MINIMUM CONSUMER PAYMENT SCHEDULING AND PRICING IN ELECTRICITY MARKETS

Transcription

1 MINIMUM CONSUMER PAYMENT SCHEDULING AND PRICING IN ELECTRICITY MARKETS Yongun Ren, Francsco D. Galana Department of Electrcal an Computer Enneerng McGll Unversty, Canaa Abstract We examne two scheulng an prcng approaches n Pool-base electrcty markets from the pont-of-vew of consumer payment. These approaches are contraste wth the more common Pool Cost Mnmzaton (PCM) generaton scheulng wth Marnal Prcng (MP). In the frst approach, the generaton scheule s obtane by solvng the PCM problem, followng whch the market prce s erve from the average cost of the generators. In the secon approach, both the generaton scheule an the market prce are obtane by solvng a Consumer Payment Mnmzaton problem (CPM) subect to a proft feasblty conton. In each of the above, both unform prce an non-unform noal prces are consere. Keywors: generator scheulng, market-clearng prce, consumer payment, marnal prcng, average cost, pay-as-b, generator proft. 1 INTRODUCTION In centrally operate electrcty markets, the generaton scheule an the market-clearng prce or prces are obtane by the applcaton of some preestablshe market rules. The nature of these rules plays a crucal role n how the market an ts partcpatng agents behave. A set of market rules wely use toay s one where the generaton scheule s foun by some form of Pool Cost Mnmzaton (PCM) of the combne generaton offers, whle the market-clearng prce s etermne by the marnal cost (or costs) of the total generaton followng the theory of Marnal Prcng (MP) [1,, 3]. The obvous avantage to socety of usng the PCM generaton scheule s that the power eman s satsfe whle consumng the least amount of fuel. In aton, economsts recommen Marnal Prcng as such a prce sens a correct economc sgnal, hgh prces ncatng a shortage of generaton an low prces reflectng an excess of generaton. In aton, n a perfectly compettve market [4], at the marnal prce, each nepenent generator operates at a MW output that maxmzes ts proft [3]. In ths eal stuaton, mnmum generaton cost an maxmum nvual proft are consstent wth each other. In practce however, uner PCM wth MP, Gencos that wel market power can easly manpulate the market prce, so that the noton of perfect markets s not realstc. Moreover, uner the same market rules t s possble for some Gencos to operate at less than maxmum proft or even at a negatve proft [8, 10, 11]. To ncte such Gencos to prouce, they must therefore receve a monetary uplft, a step that contrbutes to the eteroraton of the overall economc effcency []. Another mportant pont s that PCM oes not mnmze the true cost to socety but only the offere cost. If Gencos wth market power game, as they surely o, the offere cost an the true cost are qute fferent [5]. Recently, the possblty of alternatve prcng approaches to unform marnal prcng has receve ncrease attenton [6]. The concluson from a Blue Rbbon Panel [7] s that methos such as pay-as-b prcng woul o consumers more harm than goo, a supposton that requres further quanttatve analyss. It s clear that rrespectve of the prcng rules, gamng, market power an colluson play a maor role on the bottom lne, namely the payments by consumers an the profts of the proucers. Therefore, to compare one set of market rules wth another, two basc steps are neee: One s to evelop a mathematcal moel escrbng how generator offers are converte nto a generator scheule an market prces. Ths part of the overall market moel must be precse an unambguous, an hopefully goo for socety. The secon step s to conuct a stuy of these market rules subect to a hypothetcal moel of the forecaste eman, as well as of the gamng behavour, market power an colluson by the competng agents. Ths secon moel an ts conclusons are prone to conserable uncertanty. In ths paper, as part of the frst step, we nvestgate two alternatve scheulng an prcng methos to the stanar PCM wth MP approach. These are: - PCM Scheulng wth Consumer Payment Mnmzaton (CPM) Prcng; - Consumer Payment Mnmzaton Scheulng an Prcng. Uner the frst metho, the generaton scheule s etermne by PCM, but the market prce (or prces) s (are) foun by mnmzng the total consumer payment subect to a set of Genco proft constrants. In the secon metho, we obtan both the generaton scheule an the market prce (prces) that yel the mnmum total consumer payment. Proft constrants are also mpose here. These methos efne two new sets of market rules whose prces ve preferental treatment to the consumers, yet guarantee non-negatve generaton profts. The bng an gamng stratees that woul result from these new market rules are not analyze n ths paper.

