AN ITERATIVE ALGORITHM FOR PROFIT MAXIMIZATION BY MARKET EQUILIBRIUM CONSTRAINTS Andrés Ramos Marano Ventosa Mchel Rver Abel Santamaría Unversdad Pontfca Comllas IBERDROLA DISTRIBUCIÓN S.A.U. Alberto Agulera 23 Calderón de la Barca 16 28015 Madrd, SPAIN 03004 Alcante, SPAIN andres.ramos@t.upco.es asantamara@berdrola.es Abstract The electrcty ndustry has undergone sgnfcant restructurng toward deregulaton and competton durng the last years. In ths perod, new methodologes have been appearng for helpng n modelng the operaton of the electrc companes n market-based envronments. So far, Cournot-based equlbrum models are the most wdely used. The approach used n ths paper s ntended for modelng the medum-term operaton of the system. It s based on the Cournot conjecture, so the frms offer to the electrcty market the quantty that maxmzes ther proft. Ths model s formulated as a classc producton cost model where some market equlbrum constrants have been added. These constrants reproduce the frst-order optmalty condtons of strategc companes. All the constrants needed to represent the detaled operaton of the generatng unts are also taken nto account. In ths model an teratve algorthm has been mplemented to accurately compute the system margnal cost ncluded n the market equlbrum constrants. Besdes ths, a refnement algorthm for hydro schedulng has also been ntroduced. Results for a case study of realstc sze are presented. Keywords: electrcty market equlbrum, generaton schedulng, medum-term, hydro schedulng. 1 INTRODUCTION Kahn [4] and Hobbs [3] have recently made a classfcaton and revew of the current developments made n modelng the electrcty markets. The two man roads followed by researchers are based on the Cournot approach or on the supply functon equlbrum approach. Cournot-based models consder that frms compete only n quanttes whle the prce s derved from the demand functon. Ths assumpton s called the Cournot conjecture, see [7]. In the second approach, frms not only compete n quantty but also n prce. A great number of olgopoly market models that try to represent frms' medum and long-term behavor are somehow based on the Cournot equlbrum. Borensten and Bushnell [1] were the frst to explore the market equlbrum of an electrcty market. They modeled the Calfornan market under the Cournot framework, where the companes were consdered strategc or frnge dependng on ther characterstcs. The market equlbrum was determned usng an teratve algorthm that sets the strategc frm s producton at ts optmal level keepng constant the output of the other strategc frms. The process s repeated untl a Nash equlbrum s reached,.e., there s no ncentve to any strategc frm, takng ts compettors outputs as gven, to modfy ts output unlaterally. Later, Bushnell [2] extends ths approach by consderng hydro nflows management that nvolves multple perod decsons. Scott and Read [6] developed another dfferent approach that used a stochastc dynamc programmng model to obtan the market equlbrum for a hydrothermal system. The state varables were the system margnal prces. Recently, a new model has been proposed by Ventosa et al. [8] to determne the market equlbrum by drect formulaton of the optmzaton problems of each frm plus the demand functon that couples all these problems. Ths model can be solved as a mxed complementary problem (MCP) where, recently, solvers are becomng avalable. The contrbuton of ths paper s the clear and compact formulaton of the market equlbrum problem that allows to be used for realstc case studes. Ths paper proposes the representaton of the market equlbrum among frms by ntroducng a set of constrants nto a detaled producton cost model, see Ramos et al. [5]. These market constrants model the behavor of generaton agents under an olgopoly competton. By ntroducng these constrants the generaton agents maxmze ther profts (market revenues mnus varable operaton costs). Ths approach has the advantage of usng any classc producton cost method for modelng electrc systems. However t has a drawback, the market constrants depend on the system margnal cost, that cannot be, n general, smultaneously calculated wthn the optmzaton procedure. In ths paper we further develop an teratve procedure to avod t. Another teratve procedure s presented to deal wth the hydro nflows management for the strategc companes. Under the Cournot approach there are stll some ssues that mert further nvestgaton. The senstvty of the results to parameters such as the slope and elastcty of the demand functon should be examned. Moreover, further refnement of the soluton wll be needed to also consder stochastcty n the compettor behavor. The paper s organzed as follows. Secton 2 descrbes the man characterstcs of the model. Secton 3 analyzes the advantages and dsadvantages of the approach wth respect to the mxed complementary problem. Secton 4 presents the case study used to show the capabltes of the model to represent the market. Fnally, secton 5 provdes the conclusons.
