AN ELECTRICITY MARKET ANALYSIS METHOD BASED ON PROBABILISTIC PRODUCTION COSTING TECHNIQUE

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1 AN LCTRICITY MARKT ANALYSIS MTHOD BASD ON PROBABILISTIC PRODUCTION COSTING TCHNIQU Jog-Bae Park *, Joog-R Sh *, Yu-Wo Jeog *, Gyu-Ha Choe * ad Kwag Y. Lee ** * Det. o lectrcal g., Kokuk Uversty, Seoul, 4-70, Korea ** Det. o lectrcal g., The Pesylvaa State Uversty, PA, USA Abstract: Ths aer resets a ecet algorthm or evaluatg the rot ad reveue o geeratg uts a comettve electrcty market based o the robablstc roducto costg techque. The accurate evaluato o the rot ad reveue o geeratg uts or log-term ersectves s oe o the most mortat ssues a comettve electrcty market aalyss. For ecet calculato o the rot ad reveue o geeratg uts uder the equvalet load durato curve LDC, a ew aroach to gure out the margal lats ad the corresodg market clearg rces durg a tme erod a robablstc maer s develoed. The mathematcal ormulato ad llustratve alcato o the suggested method s reseted. Coyrght 005 IFAC Keywords: Probablstc Producto Costg Techque, quvalet Load Durato Curve, ected Prot ad Reveue valuato, Margal Plat..INTRODUCTION The demse o the regulated electrcty dustry ad the emergece o comettve markets s chagg the way that electrcty s ad wll be rced ad s makg creasgly dcult or market artcats to arase the rosects or the uture electrcty markets Murray, 998; Hut ad Shuttleworth, 996. Due to troducto o cometto the electrcty dustry, a ew ramework s requred to measure the mact o geerator vestmets terms o a geerato comay or regulatory body Park, et al., 00; Chuag, et al., 00. These ecoomc studes clude the evaluato o eected reveue, cost, ad rot o geeratg uts ad relablty measures durg a tme erod a log-term ersectve. The robablstc roducto costg techque, rst troduced by Balerau, et al. 967 ad Booth 97 ollowed by several ovel aroaches Stremel, et al., 980; Schek, et al., 984; Lakshm, et al., 99, bascally emloys the equvalet load durato curve LDC ramework to cosder the robablstc characterstc o demad eressed load durato curve LDC ad the radom outage o geeratg uts smultaeously, whch s the same as the covoluto rocess o radom varables Wag ad McDoald, 994. The robablstc roducto costg method has bee wdely aled to electrc utlty lag, mateace schedulg, ad ower system oeratoal roblems sce t ca rovde lots o useul ormato such as the eected geerato ad cost o each ut, the total oeratg eese o a alteratve, ad relablty measures, etc. However, some addtoal ormato s requred or the alcato o the roducto costg techque to comettve market aalyss such as market clearg rce MCP, eected reveue ad rot o each ut durg a tme erod, etc. I ths aer, we roose a ecet rot evaluato algorthm or each geeratg ut drectly alcable to comettve electrcty markets. Also the elct eresso o eected geerato o ut- whe ut- s the margal lat s obtaed a recursve orm based o the moded robablstc roducto costg techque, whch ca roduce the same results rom the chroologcal aroach. The suggested method ca dramatcally reduce the comutato tme to obta the requred ormato such as eected reveue, cost, ad rot o each ut whe comared to the chroologcal smulato algorthm Perera, et al., 99. Thereore, the method ca useully ad ecetly be used the ecoomc studes o acty vestmet ad develomet o mateace schedulg strateges comettve electrcty market evromets.. BASIC ASSUMTIONS. Geerator ad Demad Model I ths aer, the geerators are modeled wth -state reresetato.e., o-state wth ull acty ad ostate wth zero acty ad a ower system s oly comosed o thermal geeratg uts wthout uel lmtatos. Also, the oeratg costs o each geeratg ut are regarded as costat values whereas the values are varyg wth the outut levels a real-world roblem. The demad elastcty or the market rce ad the stochastc characterstc o demad are eglected ths aer.

