PRICE-BASED MARKET CLEARING UNDER MARGINAL PRICING: A BILEVEL PROGRAMMING APPROACH

Transcription

1 RICE-BSED MRKE CLERING UNDER MRGINL RICING: BILEVEL ROGRMMING ROCH Ricardo Fernández-Blanco José M. rroo edro J. Muñoz Natalia lguacil bstract Maret clearing in restructured ower sstems is mostl imlemented through an offer-based setting with the goal of maximizing social welfare. his aroach leads to sound results from an economic iewoint when generation offers reflect true roduction costs. Howeer, offers ma significantl differ from actual costs in ractice, thus ielding undesired distortion. Under a marginal ricing scheme, this aer resents a general bileel rogramming formulation for alternatie maret-clearing rocedures deendent on maret-clearing rices rather than on offers. he resulting nonlinear mixed-integer bileel rogramming formulation is transformed into an equialent single-leel mixed-integer linear rogram suitable for efficient off-the-shell software. he bileel formulation is inestigated through a articular instance of rice-based maret clearing drien b consumer ament minimization. his roblem has recentl receied considerable attention due to the oen challenges osed from both modeling and comutational ersecties. Numerical results are roided to illustrate the erformance of the roosed aroach. Kewords: Offer cost minimization, bileel rogramming, marginal ricing, consumer ament minimization, rice-based maret-clearing rocedure 1 INRODUCION In the framewor of the nowadas cometitie ower industr this aer considers a ool-based electricit maret for energ. he Indeendent Sstem Oerator (ISO) receies energ offers from roducers and energ bids from consumers, and determines, for eer hour, the maret-clearing rice, the ower roductions, and the consumtion leels. he maret-clearing rocedure used b the ISO is ticall formulated as a unit commitment roblem [1] in which generation offers relace roduction costs and consumtion bids are taen into account. In this roblem, the objectie function to be maximized is referred to as the declared social welfare. When the demand is inelastic, the objectie function onl includes generation offer information and the resulting roblem becomes an offer cost minimization. Maret-clearing rices resulting from the unit commitment roblem are used for maret settlement. Under the assumtion of erfect cometition, the otimal maret-clearing solution is equal to the equilibrium solution where the marginal alue to consumers is equal to the marginal cost to roducers. his equilibrium maximizes the social welfare, which is the sum of the consumer surlus and the roducer surlus. s a consequence, suliers hae no incentie to offer different from their roduction costs. hus, the declared social welfare based on generation offers reflects the true social welfare and maximizing social welfare is commonl acceted as the right goal in a maret setting. Howeer, maret-clearing rocedures based on declared social welfare maximization are characterized b seeral ractical shortcomings [2]: - he assumtion of erfect cometition does not hold in real-life ower marets. Consequentl, generation offers ma not reflect actual costs and the otimization might not maximize the true social welfare or accuratel reflect the natural behaior of maret articiants. - he scoring rule, which is used to select the acceted offers and bids, and the ament rule are different. While the former is based on offer cost minimization, the latter is determined in terms of maret-clearing rices. his discreanc leads to distortion since the solution does not necessaril constitute an equilibrium where the objecties of maret articiants are maximized. his aer analzes an alternatie te of maretclearing rocedure for energ trading in an electricit ool. he alternatie maret clearing allows considering the ament rule through the incororation of maretclearing rices in the roblem formulation. his rocedure is hereinafter referred to as rice-based maret clearing. Examles of rice-based maret clearing are (i) rocedures based on consumer ament minimization [3]-[6], where maret-clearing rices aear in the objectie function, and (ii) reenue-constrained maretclearing rocedures [3], [4], where maret-clearing rices aear in the constraints. Moreoer, new rocedures exlicitl including generation and consumer surlus maximization [7] would fall within this class of roblems. 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