2 MODEL DESCRIPTION Producton cost models are used for many purposes: hydro and mantenance schedulng, fuel purchase, pumped-hydro operaton, thermal unt commtment or economc plannng. They are able to represent n detal the techncal and economc constrants that nfluence the system operaton under a cost mnmzaton objectve functon. The mathematcal methods used to solve the optmzaton problem range from dynamc programmng, lagrangean relaxaton, and Benders decomposton of a drect mxed nteger problem formulaton. Two mportant features of these models are kept n the current market orented approach: A detaled representaton of system operaton Use of generator output levels as decson varables Whle keepng the above characterstcs, n ths approach we ntroduce the market equlbrum constrants that represent the proft maxmzaton objectve of each frm. 2.1 Producton cost models The objectve functon to be mnmzed corresponds to the total varable costs subject to the operaton constrants. These can be classfed nto nter- and ntraperod dependng on the perods that are nvolved n. Inter-perod constrants are assocated wth the coordnaton of lmted producton resources (mnmum quotas of domestc fuel consumpton, hydro nflows, and seasonal pumpng, storage and generaton). Intra-perod constrants deal wth the system operaton n each perod (balance between generaton and demand, thermal unt commtment, weekly/daly pumpng and storage and all the generaton lmts). Schematcally, ths classcal producton cost model s outlned n the followng table, consderng only the whte areas. The ntroducton of market equlbrum constrants, whch are to be dscussed n the followng secton, mples only some mnor modfcatons to the prevous optmzaton problem. The shaded areas correspond to the new market equlbrum constrants. Mnmzaton of Sum of total varable costs for each perod, subperod and load level + costs of unserved demand Subject to Inter-perod constrants Mantenance schedulng Hydro schedulng + seasonal pumpng Domestc fuel schedulng Intra-perod constrants Balance between generaton and demand Thermal unt commtment constrants Weekly/daly pumpng and storage Generaton lmts Market equlbrum constrants Margnal revenues = margnal cost for each frm Varable cost of each frm as a functon of commtted unts System margnal prce as a functon of the demand Under a market competton framework the demand reacton s modeled by the demand functon,.e., the demand response to the energy margnal prce. Then, the market equlbrum s obtaned by maxmzng the total surplus (consumer s plus producer s surplus). That s equvalent to mnmze the area under the supply curve on the left of the equlbrum output and the demand curve on the rght of ths quantty, as can be seen n fgure 1. Cleared Prce Prce Demand Curve Consumer s surplus Producer s surplus Varable costs Output Supply Curve Non-served demand costs Fgure 1. Utlty functon. Market Equlbrum Quantty 2.2 Market equlbrum constrants The market equlbrum constrants model the behavor of the market generaton agents. Ther objectve s to maxmze ther profts. The producer surplus for a gven load level s calculated as the dfference between revenues and costs. Revenues for each frm are calculated as the short run margnal prce ( SMP ) tmes the power produced by the frm, P. proft = SMP P C( P (1) where C( P s the frm s total varable cost as a functon of P. The market equlbrum constrants represent the frst-order optmalty condtons of each frm under ts proft maxmzaton objectve. For each frm n each load level the dervatve of the proft wth respect to the power generated by the frm s equal to zero, proft P = 0. SMP SMP + P MC( P = 0 (2) P Where MC( P s the frm s margnal cost as a functon of P, SMP P s the change n the SMP due to a change n the output of the frm, correspondng to the slope of the prce-demand curve, whch s negatve. The frst two terms of (2) form the margnal revenue of the frm and the last term correspond to the margnal cost. So (2) s equvalent to margnal revenue = margnal cost (3) Equaton (2) can be alternatvely expressed as the generaton level that, for each frm, maxmzes ts proft as a functon of the SMP, ts margnal cost and the slope of the demand functon SMP MC( P P = (4) SMP P
