E3 System otrol Overvew ad Ecoomc Dspatch alculato 5 Module E3 Ecoomc Dspatch alculato rmary Author: Gerald B. Sheble Iowa State Uversty Emal Address: gsheble@astate.edu o-author: James D. Mcalley Iowa State Uversty Emal Address: jdm@astate.edu Last Update: 8//99 rerequste ompeteces: Matrx Algebra artal Dervatves ad Sychroous Geerator Operato foud G. Module Objectves:. Model the geerator cost rate as a fucto of geerator output. Apply the Karush-Kuh-Tucker (KKT) codtos solvg mult-varable costraed optmzato problems. 3. Solve the ecoomc dspatch problem by applyg graphcal ad Newto approaches. 4. Idetfy the meag of cremetal cost ad how t relates to Lagrage multplers. E3. Itroducto The daly operato of the electrc trasmsso grd s prmarly cocered wth the balace of satsfyg the demad for electrcty wth the supply. Ths s accomplshed keepg md adherece to all rules of physcs ad acceptable operato for securty ad relablty whle smultaeously mmzg the cost of electrcty producto. The ma objectve of ths materal s to descrbe the calculato procedures used allocatg demad amog avalable uts at mmum cost to the geerato frm. I preparato for achevg ths objectve we wll frst preset the operatoal ad fuctoal structure whch the calculato s doe. We wll descrbe how to model geerator costs. We wll the study aalytcal procedures used mmzg multvarate fuctos uder costrats. At ths pot we wll be ready to solve the ecoomc dspatch problem. E3. Operatoal Structure The work of the Idepedet System Operator (ISO) revolves aroud of a faclty called a Eergy otrol eter(e). Through the E the ISO operates the trasmsso grd to provde maxmum access to all members of the system wth the establshed operatoal gudeles. The ISO cossts of system operators of operatoal plaers ad of operatoal audtors workg to track all schedules ad all accouts as to the plaed operato ad to the actual operato. The system operators are resposble for the swtchg operatos to solate equpmet for safety or for mateace. The system operators are also resposble for the cotrol of the geerato to mplemet the cotracted schedules. The operatoal plaers establsh schedules accordg to the cotracts ad bds offered by the GENOs TRANSOs ad DISTOs. Ths provdes that the trasmsso grd ca operate wth the establshed operatoal gudeles. The system operators mplemet the plaed schedules ad adjust the auxlary servces to meet the actual grd requremets. The ed result s a log of all operatoal evets to mplemet the schedules ad all devatos from plaed schedules. The operatoal audtors verfy all schedule complace adjust ay log etres for complete or mssg data compute ay devatos from the plaed schedules log ay eeded operatoal chages for future plas ay devatos ad the requred remedes ad documet all restrctos to the trasmsso grd trasfer capablty. The All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 6 computer capablty to support the above fuctos s a major reaso for the research ad developmet ow occurrg aroud the world. There s a ISO for each locato operated as a cotrol area. The cotrol areas are tercoected wth the two major regos of the Uted States: East ad West. Each cotrol area s to operate as a whole absorbg ay demad chages wth the cotrol area effectvely solatg each area. The areas are tercoected to provde the capablty to trade resources betwee areas whe ecoomcally feasble ad to provde resources as system eeds chage utl the commucato ad cotrol systems ca respod. The chages o the electrcal grd travel early at the speed of lght. Thus chages must be accommodated mmedately. The commucato ad cotrol systems respod o the order of secods far too slow to respod to chages as they occur. Thus the system has to be desged to heretly properly respod to chages as a atural respose. E3.. NER Gudeles The securty ad relablty of the preset electrc power grd s preserved by the cosesus of all electrc utltes through the Natoal Electrc Relablty oucl (NER). NER s dvded to several regoal groups to oversee the complace of each compay to the agreed operatoal requremets. NER s resposble for the stadards geerato ad evaluato of operatg ad plag stadards. Two fudametal deftos provded by NER are as follows : Adequacy s the ablty of the electrc systems to supply the aggregate electrcal demad ad eergy requremets of customers at all tmes takg to accout scheduled ad reasoably expected uscheduled outage of system elemets. Securty s the ablty of the electrc system to wthstad sudde dsturbaces such as electrc short crcuts ad uatcpated loss of system elemets. E3.3 Eergy Maagemet System Overvew The eergy cotrol ceter (E) s a faclty where the system operators ca aalyze ad operate the trasmsso grd through a set of software applcatos called a eergy maagemet system (EMS). There are two basc EMS fuctos; securty motorg ad cotrol / dspatch. I securty motorg the state of the power system s classfed to oe of the followg: secure alert alarm compromsed. A secure state s whe the system s operatg as plaed wth o mmedate probable problems. A alert state s whe the system s operatg as plaed wth mmedate problems probable. A alarmed state s whe the power system s operatg outsde acceptable levels. A compromsed state s whe the power system s operatg outsde allowable levels ad outsde acceptable schedule devatos. Network aalyss fuctos operate perodcally to determe the state of the power system (e.g. every fve mutes) for