2 he resence of maret-clearing rices in the roblem formulation comlicates the solution of the otimization roblem since maret-clearing rices ma themseles result from an otimization rocess. Seeral definitions for the maret-clearing rice are aailable [1], being the two most commonl adoted (i) the rice of the highest acceted generation offer, which guarantees reenue adequac for suliers, and (ii) the marginal rice, which roides aroriate economic signals. While the former has receied considerable attention in rice-based maret clearing [4], [5], and references therein, few wors hae considered marginal ricing [3], [6], [7]. In [3], ament minimization under marginal ricing was addressed b a heuristic aroach. In [6], Zhao et al. first formulated consumer ament minimization under marginal ricing as a bileel rogramming roblem, which was soled b augmented Lagrangian relaxation and surrogate otimization. In [7], agent-based simulation was used to analze an auction drien b consumer and generation surlus maximization based on marginal maret-clearing rices. Within the framewor of marginal ricing, the main urose of this aer is to formulate a general ricebased maret-clearing rocedure as a bileel rogramming roblem [8], of which the consumer ament minimization reorted in [6] is a secial case. Bileel rogramming is aroriate to model roblems where one agent, the leader, otimizes its objectie function (uer-leel roblem) considering that a second agent, the follower, will react b otimizing its own objectie function (lower-leel roblem). hese models are releant in those situations where the actions of the follower affect the decision maing of the leader. his is the case in rice-based maret clearing: the selection of acceted bids and offers (uer-leel roblem) deends on maret-clearing rices (lower-leel roblem), which are in turn determined based on the set of acceted bids and offers. Under marginal ricing, maret-clearing rices are the Lagrange multiliers or dual ariables associated with the ower balance equations in an economic disatch roblem [1]. nother salient feature of this aer with resect to [6] is the roosal of an equialent single-leel mixedinteger linear formulation based on the Karush-Kuhn- ucer (KK) otimalit conditions, dualit theor, and integer algebra results. he main adantages of exressing the original bileel otimization as an equialent single-leel mixed-integer linear rogram is the guaranteed conergence to the otimal solution in a finite number of stes and the read aailabilit of efficient commercial branch-and-cut software. he main contributions of this aer are threefold: - he ISO is roided with an otimization-based framewor for maret clearing exlicitl accounting for marginal maret-clearing rices in the roblem formulation. - Bileel rogramming is roosed as a suitable solution methodolog. - he alicabilit of bileel rogramming is illustrated with the solution of an instance of ricebased maret clearing, namel the consumer ament minimization roblem. he remainder of the aer is organized as follows. Section 2 roides a general formulation for rice-based maret clearing under marginal ricing. Section 3 resents the solution methodolog. Section 4 describes the alication of the roosed framewor for consumer ament minimization. In Section 5, numerical results illustrate the erformance of the roosed aroach. Finall, releant conclusions are drawn in Section 6. 2 ROBLEM FORMULION he rice-based maret-clearing rocedure can be formulated in a comact wa as: ( x,, ) Min F (1) x,, x X (2) G e x, = (3) ( ) 0 ( x, ) 0 ( x,, ) 0 G i (4) H, (5) where x reresents the ector of scheduling ariables (status of generating units, start-u costs, etc.), is the ector of disatching ariables (generation leels, networ-related ariables, etc.), is the ector of maret- F x,, is the objectie function (de- clearing rices, ( ) clared social welfare, consumer ament, consumer and generation surlus, etc.), X is the feasibilit set for scheduling ariables x (integralit constraints, start-u cost constraints, minimum u and down times, etc.), G e ( x, ) balance equations, G i ( x, ) denotes the equalities associated with ower models all inequalit constraints inoling ariables x and (ower limits, ram- H x,, reresents all remaining ing limits, etc.), and ( ) inequalit constraints inoling maret-clearing rices such as those imosing reenue adequac. Under marginal ricing, the ector of maretclearing rices is defined as the ector of marginal costs. Mathematicall, can be deried from the following economic disatch roblem for the otimum ector of scheduling ariables x : Min f ( x, ) ( x, = 0 : ( ) ( x, ) 0 : ( μ) G (6) e (7) i, (8) G 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

