Generation Reliability Evaluation in Deregulated Power Systems Using Game Theory and Neural Networks

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1 Smart Grd ad Reewable Eergy, 212, 3, Publshed Ole May 212 ( 1 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks Hosse Harooabad 1, Hassa Barat 2 1 Electrcal Departmet, Islamc Azad Uversty (Islamshahr Brach), Tehra, Ira; 2 Electrcal Departmet, Islamc Azad Uversty (Dezful Brach), Dezful, Ira. Emal: harooabad@au.ac.r, barat216@gmal.com Receved March 3 th, 212; revsed Aprl 27 th, 212; accepted May 4 th, 212 ABSTRACT Deregulato polcy has caused some chages the cocepts of power systems relablty assessmet ad ehacemet. I the preset research, geerato relablty s cosdered, ad a method for ts assessmet s proposed usg Game Theory (GT) ad Neural Networks (NN). Also, due to the stochastc behavor of power markets ad geerators forced outages, Mote Carlo Smulato (MCS) s used for relablty evaluato. Geerato relablty focuses merely o the teracto betwee geerato complex ad load. Therefore, the research, based o the behavor of players the market ad usg GT, two outcomes are cosdered: cooperato ad o-cooperato. The proposed method s assessed o IEEE-Relablty Test System wth satsfactory results. Loss of Load Expectato () s used as the relablty dex ad the results show geerato relablty cooperato market s better tha o-cooperato outcome. Keywords: Power Market; Geerato Relablty; Game Theory (GT); Neural Networks (NN); Mote Carlo Smulato (MCS) 1. Itroducto Power systems have evolved over decades. Ther prmary emphass s o provdg a relable ad ecoomc supply of electrcal eergy to ther customers [1]. A real power system s complex, hghly tegrated ad almost very large. It s dvded to approprate subsystems or fuctoal zoes that ca be aalyzed separately [1]. Ths paper deals wth geerato relablty assessmet (Herarchcal Level I-HLI) power pool market, ad the trasmsso ad dstrbuto systems are cosdered relable ad adequate as show Fgure 1. Most of the methods used for geerato relablty evaluato are based o the loss of load or eergy approach. Oe of the sutable dces that descrbes geerato relablty level s Loss of Load Expectato (), that s the tme whch load s more tha the avalable geerato capacty. Ge. 1 Ge. 2 Ge. Relable Trasmsso & Dstrbuto Systems Load Fgure 1. Power pool market schematc for HLI relablty assessmet. Geerally, the relablty dces of a system ca be evaluated usg oe of the followg two basc approaches [1]: Aalytcal techques, Stochastc smulato. Smulato techques estmate the relablty dces by smulatg the actual process ad radom behavor of the system. Sce power markets ad geerators forced outages have stochastc behavor, Mote Carlo Smulato (MCS), as oe of the most powerful methods for statstcal aalyss of stochastc problems, s used for relablty assessmet ths research. Sce the begg of the 21 st cetury, may coutres have bee tryg to deregulate ther power systems ad create power markets [2,3]. I the power markets, the ma fucto of players s ther ow proft maxmzato, whch severely depeds o the type of the market. As a result, geerato relablty assessmet depeds o the market s type ad characterstcs. Relablty problems have bee evaluated power markets durg the last decade [4,5]. Ths paper deals wth geerato relablty power pool markets usg game theory (GT) ad Neural Networks (NN). GT s the mathematcal study of teracto amog depedet, self-terested agets [6]. That s, where the actos of oe aget affect the payoff (utlty or proft) of aother aget a way that affects the choce of best acto by Copyrght 212 ScRes.