the plaed schedules ad operatg codtos. Alteratvely the operator may aalyze the power system uder a hypothetcal stuato to determe the state of the power system o a demad (or as eeded) bass. The cotrol ad dspatch of the geerato s drected through the eergy maagemet system. The fuctos whch mplemet ths drecto are the Automatc Geerato otrol (AG) ad the Ecoomc Dspatch calculatos (ED). These fuctos determe the set pots of the goverors ad allocate the demad amog geerators. AG operates cotuously as a feedback cotrol system that seses stataeous frequecy devatos caused by power mbalace ad adjusts geerator MW output to compesate the power mbalace. Ths s accomplshed by the acto of the goveror at each geerator (see module G). ED operates such more slowly sesg steady-state frequecy ad tele flow devato every 3-5 mutes ad readjustg all geerator MW outputs accordgly. We wll focus o the ED approach ths module. A fudametal part of ths approach s the cost of geerators electrcal eergy. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 7 E3.4 osts of Geeratg Electrcal Eergy The costs of electrcal eergy geerato arse maly from three sources: faclty costructo owershp costs ad operatg costs. The last s the most sgfcat porto of power system operato ad ths secto we focus o ths aspect. E3.4. Operatg osts These costs clude the costs of labor but they are domated by the fuel costs ecessary to produce electrcal eergy (MW) from the plat. Some typcal average costs of fuel as of are gve the followg table for the most commo types use today. Table E3. Fuel Type $/MBTU oal.6 Ol 3.34 Uraum.65 Natural Gas 3.56 These values do ot reflect the actual costs of producg electrcal eergy because substatal losses occur durg producto. Some power plats have overall effceces as low as 35%; addto the plat effcecy vares as a fucto of the geerato level g. We llustrate ths pot what follows. We represet plat effcecy by η. The η eergy output/eergy put. We ca actually obta η as a fucto of g by measurg the eergy output of the plat MWHRS ad the eergy put to the plat MBTU. For example we could get the eergy output by usg a wattmeter to obta g as a fucto of tme ad the compute the area uder the curve for a terval ad we could get the eergy put by measurg the coal toage used durg the terval ad the multply by the coal eergy cotet MBTU/to). The η s proportoal to the rato of MWHR/MBTU; a plot of ths rato versus g would appear as Fgure E3.. Fgure E3. lot of MWhr/MBTU (proportoal to effcecy) vs. Geerato (g) Fgure E3. dcates that effcecy s poor for low geerato levels ad creases wth geerato but at some optmum level t begs to dmsh. Most power plats are desged so that the optmum level s at or close to the rated output. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 8 The heat rate curve s smlar to Fgure E3. except that the y-axs s verted to yeld MBTU/MWhrs whch s proportoal to / η. Ths curve s llustrated Fgure E3.. Heat rate s deoted by H. Sce the heat rate s depedet o operatg pot we wrte H H ). Some typcal heat rates for uts at maxmum output are ( ( g MBTU/MWhrs) 9.5 for fossl-steam uts.5 for uclear uts ad 3. for combusto turbes []. Fgure E3. lot of Heat Rate (H) vs. Geerato (g) We are prmarly terested how the cost per MWHR chages wth g. We assume that we kow K the cost of the put fuel $/MBTU. Defe R as the rate at whch the plat uses fuel MBTU/hr (whch s depedet o g ) ad as the cost per hour $/hour. The R g H( g ) ad (R)(K) g H( g )K. A plot of vs. g s llustrated Fgure E3.3. Fgure E3.3 lot of ost per Hour () vs. Geerato (g) The desred $/MWHR characterstc called the cremetal cost curve for the plat ca be obtaed by dfferetatg the plot Fgure E3.3. The cremetal cost curve s show Fgure E3.4. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 9 Example E 3. Fgure E3.4 lot of Icremetal ost (I) vs. Geerato (g) A MW coal-fred plat uses a type of coal havg a eergy cotet of BTU/lb (the coverso factor from joules to BTU s 54.85 joules/btu). The coal cost s $.5/MBTU. Typcal coal usage correspodg to the daly loadg schedule for the plat s as follows: Table E3. Tme of Day Electrc Output (MW) oal Used (tos) :am-6:am 4 5. 6:am-:am 7 94.5 :am-4:pm 8 56. 4:pm-:am 7. For each of the four load levels fd (a) the effcecyη (b) the heat rate H (MBTU/MWhr) (c) the cost per hour ($/hr). Also for the loadg levels of 4 7 ad 8 MW use a pecewse lear plot of F vs to obta cremetal costs. Soluto Let T be the umber of hours the plat s producg MW whle usg y tos of coal. sec 6 watts T 36 (a) η hr MW BTU lb joules 54.85 ytos lb to BTU Note that the above expresso for effcecy s dmesoless. (b) BTU lb MBTU ytos 6 H lb to BTU T 36 3.4 Note that H ad the above expresso has uts of η 54.85 η MBTU/MWhr. (c) (R)K where R s the rate at whch the plat uses fuel ad K s fuel cost All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato $/MBTU. Note from uts of ad H that R ()(H) ()(H)(K) where H s a fucto of. Applcato of these expressos for each load level yelds the followg results: Table E3.3 T (hrs) (MW) y (tos) Effcecy H (mbtu/mwhr) ($/hr) 6 4 5..33.5 63 4 7 94.5.4 8. 85 6 8 56..44 7.8 936 8 7..4 8. 