2 In aton to evelopng the theory of both sets of market rules, these are compare wth the stanar PCM wth MP approach. A number of smulatons are also presente to quantfy these comparsons. ASSUMPTIONS In ths paper, to smplfy the presentaton an to focus on the man eas, only the sngle tme pero unt commtment scheule s consere. Ths smplfe moel s suffcently rch to brng out a number of nterestng propertes an comparsons among the three sets of market rules beng examne. In aton, we assume that the eman s nelastc an that the network s loss-less wthout transmsson congeston. Each nepenent generaton company (Genco) submts to the system operator an offer to supply power accorng to a quaratc generaton cost curve, 1 C ( P ) C a P b ( P ) = (1) subect to upper an lower generaton lmts, P P P () mn max The constant, C 0 $/h, represents the no-loa fxe 1 cost offer of Genco,whletheterm, ap + bp, s the varable cost offer. In aton, we note that the offer s not necessarly equal to the true cost. 3 POOL COST MINIMIZATION (PCM) SCHEDULING WITH MARGINAL PRICING (MP) Mnmum cost scheulng wth prcng base on system marnal cost has been wely mplemente n electrcty markets. The basc PCM moel can be escrbe as a mxe-nteger optmzaton problem 1, mn C C ( u, P ) up, g subect to, N (3) P = P (4) N u P P u P G (5) mn max ; The bnary varable scheule on or off, the parameter u ncates whether unt s s the system P eman, an G s the set of Gencos avalable for scheulng. In aton, we have efne, 1 C u P uc a P b P (, ) = ( ) (6) As s well known, the system marnal cost (SMC) s equal to the Lagrange multpler assocate wth the power balance equaton (4), enote by λ,thats, C λ = = SMC (7) P When the centralze pool sets λ as the electrcty prce for consumers an supplers, ths prcng metho correspons to Marnal Prcng (MP). 4 PCM SCHEDULING WITH CONSUMER PAYMENT MINIMIZATION (CPM) PRICING In ths approach, the generaton scheule s obtane by frst solvng the Pool Cost Mnmzaton (PCM) unt commtment sub-problem escrbe n secton 3 above. However, the market prce s not efne by the system marnal cost, λ, but by solvng the mnmum consumer payment problem uner the ven PCM generaton scheule, all subect to nonnegatve Genco proft constrants. Below we conser two varatons of ths PCM Scheulng wth CPM Prcng metho, namely, () Unform noal prces; () Non-unform noal prces. 4.1 Unform Noal Prces Generaton Scheule As escrbe above, we scheule the generaton levels by solvng the PCM problem, whch s entcal to the moel efne by equatons (3) to (6). The optmum PCM generaton scheule s enote by u, P ; G. ( ) 4.1. Market Prce Here, the market prce s efne as the unform prce,, that mnmzes the overall consumer payment subect to the PCM generaton scheule an to non-negatve nvual Genco proft constrants, that s, { } Mn P (8) subect to, pr P C ( u, P ) 0; G (9) 1 Ths stuy apples to systems wth nelastc loas. Thus, mnmzng cost s the same as maxmzng socal welfare.. 0 (10)

