel bought but ot coumed vrtually tored (uder the form of credt for a pre-et perod. Durg th perod at aytme, the fuel ca be recovered by the plat. h ow a mae-up claue. Operatoal flexblty very attractve for thermal power plat ay LM maret, for the ame reao a tated here to tae advatage of LM prce varablty. uel producer do ot le up ad dow ue of ther fuel. h motvate o cotact. o may or may ot have mae-up claue whch tore what ot ued. o cotract wthout mae-up claue are commo.

3 here are ma type of fuel for thermal power plat, wth % of US electrcty upply: Coal (5% 008, 45% 00, 4% 0 atural ga (7% 008, 3% 00. 5% 0 etroleum (le tha % throughout Although petroleum ot ued at a gfcat level to upply power plat o a percetage ba of atoal eergy (<%, there are ome area where t a bt hgher (e.g., ew Eglad, 008, ol-fred ut compred 4.9% of total capacty ad % of eergy producto []. It clear from the above uote that uppler of thee fuel ca operate more effcetly f they ca obta tae-or-pay (o cotract, effectvely reducg ther ucertaty. A o cotract where the buyer pay for a mmum amout of fuel whether t tae or ot. he prce pad uder o-delvery may be eual to or le tha the prce pad for delvery. he o cotract eable the uppler to pla more precely regard to fuel producto. Mot cotract alo clude a celg o how much fuel ca be ued a certa perod..0 Mmzg cot for fleet wth a eergy cotraed ut We frt coder that a ower of a fleet of power plat, oe of whch are eergy cotraed, wat to mmze ther cot over a tme perod. he plat are bae-loaded ad o guarateed to be rug. otatoally, we follow W&W (ecto 6. accordg to: ( : fuel cot rate for ut durg terval. : power output for ut durg terval. R : total demad perod. : umber of hour th perod. 3

4 We dere to olve the followg optmzato problem: R m ubect to 0, (,..., where the obectve fucto the total cot (ot cot rate over all perod, ad the eualty cotrat the reuremet that ay oe perod, the upply mut eual demad. We are ot accoutg for geerator power producto lmt at th pot. ow let coder f there oe mache (or a group of mache for whch the ower ha egaged a o cotract wth a celg. Mathematcally, the mplet cae whe the mmum tae eual the celg. h mea that over the tme perod, the ut mut ue exactly a partcular amout of fuel. Let call th amout of fuel O. O wll have ut of RE ( raw eergy ad wll be dfferet for each fuel type: Coal: RE=to Ga: RE=ft 3 Ol: RE=bbl (barrel=4 gallo Let the ut uder the o cotract be ut =+ (o that we have ut wth ulmted fuel cotract. Defe a the fuel put rate for ut the th tme perod. Wherea ut of O are RE, the ut of are RE/hr. ote that a fucto of,.e., = (. RE/hr. How to get th fucto = (? ( 4

5 Defe K f a the eergy cotet of fuel, MBU/RE. ypcal value for K f are: uel K f (MBU/RE Coal athracte 5.64 MBU/to Coal - btumou 3.5 MBU/to Coal - lgte.7 MBU/to atural ga 0.00 MBU/ft 3 etroleum (tadard preure 5.88 MBU/bbl Aother way to th about th that the below uatte of fuel provde MBU: uel /K f (RE/MBU Other ut Coal athracte to/mbu 78lb/MBU Coal - btumou to/mbu 86lb/MBU Coal - lgte to/mbu 88lb/MBU atural ga 980 ft 3 /MBU 0ft 0ft 0ft/MBU (tadard preure etroleum 0.7 bbl/mbu 7 gallo/mbu Recall that H, the heat rate, MBU/MWhr ad that t a fucto of the power geerato level,.e., H=H(. he the fuel per MWhr wll be H( /K f, ad multplyg th by the power geerato level reult the fuel rate,.e., H( ( K ( f Bac to our mplet cae where the mmum tae eual the celg, the ummato of fuel ue by ut over all tme terval mut eual the fuel reuremet O, that O (3 5