5 d To obta cremetal cost I we plot vs. ad the get a approxmato o the dervatve by assumg d a pecewse lear model as show Fgure E3.5. Fgure E3.5 alculato of Icremetal ost E3.4. Faclty ostructo osts ad Owershp osts ostructo costs clude the costs of all ecessary labor ad materals ecessary to pla ga regulatory approval ad costruct ew geerato facltes. I the past utltes were able to mmze these costs by buldg fewer but larger facltes due to ecoomes of scale. Ths s o loger the case for the followg reasos: Smaller plats ca be bult more quckly ad ther costructo costs are cosequetly subject to less ucertaty. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato Smaller plats ca be located closer to load ceters. Ths attrbute decreases system losses ad teds to be advatageous for system securty. ogeerato facltes are attractve because of ther hgh effcecy. They typcally have lower ratgs as a result of ther depedecy o the dustral steam processes supportg or supported by them. lats fueled by reewable eergy sources (bomass wd solar ad depedet hydro) are attractve because of ther low operatg costs ad evrometal appeal. They also ted to have lower ratgs. Owershp costs are ot related to how much the plat s used. They arse smply because the plat exsts ad they clude mateace ad captal costs. aptal costs clude surace deprecato taxes ad admstratve expeses. These costs are sometmes called exstg facltes costs or embedded costs. E3.5 Optmzato Overvew wth Ecoomc Dspatch Examples Optmzato problems occur may dfferet felds. There s fact oe feld amely operatos research whch s dedcated etrely to the study of posg ad solvg optmzato problems. erhaps the most commo applcato s to detfy the least expesve way of satsfyg a demad. The arle telephoe ad maufacturg dustres are good examples of dustres that make heavy use of optmzato. Aother good example s of course the electrc power dustry. Module G. defed a model for represetg the operatg costs of geerato. Here we approxmate the cost rate vs. geerato (Fgure E3.5) curve usg a quadratc fucto. E3.5. Itroducto Ecoomc dspatch s the process of allocatg the requred load demad betwee the avalable geerato uts such that the cost of operato s at a mmum. Oe-dmesoal mmzato problems are covered a basc calculus course. The Ecoomc Dspatch problem s a more geeral type of optmzato problem. We wll see that the Ecoomc Dspatch problem s a o-lear multvarable costraed optmzato problem. Nolear optmzato techques ca be dvded by type: ucostraed search learly costraed search quadratc objectve programmg covex programmg separable covex programmg ocovex programmg geometrc programmg fractoal programmg etc. It s easer to classfy the techques by the type of problem to be solved: a. Lear objectve fucto lear costrats b. Nolear objectve fucto lear costrats c. Nolear objectve fucto olear costrats d. Lear objectve fucto olear costrats The type a problem s most ofte solved wth Lear rogrammg techques based o the Smplex method. Approxmatg the olear objectve fucto ofte solves the type b problems. If the olear objectve fucto s of a defte form the a specalzed techque may be used. If the objectve fucto s a quadratc fucto the Quadratc rogrammg s approprate. If the objectve fucto s pece-wse lear the a separable fucto s approprate. Fuctoal characterstcs such as covexty may smplfy the techque ad correspodgly accelerate covergece to the optmal soluto. The Reduced Gradet method s best for geeral problems of ths type. The type "c" problems are the hardest to solve. Typcally uless the fuctos demostrate smplfyg characterstcs the olear fuctos are approxmated or a alteratve sequece of approxmatg problems s solved. Whe a alteratve sequece of approxmate problems s solved t s assumed that the fal approxmate problem replcates the orgal problem. The Geeral Reduced Gradet method s best for geeral problems of ths form. The type "d" problems are almost as hard to solve as the type "c" problems sce there s oly oe objectve All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato fucto ad may costrats. The o-learty of the may overshadows the learty of the oe. The ovex Smplex (L) method s best for geeral problems of ths form. The soluto methods preseted ths text are the aalytcal method ad the graphcal method (also kow as the LaGraga Relaxato method). The aalytcal techque solves the optmalty codtos as a set of smultaeous equatos to fd the soluto. The graphcal techque uses the codtos of optmalty for estmatg where the soluto should be moves to that soluto pot ad the re-estmates where the soluto should be. If the soluto pot s where t was predcted the the process has foud the optmal soluto. The followg sectos preset the basc prcples upo whch these soluto techques deped. The Ecoomc Dspatch problem wll be used to llustrate the smlartes ad the dffereces betwee the techques. Note that uderled letters are used to deote vectors (x) or arrays (A). The decso varables x wll use the subscrpt ; the m equalty costrats wll use the subscrpt j ; the r equalty costras wll use the subscrpt k. The objectve fucto wll be represeted by f ad the costrat(s) by h (equalty) ad g (equalty). Decso varables are the parameters that ca be chaged through cotrol ad commucato systems. All other varables are depedet o decso varables. The relatoshp betwee the decso varables ad the depedet varables are foud the costrats. The objectve fucto descrbes the mprovemet as a fucto of the decso ad depedet varables. Iequalty costrats typcally represet the lmtatos of equpmet (e.g. maxmum capacty). Equalty costrats ormally represet physcal laws (e.g. coservato of eergy). E3.5. Geeral