3 Equaton (9) guarantees that no Genco earns negatve proft. Rewrtng (9) for P > 0,weobtan, C( P) = AC ( ) P (11) P where we efne AC( P) as the average cost of Genco. Therefore, the proft constrant s equvalent to mposng that the prce be greater than the average cost of each generator. Snce the obectve s to get the mnmum prce, the system prce must be equal to the hghest average cost over all generators. That s, = (1) max [ AC ( P)] N, P > 0 4. Non-unform Noal Prces In ths sub-secton, the market rules are extene to permt non-unform prces between generators an loas. The moel allows for non-unform Genco prces over all buses wth a generator, enote by, G, however all loas are charge a sngle unform prce,. These prce varables are efne by mnmzng the overall consumer payment, (13), subect to the PCM generaton scheule, ( u, P) = ( u, P) ; G, an to non-negatve nvual Genco proft constrants, (14). In aton, the total consumer payment must be equal to or greater than the total generaton revenues as stpulate by nequalty (15). Ths s necessary, to ensure that the pool collects enough revenue from the loas to be able to pay the generators. In the unform prce moel, ths constrant nee not be explctly expresse snce t s automatcally satsfe. The market prces satsfy, subect to, Mn P (13) P C ( u, P ) 0; G (14) P P 0 (15) G 0an 0; G (16) The above optmzaton s a lnear program n the contnuous varables an (, G). We note that the prce pa to a Genco an the prce charge to a loa may be fferent even though they are at the same bus. 5 CONSUMER PAYMENT MINIMIZATION SCHEDULING AND PRICING The obectve of ths set of market rules s to scheule the generators so that the customers obtan the lowest prce for electrcty, whle the Gencos, at a mnmum, cover ther offere costs. Unlke n the PCM scheulng wth CPM prcng metho, here both the output levels of the Gencos an the market prces are ecson varables. We also conser two varatons uner ths metho: () Unform noal prces; () Non-unform noal prces. 5.1 Unform Noal Prces The CPM scheulng an prcng problem wth unform prces [9] s efne by,, Pg, u { } Mn P subect to, G, an to, (17) P C ( u, P ) 0 (18) up P up (19) mn max P = P (0) G 0 (1) We note that ths s a mxe-nteger non-lnear optmzaton problem n the varables an ( u, P); G. Its soluton generates the unform market prce,, as well as the corresponng generaton scheule, P. Uner the unform prce CPM Scheulng an Prcng metho there exst some loang levels for whch the optmum generaton scheule s not unque. In these cases, one can replace the pure mnmum consumer payment obectve functon by another equal to a combnaton of consumer payment an generaton cost. Ths mxe consumer payment-generaton cost mnmzaton always yels a unque generaton scheule [9]. The non-unform prcng metho escrbe next however always yels unque scheule an prces. 5. Non-unform Noal Prces The formulaton of CPM Scheulng an Prcng wth non-unform prces s as follows, Mn, up, g P ()

4 subect to, an, P C ( u, P ) 0; G (3) up P up G (4) mn max ; P = P (5) G P P 0 (6) G 0an 0; G (7) where ; G s the Genco prce at bus an s the loa prce over all buses. Both the revenue constrant (6) an the proft constrants (3) are always actve. Ths means that Gencos always make zero proft, n other wors ths s a payas-b prcng strategy. Moreover, ths scheme becomes entcal n both prces an scheule as the PCM Scheulng wth Non-Unform CPM Prcng. 6 NUMERICAL EXAMPLES In ths secton, we present smulaton results that llustrate the Consumer Payment Mnmzaton methos propose above. A numercal example consstng of a three-genco system wth a eman of 1000 MW s use to llustrate a number of comparsons. Table 1 shows the offere cost ata of the three Gencos, locate at buses 1,, an 3. The eman strbuton s shownntable. We recognze that the Gencos woul submt fferent offers uner fferent sets of market rules. However, snce we are not analyzng gamng stratees, n orer to compare ther behavour we have use the samesetofoffersforallmarketrules. All the smulatons n ths secton were carre out on a Pentum III PC usng CPLEX 7.0 uner GAMS. A pecewse-lnear convex approxmaton of the nonlnear constrants s use n the CPM scheulng an prcng moel [1]. C 0 ($/h) a ($/MWh) b ($/MW h) P g max (MW) P g mn (MW) G G G Table 1: Offere Genco ata Bus Deman (MW) Table : Deman strbuton Table 3 shows the results of the case uner PCM Scheulng an Marnal Prcng. We note that although t s possble uner ths scheme for some Gencos to earn negatve proft, n ths partcular example ths s not the case Total P g MW C K$/h AC $/MWh =IC $/MWh P g K$/h P g- C K$/h P MW P K$/h Table 3: PCM Scheulng wth Marnal Prcng 6.1 PCM Scheulng wth CPM Prcng uner Unform Prce. The results for ths case are shown n Table 4. Here, the Market Prce, = $/MWh, s efne by the average cost (AC) of Genco 1 whch has the maxmum average cost over all Gencos. Genco 1 therefore earns zero proft whle the other two Gencos earn postve profts. Comparng ths prcng strategy wth marnal prcng (see Table 3), we see that the market prce s lower uner CPM prcng, that s, = $/MWh versus = $/MWh. The consumers therefore pay less, whle the proucers also earn less Total P g MW C K$/h AC $/MWh $/MWh P g K$/h P g-c $/h P MW P K$/h Table 4: PCM Scheulng wth CPM Prcng uner Unform Prce. 6. PCM Scheulng wth CPM Prcng uner Nonunform Prces. These results are shown n Table 5. As can be seen, the Genco prce at buses where there s a Genco s exactly the average cost of that Genco, so that t earns what t offers. Comparng the non-unform case n Table 5 wth the unform case n Table 4, we see that n nonunform case the consumers pay a sgnfcantly lower amount. Concurrently, the Gencos earn sgnfcantly lower profts Total P g MW C K$/h AC $/MWh g $/MWh gp g K$/h gp g-c K$/h P MW $/MWh P K$/h Table 5: PCM Scheulng wth CPM Prcng uner Nonunform Prces.