These constrants lmt the power offered by each company as a functon of the system margnal cost SMP, the own frm margnal cost MC( P and the slope of the demand functon SMP P. The prevous market equlbrum model mplctly assumes that there are no operatng constrants. Constructng the lagrangean and then formulatng the Karush-Kuhn-Tucker (KKT) frst order optmalty condtons can solve the frm s proft maxmzaton problem subject to the operatng constrants. In the KKT equaton correspondng to the frst order dervatve of the lagrangean wth respect to the frm s output we can neglect the terms all postve assocated to the dervatve of the operatng constrants and then the constrant (4) becomes SMP MC( P P (5) SMP P The nequalty sgn can be understood ntutvely. The objectve functon of cost mnmzaton (or, equvalently, perfect competton) leads each frm s output to levels greater than those of the proft maxmzaton problem that appears n an olgopoly market. The nequalty n (5) acts therefore as a constrant on the output levels of the frms, keepng them below perfect compettve levels. The market-clearng prce SMP s represented n two dfferent ways. As a constrant, t s modeled by a lnear functon of the electrcty demand. However, n the objectve functon (.e., the term of non-served demand costs), SMP s transformed nto a decreasng stepwse functon (wth the same slope of the lnear functon) where each step s a fcttous demand bd. Ths second modelng approach s used to avod nonlneartes n the objectve functon. SMP SMP = SMP0 + P (6) P 2.3 Iteratve computaton of system margnal cost However, the frm s margnal cost MC( P nvolved n equaton (5) cannot be drectly calculated n the optmzaton problem. The reason s based on the dscrete nature of the commtment decsons and on the mnmum load of thermal unts that has to be produced once the unt s commtted. An teratve algorthm s mplemented over the producton cost model to determne t. Ths algorthm acheves the smultaneous proft maxmzaton for all the frms. The algorthm begns computng analytcally the MC( P at each teraton (takng nto account all the operaton detals) as the lowest margnal cost of each generatng unt commtted of company n ths perod. Then, the proft maxmzaton problem wth market equlbrum constrants model s updated and solved usng ths MC( P. After that, a new MC( P s computed. When the dfference among two successve MC( P n under a threshold the algorthm stops, see fgure 2. Graphcally, the effect of the teratve algorthm n the producer surplus can be seen n fgure 3. In general although not for every load level, the frm s output under market competton s lower than under cost mnmzaton so the algorthm begns wth the compettve market levels. Then, the output of each frm s reduced n each teraton. Introduce mnmum MC (P) Solve the market equlbrum constrants model Compute new MC (P) * DIFF = MC (P) *- MC (P) DIFF 0 YES NO Elmnate MC (P) Fgure 2. Iteratve algorthm for computng the MC ( P ). Producer Surplus Market Equlbrum Iter 4 Iter 3 Iter 2 Iter 1 Output Fgure 3. Evoluton of producer surplus n the teratve algorthm. 2.4 Hydroelectrc schedulng The hydro operaton determned by the model may stll have multple optma from the pont of vew of cost mnmzaton. In ths model, the hydro producton s consdered to have zero cost for any hydro plant. But the strategc companes may ncrease ther profts by exchangng energy from load levels of low system margnal cost to other wth hgher margnal values wthout volatng the market equlbrum constrants. Ths stuaton may happen when the market constrants are not bndng n all the load levels. Then, a refnng teratve procedure has been mplemented to acheve market equlbrum for the