6 Our ew problem, the, wll be exactly a our old problem a poed (, wth three excepto.. Add the cotrat (3.. Add fuel cot of our eergy-cotraed ut to the obectve fucto. 3. Iclude the power balace euato geerato correpodg to the eergy-cotraed ut, for each tme perod. herefore, the problem tatemet for the cotraed eergy problem become: m ubect to R O ( 0, (,..., he above problem tatemet ca be mplfed, however, by recogzg that the eergy-cotraed ut mut utlze exactly O of fuel, ad ce the cot of geerato domated by the fuel cot, the lat term the obectve fucto wll be a cotat. Sce cotat the obectve fucto do ot affect the deco to be made (.e., the oluto, a dcated by the deco varable, the we may ut remove that term, reultg the followg reved problem tatemet: m ubect to R O ( 0,,..., (4 (5 6

7 Let mae the followg defto, correpodg to the two eualty cotrat of (5: ( ( R ( 0, O 0,..., he, the problem tatemet ca be wrtte more cocely a: m ubect to ( ( he Lagraga become: L(,,, ( ( ( R 0, 0 O ( (,..., ( ow we ca wrte the optmalty codto. he frt optmalty codto we wll coder (6 (7 Sce $/hr, whe multpled by hr gve ut for the Lagraga of $. h mea λ mut be $/MW ad γ mut be $/(fuel-ut, where fuel-ut are ft 3 (for atural ga or to (for coal. (8 7

8 8 0 ( ( O R L (9a erformg the dfferetato, we obta 0 ( L (9b ow coder the optmalty codto 0 ( ( O R L (0a erformg the dfferetato, we obta 0 ( L (0a ally, the optmalty codto tag dervatve wth repect to the Lagrage multpler mply gve u thoe cotrat bac aga. I ummary, the optmalty codto reult the followg euato:

9 L me Iterval: me Iterval L : ( 0, ( 0,,...,,..., ( L me Iterval: me Iterval L : ( 0 ( 0 ( otce that f γ pecfed, the the euato of (, (, ad (3 from each tme perod are depedet ad each tme perod ca be olved by lambda-earch. L me Iterval: me Iterval L L : ( R R O (3 (4 We wll ext coder two dfferet way of how to olve thee euato. 3.0 uel chedulg oluto by gamma earch he approach here wll ue a er ad a outer earch mecham. he er earch mecham wll be a tadard lambda terato for each tme perod to provde u wth all geerato level for each tme perod. h earch wll be codtoed o a aumpto regardg cotraed fuel, repreeted by a 9

10 partcular value of gamma (γ. Each complete oluto of the er earch wll reult a total fuel uage correpodg to the choe value of gamma. he outer earch mecham wll be a earch o gamma to fd the certa value whch reult the fuel uage beg ut eual to the dered value O. h approach llutrated g. 6.3 of W&W. We wll how t, but frt we eed to recall how to perform the lambda-earch... Implemetato of the er earch mecham reure expreg the power output for each mache each tme terval a a fucto of lambda for that tme terval. rom (, we ca wrte for tme terval : L ( me Iterval : 0,,..., (5 If we aume ( uadratc, that ( a b c,..., (6 the ( b c,..., (7 Subttuto of (7 to (5 reult L b c 0,,..., (8 Solvg (8 for reult - b c,..., (8 h wll gve u the geerato level for all of the o-eergycotraed ut but ot for the eergy-cotraed ut. o obta the geerato level for the eergy-cotraed ut, we apply ( to tme terval. L ( 0 (9 0

11 We eed a expreo for (, but th ut euato (, repeated here for coveece: H( ( K ( f where K f the eergy cotet of fuel, MBU/RE (a we have ee. Recall the cot-rate curve (ee ote o cot curve wll be C C ( KH( (0 where K the prce of the put fuel $/MBU. Comparg ( ad (0, we ee that the expreo for ad C dffer oly by the cotat K f /K. he pot of th that the form of wll be the ame a the form of C. Sce C ut the cot-rate fucto for a ut, t wll geeral have a uadratc form. hu, we may repreet lewe,.e., for tme perod, ( a b c ( Oe hould be aware, however, that the coeffcet ( dffer from the coeffcet of ut cot-rate fucto by K f /K. Dfferetatg ( wth repect to reult ( b c ( Subttuto of ( to (9 reult L b c 0 (3 Solvg (3 for reult b c (4 I ummary, we have the followg euato to expre geerato level at each ut each tme perod: - b,..., (8 c