3 where f () is the objectie function, which ma be different from that of the rice-based maret-clearing rocedure F () ; the ector of maret-clearing rices is equal to the ector of Lagrange multiliers or dual ariables associated with the ower balance equations (7); and µ is equal to the ector of Lagrange multiliers or dual ariables associated with constraints (8). Note that under marginal ricing, the definition of is an otimization roblem itself. hus, the rice-based maret-clearing rocedure can be recast as the following bileel rogramming roblem: x (, ) Min F x, (9) x X (10) H x,,, (11) ( ) 0 where Min f and ( x, ) ( x, = 0 : ( ) ( x, ) 0 : ( μ) G are obtained from: (12) e (13) i. (14) G he aboe bileel roblem consists of an uer-leel otimization (9)-(11) and a lower-leel otimization (12)-(14). he uer leel controls the ector of scheduling ariables x. Lower-leel decision ariables comrise the ector of disatching ariables whereas the ector of maret-clearing rices is also associated with the lower-leel roblem. he goal of the uer leel is to minimize the ricebased maret-clearing objectie function (9) ealuated at the otimal alues of the lower-leel ariables, and, subject to exressions exclusiel constraining uer-leel decision ariables x (10), to a set of constrained functions with and as arameters (11), and to the lower-leel otimization (12)-(14). he lower-leel otimization is identical to the economic disatch roblem (6)-(8) associated with the otimal uer-leel decision ariables x. In general, roblem (9)-(14) is a mixed-integer nonlinear bileel rogramming roblem. It is worth noting that the lower-leel objectie function (12) is ticall iecewise linear whereas constraints (13)-(14), including ower balance equations as well as generation and networ limits, are usuall linear. s a consequence, the lower-leel roblem (12)-(14) is arameterized in terms of the uer-leel decision ector x in such a wa that the lower-leel roblem is linear and thus conex. s described next, this feature allows the transformation of roblem (9)-(14) into an equialent single-leel mixedinteger roblem. 3 SOLUION ROCH o conert the original bileel formulation (9)-(14) into an equialent single-leel roblem, the lower-leel otimization is first relaced b its KK conditions. Consider the Lagrangian function associated with the lower-leel roblem for a gien uer-leel ector x : e (,,, μ) = f ( x, ) G ( x, ) L x i μ G ( x, ). (15) he otimal solution to (12)-(14) must satisf the KK necessar otimalit conditions: e (,,, μ) f ( x, ) G ( x, ) L x = ( x, ) μ = 0 i G (16) μ 0 (17) ( x, 0 ( x, ) 0 G ( x, ) i 0 e = (18) G i (19) G = μ, (20) where (16)-(17) are the dual feasibilit constraints, (18)-(19) are the rimal feasibilit constraints, and (20) exress the comlementar slacness conditions. Hence, roblem (9)-(14) can be recast as the following single-leel equialent:,, μ ( x,, ) Min F (21) x, x X (22) H x,, (23) ( ) 0 ( x, ) e i G ( x, ) G ( x, ) f μ = 0 (24) μ 0 (25) ( x, ) 0 ( x, ) 0 G i ( x, ) 0 G e = (26) G i (27) = μ. (28) roblem (21)-(28) falls into the categor of what is nown in the literature as a mathematical rogram with equilibrium constraints (MEC) [8]. s can be seen, the ectors of Lagrange multiliers and µ become decision ariables of the resulting single-leel equialent. In 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

