Chapter 10 Counterexamples to Commonly Held Assumptions on Unit Commitment and Market Power Assessment

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1 Chapter 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet ad Market Power Assessmet Wolfgag Gatterbauer ad Marija Ilić 10.1 Cetralized Versus Decetralized Uit Commitmet (UC) This first subsectio disproves the commoly held assumptio that, i theory ad uder the coditio of perfect iformatio, decetralized ad cetralized UC would lead to the same power quatities traded ad, hece, to the same optimal social welfare [1]. We see that, eve i the absece of ay ucertaities, idepedet optimizatio of the idividual performace objectives by the decetralized market participats ca lead to lower efficiecy tha cetralized miimizatio of total operatig cost The Stadard Argumet: Cetralized UC is idetical to Decetralized UC Mathematically, a cetralized ecoomic dispatch is the problem of miimizig the total geeratio cost, usig the quatities produced by each of the possible geerators as decisio variables such that total geeratio equals total demad Q D [1]. Usig the variables Q i ad C i for the quatities produced ad the cost icurred 1 This result cocers short-term supply optimizatio for a give demad ad does ot cosider log-term ivestmet issues. W. Gatterbauer ( ) Tepper School of Busiess, Caregie Mello Uiversity 5000 Forbes Aveue gatt@cmu.edu M. Ilić Departmet of Electrical ad Computer Egieerig, Caregie Mello Uiversity, 5000 Forbes Ave, Pittsburgh, PA 15213, USA milic@ece.cmu.edu M. Ilićetal.(eds.),Egieerig IT-Eabled Sustaiable Electricity Services, Power Electroics ad Power Systems 30, DOI / , Spriger Sciece+Busiess Media New York

2 262 W. Gatterbauer ad M. Ilić by geerator i, respectively, ad the variable for the total umber of available geerators, we ca write the problem as mi Q C i (Q i ) s.t. Q i = Q D. This basic versio of a ucostraied ecoomic dispatch remais idetermiate. A ecoomically motivated coditio for solvig this problem is the equal icremetal coditio δc 1 δq 1 =...= δc δq = λ. The term λ is kow as the short-ru margial cost (SRMC) ad, at the optimum, all uit margial costs are equal to it. Usig u i as a biary variable that determies whether the geeratio uit i is tured o or off at a give momet, the basic cetralized uit commitmet problem without start-up costs or miimum up/dow time costraits is mi u,q u i C i (Q i ) s.t. Q i = Q D. Followig the Lagragia relaxatio method, we first form the Lagragia fuctio L(u,Q,λ )= u i (C i (Q i ) λ Q i )+λ Q D. By miimizig this last equatio over Q, we obtai the covetioal ecoomic dispatch equal icremetal coditio from above which permits us to solve for Q i terms of λ, the system icremetal cost. Rewritig the Lagragia as L(u,λ )= u i (C i (Q i (λ )) λ Q i (λ )) + λ Q D ad usig the Lagragia method to miimize L(u,λ ) with respect to u gives us the switchig curve law or the average cost rule u i = { 1 ifc i λ Q i < 0 0 ifc i λ Q i > 0,

3 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet that is, the uit i is tured o if the average cost C i Q i < λ ad off otherwise. Oce o, a covetioal ecoomic dispatch is used to adjust to demad chages if these are moitored more frequetly [1]. With competitive bilateral trasactios takig place i a decetralized ecoomic dispatch ad each party s objective beig the maximizatio of its idividual profit, the decetralized problem is maxπ i (Q i ). Q i Here π i (Q i )=PQ i C i (Q i ) stads for the profit made by the market participat i through some sort of tradig process, give price P. Thus, uder perfect coditios, whe the market coverges to a sigle electricity price, oe ca maximize π i by settig margial cost equal to price: δc 1 δq 1 =...= δc δq = P. The process of bilateral decisios will stabilize P at the system-wide ecoomic equilibrium uder a perfect iformatio exchage amog all market participats. This result is simply obtaied by each market participat optimizig its ow profit for the assumed exogeous market price P [1]. I the decetralized uit commitmet settig, all geerator owers are assumed to be price takers i a competitive market place. Each participat makes a uit commitmet decisio typically for each hour oe day ahead, before kowig the actual spot price. After the spot price of a respective hour is kow, the geerator decides how much power to sell i order to maximize profit. The oly cotrol for geerator i is u i, whether to tur o or off at a give hour. The expected geeratio level ˆQ i may be regarded as a fuctio of the cotrol u i ad the expected price ˆP. I the case of determiistic prices ad igored start-up costs ad must-ru time costraits, a geerator s profit while o is ˆπ i,o = ˆP ˆQ i C i ( ˆQ i ). The geerator will tur o oly if ˆπ i,o > 0, which is equivalet to C i( ˆQ i ) < ˆP, which is exactly the average cost rule used for coordiated ˆQ i uit commitmet [1]. Based o this derivatio, curret teachig is that, uder perfect market assumptios ad whe eglectig start-up costs ad itertemporal time costraits, idividual power producers would schedule the same power uits i a decetralized market as would a cetral system operator i a coordiated market. Thus, both cetralized ad decetralized UC should lead to the same power quatities traded, the same miimum operatig cost, ad, with give ielastic demad, to the same total social welfare optimum. The performace objectives of the idividual market participats (to maximize profits) ad the objective of a cetralized etity (to maximize social welfare by miimizig operatig cost) would the be equivalet [1].