12 c b (4 h wll allow u to perform lambda terato. ote (4 a fucto of gamma. Oe thg rema, however. It wll be ueful to ow how to update lambda followg each terato. o obta a lambda update formula, frt recall the toppg crtero for lambda terato whether the geerato meet the load. h expreed by euato (3, whch, for terval, R 0 L (5 Solvg for R, we obta R (6 ow ubttute (8 ad (4 to (6 to obta R c b c b - (7 he we ca dfferetate (7 wth repect to λ to obta R c c (8 We may approxmate (8 a R c c (9 Solvg (9 for λ, we obta R c c (30

13 he Lambda terato method llutrated g.. Oberve that the flow chart of g. aume that gamma ow; alo the toppg crtero baed o the dfferece betwee demad & ge: R R GUESS γ = GUESS λ ew old - b,..., c c R c b c a b c R R o I R δ? Ye SO Ye I =? o =+ g. 3

14 So g. llutrate the er loop of the fuel chedulg oluto. What we eed to do ow to detfy how to adut gamma. o do that, let beg by recallg where gamma came from. We recall the Lagraga, (8, repeated here for coveece: L(,,, ( R ( ( O ( ( Here we ee that that γ the Lagrage multpler o the fuel cotrat. o get a lttle better feel for γ, recall the fuel cotrat (3: Recall ( ( O Subttuto of ( to (3 reult O (8 (3 a b c ( a b c (3 What we are after here the depedece of O o γ. o get th, recall (4: 4

15 b c (4 I wll ot go through the detal here, but rather ut artculate the procedure, whch to ubttute (4 to (3 ad the dfferetate to obta O / γ. he, mag the approxmato O O we may derve that O (3 I wll clude my had-wrtte dervato of (3 the appedx to thee ote. We mae three commet at th pot.. rom (3, we oberve that f γ>0, the γ alway oppote g to. hat, Icreae γ to decreae fuel uage of Ut. Decreae γ to creae fuel uage of Ut.. Aaly of ut (ee Lagraga dcate γ $/RE (recall RE Raw Eergy Ut. h dcate that γ le a fuel prce. h cotet wth commet # thg of ellg fuel, f we rae the prce, the demad decreae ad f we lower the prce, the demad creae. 3. We hould recogze, however, that γ ot the fuel prce. he fuel prce, alo wth ut of $/MBU, gve by K ee (0. So what γ? c 3 5

16 hg term of optmzato, a a Lagrage multpler, we ow that γ repreet the chage the obectve fucto of creag the rght-had-de of the correpodg cotrat by ut. Sad term of th problem, γ the addtoal cot of reurg the ue of oe addtoal RE ut of fuel at ut durg the ext tme perod. A teretg pot made (pg. 74 ad proved (Appedx of Chapter 7 that the addtoal ut of fuel could be reured ay tme terval. Sad aother way, γ cotat over tme. he chapter 7 appedx alo how, however, that γ wll vary a partcular tme perod f both the below are true. he problem clude cotrat o how much fuel ca be ued a partcular tme perod ( addto to the total amout of fuel to be ued over all tme perod, whch O. h ca happe for coal-fred plat due to the lmtato of coal tored o te. he cotrat for the partcular tme perod bdg. Ug (3, we are ow a poto to draw the etre fuelchedulg flow chart, a gve g. a. 6

