Pricing and operation in deregulated electricity market by noncooperative game

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1 Electric Power Systes Research 57 (21) Pricig ad operatio i deregulated electricity arket by ocooperative gae L. Geerli a, *, L. Che b, R. Yokoyaa a a Tokyo Metropolita Uiersity, Miaiosawa 1-1, Hachioji, Tokyo 192-3, Japa b Osaka Sagyo Uiersity, Nakagaito 3-1-1, Daito, Osaka , Japa Received 7 Jue 2; received i revised for 3 October 2; accepted 9 October 2 Abstract Pricig structure is becoig cosiderably iportat for both electric utility idustries ad their custoers. This paper derives a operatio rule for a arket odel with a electric utility ad idepedet power producers (IPPs) as players of the ocooperative gae. The derived operatio rules reflectig the copetitio ca be viewed as a etesio of the covetioal equaliig icreetal cost ethod for the deregulated power systes. As idicated i this paper, the prices of electricity for purchases ad sales are equal to the icreetal costs of the geerators of IPPs but are geerally cheaper tha the icreetal cost of the geerators belogig to the utility. To eaie the proposed approach, several systes are used as the deostrated eaples i this paper. 21 Elsevier Sciece B.V. All rights reserved. Keywords: Gae theory; Equaliig icreet cost; Pricig; Power systes; Deregulatio; Market 1. Itroductio Electricity arkets are eperiecig widespread chages that are sigificatly alterig the idustry. Whe ew ecooic echaiss are beig cosidered for use i the electricity arket, there is uch talk of gaig ad whether the echais will be gaed [1]. Due to the eergece of idepedet power producers (IPP), as well as the chagig structure of the electricity supply idustry, the electric power idustry has etered a icreasigly copetitive eviroet uder which it becoes ore realistic to iprove ecooics ad reliability of power systes by elistig arket forces [4,8,9]. For eaple, to aiie his payoffs, a player (e.g. a utility) seeks to displace epesive geeratio by iportig power fro eighborig players with lower cost eergy. Likewise, a player (a IPP or a utility) with ecess geeratio capacity ca choose to eport power ad receive a iediate retur o its ivestet [1]. Up to ow, a cosiderable uber of literatures addressig the copetitio ad deregulatio issues have bee published [4,8,7,9 12]. * Correspodig author. This paper ais to propose a gae theoretical egotiatio ethod to provide ratioal decisios of the power echage for purchases ad sales betwee the utilities ad IPPs, as well as prices of the electrical eergy uder copetitive eviroet, i cotrast to the covetioal equaliig icreetal cost rule. Geerally, the gae theory is divided ito cooperative gae ad ocooperative gae, depedig o whether or ot the decisios are based o the uaious agreeet aog all players [5,1]. For the ocooperative gaes, the Nash equilibriu solutio, saddle solutio, iia priciple, Stackelberg strategy ad others are ostly used to ake the decisios, i cotrast to core solutio, ucleolus, Shapley value ad Nash bargaiig solutio, which are the solutios of cooperative gaes [5,6]. Although the solutios of cooperative gaes belog to the Pareto set fro the viewpoit of social welfare, they geerally are ot easily realied due to requireet of uaious agreeet aog all players uder copetitive eviroet, coparig with decisios of ocooperative gaes, which are actually coproise solutios for players. It is well kow that the equaliig icreet cost rule for a siplified etwork odel is ot oly siple but also optial due to the optiality of social welfare, /1/$ - see frot atter 21 Elsevier Sciece B.V. All rights reserved. PII: S (1)94-3