strategc companes takng care of hydro operaton detals. The algorthm can be dvded n the followng steps: 1. Obtan an ntal market equlbrum by solvng the optmzaton problem 2. Select a load level L1 wth hgh system margnal prce where the market equlbrum constrants are not bndng for some strategc company 3. Select a load level L2 wth lower system margnal prce where the same company can decrease ts producton 4. Fnd a frnge company that can decrease ts producton n load level L1 and smultaneously decrease n load level L2 5. Exchange the hydro generaton of the strategc and frnge companes among load levels takng nto account the techncal hydro constrants (.e., maxmum and mnmum output, maxmum and mnmum reserve levels, etc.) and any other constrant (.e, frm market share) 6. If two load levels wth dfferent system margnal prces can be selected go to step 2. In other case go to end Ths algorthm refnes the hydro unts operaton of the strategc companes. It has observed that ths algorthm ntroduces only mnor changes n the output of the strategc companes n the cases tested so far. 3 COMPARISON WITH A MCP APPROACH The approach presented has two man advantages. One s the realstc modelng of the electrc system that allows ncludng bnary unt commtment decsons. The other s the use of robust and effcent soluton methods, those of MIP problems. Convergence of the proposed method s not theoretcally guaranteed. However, cases tested so far have proven to be robust. Qualtatvely, t s nterestng to analyze the complementary features of the cost mnmzaton sde, stll explctly represented n the model, and the frm s proft maxmzaton objectve, whch are ncorporated mplctly through the market equlbrum constrants. Whle the later determnes, for each strategc frm consdered, an output level that maxmzes ts profts, t s the cost mnmzaton, whch decdes the specfc unt commtment that acheves that output level. It wll do so by lookng for the cheapest commtment of ther thermal unts and the cheapest hydro schedulng, exactly as each frm would have done f ts output requrements had been set exogenously to the model. Wth ths approach, where market behavor and operatng constrants are smultaneously consdered, the equlbrum soluton accounts for all the techncal operatng constrants modeled, thereby achevng a realstc system dspatch. The MCP approach has alternatve advantages. One s a compact problem formulaton. The other s the possblty of ntroducng nonlnear constrants. However, as an NLP-based approach t cannot ntroduce bnary varables and the soluton procedure s less effcent and robust than LP-based methods. Theoretcally, optmallty s guaranteed only wth lnear constrants. Both algorthms acheve the same results under the same underlyng modelng assumptons (.e., contnuos varables and lnear constrants). Table 1 summarzes the prevous comparson between the alternatve methods. Qualtatve MIP Approach Optmal soluton not guaranteed n every case Algorthmc Bnary varables are allowed. Constrants must be lnear Soluton process Soluton method s effcent and robust MCP Approach Optmalty guaranteed and soluton unqueness n case of lnear constrants Only contnuous varables. Constrants can be non lnear Soluton method s slower and dependng on the ntal value Table 1. Comparson between MIP and MCP approaches. 4 CASE STUDY In ths secton we present a representatve case study, whch was developed to test the model results. 4.1 Tme scope The tme scope s dvded nto 3 perods, each one representng one month, 2 subperods correspondng to week and weekend days and 3 load levels for weekdays and 2 load levels for weekend days. 4.2 Demand The demand functon s represented by a lnear functon of the prce wth a slope of Mpta/GWh/GW at each load level. 4.3 Supplers The thermal generaton system s composed of 18 unts, each one wth dfferent varable costs. Ther man characterstcs are presented n table 2. The last two columns of table 2 represent the constant and lnear terms, respectvely, of the straght lne that models the heat consumpton of each thermal unt. Frm Pmax Pmn Fuel cost [pta/mcal] Constant Varable cost [Mcal/h] Lnear Varable cost [cal/wh] T1 E1 140 50 1.38 20 3.21 T2 E2 140 80 1.38 20 3.26 T3 E3 140 50 1.38 30 3.28 T4 E1 140 40 1.38 36 3.30 T5 E2 140 50 1.38 30 3.33 T6 E3 110 50 1.62 25 3.35 T7 E1 100 45 1.62 25 2.85 T8 E2 100 45 1.62 25 2.90 T9 E3 100 60 1.62 20 2.91 T10 E1 130 50 1.62 22 3.01 T11 E2 110 50 1.62 25 3.09 T12 E3 110 60 1.62 28 3.12 T13 E1 140 120 1.68 20 2.92 T14 E2 140 110 1.68 27 3.00 T15 E3 140 75 1.68 35 2.89 T16 E1 250 55 1.68 35 2.90 T17 E2 270 90 1.68 26 2.95 T18 E3 270 85 1.68 26 3.23 Table 2. Thermal generaton system.