17 GUESS γ = GUESS λ ew old - b,..., c c R c b c a b c R R o I R δ? Ye O O t Ye I =? o =+ I O <ε? Ye o SO c O 3 g. a 7

18 4.0 Compote geerator producto cot fucto (W&W, 6.3 Oe problem wth the oluto procedure troduced Secto 3.0 that a lambda terato mut be doe for every tme terval. If there are a large umber of ut (00, each lambda terato ca tae ome tme, ad of thee ca be very computatoal. We ugget a alterate procedure th ecto. he dea to obta a compote cot curve for the o-eergy-cotraed ut, ad the the lambda terato oly for two ut rather tha. Recall that whe we troduced the ut commtmet problem, we geerated a compote cot curve for 4 ut.we dd th aalytcally by ettg cremetal cot expreo eual, ug the power balace euato, ad olvg for each ut geerato level. h wa mey for four ut ad would be tractable for 00 ut. A alteratve procedure to develop the followg table, a et of umercal data for all o-fuel cotraed ut, =,,. λ S =Σ g S =Σ ( g λ m λ where d m m,,..., dg, d,,..., d gg,m g gg, ad each g the table foud from λ=d ( g /d g. If a ut ht a lmt, t output g ad cot ( g are held cotat. ote that the above value of λ are ued mply to get the compote cot fucto ad are ot the ame a the λ value the algorthm (whch are multpled by per e. (9b. A curve-fttg approach ca the be ued to obta the compote cot-rate fucto S ( S. Oce th doe, the the algorthm of 8

19 g. a appled, except that there oly o-eergy cotraed ut, a how g. b (yellow boxe dcated chage. GUESS γ = GUESS λ ew old S - b c S S R c c S b c a b c R R S o I R δ? Ye O O t Ye I =? o =+ Ye I O <ε? o SO c O 3 9

20 5.0 uel chedulg gradet oluto for optmalty Recall the optmalty codto from the Lagraga. Repeatg (9b, L ( 0 (9b ad (0a, L ( 0 (0a Solvg for λ each of thee euato, we obta ( (33 ( (34 Euatg (33 ad (34, we obta ( ( (35 Solvg for γ reult ( ( (36 Oberve the umerator ad deomator of (36: umerator the Icremetal uel Cot ($/MW-hr for ofuel cotraed ut durg terval. Deomator Icremetal uel Rate (RE/MW-hr for the cotraed ut durg terval. Our above developmet how that, for optmalty, th rato mut be cotat for all tme terval =,,. h cotet wth our prevou obervato that γ hould be cotat over tme. We ca formulate a algorthm baed o th fact, a llutrated g. 3, adapted from g. 6.7a your text, but we eed a feable chedule. 0

21 GEERAE COMOSIE RODUCIO COS UCIO OBAI EASIBLE SCHEDULE (ROM IG. 4 SUCH HA O (,,..., ote #5 ad ec 6 decrbe g. 4 to obta a feable oluto. CALCULAE OAL COS O O-COSRAIED UIS: total CALCULAE γ OR ALL IERVALS ( / ( SELEC IERVALS HA GIVE MAX AD MI γ. HIS MEAS O SELEC + AD - SUCH HA γ IS MAXIMUM OR =+ AD MIIMUM OR =-. =(γ + -γ - 0 WHERE 0 IS A CHOSE SMALL SE ADJUS I IERVALS + AD -. ICREASE + O MAKE γ + DECREASE: = + / ; =+ DECREASE - O MAKE γ - ICREASE: = + / ; =- CALCULAE EW VALUES O γ + ad γ - CALCULAE total (ee ote O total ε? YES DOE g. 3: Gradet oluto (ote # tell why t gradet ote #3 expla the dmeoalty problem here. ote # derve the eceary expreo.

22 We mae four commet about the method llutrated g. 3.. Your W&W text dcate o pg. 83 that the method may be called gradet method becaue treated a a vector ad the γ value dcate the gradet of the obectve fucto wth repect to. You ca oberve that th the cae from (36, ug the compote cot-curve,.e., ( ( ( ( ad otg that t mut be the cae that S =-, o that ( ( ( ( (37 (38 howg that γ may be terpreted a a etvty of the chage obectve fucto to a chage the amout of fuel ued tme terval. If we th of all of a a vector, e.g., the we may wrte a the gradet to.