2 134 L. Geerli et al. / Electric Power Systes Research 57 (21) as far as each participat i operatios of a power syste is copletely cooperative [14]. However, for the deregulated power systes, this rule ay ot hold because participats ited to pursue their ow beefits, thereby ay ot always take cooperative attitude. I this paper i order to copare with the equaliig icreetal cost rule, we aily use the Stackelberg strategy ad the Nash equilibriu solutio to aalye the egotiatio process betwee utilities ad IPPs sice they are widely recogied as ratioal decisios for copetitive arkets i ters of aio [5,1]. I Sectio 2, we first derive a coo price of electricity for purchases ad sales betwee a utility ad IPPs whe assuig that all of IPPs costruct a coalitio who egotiates with the utility for bargaiig. By the Stackelberg strategy [6], the Sectio 2 also provides very clear ad siple operatio rules, which are siilar to the well-kow equaliig icreetal cost [8] but differet due to the fact of copetitive eviroet. I Sectio 3, we give the prices of electricity, as well as the operatio decisios for the situatio whe each IPP egotiates with the utility idepedetly, which eas that IPPs sell their electricity at differet prices depedig o their egotiatios with the utility. I Sectio 4, a Nash equilibriu [5] is used to aalye the egotiatio betwee IPPs ad the utility, which shows that the Nash equilibriu ad the Stackelberg strategy have the differet solutios whe assuig that the purchased electric eergy depeds o its price (or is a fuctio of the price). Sectio 5 eteds the results of the earlier sectios by takig accout of trasissio etwork. As will be show i this paper, all obtaied results ca be viewed as a etesio of the equaliig icreetal cost fro regulated power systes to copetitive power arkets. I Sectio 6, we derive a optial solutio by aiiig total welfare of electric power idustry society, for which the operatio rule obviously coicides with that of the covetioal equaliig icreetal cost. To deostrate the proposed odel, several eaples are siulated ad used to copare with the covetioal approaches [3] i Sectio 7. Fially, we give several geeral rearks to coclude this paper i Sectio Negotiatio betwee a utility ad a coalitio of IPPs This sectio derives the price of electricity ad operatio decisios o the basis of the ocooperative gae [6] by supposig that all of IPPs for a coalitio who joitly egotiates with a utility. That eas all of the electricity sells at the sae price fro IPPs to the utility. Sice the price of electricity for sales directly iflueces the payoffs of IPPs, it is reasoable for the utility to assue that the quatity of the purchased electricity eergy fro IPPs depeds o its price. The basic structure of power arket i this paper is that there are a utility with trasissio etwork ad several IPPs without the trasissio etwork. Note that all results of this paper ca straightforwardly be eteded to the ulti-utility egotiatio gae, although oly oe utility is aalyed i this paper for the sake of siplificatio [3]. Let Z=( 1,, ) be supply power vector of the utility, where j,for,, stads for real power output (MWH) of geerator-j, ad is total uber of geerators for the utility. Defie g j ( j ) for,, as the icreetal cost ($ per MWH) of geerator-j of the utility. The j g j ()d is the cost fuctio ($) for geerator-j of the utility. Assue that there are IPPs ad defie X i =( i1,, ii ) to be supply power of IPP-i= 1,,, where ij for,, i represets real power output (MWH) of geerator-j ad i is the total uber of geerators for the IPP-i. Letf ij ( ij )for,,;,, i be the icreetal cost ($ per MWH) of geerator-j for the IPP-i, ad f ij / ij. The ij f ij ()d is the cost fuctio ($) for geerator-j of the IPP-i. Sice this sectio aalyses, the egotiatio betwee a coalitio of IPPs ad a utility, all of electricity is assued to be sold at the sae price fro IPPs to the utility. The let y be the price ($ per MWH) (or the icreetal cost) at which the utility purchases the electricity fro IPPs. Let d be total dead (MWH) of cosuers ad h(d) is earig ($) of the utility fro sellig electricity to custoers ad is ot related with Z,X i. The IPPs will sell electric power to utility by aiiig their profits, which ca be forulated as the followig probles [3,15]: IPP i: (,,) Maiie y i ij i ij ;,, i ij f ij ()d (1) where ij f ij ()d is the cost fuctio ($) for geerator-j of the IPP-i, ad IPP-i is assued to decide ij, j= 1,,. O the other had, the utility iteds to aiie its payoffs by utiliig eistig geerators ad lowerig the purchasig price of electricity fro IPPs, which ca be described as the followig proble: Utility: Maiie y, j ;, h(d) y i ij (y) j g j ()d (2) s.t. i ij (y)+ j =d (3) where Eq. (3) eas that the utility has the obligatio to serve all custoers load, ad the utility is assued to decide y, i,,...,. Note that we use a presued coditio i Eqs. (2) ad (3), i.e. the purchased electric power ij (y) is supposed to be a fuctio of y. Sice the

3 L. Geerli et al. / Electric Power Systes Research 57 (21) price of electricity for sales directly iflueces the payoffs of IPPs, it is reasoable for the utility to assue that the quatity of the purchased electricity eergy fro IPPs depeds o its price. Notice that ij (y) isot a predeteried fuctio but a iplicit fuctio of y, which is decided by the gae at the equilibriu. The optiality coditios for Eq. (1) is straightforward by differetiatig the objective fuctio with respect to ij, i.e. y=f ij ( ij ) (,,;,, i ) (4) Accordig to the Iplicit Fuctio Theore ad Eq. (4), there eists a differetiable fuctio ij (y) atthe eighborhood of whe df/d. Differetiatig Eq. (4) agai, we have the followig relatios d ij dy = 1 (,,;,, df ij ( ij )/d i ) (5) ij O the other had, for Eqs. (2) ad (3) of the utility, we establish a Lagragia fuctio L(Z,y,)=h(d) y + i ij (y)+ i ij (y) j g j ()d j d (6) Note that h(d) is a costat, the the optiality coditios for Eqs. (2) ad (3) ca be obtaied by L/ j for,,, L/y ad L/ =g j ( j ) (,,) (7) i i ij (y)+(y ) ij (y)+ i d ij = (8) dy j =d (9) Substitutig Eq. (5) ito Eq. (8) ad cobiig Eq. (4) with Eq. (9), we have the solutio [6] y=f ij ( ij ) (,, i ) (1) =g j ( j ) (,,) (11) y= i i ij i 1/df ij /d ij (12) ij + j =d (13) which are actually coditios of the equilibriu of the gae. Observig Eqs. (1) ad (11), the geerators for IPPs ad the utility obviously have the sae icreetal cost y ad, respectively. As show i Eq. (12), however, their icreetal costs are differet due to copetitio betwee the utility ad the coalitio of IPPs. Sice df ij /d ij is assued to be positive, the icreetal cost for utility is geerally larger tha that for IPPs accordig to Eq. (12). That iplies that the geerators of IPPs have to supply power ecooically at the operatio poit, coparig to those of the utility because the utility ows the trasissio etwork ad is geerally burdeed with the obligatio to serve the custoers loads. The price y of electricity for purchases ad sales is take as the sae value of the equaliig icreetal cost for IPPs as idicated i Eq. (1). It is evidet that the discrepacy betwee the two icreetal costs arrows if df ij /d ij have sall value, which eas that the IPPs ca beefit ore if they have geerators with saller f ij / ij (i.e. flat cost fuctio). Therefore, Eqs. (1) (13) ca be viewed as a eteded equaliig icreetal cost rule for the deregulated power systes. Note that we ca have siilar equatios as Eqs. (1) (13) whe takig the upper ad lower liits of each variable ito cosideratio, eve though ore Lagraga ultipliers have to be itroduced. I this paper, we assue that the utility decides the purchased prices based o the outputs of IPPs while IPPs deterie their outputs depedig o the purchased prices. Therefore, this odel aily targets the syste where the utility is large eough coparig to each IPP ad alost doiates the arket. 