2 9 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks the affected aget. I Secto 2, fudametals of GT ad ts applcato to ecoomcs are dscussed. Secto 3, deals wth the algorthm for geerato relablty assessmet power markets usg GT, ad fally, case study results are preseted ad dscussed Secto Olgopoly Power Markets ad GT Cocepts Geerally, ecoomsts dvde the markets to four groups [7]: Perfect competto market, Moopoly market, Moopolstc competto market, Olgopoly market. Olgopoly market s a market whch the umber of buyers s small, whle the umber of sellers could be large. Ths market dffers from perfect competto market, because each frm s large eough to have a sgfcat effect o the market. Also t dffers from moopoly market, because there s more tha oe frm the market. It dffers from moopolstc competto market too, because ts products are smlar together. Ulke a pure moopoly or perfect compettve frm, most frms must cosder the lkely resposes of compettors whe they make strategc decsos about prce advertsg expedture, vestmet ew captal ad other varables. The ma questo of each player s: If I beleve that my compettors are ratoal ad act to maxmze ther ow profts, how should I take ther behavor to accout whe makg my ow proft-maxmzg decsos? Oe of the solutos, whch ecoomsts use to aswer ths questo, s GT. The applcato of GT has bee a mportat developmet mcroecoomcs [7]. Oe ca mage two dfferet outcomes. Frst, the frms mght get together ad form a cartel, coordatg ther behavor as f they are a sgle moopoly. Secod, they mght behave depedetly, each tryg to maxmze ts ow proft whle somehow takg accout the effect of what t does o what the other frms do. Ths paper deals wth geerato relablty evaluato of these. Market demad curve has egatve gradet, ad the amout of demad decrease s explaed by prce elastcty of demad. Ths dex s small for short terms, ad bg for log terms; because loger terms, customers ca better adjust ther load relatve to prce [7]. Demad fucto s geerally descrbed as P = a b Q. Therefore, prce elastcty of demad s explaed as: dq E d dp 1 b (1) Let s suppose load forecasted by dspatchg ceter s a depedet power from prce that equals to Q. Therefore, demad fucto ca be obtaed as: Q Q (2) P a bq bq bq E d E d Also Total Reveue (TR) s obtaed as: 2 TR P Q a Q b Q (3) 2.1. Cooperatve Behavor: The Cartel Suppose all the frms decde to cooperate ther mutual beeft: They calculate ther costs as f they were a sgle large frm, produce the quatty that would maxmze that frm s profts, ad dvde the gas amog themselves by some prearraged rule. Ths codto s lke a sgle moopoly market. Such a cartel faces a fudametal problem; t must somehow keep the hgh prce t charges from attractg addtoal frms to the market. The cartel may try to deter etry wth the threat that, f a ew frm eters, the agreemet wll break dow, prces wll pluge ad the ew frm wll be uable to recoup ts vestmet. How mght the cartel alter the stuato? Oe way would be to crease the etrace cost hgh eough. Therefore here, t's cosdered that there s o ew frm that eters the moopoly market. Offer curve of a frm, s part of the margal cost (MC) curve that s more tha mmum average varable cost [7]. Also total offer curve of all frms s obtaed from horzotal sum of each frm s offer curve. Ths s a mert order fucto. I ecoomcs, f sale prce a market becomes less tha mmum average varable cost, the compay wll stop producto because t wll ot be able to cover ot oly the fx cost but eve the varable cost [9]. Due to the chagg effcecy ad heat rate of power plats, margal cost s less tha average varable cost (AVC). Therefore, power plats, AVC replaces MC ecoomc studes [8]. I a moopoly market, the moopolst cosders the producto level that maxmzes hs proft. It has bee prove that the moopolst cosders the level of producto whch margal cost of each frm (ad total margal cost of all frms) equals to the margal reveue (MR) of the moopolst [7]: MC1 MC2 MC MR (4) where: TR P Q Q 2Q MR a 2b Q (5) Q Q E E A typcal total offer ad margal reveue curves are show Fgure No-Cooperatve Behavor: Nash Equlbrum Aother outcome olgopoly power market s to assume d d Copyrght 212 ScRes.