4 264 W. Gatterbauer ad M. Ilić The Couterexample: The Tragedy of Decetralized UC We ext prove that cetralized ad decetralized UC are, i geeral, ot ecoomically equivalet. Uder certai coditios, some geerators would ot selfschedule to prevet loss; those uits, however, would be scheduled by a PoolCo-type market (or a social plaer ) to miimize overall operatig cost ad, i tur, would receive fixed operatig costs to prevet them from loss. These situatios are missed by the previous argumet, as the average cost rule does ot always lead to the total social welfare optimum that a cetralized operator strives for. To prove this claim by example, we derive the strict coditios uder which this situatio takes place i the case of two geerators with quadratic operatig cost C i (Q i )=a i Q 2 i + b iq i + c i. We cosider oly oe particular hour for which demad Q D is give. I the case of ielastic demad, the total social welfare optimum is attaied by miimizig total operatig cost. Without loss of geerality, we cosider geerator 2 to be the oe that would ot self-schedule i order to avoid loss whe biddig margial costs. Three coditios must hold simultaeously i order to produce the specific situatio: 1. Geerator 1 makes profit, idepedet of geerator 2 participatig durig the hour or ot: C 1(Q 1 ) Q 1 < P. 2. Geerator 2 icurs loss if it is scheduled ad receives o extra paymet: C 2 (Q 2 ) Q 2 > P. 3. The total cost for satisfyig the give load is smaller if both geerators operate istead of oly geerator 1: C 1 (Q 1 )+C 2 (Q 2 ) < C 1 (Q D ). Performig some mathematical operatios that are described i more detail i [2] ad [3], we obtai the followig three coditios o the demad which, if fulfilled, lead to differet uits scheduled ad, hece, differet ecoomic outcomes from either cetralized or decetralized UC: Q D > Q miπ1 := max (a 1 + a 2 ) c1 a 1 + b 1 b 2 2 c1,, a 2 Q D < Q maxπ2 := Q D > Q mic := (a 1 + a 2 ) c2 a 2 + b 2 b 1 2, a 1 (a1 + a 2 )c 2 + b 2 b 1 2. a 1 Table 10.1 ad Fig provide a illustrative example with umbers ad graphs. Parameters for geerator 2 stem from a best quadratic fit to heat rate data a 1

5 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet Table 10.1 Example geerator parameters (a), resultig demad coditios (b), ad higher prices ad total operatig cost with decetralized istead of cetralized UC for example demad of 800 MW (c) G 1 G 2 a [$/MW 2 /h] b [$/MWh] c [$/h] (a) a Q miπ1 [MW] 361 Q maxπ2 [MW] 915 Q mic [MW] 652 (b) b D.UC CUC P [$] G 1 G 1 +G 2 G 1 G 2 Q i [MW] C i [$/h] 14,528 13,874 8,197 5,677 π i [$/h] 4, , (c) Margial Costs MC [$/MWh] MC 1 MC 2 Costs C [$/h] C 1 C 2 C 1+2 MC Idividual Load Q i [MW] Idividual Load Q i [MW] c d Costs C ad Reveues R [$/h] C 1 R 1 C 2 R 2 Total Operatig Cost C [$/h] C C.UC C D.UC C 1+2 C Total Load Q D [MW] Total Load Q D [MW] Fig Situatio with cetralized UC leadig to lower operatig cost tha decetralized UC: Market supply fuctios (a), geerator operatig cost fuctio (b), idividual operatig costs ad reveues followig margial cost bids (c), ad total operatig cost (d)