3. Negotiatio betwee a utility ad idividual IPPs This sectio gives a ocooperative solutio by supposig that each IPP idividually egotiates with the utility [3]. That eas each IPP sells its electricity to the utility at the price differet fro those of other IPPs. Let Y=(y 1,,y ) where y i for,..., represets the price at which the utility purchases the electricity fro IPP-i. All other defiitios are the sae as those i Sectio 2. Assuig that IPP-i ad the utility decide ij ( j= 1,, j )ady,z, respectively, the we have the followig probles for IPPs: IPP i: (,,) ad for the utility: Utility: Maiie Y,Z j Maiie ij ;, i y i g j ()d i ij i ij h(d) y i i ij (y i ) s.t. i ij (y i )+ j =d f ij ()d Hece i the sae way as the derivatios of the Sectio 2, we have the followig coditios for a ocooperative solutio

4 136 y i f ij ( ij ) (,,;,, i ) (14) =g j ( j ) (,,) (15) y i = i i ij L. Geerli et al. / Electric Power Systes Research 57 (21) (,,) (16) i 1/df ij /d ij ij + j =d. (17) Observig Eq. (15), it is evidet that the geerators for the utility have the sae icreetal cost, which coicides with the covetioal results. However, accordig to Eq. (14), the geerators for IPPs have differet icreetal costs because IPPs idividually egotiate with the utility by usig their ow strategies, while the equaliig icreetal cost is still hold for the geerators withi a IPP. The price y i of electricity for purchases ad sales are also differet for each IPP fro Eq. (14). Eq. (16) idicates that the flatter the fuel cost fuctios of the geerators for IPP-i is (i.e. the saller df ij /d ij is), the less the discrepacy of the icreetal costs betwee the utility ad IPP-i is liable to be. O the other had, Eqs. (14) ad (16) also show that the cheaper the icreetal cost of geerators for the IPP is, the ore a IPP ca sell power. Note that we ca derive a siilar o cooperative solutio whe assuig every geerator i IPPs sells power at its price y ij, aalogue to the derivatio of Eqs. (14) (17). Therefore, Eqs. (14) (17) ca also be viewed as a eteded equaliig icreetal cost rule for the deregulated power systes. 4. Nash equilibriu I this sectio, we oly aalye the egotiatio betwee a utility ad idividual IPPs by usig Nash equilibriu [5,3]. Therefore, we use the sae defiitio described i Sectio 3. IPP-i assues that the price y i is a fuctio of the electric power X i sellig to the utility. Assuig that IPP-i decides ij for,, j, the the strategy of IPP-i ca be forulated as: IPP i: (,,) i ij Maiie y i (X i ) i ij ;,, i f ij ()d O the other had, the utility assues that the purchased electric power ij, fro IPP-i is a fuctio of y i. The the strategy of the utility ca be described as: Utility: Maiie Y,Z ij h(d) y i i ij (y i ) j g j ()d s.t. i ij (y i )+ j =d where the utility decides Y ad Z. Supports y i / ij 1/d ij /dy i, we have the followig Nash equilibriu solutio [5]. y i = 1 i ij ij f ij ()d+ i ji ij y i () i ij (,,;, i ) (18) =g j ( j )(,,) (19) y i = i i ij (,,) (2) i i ij /f ij y i ij + j =d (21) where y i () is the electricity price whe X i =. The equatios above are obviously differet fro the Stackelberg strategy show i Eqs. (14) (17). For the Stackelberg strategy, as idicated i Eq. (14), each geerator i IPPs operates at equaliig icreetal cost, which is also take as the purchased price, while the purchased prices i the Nash gae are deteried by ot oly the icreetal costs but also outputs of geerators as show i Eq. (18). Besides, coparig Eq. (16) with Eq. (2), the discrepacies of the icreetal costs betwee the utility ad IPPs are also differet for the Stackelberg strategy ad the Nash gae. Although this sectio oly cosiders the egotiatio betwee the utility ad idividual IPPs, the Nash equilibriu solutio ca be derived as the sae way for the egotiatio betwee the utility ad a coalitio of IPPs by assuig y/ ij 1/d ij /dy. 5. Cosideratio of trasissio etwork This sectio oly deals with the egotiatio betwee the utility ad idividual IPPs. Assue that oly the utility ows trasissio etwork. Let H(Z,X 1,...,X ) be a trasissio loss fuctio (MWH), the we have the followig ocooperative gae [3,15]: IPP i: Maiie y i (,,) ij ;=1,, i Utility: j Maiie Y,Z g j ()d i ij i ij h(d) y i i ij (y i ) f ij ()d