3 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks 91 P (mlls/kwh) Total offer curve MR curve Fgure 2. Typcal total offer ad MR curves. Q (kw) that the olgopoly frms make o attempt to work together. Perhaps they beleve that agreemets are ot worth makg because they are too hard to eforce, or that there are too may frms for ay agreemet to be reached. I such a stuato, each frm tres to maxmze ts proft, depedetly. If each frm acts depedetly, the result s a Nash Equlbrum (NE). Let s be the strategy of player, ad s be the vector of strateges of all other players. Lets u s, s be the payoff to player I, the NE s a vector s, s such that [6]: u s, s u s, s ; s, (6) That s, a NE s a outcome whch each player chooses hs strategy to maxmze hs payoff, gve the equlbrum strateges of all other players. There are frms, each sellg a detcal product o a market wth verse demad fucto P(Y); where, Y y s aggregate output. Frm has cost fucto j1 C y. Frms choose output, ad choces are made smultaeously. The problem for the frm s: whch ca be rewrtte as: max P yj y Cy (7) y j1 max Py yjy Cy (8) y j Sce decsos are made smultaeously, frm s choce caot affect the choces of other frms. Thus, frm perceves correctly that y yj ; j. Thus, the best choce for the frm s obtaed usg: Py y ypc y (9) j Equato (9) ca be terpreted as a best respose fucto or reacto fucto for the frm. It specfes the best choce for the frm respose to (or reacto to) the choces by other frms. Ths termology s somewhat msleadg sce the frm does ot respod to the actos of other frms a sequetal sese (sce all frms act smultaeously); rather frm respods to what t expects other frms to do. How are those expectatos formed? The frm expects all the other frms to play the strategy (output choce) that s a best respose to ts choce. Therefore, Courot-Nash equlbrum y s characterzed by: Py y y P C y j (1) Ths s just a MR = MC codto. I other words, Courot model, each frm, supposg that other frms cotue ther preset productos, acts as a moopolst. Therefore, demad curve for the frm s obtaed as: where: P ab Q1Q 1Q1Q bq (11) a bq; a ab Q1Q 1Q1 Q (12) Usg (11), TR ad MR of frm are obtaed as (13) ad (14), respectvely: TR P Q a bq Q (13) 2 TR a Q bq MR a 2b Q (14) Q Q Therefore, a olgopoly power market wth o-cooperatve behavor, geerated power of each power plat s obtaed usg the followg soluto smultaeously: MR MC; 1 Subject to: (15) PG PG PG ; j 1 m jm jmax 3. Algorthm of Geerato Relablty Assessmet Usg GT ad NN Geerato relablty of a power system depeds o may parameters, especally o reserve marg, whch s defed as [9]: Istalled Capacty Peak Demad RM % (16) Peak Demad Now let s evaluate geerato relablty for the metoed two outcomes: cooperatve ad o-cooperatve. Copyrght 212 ScRes.

4 92 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks 3.1. Geerato Relablty Assessmet Cooperatve Codto (Moopoly Market) As explaed before, cooperatve outcome, the market acts as a moopoly market. The algorthm of geerato relablty assessmet moopoly power pool market usg Mote Carlo smulato s as follows: 1) Calculate total offer curve of the power plats. 2) Select a radom day ad ts load (Q ), ad calculate MR curve usg (5). 3) The power plats, selected for geerato the selected day, are determed from the tersecto of the power plats total offer curve ad MR curve wth regard to the reserve marg. 4) For each power plat selected the prevous step, a radom umber betwee - 1 s geerated. If the geerated umber s more tha the power plat s Forced Outage Rate (FOR), the power plat s cosdered as avalable the metoed terato; otherwse, t ecouters forced outage ad thus ca ot geerate power. Ths process s performed for all power plats usg a depedet radom umber geerated for each plat. Fally, sum of the avalable power plats geerato capactes s calculated. If the sum becomes less tha the tersecto of the power plats total offer curve ad demad expoet curve, we wll have terrupto the terato, ad therefore, wll crease by as much as oe ut; otherwse, we wll go to the ext terato. The algorthm of avalable geerated power ad calculato for each terato MCS s show Fgure 3. 5) The Steps 2 to 4 are repeated for calculato of the fal Geerato Relablty Assessmet No-Cooperatve Codto (Courot-Nash Equlbrum) The algorthm of geerato relablty assessmet olgopoly power pool market for o-cooperato outcome usg MCS s as follows: 1) Select a radom day ad ts load (Q ), ad calculate demad curve cross of bass ad gradet usg (2). 2) Calculate the power plats geerated powers the selected day usg (11)-(15). Also, the amout of total demad s obtaed usg sum of plats geerated powers regardless of reserve marg. 3) For each power plat selected the prevous step, wth regard to the reserve marg, a radom umber betwee - 1 s geerated. If the geerated umber s more tha the power plat s Forced Outage Rate (FOR), the power plat s cosdered as avalable the metoed terato; otherwse, t ecouters forced outage ad thus ca ot geerate power. Ths process s performed for all power plats usg a depedet radom umber geerated for each plat. Fally, sum of the avalable power = + 1 Select the frst geerator ( = 1); Avalable Geerated Power (AGP) = ; = N N does ot chage Geerate a radom umber betwee [-1] (U ) U >= FOR AGP = AGP + PG Y = NG Total AGP calculato AGP < Load = + 1 Fgure 3. The algorthm of avalable geerated power ad calculato for each terato usg MCS. plats geerato capactes s calculated. If the sum becomes less tha the tersecto of the power plats total offer curve ad demad expoet curve, we wll have terrupto the terato, ad therefore, wll crease by as much as oe ut; otherwse, we wll go to the ext terato. 4) Steps 1-3 are repeated for calculato of the fal Geerato Relablty Evaluato Usg NN Now, to create a uque structure, a four-layer Perceptro NN s used for relablty evaluato. The umber of euros each layer s 2, 15, 1 ad 1, respectvely (Fgure 4). All euros the frst, thrd ad last layers have POSLIN trasfer fucto, ad the secod layer has TANSIG trasfer fucto. Three puts of the NN clude: C: A umber that shows the kd of outcome (1 for cooperatve, ad 2 for o-cooperatve outcomes), E d : Prce elastcty of demad, RM: Reserve marg. Also, the NN s output s dex. Y N Copyrght 212 ScRes.