23 . he ext-to-lat tep the algorthm of g. 3 dcate CALCULAE total (ee ote. h repreet the chage total cot (ad ot the chage total cot rate correpodg to the adutmet fuel uage made the prevou tep of the algorthm ( ADJUS I IERVALS + ad -. h calculato doe baed o the followg: ( ( ( ( ( ( ( ( But S+ =- +, ad S- =- +. Mag approprate ubttuto reult ( ( ( ( ( ( We mplfy the otato here a follow: (39 ow recall that γ + too hgh ad γ - too low, o we eed to decreae γ + ad creae γ -, whch we do by creag + ad decreag -. he fuel creae + mut eual the fuel decreae -, therefore, ad recallg that the fuel rate, we have that the fuel creae gve by the followg value (choe to be the ame g a + +, whch mut be potve cotet wth the bullet above whch dcate we mut creae t+. (40 herefore,, ( ( (4 3

24 Subttutg (4 to (39 reult Brgg the - over to the other de, we get: (43 ad you ca recogze the left-had-de a total that reured by the ext-to-lat tep the algorthm of g. 3. Commet: If the algorthm to coverge, that, f the total get maller wth each terato, the the left-had-de of (43 mut be egatve. We ee th mut be the cae by pectg the rghthad-de of (43 ce wa choe potve (ee (40 ad ce γ + >γ - by defto. Commet: If all tme terval are choe of eual durato,.e., f + = -, the (43 become (44 (4 3. he flow chart tep =(γ + -γ - 0 WHERE 0 IS A CHOSE SMALL SE ot dmeoally correct a t tad, becaue gamma ha ut of $/RE, ad whe multpled by RE, gve $, cotet wth the above dcuo regardg (43. You ca aume, however, that the relato really =[(γ + -γ - /][ 0 ], where the ha the ame ut a γ. he we oberve that f Δ 0 ha ut of RE, the o wll Δ. Bacally, th relato ut tellg u that f we wat to correct two terval - ad + for ther fuel (or water uage, we hould chooe a amout of fuel (or water to hft that proportoal to the dfferece betwee the two terval gamma value. 4

25 4. Oberve that toppg crtero to chec to ee f total chage gfcatly. 5. he ecod tep the algorthm of g. 3 dcate that t aume a feable (but ot ecearly optmal chedule that the fuel ue reuremet met ad the geerato dpatch of each tme terval locally optmal meag that t would be the optmal dpatch f we codered oly that tme terval. he problem at had : from where do we obta a feable chedule? h the topc of the ext ecto. 6.0 uel chedulg gradet oluto for feablty h approach llutrated g. 4. 5

26 GEERAE COMOSIE RODUCIO COS UCIO OBAI EASIBLE SCHEDULE IGORIG UEL COSRAI. BES AROACH IS O SOLVE ECOOMIC DISACH OR EACH ERIOD. ( (,,..., CALCULAE OAL UEL USED BY UI ' O CALCULAE γ OR ALL IERVALS ( / ( O (potve ID * WIH MI γ AD DECREASE, ICREASE S, O DECREASE UEL USE: = - for =* O - O <ε? O O - O <0? YES YES (egatve DOE (Go to g. 3. ID * WIH MAX γ AD ICREASE, DECREASE S, O ICREASE UEL USE: = + for =* CALCULAE γ OR =* ( / g. 4 ( 6

27 wo mportat obervato may be made from g. 4:. he ecod bloc from the top dcate that the algorthm beg wth a feable chedule for the problem wthout the fuel cotrat. he bet way to obta th from a ecoomc dpatch for each perod.. he thrd bloc obta the fuel ued by the partcular choe chedule. h computed by (3, repeated here for coveece: O a b c (3 Your W&W text provde a example fuel chedulg problem, whch olved by gamma earch (Example 6B, pg ad by the gradet approach (Example 6C, pg leae revew thee two example ad ow how to wor them. [] R. Chabar, M. erera, S. Gravlle, L. Barroo, ad. Ilad, Optmzato of uel Cotract Maagemet ad Mateace Schedulg for hermal lat uder rce Ucertaty, roc. of the ower Sytem Coferece ad Expoto, Oct. 9-ov, 006. [] H. Chao, Itegrato of atural Ga ad Electrcty ew Eglad ad the ret of US, preetato at the 008 AEX Coferece Sydey, Autrala, October 3-4,