4 addition, the following tes of nonlinearities are resent in roblem (21)-(28): - Nonlinear exressions (23) ticall inoling roducts of maret-clearing rices and ower oututs. hese sets of nonlinear roducts can be transformed into equialent mixed-integer linear rogramming exressions based on the KK conditions as described in [9]. - roducts of scheduling ariables x, ticall binar, and continuous ariables and µ in (24). hese nonlinearities can be equialentl formulated as linear exressions using well-nown integer algebra results [10]. - roducts of Lagrange multiliers and lower-leel decision ariables in the comlementar slacness conditions (28). s shown b Fortun-mat and McCarl [11], comlementar slacness conditions can also be formulated as mixed-integer linear rogramming exressions. It is worth mentioning that under the assumtion of linearit of the lower-leel roblem, comlementar slacness conditions (28) can be relaced b the equalit associated with the strong dualit theorem [10], thereb leading to a more effectie linearization in terms of comutational erformance. hus, after some algebra, roblem (21)-(28) is recast as a mixed-integer linear rogramming roblem suitable for commerciall aailable branch-and-cut software. 4 LICION o illustrate the aboe bileel rogramming framewor, we now consider an instance of rice-based maret clearing, namel the ament minimization roblem resented in [5], hereinafter referred to as M. s done in [5], we assume that (i) the demand is inelastic, (ii) generation offers comrise a single bloc, and (iii) minimum u and down times, raming limits, transmission networ, and ancillar serices are ignored. Notwithstanding, these simlifications do not alter our main conclusion, and results could be extended to include all of them at the exense of an increased notational comlexit. For quic reference, M is formulated as: Minimize, s,, K D s K ( ); j s C j j, 1 s (29) (30) 0; j (31) { 0,1} ; j (32) = D ; K (33) ; j (34) = K D C ;, (35) where is the ower outut of unit j in eriod ; s is the start-u cost of unit j in eriod ; is a binar ariable that is equal to 1 if unit j is scheduled on in eriod, being 0 otherwise; is the maret-clearing rice in eriod ; K is the index set of time eriods; D is the demand in eriod ; J is the index set of generating units; C j is the coefficient of the start-u cost of unit j; and are the lower and uer bounds for the ower outut of unit j in eriod, resectiel; and C reresents the offer cost coefficient of unit j in eriod. he objectie function (29) reresents the total ament b consumers and comrises two terms: the ament for energ and the ament for generation startus. Constraints (30) and (31) model the start-u costs. Constraints (32) imose the integralit of ariables. Constraints (33) are the ower balance equations. Constraints (34) set the lower and uer bounds for the ower oututs. Finall, constraints (35) model the marginal ricing setting. For the sae of unit consistenc, hourl time eriods are considered. ccording to Section 2, the bileel rogramming formulation for roblem (29)-(35) is: Minimize s, K D K ( ); j s C j j, 1 s s (36) (37) 0; j (38) { 0,1} ; j, (39) where maret-clearing rices Minimize K ( ) C are obtained from: (40) = D : ; (41) ( γ, θ ); j, : (42) 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

5 and where γ and θ are the Lagrange multiliers associated with the lower and uer bounds for the ower outut of unit j in eriod, resectiel. he uer-leel otimization (36)-(39) determines the scheduling ariables and the corresonding start-u costs s that minimize the consumer ament. In contrast, the lower-leel roblem (40)-(42) determines generation ower oututs and the maretclearing rices associated with the uer-leel scheduling ariables b soling a multieriod economic disatch based on the minimization of the generation offer cost. ling the methodolog described in Section 3, the bileel rogramming roblem (36)-(42) is transformed into the following equialent single-leel mixed-integer linear rogram: Minimize D s (43),a, a,b,b, K s,, γ, θ, K ( ); j s C j j, 1 s (44) 0; j (45) { 0,1} ; j (46) γ θ = C ; j (47) γ 0; j (48) θ 0; j (49) K a = D ; K (50) K ; j (51) D b K = a K C (52) = γ a ; j (53) 0 a γ ; j (54) ( 1 ) γ ; j 0 a (55) b = θ b ; j (56) θ b 0; j (57) ( 1 ) θ b 0; j, (58) where a reresents the nonlinear roduct b reresents the nonlinear roduct θ, γ, a and b are auxiliar continuous ariables used in the linearization of the aboe nonlinear terms, γ is the uer bound for the dual ariable lower bound for the dual ariable θ. Exressions (43)-(46) corresond