5 L. Geerli et al. / Electric Power Systes Research 57 (21) Table 1 Coefficiets of geerators for 8-geerator power syste Geerator uber a b c Capacity (MW) Utility Uit Uit Uit Uit IPP IPP IPP IPP IPP Socially optial behavior Net, we will show that the covetioal equaliig icreetal cost rule optiies the social welfare ad is also a Pareto solutio. Cosider a power idustry society icludig a utility ad several IPPs. The the proble to aiie the welfare of the society ca be forulated as follows [2]: Society: ij Maiie X i,z;,, j f ij ()d h(d) i g j ()d s.t. t=1 i ij (y i )+ j H(Z,X 1,,X )=d. s.t. i ij (y i )+ j H=d where H=H(,X 1 (Y),...,X (Y)). The i the sae way as the derivatios of the Sectio 3, we have a ocooperative solutio, which ca be cosidered as a eteded equaliig icreetal cost rule y i f ij ( ij )(,,;,, i ) (22) = g j( j ) (,,) (23) 1 H/ j 1 i H/ ij /df ij /d ij y i = i 1/df /d ij ij i ij i 1/df ij /d ij (,,) (24) i ij + j H(Z,X 1,,X )=d. (25) Obviously, for the geerators of the utility, Eq. (23) coicides with the covetioal ecooic dispatch with the cosideratio of the etwork loss. Geerally, the trasissio loss H, which is epressed i quadratic fors of the geeratio powers ca be estiated by the B-coefficiet ethod [13]. Thus the optiality coditios satisfy = g k( k ) = f ij( ij) 1 H/ k 1 H/ ij (,,,,, i, k=1,,) (26) i ij + j H(Z,X 1,,X )=d. (27) Eq. (26) is the equaliig icreetal cost with the cosideratio of the trasissio loss for all of the geerators i the society, which eas that the best solutio to the ecooic dispatch proble is the equaliig icreetal cost if there is o copetitio but cooperatio. 7. Nuerical siulatio 7.1. A 8-geerator power syste A power syste with 8 theral geerators show i Table 1 is used to illustrate the applicatio, where the utility ows four theral geerators ad four IPPs have oe geerator, respectively. This paper assues that all of cost fuctio for geerators, i.e. g j ()d=a j 2 + b j +c j or f ij ()d=a ij 2 +b ij +c ij, has the quadratic fors, as show i Tables 1 ad 2. The Table 2 Coefficiets of geerators for 4-geerator power syste Geerator uber a b c Capacity (MW) Utility Uit Uit Uit Uit Uit IPP IPP IPP IPP

6 138 L. Geerli et al. / Electric Power Systes Research 57 (21) Table 3 Siulatio results for 8-geerator power syste a Dead Coalitio Idividual 45 (MW) Output (MW) Cost ($ per h) Beefit I ($ per Output (MW) Cost ($ per h) Beefit IC ($ per ($ per h) MWH) ($ per h) MWH) Utility IPP IPP IPP IPP Total a IC: icreetal cost. Table 4 Siulatio results for 4-geerator power syste a Dead Coalitio Idividual 21 (MW) Output (MW) Cost ($ per h) Beefit IC ($ per Output Cost Beefit ($ per h) MWH) (MW) ($ per h) ($ per h) IC ($ per MWH) Utility Uit Uit Uit Uit Uit IPP IPP IPP IPP Total a IC: icreetal cost. siulatio results are suaried i Table 3. Accordig to Table 3, the electricity prices of IPPs are lower tha the operatio cost of the utility because the utility has the obligatio to serve all custoers load. While the total outputs of geerators for IPPs ad the utility are the sae for both coalitio ad idividual. Actually, it ca be prove that total outputs of IPPs are the sae for the egotiatios of Sectios 2 ad 3 as far as the cost fuctios j f ij ()d ad j