Optmzato roblem Statemet The geeral form of a olear programmg problem s to fd x so as to: M f (x) (E3.) subject to: g (x) b h (x) c ad: x where f g ad h are gve fuctos of the decso varables x. Note that the codto x ca be satsfed by approprate defto of decso varables. Ths text does ot attempt to survey the geeral optmzato problem. Ths s a large research area wth may texts approprate for further study. osderable research s cotug ths area ad wll cotue for some tme. There are o absolutes the area of Nolear Optmzato. revous ad ew techques ca oly be assessed by tral ad error. revously judcated good techques may o loger be approprate as ew costrats or parameter chages are eeded. Fortuately at least the ecessary codtos for a optmum to exst ca be detfed most of the tme. E3.5.3 KKT odtos ad LaGraga Multplers The frst step s to form the LaGraga fucto of (E3.): F T ( x λ ) f ( x) λ h( x) [ b] T [ c] g( x) (E3.) where λ ( λ λ ) ad ( r) K λ m are called dual varables. The LaGraga fucto s smply K the summato of the objectve fucto wth the costrats. It s assumed that f h ad g are cotuous ad dfferetable ad that f s covex. Gve that x s a feasble pot the codtos for whch the optmal soluto occurs are: All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 3 F x F λ j j m (E3.3a) (E3.3b) [ gk( x) b] k r k (E3.3c) x (E3.3d) These codtos are kow as the Karush-Kuh-Tucker (KKT) codtos or more smply as the Kuh-Tucker (KT) codtos. The KKT codtos state that for a optmal pot ) The dervatves of the LaGraga wth respect to all decso varables must be zero (Eq. E3.3a). ) All equalty costrats must be satsfed (Eq. E3.3b). 3) A multpler k caot be zero whe ts correspodg costrat s bdg (Eq E3.3c). 4) All decso varables must be o-egatve at the optmum (Eq.E3.3d). Requremet 3 correspodg to E3.3c s called the complemetary codto. The complemetary codto s very mportat to uderstad. If x occurs o the boudary of the k th equalty costrat the g k (x) b k. I ths case g k ( x) hm ( x) k λm Eq.(E3.3c) allows k to be o-zero. Oce t s kow that the k th costrat s bdg the the k th costrat ca be moved to the vector of equalty costrats;.e. g k (x) ca the be reamed as h m (x) ad k as λ m. O the other had f the soluto x does ot occur o the boudary of the k th equalty costrat the (assumg x s a attaable pot) g k (x) - b k <. I ths case Eq. E3.9c requres that k ad the k th costrat makes o cotrbuto to the LaGraga. It s mportat to uderstad the sgfcace of ad λ. The optmal values of the LaGraga Multplers are fact the rates of chage of the optmum attaable objectve value f(x) wth respect to chages the rght-had-sde elemets of the costrats. Ecoomsts kow these varables as shadow prces or margal values. Ths formato ca be used ot oly to vestgate chages to the orgal problem but also to accelerate repeat solutos. The margal values λ j or k dcate how much the objectve f(x) would mprove f a costrat b j or c k respectvely were chaged. Oe costrat ofte vestgated for chage s the maxmum producto of a plat. Ths s the lmt of optmzato theory to be preseted. The terested reader s refereced to oe of the texts the refereces [-8]. E3.6 Ecoomc Dspatch Formulato Ecoomc Dspatch s the process of allocatg the requred load demad betwee the avalable geerato uts such that the cost of operato s mmzed. There have bee may algorthms proposed for ecoomc dspatch: Mert Order Loadg Rage Elmato Bary Secto Secat Secto Graphcal/Table Look-Up ovex Smplex Datzg-Wolf Decomposto Separable ovex Lear rogrammg Reduced Gradet wth Lear ostrats Steepest Descet Gradet Frst Order Gradet Mert Order Reduced Gradet etc. The close All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 4 smlarty of the above techques ca be show f the soluto steps are compared. These algorthms are well documeted the lterature. We wll use oly the aalytcal ad the graphcal (LaGraga Relaxato) techques. Ecoomc Dspatch s also the most tesve part of a Ut ommtmet program. A Ecoomc Dspatch algorthm expeds approxmately sevety (7) percet of the computer tme of a Ut ommtmet program. Thus the selecto ad mplemetato of a Ecoomc Dspatch algorthm s a cetral ssue of ay Ut ommtmet research. Also sce Ecoomc Dspatch executes approxmately oce every fve mutes each eergy cotrol ceter ay computato reducto has a sgfcat mpact. Thus t s ecessary for the selecto of the best method for Ecoomc Dspatch. Ths text s drected to troduce the optmzato algorthms the geeral lterature. Thus the followg does ot address the selecto of the best method to use for Ecoomc Dspatch for a gve problem or data. However the followg does provde a excellet startg pot. E3.6. Geerato Models The electrc power system represetato for Ecoomc Dspatch cossts of models for the geeratg uts ad ca also clude models for the trasmsso system. The geerato model represets the cost of producg electrcty as a fucto of power geerated ad the geerato capablty of each ut. Ths model was dscussed secto 3.4 of ths module. We ca specfy t as:. Ut cost fucto: OST ( ) (E3.4) where OST producto cost (uts of $/hr). Ut capacty lmts: eergy to cost coverso curve producto power (E3.5) where m m geerator level max max geerator level E3.6. Trasmsso Model The geeral trasmsso model used for ED represets the balace betwee power suppled ad power cosumed wth ad delvered from the area of the tercoecto for whch the calculato s beg doe. Ths area of coecto s hereafter referred to as the cotrol area. I geeral the we may wrte where s the power geerato at ut D s the total power demaded the cotrol area LOSS s the total power loss the cotrol area ad te s the total power flowg out of the cotrol area to other tercoected cotrol areas. If the power s flowg the te s a egatve umber. Ut commtmet s the procedures used to determe whch geerato uts should be coected to the grd. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 5 D Loss te (E3.6) Let s assume that for a gve demad D ad te flow te the losses are fxed. Ths s a approxmato because realty the losses wll chage depedg o how power demad s allocated to the varous geerators. We accept ths approxmato here order to keep the dscusso basc. We ote that (E3.6) represets a equalty costrat. It s sometmes called the power balace costrat. E3.6.3 Formulato of the LaGraga We are ow a posto to formulate our optmzato problem. Stated words we desre to mmze the total cost of geerato subject to the equalty costrats o dvdual uts (E3.5) ad the power balace costrat (E3.6). Stated aalytcally we have: Mmze: ( ) Subject to: D Loss te T (E3.7) We ote that ths optmzato problem s the same form as E3. f we recogze these smlartes: T x ( ) f ( x) D LOSS h te c g ( x) b The equalty costrat h(x) c for the geeral case was allowed to cota multple costrats. Here the ED problem we see that there s oly oe equalty costrat.e. h ad c are both scalars. Ths mples that λ s a scalar also. The LaGraga fucto the s: F ( λ ) ( ) λ [ ] T [ ] [ ] (E 3.8) All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 6 All materals are uder copyrght of owerlear. opyrght all rghts reserved Here we ote that r ( s the umber of geerators) because there are equalty costrats for each decso varable : the maxmum ad mmum levels of geerato. E3.6.4 KKT odtos Applcato of the KKT codtos to the LaGraga fucto of E.3. results : ( ) F λ (E3.9) te LOSS D F λ (E3.) ( ) [ ] [ ] [ ] [ ] [ ] [ ] [ ]... Τ b x g (E3.) The KKT codtos provde us wth a set of equatos that ca be solved. The ukows these equatos clude the geerato levels ad the LaGrage multplers λ...... a total of (3) ukows. We ote that E3.9 provdes equatos E3. provdes oe equato ad E3. provdes () equatos. Thus we have a total of (3) equatos. E3.6.5 KKT odtos for a -Ut System To llustrate more cocretely let s cosder a smple system havg oly two geeratg uts. The LaGraga fucto from E3.8 s: ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] F T λ λ (E3.) The KKT codtos from E3.9 E3. ad E3. become: ) 3. 3 ( E ) 35. 3 ( E
( E 3. 4 ) E3 System otrol Overvew ad Ecoomc Dspatch alculato 7 F F F λ ( ) ( ) λ T Τ λ [ h( x) c] [ ] [ ] (E3.3) (E3.4) (E3.5) We see that there are seve ukows: λ There are also seve equatos. Example E 3. Let s ow provde umercal data for the two-ut problem. The cost-curves are approxmated usg quadratc fuctos. I geeral the form of these fuctos s gve by: ( ) a( ) b c (E3.6) where a s the quadratc term b s the lear term ad c s the costat term. These terms together wth the mmum ad maxmum geerato specfcatos for each geerator are gve the table below. The total geerato to be allocated s T D LOSS TIE 4MW Table E3.4 Dspatch Data for Example ase Ut Ut Geerato Specfcatos: Mmum Geerato MW MW Maxmum Geerato 38 MW MW ost urve oeffcets: Quadratc Term.6.9 Lear Term.87.47 ostat Term.3 74.74 All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 8 The LaGraga fucto s: F λ.6 λ ( ).87( ).3.9( ).47( ) [ 4] [ ] [ 38] [ ] [ ] 74.74 (E3.7) The KKT codtos are the gve by: F.3 F.38.47 λ F λ 4 Τ ( ).87 λ ( ) [ g( x) c] [ ] [ 38] (E3.8) (E3.9) (E3.) (E3.) [ ] [ ] (E3.) E3.7 Soluto rocedures We wll study two soluto procedures. The frst oe s aalytcal ad the secod oe s graphcal. We wll llustrate both soluto procedures by extedg Example E3.. E3.7. Aalytcal Soluto Oe otes that Eq. (E3.8) (E3.9) ad (E3.) are lear the ukows. However Eq. (E3.) ad (E3.) are ot lear due to the product terms cosstg of the LaGrage multplers ad the varables. I geeral solvg lear equatos s easy whle solvg o-lear equatos s ot. We desre a soluto approach where we ca apply the mathematcs of lear equatos. Recall from that Eqs. (E3.) ad (E3.) are derved from the complemetary codto. Ths codto requres that (E3.) or 38 All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 9 ad (E3.) ad or or Our soluto procedure s based o the followg dea: For each equato assocated wth the complemetary codtos we ca guess whch term s zero. We the solve the resultg set of equatos (E3.8 E3.9 ad E3.) ad check to see f the soluto satsfes the orgal equalty costrats. If t does our guess was correct. If t does ot we make aother guess ad try aga. The most atural startg guess s that all equalty costrats are o-bdg meag the soluto has all geerato levels wth (but ot at) ther assocated lmts. The mplcato of ths s that ad Example E 3.3 For the two-ut system supplyg 4 MW the KKT codtos reduce to.3.87 λ ( ) ( ).38.47 λ 4 We ca rewrte these equatos as.3( ) ( ) λ.87 ( ).38( ) λ.47 4 (E3.3) (E3.4) I matrx form ths set of equatos s represeted by.3.38.87.47 λ 4 (E3.5) Solvg ths matrx equato (usg MATLAB) we have.9 79.7 λ 9.4 where.9 MW 79.7 MW ad λ 9.4 $/MWhr. From table E3.4 we ca see that All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 35 We have guessed correctly ad foud the soluto. Example E 3.4 It s mportat at ths pot to uderstad the meag of λ 9.4 $/MWhr. Ths s the system cremetal cost. It dcates how the total system costs would chage f we creased the demad by MW for the ext hour. We ca check ths terpretato by computg the total system cost at 4 MW ad aga at 4 MW. At 4 MW we have.9 ad 79.7. Therefore Total costs are T ( ).6(.9).87(.9) ( ) 378.53 $ / hr ( ).9( 79.7).47( 79.7) ( ).5 $ / hr.3 74.74 ( ) ( ) 378.53.5 498. 