6 266 W. Gatterbauer ad M. Ilić of the thermal plat Morro Bay 4 published i [4] ad a assumed fuel price of 2 $/MBtu. The parameters were slightly chaged for geerator 1 i order to create a sample situatio Discussio The literature gives several examples of cases i which idividual objective fuctios are ot aliged with those of the overall social welfare. The most ofte cited example was give by Hardi i The Tragedy of the Commos [5]. Aother oe is Braess article o traffic etworks [6] i which he gives a example i which drivers attempt to miimize their trasit times leads to icreased cogestio ad icreased traffic times for all participats. Braess paradox has become a importat issue i the cotext of queuig etworks [7]. I power systems, however, the commoly held assumptio is still that, at least i theory, a cetralized ad a decetralized UC should lead to the same power quatities traded ad to the same optimal social welfare. The performace objectives of the idividual market participats are cosidered equal to the oe of miimizig total operatig cost [1, 8, 9]. The importat implicatio of the example give before is that, eve i the absece of load ucertaities ad itertemporal costraits, decetralized UC does ot ecessarily lead to the same maximized welfare as cetralized decisio makig. The reaso is that, uder certai circumstaces, several geerators ca supply the load at a lower overall cost tha the subset of geerators that would make positive profits i a market settig if switched o durig the hour. I the Pesylvaia New Jersey Marylad ad the New York electricity markets, the ISO (Idepedet System Operator) offers a volutary uit commitmet service, based o three-part bids, allowig geerators to bid actual operatig costs more precisely ad permittig a more efficiet uit commitmet. Geerators may also self-schedule their ow uits, but they may also allow the ISO to determie the most ecoomic uit commitmet for their plats. Participatig geerators are guarateed recovery of their start-up ad miimum geeratio costs i the evet they fail to recover these costs from the prices received i the ISO-coordiated markets [10, 11]. This mechaism elimiates the ucertaity of whether a geerator will be committed oly to lose moey, ad it allows for a more efficiet dispatch. The quadratic cost curve example shows how a PoolCo-type market would work more efficietly tha a power exchage (for which the oe-part bids result i some iefficiecy). It is importat to ote that the coclusios here focus o the short ru, i that they do ot take ito accout the log-term motivatioal effects of a decetralized commitmet o ivestmet decisios ad the possible etry of ew firms or geeratig plats. The literature gives several qualitative argumets why a decetralized commitmet process might be preferable despite the better overall efficiecy of the cetralized process [11, 12].

7 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet Margial Cost Biddig ad Market Power This secod subsectio illustrates that a geerator ower s optimum bid sequece for a cetralized wholesale market uder a decetralized UC regime is geerally above margial cost eve i the complete absece of market power. This result challeges ecoomic literature statig that market prices above margial cost would uambiguously idicate gamig ad the abuse of market power A Illustrative Model We will use a simple model to show how prices above margial cost arise i a decetralized UC scheme as a atural cosequece of the decetralized decisio process ad itertemporal costraits. We deploy the dyamic programmig formulatio from [13] for calculatig a geerator s optimal biddig strategy i the presece of a price forecast with give stadard variatio. For simplificatio, we cosider a geerator whose margial operatig cost is costat over the output rage: MC = b. The ower ca offer electricity by submittig a bid to a cetralized market for each hour ad is scheduled if the bid price turs out to be lower tha or equal to the market price. We eglect the case of the geerator beig the margial uit ad scheduled for less tha full output. Because of the costat margial cost, the most efficiet way to operate the geerator is to either produce full output Q G or othig ad to use a flat bid curve. I additio to variable costs, the geerator icurs fixed operatig cost c for every hour of operatio regardless of whether it is producig electricity or ot ad also start-up cost c u ad shutdow cost c d.as itertemporal costrait, oce the geerator is switched o, it has to remai i that state for at least 2 hours, durig which it icurs the fixed operatig cost. If the geerator gets scheduled for 1 hour but ot for the other, it still icurs the fixed operatig cost c for the secod hour as well. Hece, the geerator has to iteralize these itricacies whe it is biddig ito a hourly market. The geerator does ot kow the market prices whe biddig but has some kowledge about the probability distributio of the prices, which are cosidered to be exogeous variables, ot iflueced by the behavior of the geerator. We ow cosider the specific situatio i which oly 2 successive hours have price distributios above MC (Fig. 10.2a). The problem of fidig the optimal bids is drastically simplified ad ca be solved i a closed form. I this special example, the sum of the fixed costs ca be uited ito oe term total icurred fixed cost c tot = c u + c d + 2c which will be icurred oce the geerator starts up. This aggregatio does ot chage the optimal strategy but simplifies the formulatio. Fixed ooperatig costs, such as suk capital costs, which are icurred regardless of the geerator producig output or ot durig 1 h, do ot affect the optimal decisio ad ca be disregarded. For the umerical calculatio, we assume that prices ca have oly a