to the uer-leel γ, and θ is the roblem while constraints (47)-(58) equialentl relace the lower-leel roblem. Constraints (47)-(49) are the dual feasibilit constraints, (50)-(51) reresent the rimal feasibilit constraints, and (52)-(58) are related to the linearization of the strong dualit equalit. Constraint (52) is the linear exression where the objectie functions of the rimal and dual roblems are equated. Constraints (53)-(55) and (56)-(58) model the linearization of the roduct terms tiel. γ and 5 NUMERICL RELS θ, resec- he bileel rogramming framewor for M has been tested on two case studies described in [5]. For both cases, the results of M are comared with those achieed b a conentional maret-clearing rocedure based on the minimization of the sum of the generation offer costs and start-u costs. his cost minimization roblem is denoted as CM. he model has been imlemented on an Intel Core i7, 1.73-GHz rocessor with 8 GB of RM using Clex 12.0 under GMS. 5.1 Illustratie Examle he first examle considers two hours and four generating units which are initiall scheduled off. able 1 shows the data for both eriods. his small examle is useful to highlight the differences in the results ielded b M and CM. Hour 1 Hour 2 Unit D 1 = 100 MW j1 j1 1 C j C j [MW] [MW] [$/MWh] [$] D 2 = 150 MW Unit j2 j2 C j2 C j [MW] [MW] [$/MWh] [$] able 1: Data for the illustratie examle. For this illustratie examle, the comuting time required to achiee the otimal solution to M using the single-leel equialent was less than 1 s. he otimal solutions to CM and M are summarized in able 2. Under conentional maret clearing, generators 1, 2, and 3 are disatched at maximum caacit in both eriods whereas generator 4 is not sched- 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

6 uled. Generator 3 is the marginal unit in both eriods thereb setting the corresonding maret-clearing rices equal to its offer cost, i.e., $65/MWh. his solution costs $6050 and ields a ament equal to $ CM M Unit Hour Hour [MW] [$/MWh] Offer Cost Start-u Cost [$] ament [$] able 2: Results for the illustratie examle. In contrast, M results in a different schedule in both eriods. While generators 1 and 2 do not exerience an change in their generation leels with resect to the solution to CM, generators 3 and 4 exchange their schedules and ower disatches. s a consequence, generator 4 becomes the marginal unit in both eriods, setting both maret-clearing rices at $30/MWh. he otimal ament is equal to $9300 and the associated cost is equal to $6400. In other words, a 42.9% reduction in ament is attained at the exense of a 5.8% increase in cost. In this case, the maret-clearing rice in each eriod under marginal ricing is identical to the corresonding highest acceted offer. Note, howeer, that both ricing schemes lead in general to different maret-clearing rices irresectie of the objectie function being minimized. s an examle, let the demand at hour 1 be reduced so that it belongs to the interal (50, 55] MW. t the otimal solution to either CM or M, generator 2 would be disatched at its minimum ower outut while generator 1 would be the marginal unit b suling the remaining demand. Under marginal ricing, the maretclearing rice would be the offer cost of generator 1, i.e., $10/MWh, whereas the highest acceted offer is that of generator 2, i.e., $20/MWh. 5.2 Medium-Sized est Case he second case stud considers 25 generating units and 24 hours. he hourl sstem demand is shown in able 3. Data for generators are resented in able 4. It is assumed that generators do not modif their resectie sul offers oer the time san. In addition, all units are initiall scheduled off excet units 1-8. he execution of Clex was stoed when the alue of the ament was below a secified threshold or when this number aeared to hae reached a lower bound. With these stoing criteria, able 5 shows the results attained b the roosed aroach for M. he comuting time required b this solution was equal to 78 s. able 5 also lists the results corresonding to the otimal solution to CM. s can be seen, a 7.0% reduction in ament is achieed b slightl increasing the cost b 2.6%. Hour D [MW] Hour D [MW] able 3: Sstem demand for the medium-sized test case. Unit C C j [MW] [MW] [$/MWh] [$] able 4: Generation data for the medium-sized test case. M CM 0ament [$] Offer Cost Start-u Cost [$] able 5: Results for the medium-sized test case. 17 th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011