5 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks 93 tos may be chaged aother power system based o the power system s parameters. [Days / Secod half of year] 1 Moopoly Courot-Nash RM=% RM=4.8% RM=9% RM=13% Fgure 5. values for the frst study usg MCS. Fgure 4. Proposed NN for geerato relablty assessmet. Parts of the MCS results, obtaed from the metoed algorthm, are used for NN trag. 4. Numercal Studes IEEE-Relablty Test System (IEEE-RTS) s used for case studes. Data for IEEE-RTS ca be foud [1]. I all case studes, the followg assumptos are appled: 1) All case studes are smulated for the secod half of the year based o the daly peak load of the metoed test system. 2) All smulatos are doe wth teratos. 3) Each study s smulated for four dfferet reserve margs (%, 4.8%, 9% ad 13%). 4) All scearos are smulated for two prce elastcty of demads (.1 ad.1). 5) I Courot-Nash outcome, t s assumed that each power plat belogs to a depedet frm ( = m). 6) NN s traed wth TRAINLM method MATLAB software wth 1 epochs. I ths research, the NN reached.1 Mea Square Error (MSE) after trag. I the frst study, prce elastcty of demad equals.1. Based o ths assumpto ad usg MCS algorthm ad the proposed NN, values are obtaed versus dfferet reserve margs as show Fgures 5 ad 6, respectvely. I the secod study, prce elastcty of demad equals.1. Based o ths assumpto ad usg MCS algorthm ad the proposed NN, values are obtaed versus dfferet reserve margs as show Fgures 7 ad 8, respectvely. As show, both case studes, values the NN method are very smlar to those of the MCS method. Evdetly, the NN s specfcatos deped o the power system s characterstcs, ad the proposed NN s vald for the metoed power system. Therefore, NN s specfca- [Days / Secod half of year] [Days / Secod half of year] [Days / Secod half of year] 1 Moopoly RM=% RM=4.8% RM=9% RM=13% Courot-Nash Fgure 6. values for the frst study usg NN. 1 Moopoly RM=% RM=4.8% RM=9% RM=13% Courot-Nash Fgure 7. values for the secod study usg MCS. 1 Moopoly RM=% RM=4.8% RM=9% RM=13% Courot-Nash Fgure 8. values for the secod study usg NN. Copyrght 212 ScRes.

6 94 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks I both case studes, f reserve marg creases, wll decrease ad relablty wll mprove. I moopoly market, f prce elastcty creases, MR curve takes less gradet. As a result, tersecto of the power plats total offer curve ad MR curve occurs at less demad. Ths leads to the operato of fewer power plats. Therefore, all case studes moopoly market, f prce elastcty creases, wll decrease. I Courot-Nash equlbrum, f prce elastcty vares, the geerated power of every power plat vares, too. Therefore, wll dffer based o the share of every plat s geerated power ad FOR. values Courot-Nash outcome are very bgger tha those of the moopoly outcome. Because moopoly market, oly the plats, whch are selected by tersecto of the total offer ad MR curves, are servce (cosderg RM), whle Courot-Nash outcome, all of the players partcpate the market, ad the load feeds based o the plats optmum geerated power. Therefore, moopoly market, at every tme perod, oly a few plats are servce, whle Courot-Nash outcome, all of the power plats are servce based o ther optmum geerato. Sce the umber of -servce plats Courot-Nash outcome s very bgger tha moopoly market, geerato relablty moopoly market s better tha Courot-Nash outcome. It s worth otg that, sce avalable capacty of hydro plats IEEE-RTS are dfferet the frst ad the secod halves of the year, therefore, the preset work, smulatos were doe for the secod half of the year. Evdetly, the proposed method ca be utlzed for every smulato tme. 5. Coclusos Ths research deals wth geerato relablty assessmet power pool market usg GT. Due to the stochastc behavor of market ad geerators FOR, MCS was used for smulatos. Also, for creato of a uque structure for relablty assessmet, a NN was used, whch ts outputs were very smlar to the MCS results. Based o the players cooperato codtos, two outcomes (Moopoly ad Courot-Nash equlbrum) were cosdered. was used as geerato relablty dex, ad t was show that geerato relablty moopoly market s better tha Courot-Nash model. Also, moopoly market, f prce elastcty creases, wll mprove. O the other had, Courot-Nash model, f the prce elastcty of demad vares, the power plats geerated powers wll vary too, ad that s why chages. REFERENCES [1] R. Bllto ad R. Alla, Relablty Evaluato of Power Systems, 2d Edto, Pleum Press, New York, 1996, pp. 1, 1, 372. [2] L. Salvader, Electrc Sector Restructurg Italy, IEEE Power Egeerg Revew, Vol. 2, No. 4, 2, pp do:1.119/ [3] H. B. Puttge, D. R. Volzka ad M. I. Olke, Restructurg ad Reregulato of The US Electrc Utlty Idustry, IEEE Power Egeerg Revew, Vol. 21, No. 2, 21, pp do:1.119/ [4] R. Azam, A. H. Abbas, J. Shaker ad A. F. Fard, Impact of EDRP o Composte Relablty of Restructured Power Systems, IEEE PowerTech, Bucharest, 28 Jue-2 July 29, pp do:1.119/ptc [5] P. Wag, Y. Dg ad L. Goel, Relablty Assessmet of Restructured Power Systems Usg Optmal Load Sheddg Techque, Geerato, Trasmsso & Dstrbuto, Vol. 3, No. 7, 29, pp [6] K. Leyto-Brow ad Y. Shoham, Essetals of Game Theory, Morga & Claypool Publshers, USA, p. 11. [7] R. S. Pdyck ad D. L. Rubfeld, Mcroecoomcs, 5th Edto, Pretce Hall, Saddle Rver, 21. [8] S. Boreste, Uderstadg Compettve Prcg ad Market Power Wholesale Electrcty Market, Eergy Isttute, Uversty of Calfora, Berkeley, [9] Iteratoal Eergy Agecy (IEA), Securty of Supply Electrcty Markets Evdece ad Polcy Issues, IEA, Pars, 22, p. 16. [1] IEEE Relablty Test System Task Force of Applcato of Probablty Methods Subcommttee, IEEE Relablty Test System, IEEE Trasactos o Power Apparatus ad Systems, Vol. 98, No. 6, 1979, pp Copyrght 212 ScRes.

7 Geerato Relablty Evaluato Deregulated Power Systems Usg Game Theory ad Neural Networks 95 Symbol Lst MC: Margal cost (mlls/kwh) (1 mlls =.1 $) TR: Total reveue (mlls/h) MR: Margal reveue (mlls/kwh) AGP: Avalable geerated power U : A radom umber betwee [ - 1] NG: Number of power plats Q: Quatty of power (kw) P: Electrcal eergy prce (mlls/kwh) RM: Reserve marg (%) E d : Prce elastcty of demad (kw 2 h/mlls) C: Type of outcome Q : Forecasted load (kw) : Loss of load expectato (days/secod half of the year) FOR: Forced outage rate of power plats a: Demad expoet curve cross of bass (mlls/kwh) b: Demad expoet curve gradet (mlls/kw 2 h) m: Number of power plats the pool market : Number of depedet frms the pool market PG: Geerated power (kw) PG m : Mmum lmt of geerator (kw) PG max : Maxmum lmt of geerator (kw) Copyrght 212 ScRes.