2 . Market Model Geerally, the electrcty market short-term ersectves ca be modeled as a seres o auctos or the rght to serve the demad amog geeratg uts. Also, each aucto ca be characterzed as sealed-bd ad uorm rced Sheble, 999. However sce we are terested the log-term aalyss, the electrcty market model should cororate the radom outage o geeratg uts. Thereore, we have assumed that the market clearg eatures also have the stochastc characterstcs relectg the radomess o geeratg uts. Also, the oer rce o each geerator s coed as the costat values wth ull acty. The dsatch order amog geeratg uts a LDC s determed by the creasg order o ther oer rces, whch s very smlar to the dsatch stuato the covetoal robablstc roducto costg techques. The market clearg rce MCP o each aucto s determed by the oer rce o the margal geerator whle the MCP s ed wth the re-determed costat value the case o the ower shortage.. TH MATHMATICAL MODL I ths chater, we wll descrbe how to evaluate the eected eergy, cost, reveue, ad rot o geeratg uts uder a LDC. Suose the market has geeratg uts ad the geeratg uts are deed by the ascedg order o the oer rces. Let the ut- have the acty o C [], the orced outage rate o q, the oeratg cost o oc [$/h], ad the oer rce o o [$/h]. We deote the tme erod o a LDC ad the cumulatve acty utl ut- by DT [hours] ad [], resectvely. Let be the eected rot o ut-, the we ca decomose ths value wth resect to the market clearg rces. I we let ad be the ut- s eected rot whe the market clearg rces corresod to the oer rce o ut- ad the rce, mc, resectvely, the ca be rereseted by. < + Sce 0,, ca be rewrtte as. + Let us dee the eected geerato o ut- whe ut- s the margal oe ad whe the rce becomes the market clearg rce by ad resectvely. The ad are rereseted by ad 4 whch are comosed o the reveue comoet.e., o ad mc ad the cost comoet.e., oc ad oc. o oc mc oc 4 Thereore, the eected rot ca be seced oly we obta ad or all geeratg uts. Based o the covetoal covoluto theory, we ca cosder the radom outage characterstcs o geeratg uts usg equvalet load durato curve LDC. Geerally, the LDC s eressed as ollows: 0 G G + q C where G : radom varable reresetg geeratg ut-, : covoluto oerator, 0 G L G : orgal load durato curve. 5 To develo ad wth aalytc eressos or all geeratg uts, t s ecessary to dee a varat o LDC relectg the codtoal robablstc stuatos. To do ths, we have deed as the codtoal LDC uder the codtos that ut- s o-state wth robablty o ad that the geerators utl ut- are covolved. The, ca be deed as ollows: G + L G < 6 > Whe s smaller tha, ut- s rrelevat to the covoluto rocess utl ut-. Thereore, becomes. Whe s equal to, the covoluto rocess s reeated utl ut-- ad the o-state codto o ut- s hadled by multlyg. Thereore, s eressed as. I case s larger tha, the covoluto rocess s erormed by covolvg geerators utl ut- ecet ut- resultg as G + L G Based o the codtoal LDC deed 6, we ca acqure the aalytc reresetatos or the eected eergy o each geeratg ut a recursve orm. Here, we should ote that whe s assessed, there s o eed to covolve ut, +,, ad sce ut- s the margal ut ad the radomess o ut- s dvdually cosdered by the o-state codto. The ca be categorzed to cases; the case o <, the case o, ad the case o >. I case o <, the value o obvously becomes zero sce the dsatch order o ut- s receded by ut-. I case o, the.e., - should be calculated based o.e.,. sce ut- tsel becomes the