The mantenance schedulng program has determned that thermal unt 1 wll be on mantenance n perod 1, unt 5 n perod 2 and unt 9 n perod 3. The hydro generaton system s composed by 6 plants. Ther man characterstcs are presented n table 3. The columns represent the maxmum power avalable for consumpton as pump or producton as generator, the ntal and maxmum and mnmum reserve levels of each hydro reservor 1 and the performance of the pumpng-generaton cycle for a pumpng-hydro unt. Accordng wth the table only the last three hydro unts are pumpng-hydro. The natural hydro nflows, expressed n GWh, receved n each hydro reservor n each perod are represented n table 4. 5.3 5.25 5.2 5.15 5.1 5.05 5 4.95 4.9 4.85 4.8 Cost Mnmzaton M arket Constrants 0.9 Fgure 4. System margnal prce. Frm Pmax pump Pmax gener Intal reserve [TWh] Max reserve [TWh] Mn reserve [TWh] Perfrm [p.u.] H1 E3 300 80 150 35 H2 E2 270 70 132 33 H3 E1 250 60 123 31 H4 E3 176 220 50 90 22 0.65 H5 E2 160 200 45 82 20 0.70 H6 E1 112 140 30 58 15 0.68 Table 3. Hydro generaton system. 0.8 0.7 0.6 0.4 0.3 0.2 0.1 0 Cost Mnmzaton Market Constrants P1 P2 P3 H1 35 30 33 H2 30 25 35 H3 25 25 32 H4 25 20 25 H5 25 20 20 H6 50 10 10 Table 4. Natural hydro nflows. 4.4 Analyss of results The model has been run wth and wthout market equlbrum constrants. In the second case, the model represents a cost mnmzaton framework or a perfectly compettve market. In the frst case, t ncorporates the proft maxmzaton objectve of strategc frms by means of the market equlbrum constrants. As t can be observed n fgure 4 the prce for the cost mnmzaton problem moves between 4.86 and 5.16 pta/kwh whle n the proft maxmzaton problem the prce range s between 5.00 and 5.22 pta/kwh for the 15 load levels. The followng fgures represent the power generated by each company and the total system producton, expressed n GW, under cost mnmzaton or market equlbrum constrants. 0.9 0.8 0.7 0.6 0.4 0.3 0.2 0.1 0 0.9 0.8 0.7 0.6 0.4 0.3 0.2 0.1 Fgure 5. Output of frm E1. Cost Mnmzaton Market Constrants Fgure 6. Output of frm E2. 0 Cost Mnmzaton Market Constrants Fgure 7. Output of frm E3. 1 Each hydro unt s supposed to have an assocated reservor.
0 2.5 Power Markets IEEE-PES Summer Meetng Proceedngs. Vol 4, pp 2272-2277. Seatlle, USA. 2000. 2 1.5 1 Cost mnmzaton Market Constrants Fgure 8. System generaton. It can be observed from prevous fgures that the energy produced by the companes n the proft maxmzaton model s lower than those produced n the cost mnmzaton model. However, not every load level follows ths tendency. 5 CONCLUSIONS We have presented a practcal approach to model the market equlbrum under an olgopoly market competton based on the Cournot conjecture. The model presented n the paper ncludes the so-called market equlbrum constrants n a detaled producton cost model. By ths approach all the techncal operaton constrants can be consdered ncludng the bnary commtment decsons of thermal unts. The resultng optmzaton model s a MIP problem so conventonal, effcent and robust solvers can be used. Two teratve algorthms, for obtanng the system margnal prce and refnng the hydro operaton of strategc companes, have been added to the prevous market equlbrum model. The results wth ths MIP problem formulaton are equal than those obtaned by the MCP approach under the same modelng assumptons. The model has been tested wth a case study and the results are shown n the paper. 6 REFERENCES [1] Borensten, S. and Bushnell, J. An Emprcal Analyss of the Potental for Market Power n Calforna s Electrcty Industry POWER Conference on Electrcty Restructurng. Unversty of Calforna. Energy Insttute. 1997. [2] Bushnell, J. Water and Power: Hydroelectrc Resources n the Era of Competton n the Western US POWER Conference on Electrcty Restructurng. Unversty of Calforna. Energy Insttute. 1998. [3] Hobbs, B. F. Lnear Complementarty Models of Nash-Cournot Competton n Blateral and POOLCO Power Markets. IEEE Transactons on Power Systems, 16(2), pp. 194-202, May 2001. [4] Kahn, E. Numercal technques for analyzng market power n electrcty The Electrcty Journal. pp. 34-43. July 1998. [5] Ramos, A., Ventosa, M., Rver, M. Modelng Competton n Electrc Energy Markets by Equlbrum Constrants. Utltes Polcy, Vol. 7 Issue 4 pp. 233-242, Dec. 98. [6] Scott, T.J. and Read, E.G. Modellng Hydro Reservor Operaton n a Deregulated Electrcty Market Internatonal Transactons n Operatonal Research. Vol. 3 pp. 243-253. 1996. [7] Varan, H.R. Mcroeconomc Analyss. W.W. Norton & Company. New York. 1992. [8] Ventosa, M., Rver, M., Ramos, A. and García-Alcalde, A. An MCP Approach for Hdrothermal Coordnaton n Deregulated