78 Now we eed to obta total costs for T 4 MW. Aga guessg that the costrats are o-bdg (guessg that our optmzed soluto wll be wth the bouds of operato for the geerator) the KKT codtos reduce to.3.38.87.47 λ 4 (E3.6) The soluto s.83 8.7 λ 9.5 The costs for each geerator are ( ).6(.83).87(.83) ( ) 383.5 $ / hr ( ).9( 8.7).47( 8.7) ( ) 4.5 $ / hr.3 74.74 The total cost s T ( ) ( ) 383.5 4.5 58. 3 So as a result of the MW crease demad the total cost wll chage by 58.3-498.78 9.5 $/hr. Ths s agreemet wth our soluto of λ 9.5 $/hr. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato Example E 3.5 ( Bdg ostrat ) Now let s vestgate what happes whe our tal guess s correct. osder a total demad of T 55 MW. Guessg that all costrats are o-bdg the KKT codtos reduce to.3.87.38.47 λ 55 wth a soluto of 3.7 48.9 λ.84 (E3.7) Because geerator must geerate betwee MW ad MW we see that s out of rage. So we must guess aga. However our ext guess should ot be made arbtrarly. The fact that s above ts geerato lmt wth costrats gored suggests that t s a less expesve ut. Accordgly we should try to extract as much power from t as possble. So let us set MW. Because ths example cludes oly two uts the soluto may be foud qute easly: 55 MW 35 M However ths drect-substtuto approach would ot work for systems havg more tha uts sce assgg a costat value to oe of the three or more varables would stll leave two or more varables for whch to solve. I addto t does ot detfy the values of the LaGrage multplers. We wll therefore proceed wth the formal soluto approach. Settg MW mples that the costrat assocated wth [ ] s bdg. Ths meas that -double-upper-bar may ot be zero. The KKT codtos are therefore from E3.8 E3.9 E3. ad E3...3.38 ( ) ( ).87 λ.47 λ 55 (E3.8) Because we have three equatos ad four ukows we eed aother equato. We kow that -. Ths provdes our fourth equato. Rewrtg all four equatos we have.3 ( ) ( ) ( ).38( ) 55 ( ) ( ) λ.87 λ.47 (E3.9) I matrx form ths set of equatos becomes All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato.3.38.87.47 λ 55 (E3.3) The soluto s 35 λ 3.39 3.38 Example E 3.6 (The Meag of ) The uts of are the same as those of λ: $/MWhr. It s mportat to uderstad the meag of 3.38 $ / MWhr Ths s the cremetal cost of the costrat assocated wth the upper lmt of ut. It dcates the cost of creasg ths lmt by MW for the ext hour. Sce -double-upper-bar s egatve the cost s actually a savgs. We ca check ths cocluso by computg the total system costs whe the costrat s MW ad whe t s MW. For a total demad of 55 MW wth a maxmum ut- geerato capablty of MW we have 35 ad therefore Now we eed to obta the total costs for a total demad of 55 MW wth a crease the upper lmt of the secod ut by MW. To do ths we eed to re-solve the ecoomc dspatch problem. Sce we have already foud the ucostraed problem (refer to Eq. E3.7) to result 3.7 48.9 we kow that the maxmum lmt of MW lmt wll be volated. We smply eed to adjust Eq. E3.3 resultg wth a soluto of: ( ).6( 35).87( 35).3 ( ) 845.76 $ / hr ( ).9( ).47( ) 74.74 ( ) 35.47 $ / hr T ( ) ( ) 845.76 35.47 46.3 $ / hr.3.38.87.47 λ 55 349 λ 3.35 All materals are uder copyrght of owerlear. opyrght all rghts reserved 3.3
E3 System otrol Overvew ad Ecoomc Dspatch alculato 3 The costs for each geerator are the The total costs are T ( ).6( 349).87( 349).3 ( ) 83.39 $ / hr ( ).9( ).47( ) 74.74 ( ) 35.5 $ / hr T ( ) ( ) 83.39 35.5 457.89 $ / hr ( ) ( ) 83.39 35.5 457. 89 So as a result of the MW crease max the total cost wll chage by 457.89 46.3 3.34 $ / h The small dfferece betwee ths value ad -double-upper-bar s due to roud-off error ad the o-learty of the problem. E3.7. Graphcal Soluto Recall the frst KKT codto whe appled to the geeral system eq.(e3.9) show aga here for coveece: ( ) F λ If we assume that all bdg equalty costrats have bee coverted to equalty costrat so that the mu s are zero the eq. (E3.3) (above) becomes F ( ) λ B where B s the set of all geerators wthout bdg costrats. Ths equato mples that for all regulatg geerators (.e. uts ot at ther lmts) each geerator s cremetal costs are the same ad are equal to λ: ( ) ( ) ( ) L λ B Ths very mportat prcple provdes the bass o whch to apply the graphcal soluto method. The graphcal soluto s llustrated Fgure E3.6 (ote that I meas cremetal-cost-curve). The ut's data are smply plotted adjacet to each other. The a value for λ s chose (judcously) a ruler s placed horzotally across the graphs at the value of λ ad the geeratos are added. If the total geerato s equal to the total demad T the the optmal soluto has bee foud. Otherwse a ew value for λ s chose ad the process repeated. The lmtatos of each ut are cluded as vertcal les sce the soluto must ot clude geerato beyod ut capabltes. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 4 Fgure E3.6 Graphcal Soluto of ED Example E 3.7 Our two-ut problem ca be solved usg a graphcal approach as show Fgure E3.7. The λ axs s o the far rght ad s used for all uts sce all uts must have the same value for λ at the optmum soluto. The ruler yelds the geerato for each ut at the gve value of λ ad s show as a le wth sold dots at each ed. The ruler ca the be used to fd the geerato for each ut for a gve fucto of λ by movg t up ad dow. These geerato values are the added to fd the total geerato. If the total geerato s the geerato to be dspatched the the placemet of the ruler s optmal. Otherwse the ruler has to be moved up f the total geerato s too low ad dow f the total geerato s too hgh. To smply the operato ote that the total geerato for each value of λ s show o the far rght. Also the λ axs s provded at both the left ad rght had sdes for coveece. A smlar producto-costg curve s show Fgure E3.8 wth a ruler whch would move parallel wth the above ruler. The soluto dcated Fg E3.7 correspods to a loadg level of about T 4 MW λ 9.3 $/MWhr 3 MW ad 87 Mw. See f you ca verfy the solutos foud example 3.3 ( T 4 MW) ad 3.5 ( T 55 MW). λ ($/MW) (MW) (MW) T (MW) λ ($/MW) 3.5 35. 55. 3.5 -- -- -- -- -- -- T -- -- S -- -- O -- -- Y -- -- T -- S -- S 9.848 44.. A 444. Y 9.848 T -- U -- U -- L -- S -- E -- N -- N -- -- T -- M -- I -- I -- G -- E -- -- T. T -- E -- M -- L 9.6 -- 8.4 N 43.4 9.6 A -- -- -- E -- L -- M -- -- -- R -- A -- B -- -- -- A -- M -- D --. -- T -- B -- A 8.4375 6.8 I 36.8 D 8.4375 -- -- O -- A -- -- -- N -- -- -- -- -- -- -- -- -- -- 6.. 3. 6. Fgure E3.7 Ecoomc Dspatch Graphcal Soluto All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 5 λ($/mw) ($) ($) T ($) λ ($/MW) 3.5.8 4. 3.5 -- -- -- -- -- -- T -- -- S -- -- O -- -- Y -- -- T -- S -- S 9.848.59.3 A.89 Y 9.848 T -- U -- U -- L -- S -- E -- N -- N -- -- T -- M -- I -- I -- G -- E -- -- T.376 T -- E -- M -- L 9.6 --. N.496 9.6 A -- -- -- E -- L -- M -- -- -- R -- A -- B -- -- -- A -- M -- D --.83 -- T -- B -- A 8.4375.957 I.4 D 8.4375 -- -- O -- A -- -- -- N -- -- -- -- -- -- -- -- -- -- 6..5.683 6. E3.8 Summary Fgure E3.8 roducto ostg Graph The followg refereces are suggested for further readg. Ths author has used these texts. The texts by Hller ad Leberma are must readg for all studets of Operatos Research [3]. Ths text outles the applcato of Lear rogrammg to may uque problems ad eve apples specal Lear rogrammg algorthms. The texts by ooper ad Steberg [ ] ad by Smmos [6] are texts that are more advaced for Operatos Research. Lueberger's text [4] ad erre's text [5] preset the materal at a graduate level. Lasdo's text [7] presets advaced materal at a graduate level. The classc presetato of optmzato for power systems s the text by Wood ad Wolleberg [8]. Refereces. L. ooper ad D. Steberg Itroducto to Methods of Optmzato W. B. Sauders ompay hladelpha esylvaa 97.. L. ooper ad D. Steberg Methods ad Applcatos of Lear rogrammg W. B. Sauders ompay hladelpha 974. 3. F. S. Hller ad G. J. Leberma Itroducto to Operatos Research Holde-Day Ic. Oaklad alfora 986. 4. D. G. Lueberger Itroducto to Lear ad No-Lear rogrammg Addso-Wesley ublshg ompay Readg Massachusetts 973. 5. D. A. erre Optmzato Theory wth Applcatos Joh Wley & Sos Ic. New York 969. 6. D. M. Smmos Nolear rogrammg for Operatos Research retce-hall Ic. Eglewood lffs N. J. 975. 7. L. S. Lasdo Optmzato Theory for Large Systems MacMlla ublshg o. Ic. New York NY oller MacMlla ublshers Lodo Eglad 97. 8. A. J. Wood ad B. F. Wolleberg ower Geerato Operato ad otrol Joh Wley & Sos New York NY 984. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 6 R O B L E M S roblem A two-ut system s gve by the followg data ( g ).5 ( g ) ( g ) 6 ( ).5 ( ) 7 ( ) 3 g The total system demad s 5MW. The lower ad upper lmts for each geerator ut are ad 3MW respectvely. (a) Determe the optmal dspatch gorg equalty costrats (b) Ad detfy whether t s a feasble dspatch or ot (support your aswer) g g roblem Geerator cost rate fuctos $/hr for a three ut system are gve as ( ).4 5.3 5 ( ).6 5.5 4 ( ).9 5.8 Lmts o the geerato levels are 45 5 35 supply a total demad of 975 MW. 3 3 3 3 3 5. These three geerators must (a) Form the lear matrx equato ecessary to solve the ucostraed optmzato problem. (b) The soluto to the ucostraed optmzato problem s 48.9MW 35.3MW 3 86.5MW. For ths soluto (.e. gorg lmts) () ompute λ () Determe the total cost rate () How much would the total cost rate chage f the total load creased from 975 to 976 MW? (Idcate whether the total cost rate creases or decreases). (c) Form the lear matrx equato ecessary to solve the ext terato of gettg the soluto to ths problem. roblem 3 A three-ut system s gve by the followg data. The total system demad s MW. Geerator costrats are < < 55 g < < g 3 < < g 3 3 ( g ). ( g ).3 ( g ) ( g ).3 ( g ). ( g ) 3 ( ). ( ).9 ( ) 5 3 g3 g3 g3 (a) Idetfy the objectve fucto for ths optmzato problem. (b) Idetfy the LaGraga fucto assumg o costrats are bdg. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 7 (c) Idetfy the KKT codtos assumg o costrats are bdg. (d) Fd the soluto to the problem assumg o costrats are bdg. (e) Fd the soluto to the problem accoutg for ay bdg costrats. (f) Fd the total cost of supplyg the MW usg the soluto foud part (e) (g) Approxmately the total cost of supplyg the MW chage f the upper lmt o geerator was creased from 55MW to 56MW. roblem 4 A three-ut system s gve by the followg data. The total system demad s MW. Geerator costrats are < 7 < 3 < 5.3. < g < g < g ( g ).8 ( g ).5 ( g ) 5 ( g ).3 ( g ). ( g ) 3 ( ). ( ) ( ) 5 3 g3 g3 g3 (a) Set up the lear matrx equato to solve the ecoomc dspatch problem assumg all costrats are satsfed (.e. gore costrats. DO NOT solve the equato. (b) The soluto to the problem (a) s g 664.5MW g 8.MW ad g 3 53.3MW. Reformulate ths lear matrx equato to solve the ecoomc dspatch problem for ths system accoutg for ay volated costrats. Aga you DO NOT eed to actually solve the equato just set t up. together wth formato gve the part b problem (c) Usg oly the cost fucto for geerator ( ) g statemet determe the system λ for the soluto to the ucostraed problem. roblem 5 Recall that the "system λ" s the cost to the system ower of producg the ext MW over the ext hour; t s equal to the cremetal cost of a dvdual ut whe the system s ecoomcally dspatched for mmum cost ad the ut s ot at a upper or lower geerato lmt. A two-ut system s gve by the followg data. ( g ).5 ( g ) ( g ) 6 ( ). ( ) 6 ( ) 4 g g g The demad s 3MW. Wrte the KKT codtos that must be satsfed at the optmal soluto to ths problem assumg that both uts are operatg betwee ther respectve upper ad lower lmts.. Set up the lear matrx equato to solve the ecoomc dspatch problem for ths system assumg that both uts are operatg betwee ther respectve upper ad lower lmts. Do NOT solve the system of equatos. 3. The soluto to the problem () s g 8.57MW g 7. 43MW. Assumg that each ut has a mmum geerato capablty of 8 MW. (a) Idcate why the gve soluto s ot feasble. (b) Idetfy the optmal feasble soluto (c) Idetfy the cremetal costs of each ut at the optmal feasble soluto (d) Idetfy the system λ at the optmal feasble soluto (e) Would the total cost of supplyg the 3MW crease or decrease (relatve to the total cost correspodg to the optmal feasble soluto) f the mmum geerato capabltes o both uts were chaged to 79MW? All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 8 roblem 6 The system λ s the cost to the system ower of producg the ext MW over the ext hour. It s equal to the cremetal cost of a dvdual ut whe the system s ecoomcally dspatched for mmum cost ad the ut s ot at a upper or lower geerato lmt. A three-ut system s gve by the followg data. Total system demad s MW. ( g).8 ( g).5 ( g) ( g ).5 ( g ) ( g ) 3( g3). ( g3) g3 5 a) Set up the lear matrx equato to solve the ecoomc dspatch problem for ths system. DO NOT solve the equato. b) The soluto to the problem (a) s g 549.6 MW g 43. MW ad g3 7.3 MW. Assume that each ut has a maxmum geerato capablty of 35 MW. Reformulate the lear matrx equato to solve the ecoomc dspatch problem for ths system. Aga DO NOT solve the system. c) What s the cremetal cost for ut uder the codto specfed part (b)? Do you thk the system λ s greater tha or less tha ths value? 5 6 roblem 7 Geerator has a cremetal cost curve of: I ( g ).5( g ). ad lmts of: MW g MW. The geerator operates a ecoomcally dspatched system. I ths system t s foud that supplyg a addtoal 5 MW costs a addtoal $5/hr. Determe g. roblem 8 A system cossts of two geerators supplyg a load. Geerators ad have cremetal cost curves as dcated below: I.4. I ( g ) ( g ) ( ).6( ).. g g ad lmts of: MW 3 MW g g MW MW a) I ths system whe the load s 4 MW what s the dspatch of these two uts? b) I ths system whe the load s 9 MW what s the dspatch of these two uts? c) I ths system uder a certa ecoomcally dspatched scearo (a scearo dfferet tha part (a) ad (b)) t s foud that supplyg a addtoal MW costs a addtoal $5.68/hr. Determe g ad g. All materals are uder copyrght of owerlear. opyrght all rghts reserved
E3 System otrol Overvew ad Ecoomc Dspatch alculato 9 roblem 9 A two ut system has cremetal cost curves (the dervatves of the cost curves) of I. 5 ad I. 4 where ad are gve MW. The demad s 3 MW. Igorg lmts o the geerators determe the values of ad that mmze the cost of supplyg the 3 MW. roblem A two-geerator system s operatg o ecoomc dspatch ad supplyg 4 Mw of load. The total cost of supply s computed from the fal ED soluto (.e. all costrats are satsfed) ad foud to be $3/hr. From ths same fal soluto the LaGrage multplers are foud to be: Equalty costrat λ$5/mw-hr g > Mw L g < 3 Mw H g > Mw L g < Mw H -$4./Mw-hr Here the subscrpts L ad H dcate Low lmt ad Hgh lmt respectvely ad refer to the correspodg equalty costrat. For each questo below you must provde some bass or reasog for your respose. (a) What would be the (approxmate) total cost of supply f the total demad was creased to 4 MW? (b) What would be the total cost of supply f the lower lmt for geerator was creased from MW to MW? (c) What would be the total cost of supply f the upper lmt for geerator was creased from MW to MW? (d) What are the geerato levels Mw of geerators ad? (e) What s the cremetal cost for geerator? roblem A two geerator system has cost curves ($/hr) of ( ).6 5 3 ad ( ). 4 where ad are gve MW. The total demad s T 5 MW. The lmts o these geerators are < <3 ad < <3. a. Determe the ucostraed values of ad that mmze the cost of supplyg the 5 MW ad dcate whether ths soluto s feasble or ot. b. For the soluto foud (a) how much would the total cost of supply chage f the total demad creased to 5 MW for oe hour? c. Use the complemetary codto (the thrd codto the KKT codtos) to detfy the values of each Lagrage multpler assocated wth the equalty costrats. All materals are uder copyrght of owerlear. opyrght all rghts reserved