3 margal ut. The resultg eresso or s gve 7. The rst term o the rght-had sde 7.e., DT d s the total eected eergy o ut-, ad the secod term.e., DT C s the eected eergy o ut- whe ut- s ot the margal ut. Thereore, by subtractg the secod term rom the rst term, we ca obta as 7. [ d C ] DT 7 For ut-, oly the o-state s our cocer sce the ostate has o eect o. The ca deal wth ths stochastc stuato, ad, thereore, ca be obtaed as 9. DT C 9 Fg. shows ad the shaded rectagular s the eected eergy o ut- whe the rce becomes the market clearg rce uder ut- s o-state codto whch s deoted by. The shaded tragle Fg. s the eected eergy o ut- whe ut- becomes the margal oe uder the codto that ut- s o-state. Probablty Probablty - ut- - Peak+ -C Ut- Fg.. Codtoal LDC o or - Fg.. Codtoal LDC o - or Peak+ - O the other had, we should mose oe more codto whe obtag, where >, that ut- should also be o-state sce otherwse, ut- caot be the margal ut. Ths stochastc stuato ca be take care o by -. The robablty wth ut- as the margal ut uder the o-state codto o ut- ca be eressed by [ ]. Thereore, ca be calculated as 8. [ ] DT C 8 Fg. shows - where > ad the shaded rectagular corresods to the eected eergy o ut- whe ut- becomes the margal oe uder the o-state codto o ut- whch s deed by. Probablty - Thereore, the geerc eresso or cludg the rce ed stuato s; 0 DT DT C DT C [ d C ] [ ] < > 0 Cosequetly, the eected rot o ut- ca be obtaed by combg, 4, ad NUMRICAL TSTS 4. Smle Test System wth Uts Test System Descrto The develoed algorthm or the evaluato o rot o each ut has bee aled to a samle ower system wth uts whose data s descrbed Table. The dsatch order a electrcty market s determed by the oer rce gve the 5 th colum o Table. Table. Geerator data o test system ut- ut- - - Fg.. Codtoal LDC o - or Peak+ - -C Geerator Name Geerator # Geerator # Geerator # Caacty FOR 0.0 Oeratg Cost $/h Oer Prce $/h Fally, order to evaluate, the radom outages o all geeratg uts ecet ut- should be cosdered. The cosdered chroologcal demad durg hours s gve Fg. 4 wth the base load o 00[], the

4 medum o 00[], ad the eak o 500[], resectvely. Fg. 4. Chroologcal demad o test system Numercal Tests I ths secto, we wll comute the eected geerato, cost, reveue, ad rot o each geeratg ut or the test system usg the roosed algorthm. Also, to show the valdty o the suggested algorthm, the results o the roosed algorthm are comared wth those o the covetoal chroologcal aroach Perera, et al., 99. To evaluate the requred ormato a robablstc maer usg the chroologcal aroach, t s ecessary to dety all the ossble outcomes as descrbed Table. ach outcome s deed as a state deoted by a row vector whose elemet.e., -th elemet corresods to -th geerator has a bary value here, we deote o-state by ad o-state by 0. Based o the results Table, oe ca easly obta the eected geerato, cost, reveue, ad rot o each geerator each hour. The results are summarzed Table. From ow o, we wll show the eectveess ad ececy o the roosed algorthm wth the same test system. As we have dscussed earler, all ormato o each geerator Table ca be obtaed usg, 4, ad 0. We wll oly show the detals o soluto rocedures ocused o Geerator #. The codtoal LDCs or Geerator # cludg the orgal LDC ca be obtaed rom 6 ad dected Fg Pr Fg. 5. Codtoal LDCs or Geerator # The eected geerato o Geerator #.e., s comosed o 4 comoets.e.,,,,. ach comoet ca be calculated as ollows usg 6 ad 0. DT 0 0 [ d C ] [h] [ ] DT C [h] [ ] DT C DT C [ ] 7[h] [h] State,,,,0,0,,0,0 0,, 0,,0 0,0, 0,0,0 Prob Table. State eumeratos o the test system Hour- 00 Hour- 500 Hour- 00 Geer. MCP Geer. MCP Geer. MCP 00,0, ,00, ,00, ,0, ,00, ,00, ,0, ,0, ,0, ,0, ,0, ,0,0 0. 