5 6.3 CPM Scheulng an Prcng uner Unform Prce These results are shown n Table 6. One notable fference from the prevous methos s that the generaton scheule s now fferent. In fact, the most expensve generator G1 has been shut own. As ncate, both G an G3 are pa exactly ther offer an therefore earn zero proft. The consumer payment of k$/h s less than the consumer payment of k$/h uner PCM Scheulng wth CPM Unform Prcng an k$/h uner PCM scheulng wth MP. In comparson, we also note that, as expecte, the total generaton cost of k$/h s larger than uner PCM where t s k$/h Total P g MW C K$/h AC $/MWh $/MWh P g K$/h P g-c $/h P MW P K$/h Table 6: CPM Scheulng an Prcng uner Unform Prce. 6.4 CPM Scheulng an Prcng uner Non-unform Prces These results are shown n Table 7. It s nterestng to compare the generaton scheule n Table 7 wth that obtane uner PCM as shown n Table 5. Snce these scheules are entcal, t means that not only s consumer payment beng mnmze but also the total generaton cost s mnmum. However, as always, nonunform prcng yels lower consumer payments than unform prcng. Non-unform prcng uner ths metho always results n zero Genco profts; thus, ths satrue pay-as-b scheme Total P g MW C k$/h AC $/MWh g $/MWh gp g k$/h gp g-c $/h P MW $/MWh P k$/h Table 7: CPM Scheulng an Prcng uner Non-unform Prces. 7 CONCLUSIONS Scheule PCM PCM PCM CPM CPM CPM CPM CPM CPM Prcng MP (unform) (non-unf.) (unform) (non-unf.) Pool cost C C = C C = C C > C C = C 4 3 Loa λ Prce 1 < λ 3 1 < < λ 1 < 4 < = Genco λ Prces g = g = ACg g = g = ACg Genco pr pr 0 Profts pr = 0 pr 0 pr = 0 Loa < < 3 4 < 3 < Payment 1 < 1 < = Table 8: Summary of the Varous Markets (Note: Some Genco profts may be negatve uner PCM wth MP) The summary of the varous markets scusse above s shown n Table 8. As expecte, both sets of market rules, PCM Scheulng wth CPM Prcng an CPM Scheulng an Prcng, ecrease the electrcty prce pa by the consumers compare to the prce pa uner marnal prcng. Ths concluson s base on the assumpton that uner both sets of rules the Gencos submt the same offers. Further stues wll nvestgate gamng stratees n whch the generator offers are aapte to the market rules. Note that uner PCM wth MP the average prce of some Gencos scheule on may be hgher than the marnal prce (negatve proft). Such Gencos then have to receve some a hoc uplfts to be ncte to operate [10, 11], an extraornary charge that must be passe on to the consumers. In contrast, as a result of the Gencos proft constrants, ths rawback s not present uner all cases wth CPM. Uner PCM Scheulng wth CPM Unform Prcng, the market prce s equal to the hghest average cost over all generators. Therefore all Gencos earn postve proft except one, whch breaks even. Uner CPM Scheulng an Prcng wth unform prcng, wth some exceptons [9], the market prce s such that all Gencos operate at zero proft. We also note that the CPM Scheulng an Prcng wth Non-unform Prces yels the entcal generaton scheule as the PCM scheulng metho ue to that the system payment constrant an the Gencos proft constrants of ths moel are always actve. Therefore, the PCM Scheulng wth Non-unform CPM Prcng an the CPM Scheulng an Prcng wth Non-unform Prces have the same result. Here, The consumer payment uner these schemes s always the lowest an all Gencos earn zero proft. Current research s examnng bng stratees uner the above market rules as well as generalzng the market to nclue conseraton of transmsson losses an congeston.

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