7 owered b CDF ( Hourl maret-clearing rices associated with the solutions resented in able 5 are deicted in Fig. 1. his figure also shows the hourl sstem demand. s exected, maret-clearing rices resulting from either M or CM follow the shae of the demand cure. Note, howeer, that maret-clearing rices corresonding to M are less than or equal to those resulting from CM for all eriods. Maret-clearing rice ($/MWh) Hour Maret-clearing rice for CM Maret-clearing rice for M Sstem demand Figure 1: Maret-clearing rices and sstem demand for the medium-sized test case. 6 CONCLUSION his aer has formulated the rice-based maretclearing roblem under marginal ricing as a general bileel rogramming roblem, offering flexibilit in the roblem definition. he resulting bileel rogramming formulation is transformed into an equialent singleleel mixed-integer linear rogram. his transformation comrises two stes. First, the lower-leel otimization is relaced b a set of constraints based on the KK otimalit conditions and dualit theor. Subsequentl, a number of nonlinearities are conerted to linear equialents using some well-nown integer algebra results. he ultimate goal of this aer is to roide the ISO with a tool that can be used for comaratie analsis of different maret-clearing rocedures, so that informed decisions can be made. his general bileel formulation and its single-leel equialent hae been alied to an instance of ricebased maret clearing in which consumer ament is minimized. Numerical results reeal that bileel rogramming is an effectie aroach to address ricebased maret-clearing rocedures. Research is currentl underwa to sole other instances of rice-based maret clearing such as those exlicitl maximizing the surlus of consumers and roducers, as well as maret-clearing rocedures with reenue adequac constraints. Finall, further research will also be deoted to the analsis of joint energ and resere electricit marets Sstem demand (GW) CKNOWLEDGMEN he authors acnowledge the suort from the Ministr of Science of Sain under roject ENE , and from the Junta de Comunidades de Castilla La Mancha under roject I REFERENCES [1] M. Shahidehour, H. Yamin and Z. Li, Maret Oerations in Electric ower Sstems, New Yor, Wile, 2002, ISBN [2] J. M. Jacobs, rtificial ower Marets and Unintended Consequences, IEEE ransactions on ower Sstems, ol. 12, no. 2, , Ma 1997 [3] J. lonso,. rías, V. Gaitan and J. J. lba, hermal lant Bids and Maret Clearing in an Electricit ool. Minimization of Costs s. Minimization of Consumer aments, IEEE ransactions on ower Sstems, ol. 14, no. 4, , Noember 1999 [4] C. Vázquez, M. Riier and I. J. érez-rriaga, roduction Cost Minimization ersus Consumer ament Minimization in Electricit ools, IEEE ransactions on ower Sstems, ol. 17, no. 1, , Februar 2002 [5]. B. Luh, W. E. Blanson, Y. Chen, J. H. Yan, G.. Stern, S.-C. Chang and F. Zhao, ament Cost Minimization uction for Deregulated Electricit Marets Using Surrogate Otimization, IEEE ransactions on ower Sstems, ol. 21, no. 2, , Ma 2006 [6] F. Zhao,. B. Luh, J. H. Yan, G.. Stern and S.-C. Chang, ament Cost Minimization uction for Deregulated Electricit Marets with ransmission Caacit Constraints, IEEE ransactions on ower Sstems, ol. 23, no. 2, , Ma 2008 [7]. Somani and L. esfatsion, n gent-based est Bed Stud on Wholesale ower Maret erformance Measures, IEEE Comutational Intelligence Magazine, ol. 3, no. 4, 56-72, Noember 2008 [8] S. Deme, Foundations of Bileel rogramming, Norwell, Kluwer cademic ublishers, 2002, ISBN [9] C. Ruiz and. J. Conejo, ool Strateg of a roducer with Endogenous Formation of Locational Marginal rices, IEEE ransactions on ower Sstems, ol. 24, no. 4, , Noember 2009 [10]C.. Floudas, Nonlinear and Mixed-Integer Otimization: Fundamentals and lications, New Yor, Oxford Uniersit ress, 1995, ISBN [11]J. Fortun and B. McCarl, Reresentation and Economic Interretation of a wo-leel rogramming roblem, Journal of the Oerational Research Societ, ol. 32, no. 9, , Setember th ower Sstems Comutation Conference Stocholm Sweden - ugust 22-26, 2011