0,00, ,00, ,00, ,00, ,00,0 0. 0,00,0 0. 0,0, ,0, ,0, ,0,0 0. 0,0,0 0. 0,0,0 0. Table. ected geerato ad rot rom chroologcal aalyss Geerator # Geerator # Geerator # H H H Tot. H H H Tot. H H H Tot.. Ge. h Rev. $ Cost $ Pro. $

5 Thereore, becomes 475[h] whch eactly matches wth the results o the chroologcal aroach. ach rot comoet o Geerator #.e., be easly obtaed by ad 4. That s, ca o oc [$] o oc o oc mc oc [$] 0.7[$].97[$] Thereore, the total rot o Geerator # becomes 4.8[$] whch also eactly matches wth the result Table. The rots o Geerator # ad # are comuted as.86[$] ad.04[$] by the same rocedures, whch are the same as the results Table. Note that the suggested algorthm ca dramatcally reduce the umber o eumeratos or rot evaluato o each geerator. For eamle, the total umber o eumeratos ths samle study becomes 9.e., 4++ whle the chroologcal aroach requres 4.e.,. Thereore, oe ca use the suggested algorthm the real-world ower systems wthout the comutatoal burde. 4. Moded Relablty Test System Descrto o System The geerator data o the moded I 4-ode Relablty Test System RTS I Commttee Reort, 979 are reseted Table 4. The dsatch order amog the geerators a electrcty market s determed by the oer rce reseted the 5 th colum o Table 4. Note that the rce o market s.0[$/h] case o the ower shortage. The Table 4. Geerator data o moded I RTS Geerator Name Ge. # Ge. # Ge. # Ge. #4 Ge. #5 Ge. #6 Ge. #7 Ge. #8 Ge. #9 Ge. #0 Ge. # Ge. # Ge. # Ge. #4 Ge. #5 Ge. #6 Ge. #7 Ge. #8 Ge. #9 Ge. #0 Caacty FOR Oeratg Cost $/h Oer Prce $/h chroologcal demad data has bee used rom data rovded I 4-ode RTS ad the aual eak load o ths system has bee set to,000[]. Numercal Results We wll comute the eected geerato ad rot o each geeratg ut or the moded I 4-ode RTS usg the roosed algorthm. Also, to show the ececy ad eectveess o the suggested algorthm, the results o roosed algorthm are comared wth those o Mote Carlo Smulato MCS Bllto ad L, 994 as well as chroologcal aroach. The results are summarzed Table 5. I Table 5, the results o Mote Carlo Smulato are obtaed based o the ollowg rocedure: Table 5. Comarso o smulato results o each method Geerator Name Ge. # Ge. # Ge. # Ge. #4 Ge. #5 Ge. #6 Ge. #7 Ge. #8 Ge. #9 Ge. #0 Ge. # Ge. # Ge. # Ge. #4 Ge. #5 Ge. #6 Ge. #7 Ge. #8 Ge. #9 Ge. #0 Chroologcal Aroach MCS Proosed Algorthm. Ge.. Pro.. Ge.. Pro.. Ge.. Pro. GWh 0 $ GWh 0 $ GWh 0 $