8 268 W. Gatterbauer ad M. Ilić a b P t P t MC P k P P k+1 k+1 f(pk+1 ) MC P 1 P 1 + 2Δ P 1 +Δ P 2 P 2 p(p 2 ) k k+1 Hour t 1 2 Hour t Fig Margial cost ad hourly predicted prices for the ext day (a) ad assumed discrete price distributio of two relevat hours (b) limited umber of discrete values durig the two hours ad are ucorrelated: P k { P k 2Δ, P k Δ, P k, P k + Δ, P k + 2Δ} (Fig. 10.2b) Profit Optimizatio i a Competitive Market I order to fid the optimal biddig sequece, the profits for all possible combiatios of bid heights have to be compared: J = max[j (x 1,x 2 )] x 1,x 2 with {(x 1,x 2 ) (x 1,x 2 )=(P i,p j )} ad (P i,p j ) beig possible prices for the 2 hours. I order to calculate the expected profit for the bid combiatios, all possible price outcomes have to be compared: J (x 1,x 2 )= P i P i x 1 + P i P i x 1 + P i P i <x 1 P j P j x 2 p(p 1 = P i )p(p 2 = P j )((P i + P j 2b)Q G c) P j P j <x 2 p(p 1 = P i )p(p 2 = P j )((P i b)q G c) P j P j x 2 p(p 1 = P i )p(p 2 = P j )((P j b)q G c). Whereas fidig the optimal bid sequece i our example is still possible, this task becomes icreasigly itractable whe optimizig for more periods. The time for calculatio icreases expoetially with the umber of periods. Table 10.2 shows a illustrative example. The optimum bids are ot oly higher tha the margial cost but also higher tha the average operatig cost. I additio,

9 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet Table 10.2 Example of optimal bids beig above MC: Prices (a), geerator cost (b), expected profits for differet bid sequeces (c). Price distributios P 1 60 P 2 50 Δ 2 p(p k = P k ± 2Δ) 0.19 p(p k = P k ± Δ) 0.16 p(p k = P k ) 0.30 (a) Geerator MC 50 Q G 1 c 4 c u 1 c d 1 c tot 10 (b) Bid Sequece Exp. Profit (58,52) or(60,54) (58,54) (56,50) or (56,52) (60,56) or (62,56) (60,52) (c) we see that the optimum bids vary betwee differet hours ad are depedet o the assumed forecast prices ad related price dyamics Discussio May of the recet papers o assumed market power abuse i deregulated electricity markets assume that market participats bid their true margial costs i a competitive market if o market power is exerted. However, i the cotext of biddig decisios of power plats, which ot oly icur MC but also start-up ad shutdow costs ad miimum commitmet costraits, these assumptios are ot valid. Geerators bid higher tha MC, ot because they ca exercise market power, but because of itertemporal costraits ad ucertaities about prices of cosecutive hours. The literature disagrees as to what exactly costitutes market power but geerally agrees that it has to do with actively raisig the prices at which oe is willig to sell output (oe s price offer) above MC i order to chage the market price [14] ( If suppliers exercise market power, prices could be higher tha margial costs. ). MC iclude both the variable costs due to fuel ad the other variable operatig ad maiteace costs. For example, [15] states that The fudametal measure of market power is the margi betwee price ad the margial cost of the highest cost uitecessarytomeetdemad.(...)ifofirmwereexercisigmarketpower,the all uits with margial cost below the market price would be operatig. I the formulatio of this chapter, the power producer is modeled as a price taker. He has assumptios about the probability distributios of prices for certai hours. His biddig decisio does ot affect the prices ad, hece he has o market power. Nevertheless, his optimum bids deviate from MC. It is, therefore, ot market power that creates prices above MC but the ecessity to icorporate start-up ad shutdow costraits i the presece of ucertai prices. The geerator i the example respods to the simple ecoomic icetive of maximizig profits give ucertai prices. As a result, the competitive price does ot equal margial cost at peak periods uder competitio, ad therefore simple price-cost margi studies caot cofirm the exercise of market power. We thus coclude that above MC bids of geerators