g j ()d are quadratic. For the sake of the siplificatio, let h(d) = d where =8 ($ per MWH) is the electricity rate to cosuers. Sice the df ij /d ij IPP-1 are saller tha those of IPP-2, IPP-3, IPP-4, the IPP-1 supplies ore power to the utility, especially for the case of the idividual accordig to Table 3. The results also show that the total beefit for the case of the coalitio is larger tha the case of the idividual due to their joitly bargaiig with the utility A 4-geerator power syste The proposed ethod also applied to a 4-geerator power syste with oe utility ad 1 IPPs. The utility has 3 geerators ad 1 IPPs ow oe geerator, respectively. Table 4 shows the siulatio results where h(d)=8d is used. I additio to the price y, obviously the earig of the utility also depeds o the h(d) to cosuers. Accordig to Table 4, the total outputs of geerators for IPPs ad the utility are the sae for both coalitio ad idividual. The icreetal costs of the utility (12.8 ad 12.32$ per h) are higher tha the icreetal costs of IPPs (1.2 ad 1.66$ per h). As show i Table 4, total beefit for the case of the idividual is actually ore tha that for the case of the coalitio, but the beefit of each is ot all icreased due to their differet icreetal costs. 8. Coclusios Gae theory is a useful tool for the aalysis of electric power arket. The derived operatio rules reflectig the copetitio ca be viewed as a etesio of the covetioal equaliig icreetal cost ethod. Besides, we also provided the prices of electricity for purchases ad sales, ad as idicated i this paper, these prices are geerally cheaper tha the icreetal

7 L. Geerli et al. / Electric Power Systes Research 57 (21) costs of the geerators belogig to the utility, whe presuig that the purchased electric eergy is a fuctio of the prices. To eaie the proposed approach, two test systes were used as the deostrated eaples. As further research work, the pricig structures ad the wheelig for power systes with a variety of practical costraits will be ivestigated by the gae theory. Refereces [1] Gae Theory Applicatios i Electric Power arkets, IEEE Catalog Nuber: 99TP136-. [2] L. Che, J. Toyoda, Optial pricig policy ad operatio decisios for future power syste, Iteratioal Coferece o Power ad Eergy i Chia, Beijig, 199. [3] L. Che, et al., Pricig structure of arket-based power systes by gae theoretic aalysis, Bulk Power Syste Dyaics ad Cotrol TV-Restructure, August, Satorii, Greece, [4] F.C. Schweppe, et al., Spot Pricig of Electricity, Kluwer Acadeic Publishers, [5] G. Owe, Gae Theory, Acadeic Press, [6] J.B. Cru, Leader-follower strategies for ultilevel systes, IEEE Tras. Autoatic Cotrol AC-23 (1978) 2. [7] G. Shehle, Price based operatio i a auctio arket structure, IEEE Tras. Power Systes 11 (4) (1996) [8] J.S. Clayto, S.R. Erwi, C.A. Gibso, Iterchage costig ad wheelig loss evaluatio by eas of icreetal, IEEE Tras. Power Systes 5 (3) (199) [9] M.C. Caraais, N. Roukos, F.C. Schweppe, A tool for evaluatio the argial cost of wheelig, IEEE Tras. Power Systes 4 (2) (1989) [1] A. Suuki, T. Sakai, K. Ogioto, L. Che, K. Ikeda, M. Ishieki, Cost Beefit Miocatio Based o Cooperative Gae Theory, Coferece o the Developet ad Operatio of Large Itercoected SVstes, UNIPEDE, 931e3.2, [11] S. Burs, G. Gross, Value of service reliability, IEEE Tras. Power Systes 5 (3) (199) [12] R.D. Tabors, Trasissio syste aageet ad pricig: ew paradigs ad iteratioal copariso, IEEE Tras. Power Systes 9 (1) (1994) [13] H.H. Happ, Optial power dispatch, IEEE Tras. PAS 93 (1974) [14] L. Che, et al., Mea field theory for optial power flow, IEEE Tras. PWRS 12 (4) (1997) [15] L. Geerli, L. Che, Y. Kioshita, R. Yokoyaa, Negotiatio odels for electric pricig ad load dispatch i electricity arket, Tras. Ist. Electr. Eg. Jp. 12-B (2) (2)