6 Ste Determe oeratg cycles o each geeratg ut. The oeratg cycle s the orm o chroologcal u-dow-u. To obta samlg values o the TTF Tme-to-Falure ad TTR Tmeto-Rear, we have assumed that oeratg tme s eoetally dstrbuted ad rear tme s costat value evaluated by multlyg 68 hours week by FOR. Ste Comute the eected geerato ad rot o each ut durg the samlg year. These values ca be obtaed by comarg the hourly avalable acty model wth the chroologcal load model. Ste Reeat ste ad utl the terato s aroached to the reed umber o samle year. Note that we have set the umber o samle year as 5,000. Ste 4 Comute the average values o eected geerato ad rot o each ut. These values ca be evaluated by dvdg summato o eected geerato or rot o each samlg year to that o the total umber o samle year. The CPU tme o each method s rovded Table 6. The CPU tme o roosed method s remarkably low comarso wth those o MCS as well as chroologcal aroach. Table 6. Comarso o CPU tmes o each method CPU tme m Chroologcal Aroach MCS Proosed Algorthm As show Table 5 ad Table 6, we ca determe that the results o roosed algorthm are eactly deted wth those o the chroologcal smulato algorthm ad the comutato tme o roosed algorthm ca be reduced whe comared wth those o chroologcal ad MCS. 5. CONCLUSIONS Ths aer resets a ecet algorthm or evaluatg the rot ad reveue o geeratg uts a comettve electrcty market based o a varat o the robablstc roducto costg techque. I the suggested algorthm, the eected rot o each ut ca be acqured by detyg the margal lat robablstcally the vewot o the geerator. Wthout ay loss o accuracy o the robablstc chroologcal method, the suggested method ca dramatcally reduce the comutatoal comlety. Thereore, the roosed method s adequately alcable to log-term ower system lag roblems the comettve electrcty market evromets. ACKNOWLDGMNT RFRNCS Murray, B lectrcty Markets Ivestmet, Perormace ad Aalyss, lectrcty Market Servce Ltd., Wokg, Joh Wely & Sos, UK. Hut, Sally ad Graham Shuttleworth 996. Cometto ad Choce lectrcty, Joh Wley & Sos. Park, J. B., J. H. Km ad K. Y. Lee 00. Geerato easo lag a comettve evromet usg a geetc algorthm. Proceedgs o I Power geerg Socety 00 Summer Meetg, Paer No X/0. Chuag, A. S., F. Wu ad P. Varaya 00. A gametheoretc model or geerato easo lag: Problem ormulato ad umercal comarsos. I Tras. o Power Systems, Vol. 6, No. 4, Balerau, H.,. Jamoulle ad Lard de Guertech 967. Smulato de l elotato d u arc de maches thermques de roducto d électrcté coulé a dès statos de omage. Revue de la Soc. Royale Belge des lect., Booth, R. R. 97. Power system smulato model based o robablty aalyss. I Tras. o PAS, Vol. PAS-9, No., Stremel, J. P., R. T. Jeks, R. B. Babb, W. D. Bayless 980. Producto costg usg the cumulat method o reresetg the equvalet load curve. I Tras. o PAS, Vol. PAS-99, No. 5, Schek, K. F., R. B. Mrsa, S. Vassos, W. We 984. A ew method or the evaluatg o eected eergy geerato ad loss o load robablty. I Tras. o PAS, Vol. PAS-0, No., Lakshm, S. R., S. C. Trathy, K. S. P. Rao, R. Balasubramaa 99. Producto costg usg med-rad ad Wograd Fourer trasorms. I Proceedgs-C, Vol. 9, No. 6, Wag, X. ad J. R. McDoald 994. Moder Power System Plag, McGraw-Hll Book Comay, Lodo. Perera, M. V. F., B. G. Gorest, M. Morozowsk Fo, J. B. Slva 99. Chroologcal robablstc roducto costg ad wheelg calculatos wth trasmsso etwork modelg. I Tras. o Power Systems, Vol. 7, No., Sheble, G. B Comutatoal aucto mechasms or restructured ower dustry oerato, Kluwer Academc Publshers. I Commttee Reort 979. I relablty test system, I Tras. o PAS, Vol. PAS-98, No. 6, Bllto, R. ad W. L 994. Relablty assessmet o electrc ower systems usg mote carlo methods, Pleum Press, New York. Ths work was acally suorted by MOCI through IRC rogram.