10 270 W. Gatterbauer ad M. Ilić do ot ecessarily idicate the exercise of market power. Especially i times whe prices are very volatile, geerators have to bid above margial costs i order to take accout of the possibility of beig scheduled for oe hour ad ot the followig oe Summary ad Take-Aways By adoptig the perspective of a idividual market participat i a decetralized UC regime ad simulatig ecoomically optimal behavior, we could draw the followig two coclusios: 1. Coordiatio Value: Decetralized UC does ot lead to the same short-time efficiet ecoomic outcome as cetralized UC, eve i the theoretical case of perfect iformatio. The o-obvious reaso is that, uder certai load ad cost coditios, the overall miimum operatig cost to supply a give demad could be achieved with some geerators switched o which would ot operate i a completely decetralized market. The cosequece of this observatio does ot limit itself to electricity markets oly, but ca be cosidered a more geeral situatio where the performace objectives of the idividual market participats are ot equal to the oe of miimizig total productio cost, quite aalogous to the Tragedy of the Commos. 2. Market Power: Above margial costs i electricity markets are a ecessary cosequece of the decetralized decisio makig uder ucertaity ad do ot ecessarily idicate gamig i the presece of market power. The reaso is that geerators face itertemporal time costraits ot commo i other idustries. As a cosequece, ad i cotrast to ecoomic teachig, price takig geerators have to bid above margial cost i order to iteralize the ucertaities of beig o but ot sellig ay power ito the market ito their bids. The results of this chapter were published i [2]. More details ad a exteded optimizatio algorithm that also cosiders itertemporal correlatio betwee price forecast errors ca be foud i [3]. Ackowledgmets This work was partially supported by a Rotary Ambassadorial Scholarship, a fellowship from the MIT graduate studet office, a DOC scholarship from the Austria Academy of Scieces, ad by NSF IIS Refereces 1. M.D. Ilić, F. Galiaa, L. Fik (eds.), Electric Power Systems Restructurig: Egieerig ad Ecoomics (Kluwer, Dordrecht, 1998) 2. W. Gatterbauer, M.D. Ilić, Couterexamples to commoly held assumptios o uit commitmet ad market power assessmet, i Proceedigs of the 34th Aual North America Power Symposium (NAPS 2002) (IEEE, New York, 2002), pp

11 10 Couterexamples to Commoly Held Assumptios o Uit Commitmet W. Gatterbauer, Iterdepedecies of electricity market characteristics ad biddig strategies of power producers, Master s thesis, Massachusetts Istitute of Techology, May J.B. Klei, The use of heat rates i productio cost modelig ad market modelig, Electricity Aalysis Office, Califoria Eergy Commissio, April G. Hardi, The tragedy of the commos. Sciece 162(3859), (1968) 6. D. Braess, Über ei Paradoxo aus der Verkehrsplaug. Uterehmesforschug 12, (1969) 7. J.E. Cohe, F.P. Kelly, A paradox of cogestio i a queueig etwork. J. Appl. Probab. 27, (1990) 8. F. Wu, P. Varaiya, P. Spiller, S. Ore, Folk theorems o trasmissio access: Proofs ad couterexamples. J. Regul. Eco. 10(1), 5 23 (1996) 9. C.J. Day, B.F. Hobbs, J.-S. Pag, Oligopolistic competitio i power etworks: A cojectured supply fuctio approach. IEEE Tras. Power Syst. 17, (2002) 10. J. Chadley, A stadard market desig for regioal trasmissio orgaizatios. Electricity J. 14(10), (2001) 11. R.P. O Neill, U. Helma, P.M. Sotkiewicz, M.H. Rothkopf, W.R. Stewart, Regulatory evolutio, market desig, ad uit commitmet, i TheNextGeeratioofUitCommitmet Models. Iteratioal Series i Operatios Research ad Maagemet Sciece (Kluwer, Dordrecht, 2001) 12. R.D. Tabors, Ubudlig of trasmissio: The operatio ad ecoomics of a for-profit trasmissio compay, 33rd Hawaii Iteratioal Coferece o System Scieces (HICSS). IEEE Computer Society, Maui, Hawaii, vol. 4 (2000), p E. Alle, M.D. Ilić, Price-Based Commitmet Decisios i the Electricity Market. Advaces i Idustrial Cotrol (Spriger, Berli, 1999) 14. D.O.E. Eergy Iformatio Admiistratio, Electricity prices i a competitive eviromet: Margial cost pricig of geeratio services ad fiacial status of electric utilities, August 1997, DOE/EIA ftp://ftp.eia.doe.gov/electricity/0614.pdf 15. S. Borestei, J.B. Bushell, F.A. Wolak, Measurig market iefficiecies i Califoria s restructured wholesale electricity market. Am. Eco. Rev. 92(5), (2002)

Overlapping Generations

Overlapping Generations Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio