Risk-Constrained Bidding Strategy with Stochastic Unit Commitment

Transcription

1 Risk-Costraied Biddig Strategy with Stochastic Uit Commitmet 1 Tao L Member, Mohammad Shahidehpour, Fellow, ad Zuyi L Member, IEEE Abstract This paper develops optimal biddig strategies based o hourly uit commitmet i a geeratio compay (GENCO) which participates i eergy ad acillary services markets. The price-based uit commitmet problem with ucertai market prices is modeled as a stochastic mixed iteger liear program. The market price ucertaity is modeled usig the sceario approach, Mote Carlo simulatio is applied to geerate scearios, sceario reductio techiques are applied to reduce the size of the stochastic price-based uit commitmet problem, ad postprocessig is applied based o margial cost of committed uits to refie biddig curves. The fiacial risk associated with market price ucertaity is modeled usig expected dowside risk which is icorporated explicitly as a costrait i the problem. Accordigly, the proposed method provides a closed-loop solutio to devisig specific strategies for risk-based biddig i a GENCO. Illustrative examples show the impact of market price ucertaity o GENCO s hourly commitmet schedule ad discuss the way GENCOs could decrease fiacial risks by maagig expected payoffs. Idex Terms Risk, biddig strategy, stochastic price-based uit commitmet, mixed iteger programmig I. NOMENCLATURE Variables: EDR () Expected dowside risk for a give target profit i Deote a thermal uit I ( ) Uit status idicator with 1 meas o ad 0 meas off I d ( ) Idicator for providig o-spiig reserve whe off F () Uit s cosumptio of fuel for a sceario m Segmet idex Deote a ode i the sceario tree ' Deote a ode differet from ode i the sceario tree N u ( ) No-spiig reserve of a uit whe o N d ( ) No-spiig reserve of a uit whe off p m ( ) Geeratio of segmet m i liearized heat curve P ( ) Geeratio of a uit PF ( ) Payoff for a sceario R ( ) Spiig reserve of a uit RISK () Dowside risk for a sceario The authors are with the Electric Power ad Power Electroics Ceter, Illiois Istitute of Techology, Chicago, IL USA. ( litao@iit.edu, ms@iit.edu, lizuyi@iit.edu). s Deote a sceario t Hour idex TP ( ) Total geeratio offered to (positive value) or purchased from (egative value) the market by a uit TN ( ) Total o-spiig reserve offered by a uit v m ( ) Idicate whether a uit is started at segmet m of the startup cost curve, 1 meas started at segmet m ad 0 meas off x ( ) Auxiliary biary variable for oe sceario z ( ) Shutdow idicator Costats: b m ( ) Slope of segmet m i liearized heat curve f ( ) Heat rate at the miimum geeratig capacity ρ f ( ) Fuel price of a uit HN ( ) Set of odes for a hour MSR ( ) Maximum sustaied ramp rate (MW/mi for a uit NS ( ) Number of segmets for the startup fuel curve NSF ( ) Number of segmets for the piece-wise liearized heat rate curve P 0 ( ) Bilateral cotract of a uit SG 0 ( ) Icome from bilateral cotract of a uit P g ( ), P g ( ) Miimum/maximum geeratig capacity QSC ( ) Quick start capacity RU ( ), RD ( ) Rampig up/dow limit of a uit SD ( ) Shutdow cost of a uit SF m ( ) Startup fuel if started at segmet m SN ( ) Set of odes for a sceario z 0 Targeted profit π () Probability for a sceario ρ g ( ) Market price for eergy ρ sr ( ) Market price for spiig reserve ρ r ( ) Market price for o-spiig reserve II. INTRODUCTION EVISING a good biddig strategy is very crucial for a DGENCO to maximize its potetial profit [1,2]. The approaches for developig biddig strategies could be categorized ito: equilibrium models ad o-equilibrium

2 models. Equilibrium models such as Supply Fuctio Equilibrium ad Courot Equilibrium were widely applied for developig GENCOs biddig strategies ad aalyzig market power i eergy markets [2]-[5]. However, uit costraits such as miimum o/off time, rampig limits, ad startup cost were ot cosidered i most of the equilibrium models because the existece of equilibria could ot be prove whe iteger variables were used i those models. Accordigly, the simulated market equilibrium without the uit prevailig costraits could deviate largely from practical operatio. Meawhile, there may be some computatioal problems whe equilibrium models are applied to a large system with may market participats. However, equilibrium models would be very importat for aalyzig the potetial market power of a GENCO ad the optimal biddig strategy of GENCOs with market power. There are several o-equilibrium approaches i the literature for developig optimal biddig strategies. For example, a ordial optimizatio method was used i [6] to fid the good eough biddig strategy for power suppliers. The basic idea was to use a approximate model for aalyzig the impact of GENCO s biddig strategies o market clearig price. A biddig model was proposed i [7] based o a ecoomic priciple kow as cobweb theorem. The proposed model calculated the maximum biddig price ad quatity by a iterative procedure usig the GENCO s residual demad curve. Determiistic price-based uit commitmet (PBUC) was applied for developig biddig strategies i [1], [8]-[11]. However, the precisio of market price forecastig could have a direct impact o PBUC solutio. Due to electricity market dyamics, which could make it difficult to forecast market prices accurately, it would be very importat to cosider the market price ucertaity. There are several approaches to modelig the market price ucertaity. The first approach is to model directly the market price ucertaity. Without cosiderig acillary services, the stochastic uit commitmet problem i a PoolCo market with ucertai market prices was solved usig LR, stochastic dyamic programmig, ad Beders decompositio i [12]. However, the purpose of [12] was to develop policies icludig uit commitmet ad geeratio dispatch for each sceario istead of developig biddig strategies. The market price ucertaity was take ito cosideratio i [13] whe developig optimal biddig strategies i multi-markets. The secod approach is to model the ucertaity of residual demad curve. The ucertaity of residual demad curve was cosidered i [14] by applyig the sceario approach to develop optimal biddig strategies for a GENCO. Beders decompositio was employed to solve the correspodig stochastic liear program. The third approach is to model the ucertaity of competitors bids ad system loads. The ucertaity of competitors bids was modeled i [15] for developig optimal biddig strategies i eergy market. Biary expasio approach was applied to trasform the bilevel optimizatio problem ito a mixed iteger liear program, ad the resultig problem was solved by a MIP solver. I [13] ad [14], uit commitmet decisios were ot cosidered ad assumed to be give. Meawhile, models i [13]-[15] were risk-eutral i the sese that the objective was to maximize expected payoffs. The optimal decisio could expose the GENCO at a sigificat risk level because of the market price ucertaity. It would be very importat for a GENCO to maximize its potetial profit while keepig the ivolved risk at a acceptable level. There are several ways to model risks associated with a decisio. A very commo way is the mea-variace approach proposed i [16], which is the idustry s stadard model i portfolio selectio. I this approach, the risk is measured usig the variace of the expected payoff ad a utility fuctio is devised by appedig the variace of expected payoff ito the origial expected payoff fuctio. The objective of a decisio maker would be to maximize its utility fuctio. I the case of stochastic iteger program, this method is criticized for its computatioal itractability whe appedig a variace term i the objective fuctio [17]. The value at risk (VaR) approach [1] was applied to aalyze risk associated with a biddig strategy. However, a ope-loop solutio made it very difficult for a GENCO to modify its biddig strategy based o the risk level. Real optio models ad stochastic optimizatio techiques were applied i [18] to maage risk. A detailed overview of risk assessmet method i eergy tradig was give i [19]. This paper proposes a risk-costraied biddig strategy for a GENCO to devise optimal bids i day-ahead eergy market ad acillary services market. Our proposed method belogs to the o-equilibrium approach category. The problem is formulated as a stochastic mixed iteger liear programmig ad solved by commercial mixed iteger programmig (MIP) solver. The tradeoff betwee maximizig expected payoff ad miimizig risk due to the market price ucertaity is modeled explicitly by icludig the expected dowside risk as a costrait. Accordigly, the proposed procedure provides a closed-loop solutio to devisig biddig strategy. Meawhile, prevailig uit commitmet costraits are cosidered. After solvig the stochastic PBUC problem, postprocessig techiques based o margial cost are applied to refie the biddig curves. Illustrative examples show the impact of market price ucertaity o commitmet schedule of geerators ad a GENCO could sigificatly decrease the level of ivolved risk at the cost of reducig its expected payoff. This paper is orgaized as follows: the stochastic PBUC problem is formulated i sectio III ad the costructio of biddig curve is show i sectio IV. Sectio V models risk costrait. Illustrative examples ad coclusios are provided i sectios VI ad VII, respectively. III. STOCHASTIC PRICE-BASED UNIT COMMITMENT We devise biddig strategies simultaeously for eergy ad acillary services markets. If acillary services are cleared after the day-ahead eergy market, we would apply the proposed method to submit eergy biddig curves ad execute the program agai by pluggig i day-ahead eergy results. I 2

3 this way, the certaity of market clearig results i the dayahead eergy market could ehace a GENCO s profit. Market prices could be stated as locatioal margial prices (LMP or uiform market clearig prices (MCP which deped o market rules i cosideratio. Our model could be applied to either case. As market price variables are represeted by cotiuous probability distributios, it is very difficult, if ot impossible, to solve the correspodig stochastic programmig problem sice itegratio over such variables are required explicitly or implicitly. To overcome this problem, we geerally resort to approaches that could substitute the cotiuous market price variables with a set of discrete outcomes. Each possible discrete outcome of market price is called a sceario. We discuss ext the approach for geeratig market price scearios based o forecasted market prices. A. Sceario geeratio There are various approaches to geeratig scearios for stochastic programmig. Scearios are commoly geerated by samplig historical time series or statistical models such as time series or regressio models. Time series models were applied to geerate scearios for prices i electricity markets i [13]. A oliear optimizatio problem was employed i [20] to geerate sceario trees give the statistical properties of stochastic variables. A detailed literature review o sceario geeratio was preseted i [21]. I this paper, the Mote Carlo simulatio method is applied to geerate scearios. Market price ad market price variace forecasts for eergy ad acillary services are assumed to be calculated by applyig techiques such as time series ad artificial eural etwork [1]. The, Mote Carlo simulatio is executed M (i.e., a very large umber) times to geerate scearios for market prices whe the probability of each sceario is assumed to be 1 / M. B. Sceario reductio If we execute Mote Carlo simulatio 100,000 times, we would obtai 100,000 scearios ad the resultig stochastic program would be too large to solve. Accordigly, sceario reductio techiques are applied to reduce the umber of scearios i cosideratio while maitaiig a good approximatio of the statistical properties of market prices. The basic idea of sceario reductio is to elimiate a sceario with very low probability ad budle scearios that are very close. Accordigly, sceario reductio algorithms determie a subset of scearios ad calculate probabilities for ew scearios such that the reduced probability measure is closest to the origial probability measure i terms of a certai probability distace betwee the two measures [22], [23]. After reducig ad budlig scearios, a example of the reduced sceario tree for a three-stage problem is show i Fig.1. t = 1 t = 2 t = 3 Fig. 1 Sceario tree for a three-stage stochastic problem A ode i the sceario could have multiple successors but oe acestor at most. The ode without ay acestor is called the root ode ad odes without ay successors are called leaves. A sceario is defied as a path i the sceario tree from the root ode to a leave ode. For a ode i the sceario tree, we apply + to deote its predecessor. Meawhile, we apply 0 to deote the root ode i the sceario tree. Sice a small umber of scearios could result i a poor approximatio, a tradeoff exists betwee problem precisio ad problem size. A practical way is to set the umber of reduced scearios whe the objective fuctio is ot chagig sigificatly or the relative distace betwee origial scearios ad reduced scearios is withi a acceptable level [22], [23]. C. Objective fuctio for a GENCO A GENCO iteds to maximize its expected payoff as: max : π ( PF( (1) s ad the profit for sceario s is: ρ f ( F( ρ g ( TP( + ρ sr ( R( PF( = i + + ρ r ( [ Nu ( + N d ( ] SN ( + SG0 ( z( SD( (2) Sice all the iformatio (i.e., uit status, price, geeratio, etc.) for hour t is icluded i the set of odes HN (t) at that hour, we preset the equatios as a fuctio of ode istead of hour t. We preset the startup cost as a fuctio of startup fuel (MBtu), for modelig the cosumptio of costraied fuel, ad shutdow cost as fuctio of cost (dollar. However, we could also preset the startup cost as a fuctio of cost (dollar istead of fuel (MBtu). The variatio would ot impact the proposed formulatio. I this paper, we cosider thermal uits. However, other types of uits such as combied-cycle, cascaded-hydro, ad pumped-storage uits could be icluded without much difficulty [11]. We discuss the various costraits for a GENCO as follows. D. Uit costraits A GENCO would make uit commitmet decisios before submittig bids to the day-ahead market. Accordigly, the fial uit commitmet decisio should be the same for all possible scearios of market price. If a system operator, i.e., 3

4 a ISO or RTO, is resposible for commitmet decisios, decisios could be differet from oe sceario to aother. The uit-specific costraits were give i [11] such as miimum o/off time limits ad time-varyig startup costs. If the outcome of stochastic variables for a certai umber of scearios is the same, decisio variables for those scearios should also be the same. This is called oaticipativity costraits [24]. For a stochastic program, there are two ways of expressig oaticipativity costraits: sceario-based ad ode-based. I the sceario-based approach, oaticipativity costraits are eforced explicitly [13] which itroduce additioal costraits ad variables to the problem. I the ode-based approach, costraits are formulated for each ode istead of each sceario ad oaticipativity costraits are eforced implicitly as show ext. Rampig costraits: P( P( + ) RU ( (3) P( + ) P( RD( Eergy bilateral cotracts for uits: P ( TP( = P0 ( (4) where positive TP ( represets the geeratio offered to the market ad egative TP ( is the purchased geeratio from the market. The market price for a higher quality reserve (spiig reserve) should be higher tha that of a lower quality reserve (o-spiig reserve). However, this is ot always observed i practical markets because of market iefficiecies. Accordigly, a geeral case of geeratig uit costraits for supplyig eergy ad acillary services i spot market is modeled i (5): P( + R( + Nu ( Pg ( I( P g ( I( P( R( + Nu ( 10 MSR( I( (5) N d ( QSC( I d ( I d ( + I( 1 The fuel cosumptio of uit i for sceario s is calculated as: NSF ( NS( F( = [( f ( I( + pm( bm ( ) + vm( SFm ( ] SN ( m= 1 m= 1 NSF ( P( = Pg ( I ( + pm( m= 1 0 pm( pm( (6) [15]. I practical electricity markets, a market participat would submit a piecewise o-decreasig biddig curve such as that i Fig. 2. A. No-decreasig coditios I the stochastic iteger program formulated i sectio III, scearios are treated idepedetly. However, the hourly price ad geeratio quatity pairs may ot be mootoically icreasig. Accordigly, the o-decreasig coditios are eforced as: [ ρ g ( ρ g ( ' )] [ TP( TP( ' )] 0, ' HN( t), t [ ρ sr ( ρ sr ( ' )] [ R( R( ' )] 0, ' HN( t), t [ ρr ( ρr ( ' )] [ TN( TN( ' )] 0, ' HN( t), t (7) A GENCO is boud to eforce o-decreasig coditios for its total geeratio, total spiig reserves, ad total ospiig reserves if it submits biddig curves for the etire GENCO istead of its idividual uits. B. Costructio of biddig curve The problem solutio presets hourly o-decreasig price ad quatity pairs. A sample for a uit is show i Fig. 3 i which there is a large jump from the secod pair to the third. I such cases, there are two possible ways of costructig a cotiuous biddig curve as show i Fig. 4. Price($/MWh) ρ i3 ρ i2 ρ i1 Fig.2. Biddig curve for uit i p i1 p i2 p i 3 Geeratio(MW) 4 Other costraits could be modeled similarly as show i [11]. IV. CONSTRUCTION OF BIDDING CURVE Ideally, a market price i real time should correspod to oe sceario for the realizatio of possible market prices. I this sese, the geeratio obtaied i sectio III is the optimal biddig quatity for a GENCO at the correspodig market price. Accordigly, price ad geeratio pairs obtaied i sectio III could be applied to costruct a biddig curve [14], Fig.3. Biddig price ad quatity pairs

5 The first method is to use the lower geeratio ad upper price of two cotiuous poits to obtai a ew price ad quatity pair ad coect all the price ad quatity pairs to devise the biddig curve. By applyig this method, the uit may lose some reveue if the market price is betwee $18/MWh ad $25/MWh sice the uit will oly be awarded 60MW. margial cost of uit i Fig. 3 is P ($/MWh), ε g = 10MW, ad ε ρ = $1/MWh. We apply the proposed method to costruct biddig curve. The geeratio differece betwee the secod ad third poits is = 40 MW > 10MW, so we divide the geeratio rage betwee 60MW ad 100MW ito l = 3 segmets. That is, the ew geeratio poits are 70, 80, 90 MW with margial costs of 19, 20, ad 21$/MWh, respectively. The biddig curve is show i Fig Fig.4. Two atural methods for costructig biddig curve This is a coservative way of costructig biddig curves. The secod method is to use the upper geeratio ad lower price of two cotiuous poits to obtai ew price ad quatity pairs. This would be a risky method as a uit may icur losses if the market price is betwee $18/MWh ad $25/MWh ad the awarded geeratio is 100MW. Based o the two methods, we propose two other ways to costruct biddig curves. After fidig price ad quatity pairs, the uit status, o-load cost, ad startup/showdow cost are determied. The margial cost of a GENCO would determie whether or ot the GENCO should offer additioal geeratio. Accordigly, margial cost could be utilized for costructig biddig curves. Assume we have two cotiuous price ad quatity pairs m' ad m. If the geeratio differece betwee the two cotiuous price ad quatity pairs, Δ P( = P( m' ) P(, is larger tha a give tolerace ε g ad there is a price jump betwee the two pairs that is larger tha the price tolerace ε ρ, we divide Δ P( ito l segmets which is the maximum iteger umber that is smaller tha Δ P( / ε g ad calculate the margial cost of uit at P( + k ε g for k = 1,..., l. If the margial cost at ay ew geeratio poit is less tha ρ g (, the margial cost will be substituted by ρ g ( to satisfy o-decreasig coditios. Likewise, the correspodig margial cost is substituted by ρ g ( m' ) if the margial cost of ay ew geeratio poit is larger tha ρ g ( m' ). The ewly obtaied price ad geeratio pairs could be combied with the origial pairs to costruct a biddig curve. For istace, assume the Fig.5. Method 3 to costruct biddig curve Similarly, method 4 could be developed by dividig the price differece ito several segmets. I the above example, we could divide the price differece betwee secod ad third poits i Fig. 3 ito (25 18) / 1 = 7 segmets. We would obtai the ew price poits at 19, 20, 21, 22, 23, 24, 25 $/MWh with the correspodig geeratio at 70, 80, 90, 100, 100, 100, 100 MW. Here the geeratio for secod ad third poits should be o less tha that for the secod poit ad o larger tha that for the third poit to satisfy the o-decreasig coditio. I this example, the obtaied biddig curve by method 4 is the same as that show i Fig. 5. The above methods devise biddig curves i eergy market. Similar methods for biddig i acillary services market could be adopted based o opportuity cost. V. RISK CONSTRAINTS The stochastic PBUC formulatio i sectio III is a riskeutral model which is cocered with the optimizatio of expected payoff. However, a GENCO may also be cocered with its risk. Accordigly, we apply the method itroduced i [25] for aalyzig the risk associated with such decisios. A GENCO would set a targeted profit z 0 ad the risk associated with its decisio is measured by failure to meet the targeted profit. If the profit for oe sceario is larger tha the targeted profit, the associated dowside risk would be zero; otherwise, it is the amout of ufulfilled profit. That is: z0 PF( if PF( z0 RISK ( = (8) 0 if PF( > z0 This coditioal expressio is a liear costrait represeted

6 by auxiliary biary variables as: 0 RISK( [ z0 PF( ] M [1 x( ] (9) 0 RISK( M x( where M is a large umber. However, (9) may be deemed uecessary based o the approach that will be itroduced later. The expected dowside risk for a GENCO is defied as: EDR( z 0 ) = E[ RISK( ] = π ( RISK( (10) s The smaller the EDR ( z 0 ), the better it is for decisio makers sice EDR ( z 0 ) represets the profit shortfall at a targeted profit. If a decisio maker is ot satisfied with the risk level, a risk costrait could be added to the origial formulatio as: EDR ( z 0 ) EDR 0 (11) where EDR 0 is the acceptable dowside risk tolerace. A small example is give below to show the proposed method for measurig risks. Suppose we have oe thermal uit with a margial cost of $20/MWh ad a miimum o/off time of oe hour. Miimum ad maximum capacity limits are 50MW ad 100MW, respectively ad startup ad shutdow costs are igored. The iitial status of the uit is o for submittig bids to the eergy market. The possible scearios for hourly market price at a specific hour are show i Table I. The optimal solutio based o stochastic programmig i sectio III would be to commit the uit with submitted geeratio ad payoff for each sceario show i Table II. The expected payoff i this case is: = $150 If we set the targeted profit for the uit at zero, the dowside risk would be: EDR ( 0) = [0 ( 250) ] [0 ( 100) ] 0.2 = $70. That is, the uit could obtai a expected payoff of $150 with a expected dowside risk of $70. The uit which is ot satisfied with the risk level will costrai the expected dowside risk at a value such as EDR ( 0) 0 70 = 0 as discussed i sectio III. The correspodig risk-costraied solutio would be to shut dow the uit with a zero payoff ad zero expected dowside risk i each sceario. That is, the uit could reduce its risk level at the cost of decreasig its expected payoff. Accordigly, the iclusio of risk costrait could impact the optimal solutio. The costrait o expected dowside risk could result i a ifeasible solutio whe the costrait is tight (i.e., relatively low risk tolerace EDR 0 or relatively high targeted profit). Oe possible approach to settig up a reasoable targeted profit ad associated dowside risk tolerace is to choose the profit based o operatig experiece or without cosiderig risk costrait. For example, a GENCO could cosider the sceario with the highest probability as its targeted profit ad pick its risk tolerace. If the targeted profit is relatively high, the the dowside risk tolerace should also be relatively high. The techique itroduced i [25] is to successively tighte the costrait o the expected dowside risk. For istace, if the dowside risk without risk costrait is UEDR 0, we add a costrait such as EDR( z0 ) UEDR0 ad solve the resultig costraied problem with a ew expected dowside risk UEDR 1. For the ext ru, we tighte the risk costrait as EDR( z0 ) UEDR0 ad repeat this procedure util we reach a acceptable risk level. The succeedig rus i this case are viewed as idepedet ad could be solved i parallel after solvig the problem without risk costraits. We could also pealize the expected dowside risk i the objective fuctio usig a very large umber. However, the advatage of the first method is that it provides a GENCO with a choice betwee expected payoff ad risk. The secod method would miimize the dowside risk at the cost of reducig profit. However, the secod method would oly require a sigle executio while the first method would eed multiple rus. TABLE I MARKET PRICE FOR THE SMALL EXAMPLE Sceario # Eergy price ($/MWh) Probability TABLE II GENERATION AND PAYOFF FOR EACH SCENARIO Sceario # Geeratio (MW) Payoff ($) I this paper, we propose the followig procedure for cosiderig risk by combiig the two methods. 1. Solve the problem without cosiderig risk costrait; Choose a target profit z 0 ad calculate the expected dowside risk UEDR ( z 0 ). If the calculated expected dowside risk is withi the GENCO s risk tolerace, stop. Otherwise, go to step (2). 2. Pealize the expected dowside risk ito the objective fuctio by a very large umber ad solve the correspodig problem for calculatig the miimum expected dowside risk ad the correspodig expected payoff. If the miimum expected dowside risk is ot acceptable, the GENCO s targeted profit is too high ad a lower risk level i uattaiable. The GENCO could start over with a ew targeted profit. If the miimum expected dowside risk is acceptable, go to step (3). 3. Choose a acceptable risk level based o the miimum expected dowside risk, solve the risk-costraied problem, provide the commitmet schedule of uits, ad devise biddig curves. Based o the above procedure, (9) may be deemed uecessary. I the course of implemetig the above procedure, step (1) would yield a risk-eutral solutio while step (2) would yield a solutio with miimum risk level. The solutio obtaied from step (3) is a tradeoff betwee maximizig expected payoff ad miimizig risk. 6

7 Accordigly, the GENCO is i a improved positio for submittig optimal biddig strategies. The formulated problem is a mixed iteger liear program which could be solved by a commercial MIP solver. VI. NUMERICAL EXAMPLES I this sectio, a GENCO with 20 thermal uits is cosidered for illustratig the proposed method. The detailed uit data ad market prices for eergy ad capacity price are give i I this study, we assume a uiform market clearig price for all buses. The case studies i this sectio utilize CPLEX 9.0 o a Petium-4 1.8GHz persoal computer. We assume that market price has a ormal distributio. However, other distributio properties could also be cosidered. I all the case studies, we oly illustrate the biddig curve i the day-ahead eergy market. Biddig curves for acillary services could be developed similarly. Although, we itroduce biddig curves for the etire GENCO, we could devise biddig curves for each uit similarly. Case 1: Impact of market price ucertaity o PBUC I this case, we first assume that the actual market price would be the same as forecasted price without risk costrait. That is a determiistic PBUC as i [11]. Table III shows the uit schedule with a objective fuctio of $69, The idetical uits , 1006, , , 1015, are shut dow whe hourly market prices are lower tha respective margial costs. The other uits are committed whe hourly market prices are high. Uit 1007 is a relatively cheap uit which is shut dow at hours 2 to 6 whe hourly market prices are lower tha the margial cost of the uit. Table III shows that the hourly uit commitmet is very sesitive to market prices as was explored i [11]. We cosider the market price ucertaity as follows. The umber of reduced scearios is chose to be 30 sice the objective fuctio does ot chage dramatically at this umber. The proposed stochastic programmig solutio results i a objective fuctio of $62, with a executio time of 848 secods. The differece i profit with determiistic ad stochastic market prices is $ (i.e., $69, $62,278.50), which is called the value of perfect iformatio [24]. The portio of uit schedule that is differet from that i Table III is show i Table IV. Table IV shows that the idetical uits , 1006, , , 1015, are shut dow because they are relatively expesive whe hourly market prices are relatively low. The compariso of Tables III ad IV shows that whe applyig stochastic market prices, uit 1004 is shut dow for the etire schedulig period ad other uits are committed at some hours besides those i Table III. As to the commitmet of uit 1004, the market price at hours 11 to 22 is lower tha its margial cost i some scearios whe the objective fuctio is to maximize the expected payoff istead of the payoff for a specific sceario. Other uits i IV are committed whe hourly market prices are higher tha margial costs i certai scearios ad the commitmet would icrease the expected payoff. This case study shows that the market price ucertaity could have a sigificat impact o the hourly uit schedule. The market price ucertaity could lower the GENCO s profit represeted by the value of perfect iformatio ($7,554.50). TABLE III UNIT SCHEDULE UNDER DETERMINISTIC MARKET PRICE Uit Hours (0-24) TABLE IV PORTION OF UNIT SCHEDULE UNDER STOCHASTIC MARKET PRICE Uit Hours (0-24) Case 2: Establishig biddig curves Whe risk costrait is igored, biddig curves at hour 8 are show i Figs. 6 ad 7 without ad with postprocessig techiques (method 4 i sectio IV), respectively. Fig. 6 Biddig curve for hour 8 without postprocessig 7

8 Fig. 6 shows that the biddig curve is mootoically icreasig ad there is a large discotiuity betwee 700MW ad 1300MW. The discotiuity i bidig curve could icur fiacial losses to a GENCO as discussed before. I Fig. 7, biddig curve is refied by applyig method 4 i sectio IV. This curve is cotiuous betwee 700MW ad 1300MW which is more desirable as it will ot result i sigificat chages i profit betwee ay two segmets. Fig. 7 Biddig curve with postprocessig at hour 8 Case 3: Impact of risk costraits I this case, we study the impact of risk costrait o biddig curve. Table V shows the profit ad probability for each sceario without risk costrait. Table V shows that the profit for ay give sceario, which is a fuctio of hourly market price, could be quite differet from those of others. For scearios 1 ad 8, the payoff is egative because the market price is quite low i these two scearios. The GENCO would set its targeted profit at $50,000 ad those below the target are show i bold i Table V. The correspodig probability for such scearios is (i.e., ) with a expected dowside risk of $9, This risk level could be uacceptable to a GENCO. The GENCO could the apply the proposed method to cotrol the associated risk level. Usig a very large umber (10,000), the expected dowside risk is adjoied to the objective fuctio for fidig the miimum expected dowside risk. The miimum dowside risk is $4, for the targeted profit of $50,000 with a expected payoff of $52, That is, for the give targeted profit, the GENCO could ot expect a lower dowside risk. If a GENCO is still usatisfied with this risk level, it meas that the GENCO s targeted profit is too high. Accordigly, we assume the acceptable expected dowside risk level for the GENCO is $6,000. By cosiderig risk-costraied model, the objective would be $59, with a dowside risk of $6,000. Accordigly, a GENCO could reduce its dowside risk level by 37.59% ((9, ,000) / 9,611.74) at the cost of reducig its expected payoff by 4.31% ((62, ,591.40) / 62,278.50). Table VI shows the payoff for each sceario with risk costrait i which scearios with icreased profit are show i bold. The portio of uit schedule that is differet from that i Table IV is show i Table VII. The compariso of Tables V ad VI shows that the profit for scearios 1 ad 8 is o loger egative whe uits 1005, 1010, ad 1011 are shut dow at additioal hours. The additioal payoff for scearios i Table VI is at the cost of reducig payoffs for other scearios. The compariso of Tables IV ad VII shows that uits 1005, 1010, ad 1011 are shut dow at additioal hours whe the commitmet of these uits would cotribute to the maximizatio of expected payoff but would also icrease the dowside risk. The GENCO could shut dow the uits for reducig its expected dowside risk at the cost of reducig the expected payoff. It should be oted that the iclusio of risk costrait would impact hourly biddig curve. For the purpose of compariso, we oly show i Fig. 8 the biddig curve at hour 8. Compariso of Figs. 7 ad 8 shows that the risk costrait would reduce the offered geeratio based o biddig price. The major reductio is because the relatively large uits 1010 ad 1011 are shut dow at hour 8. However, the dispatch of smaller uits at hour 8 also cotributes to the reductio of total geeratio offered by the GENCO. TABLE V SCENARIO PROFIT WITHOUT RISK CONSTRAINTS # Profit ($) Probability # Profit ($) Probability TABLE VI SCENARIO PROFIT WITH RISK CONSTRAINTS # Profit ($) Probability # Profit ($) Probability TABLE VII UNIT SCHEDULE WITH RISK CONSTRAINTS Uit Hours (0-24)

9 9 Fig. 8 Biddig curve for hour 8 with risk costrait VII. DISCUSSION It should be oted that there may be couplig amog cosecutive hourly prices. For example, if the market price at oe hour is high, it is probable that the ext hour price would also be high. I this paper, we ru the Mote Carlo simulatio for idividual hours ad pla to cosider the couplig amog hourly market prices i our future studies. However, differet market price simulatio approaches would oly impact the iput to our proposed formulatio. The proposed model could be applied to a large system as our formulatio depeds o commercial MIP solvers for solvig the stochastic mixed iteger program. With the further developmets i both hardware ad software algorithms, leadig commercial MIP solvers such as CPLEX, OSL, XPRESS, ad LINDO have bee improved sigificatly for solvig very large cases [11]. Stochastic Lagragia relaxatio (LR) [26], [27] could also be applied for solvig the proposed formulatio. However, the MIP formulatio has the followig advatages over the LR approach [11]: (1) global optimality; (2) direct measure of the optimality of a solutio; (3) more flexible ad accurate modelig capabilities. We chose to apply the commercial MIP solver to solve the GENCO s problem sice the size of the problem is geerally withi the solvig capability of commercial solvers. The iput data to our formulatio iclude market prices for eergy ad acillary services ad uit techical data such as cost curve, miimum o/off times, rampig up/dow limits, etc. Market prices ad variaces could be forecasted by available forecastig techiques such as artificial eural etwork, time series model [1]. The maiteace of the iput data is easy ad flexible. The proposed formulatio is very practical which could be applied by GENCOs for submittig offers to eergy ad acillary services markets. Meawhile, the possible iclusio of arbitrage strategies [1], [27], [28] i the proposed model would provide a very practical GENCO tool for maximizig portfolios i eergy, bilateral cotracts, acillary services, fuel, ad emissio allowace markets. VIII. CONCLUSIONS A risk-costraied biddig strategy for day-ahead eergy ad acillary services markets is proposed for GENCOs. The market price ucertaity is modeled usig scearios ad sceario reductio techiques are applied to reduce the umber of scearios i cosideratio. The risk associated with the market price ucertaity is modeled usig the dowside target profit shortfall ad is icorporated explicitly as a costrait i the model. Accordigly, a closed-loop biddig strategy is costructed. After solvig the stochastic PBUC problem, postprocessig is applied based o margial costs for refiig biddig curves. Test results illustrate that it is ecessary to cosider market price ucertaity ad icorporate the impact of stochastic market price o the commitmet schedule of uits. It is also show that risk costraits would play a importat role i devisig bidig curves. A GENCO could sigificatly reduce its risk level at the cost of reducig expected payoffs. REFERENCES [1] M. Shahidehpour, H. Yami, ad Z. L Market Operatios i Electric Power Systems. New York: Wiley, [2] T. Li ad M. Shahidehpour, Strategic biddig of trasmissiocostraied GENCOs with icomplete iformatio, IEEE Tras. Power Syst., vol. 20, pp , Feb [3] A. David ad F. We, Strategic biddig i competitive electricity market: a literature survey, i Proc IEEE Power Egieerig Society Power Egieerig Cof., pp [4] T. Li ad M. Shahidehpour, Risk-costraied FTR biddig strategy i trasmissio markets, IEEE Tras. Power Syst., vol. 20, pp , May [5] B.F. Hobbs, Liear complemetarity models of Nash-Courot competitio i bilateral ad PoolCo power markets, IEEE Tras. 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Power Syst., vol. 20, pp , Nov [12] S. Takrit B. Krasebrik, ad L. Wu, Icorporatig fuel costraits ad electricity spot prices ito the stochastic uit commitmet problem, Operatios Research, vol. 48, pp , Mar.-Apr [13] M.A. Plazas, A.J. Coejo, ad F. Prieto, Multimarket optimal biddig for a power producer, IEEE Tras. Power Syst., vol. 20, pp , Nov [14] A. Baillo, M. Vetosa, M. Rivier, ad A. Ramos, Optimal offerig strategies for geeratio compaies operatig i electricity spot markets, IEEE Tras. Power Syst., vol. 19, pp , May [15] M.V. Pereira, S. Graville, M.H.C. Fampa, R. Dix, ad L.A. Barroso, Strategy biddig uder ucertaity: a biary expasio approach, IEEE Tras. Power Syst., vol. 20, pp , Feb [16] H.M. Markowitz, Portfolio selectio, Joural of Fiace, vol. 8, pp , 1952.

10 10 [17] S. Ahmed, Mea-risk objectives i stochastic programmig, Available: [18] M. Deto, A. Palmer, R. Masiello, ad P. Skatze, Maagig market risk i eergy, IEEE Tras. Power Syst., vol. 18, pp , May [19] R. Dahlgre, C. Liu, ad J. Lawarrėe, Risk assessmet i eergy tradig, IEEE Tras. Power Syst., vol. 18, pp , May [20] K. Hoylad ad S.W. Wallace, Geeratig sceario trees for multistage decisio problems, Maagemet Sciece, vol. 47, o. 2, pp , Feb [21] J. Dupačová, G. Cosigl ad S.W. Wallace, Scearios for multistage stochastic programs, Aals of Operatios Research, vol. 100, pp , Dec [22] J. Dupačová, N. Gröwe-Kuska, ad W. Römisch, Sceario reductio i stochastic programmig: a approach usig probability metrics, Mathematical Programmig Series A, vol. 3, pp , [23] N. Gröwe-Kuska, H. Heitsch ad W. Römisch, Sceario Reductio ad Sceario Tree Costructio for Power Maagemet Problems, Power Tech Coferece Proceedigs, 2003 IEEE Bologa, vol. 3, pp , Ju [24] J.R. Birge, F. Louveaux, Itroductio to Stochastic Programmig. Spriger, New York, [25] G.D. Eppe, R.K. Marti, ad L. Schrage, A sceario approach to capacity plaig, Operatios Research, vol. 37, pp , Jul.- Aug [26] S. Takrit J.R. Birge, ad E. Log, A stochastic model for the uit commitmet problem, IEEE Tras. Power Syst., vol. 11, pp , Aug [27] P. Carpetier, G. Gohe, J.C. Culiol ad A. Reaud, Stochastic optimizatio of uit commitmet: a ew decompositio framework, IEEE Tras. Power Syst., vol. 11, pp , May [28] M. Shahidehpour, T. L ad J. Cho Optimal geeratio asset arbitrage i electricity markets, KIEE Iteratioal Trasactios o Power Egieerig, vol. 5-A, o. 4, pp , Dec [29] T. Li ad M. Shahidehpour, Risk-costraied geeratig asset arbitrage, mauscript to be submitted to IEEE Tras. Power Syst., Tao Li received his B.S. ad M.S. degrees from Shaghai Jiaotog Uiversity, Chia, i 1999 ad 2002, respectively, ad his PhD degree from Illiois Istitute of Techology i 2006, all i electrical egieerig. He is curretly a Seior Research Associate i the Electric Power ad Power Electroics Ceter at IIT. His research iterests iclude power system ecoomics ad optimizatio. Mohammad Shahidehpour (F 01) is the Bodie Professor ad Chairma i the Electrical ad Computer Egieerig Departmet at Illiois Istitute of Techology. Dr. Shahidehpour is a IEEE Distiguished Lecturer ad has lectured across the globe o electricity restructurig issues ad has bee a visitig professor at several uiversities. He is a Fellow of IEEE. Zuyi Li (M 01) received the B.S. degree from Shaghai Jiaotog Uiversity, Shagha Chia, i 1995, the M.S. degree from Tsighua Uiversity, Beijig, Chia, i 1998, ad the Ph.D. degree from Illiois Istitute of Techology, Chicago, i 2002, all i electrical egieerig. Presetly, he is a Assistat Professor i the Electrical ad Computer Egieerig Departmet, Illiois Istitute of Techology. His research iterests are focused o market operatio of power systems. He is the coauthor of the book titled Market Operatios i Electric Power Systems (New York: Wiley, 2002).