Efficiency of Linear Supply Function Bidding in Electricity Markets

Transcription

1 Efficiecy of Liear Supply Fuctio Biddig i Electricity Markets Yuazhag Xiao, Chaithaya Badi, ad Ermi Wei 1 Abstract We study the efficiecy loss caused by strategic biddig behavior from power geerators i electricity markets. I the cosidered market, the demad of electricity is ielastic, the geerators submit their supply fuctios i.e., the amout of electricity willig to supply give a uit price to bid for the supply of electricity, ad a uiform price is set to clear the market. We aim to uderstad how the total geeratio cost icreases uder strategic biddig, compared to the miimum total cost. Existig literature has aswers to this questio without regard to the etwork structure of the market. However, i electricity markets, the uderlyig physical etwork i.e., the electricity trasmissio etwork determies how electricity flows through the etwork ad thus iflueces the equilibrium outcome of the market. Takig ito accout the uderlyig etwork, we prove that there exists a uique equilibrium supply profile, ad derive a upper boud o the efficiecy loss of the equilibrium supply profile compared to the socially optimal oe that miimizes the total cost. Our upper boud provides isights o how the etwork topology affects the efficiecy loss. I. INTRODUCTION I power systems, electricity markets aim to balace the demad ad the supply at all locatios ad at all times. Imbalace i the demad ad the supply will derail the power system from its ormal operatig frequecy ad may cause serious cosequeces such as blackout [1]. Therefore, electricity markets are crucial for the successful ad stable operatios of power systems. I electricity markets, a cetral coordiator, amely the idepedet system operator ISO, is itroduced to esure the balace betwee supply ad demad. The ISO forecasts a ielastic demad profile ad dispatch the power geerators to satisfy the demad at the miimal total geeratio cost. Ideally, the ISO should kow the geeratio cost fuctios of all the geerators, ad allocate the demad amog the geerators such that the total geeratio cost is miimized. I practice, the ISO elicits iformatio about the cost fuctios from the geerators i the form of supply fuctios. Specifically, each geerator submits its supply fuctio i.e., a curve specifyig the amout of electricity it is willig to supply give the uit price of electricity as its bid. Recet icidets, most otable of which is the Califoria electricity crisis [?], suggest the existece of strategic biddig behavior by the geerators. A geerator may submit a supply fuctio that does ot truly reflect its geeratio cost, i order to maximize its ow profit. Subsequetly, existig literature has started to look ito the efficiecy loss due to strategic behavior of geerators. Existig works [2][3][4] quatify the efficiecy loss by price of aarchy Y. Xiao ad E. Wei are with Departmet of EECS, Northwester Uiversity. s: xyz.xiao@gmail.com, ermi.wei@orthwester.edu C. Badi is with Kellogg School of Maagemet, Northwester Uiversity. c-badi@kellogg.orthwester.edu

2 2 PoA, defied as the ratio of the total geeratio cost at the equilibrium to that at the social optimum. By defiitio, PoA is a umber o smaller tha 1, ad a larger PoA idicates greater efficiecy loss. There has bee active research i studyig the efficiecy of supply fuctio biddig i geeral markets [2][5] ad i electricity markets [3][4][6][7][8]. The usual coclusio is that the efficiecy loss is upper bouded, ad vaishes as the umber of suppliers icreases. These works [2] [8], however, focus o markets with o uderlyig etwork structure. I electricity markets, there is a uderlyig physical etwork that limits how the supply ca match the demad. For istace, the trasmissio etwork topology ad the flow limits of trasmissio lies put costraits o the amouts of electricity ijected by the geerators ito the system. I this work, we aim to study how the etwork affects the efficiecy loss i etworked markets. 1 Our first mai result suggests that the efficiecy loss is still upper bouded whe the uderlyig trasmissio etwork is cosidered. Our secod major result provides a upper boud of the PoA, ad sheds light ito how the etwork topology affects the upper boud. Our results provide isights i cofigurig the trasmissio etwork ad placig the geerators. The rest of this paper is orgaized as follows. I Sectio II, we will describe our model of electricity markets ad defie the supply fuctio equilibrium. We aalyze the equilibrium i geeral electricity etworks i Sectio III, ad that i special radial etworks i Sectio IV. Fially, Sectio V cocludes the paper. II. MODEL A. Basic Setup Cosider a power system represeted as a graph N, E. Each ode i N has a geerator or a load or both, ad each edge i E is a trasmissio lie coectig two odes. We deote the set of odes with a geerator by N g ad the set of odes with a load by N l. We assume that the load is ielastic, 2 ad deote the ielastic load profile by d = d j j Nl R N l +. 3 The total demad is the give by D j N l d j. Each geerator has a cost c s i providig s 0 uit of electricity. We make the followig stadard assumptio about cost fuctios. Assumptio 1: For each geerator, the cost fuctio c s is strictly covex, icreasig, ad cotiuously differetiable i s [0, +. Due to physical costraits, each geerator s supply s must be i a rage [s, s ]. As i [4], we assume that o geerator is very big i its capacity i the followig sese. Assumptio 2: No geerator has a capacity that is larger tha or equal to half of the total demad, amely s < D 2, N g. The supply profile s = s Ng R Ng + must also satisfy physical costraits of the electrical etwork. First, i a power system, it is crucial to balace the supply ad the demad at all time for the stability of the system [1]. 1 Some related works study the efficiecy loss i electricity markets while cosiderig the effect of the trasmissio etwork [9][10]. However, they adopt a Courot competitio model, where the suppliers determie the amouts of supply, istead of the more practical form of biddig a supply fuctio. 2 We ca easily exted our results to the case with elastic loads. A elastic load ca be decomposed as a large ielastic load ad a geerator whose supply reduces the et load. 3 We use the otatio R + to deote the set of positive real umbers.

3 3 Hece, we eed to have N g s = D. 1 Secod, the flow o each trasmissio lie, which depeds o the supply profile, caot exceed the lie capacity. Sice the direct curret DC power flow model is commoly used i electricity markets e.g., by the ISO i ecoomic dispatch ad by geerators i supply biddig, we use the DC power flow model, where the flow o each lie is the liear combiatio of ijectios from each ode. The the lie flow costraits ca be writte as follows: f A g s + A l d f, 2 where f R E + is the vector of lie capacities, A g R E Ng ad A l R E N l are shift-factor matrices. The shift-factor matrices A l ad A g deped o the uderlyig trasmissio etwork topology ad the admittace of trasmissio lies see [1] ad [10] for more details. Note that the boud i 2 is two-sided, because the flows o each power lie ca go i either directio. I a regulated ad cetralized market, the ISO kows the cost fuctios ad ca determies the amout of electricity supplied by each geerator. It determies the optimal supply profile s that miimizes the total geeratio cost subject to the costraits metioed above. We summarize the optimizatio problem to solve as follows: max s N g c s 3 s.t. N g s = D, s s s, N g, f A g s + A l d f. To avoid triviality, we assume that the feasible set of power geeratio is o-empty ad is ot a sigleto. Assumptio 3: There exists a strictly feasible allocatio of power geeratio s. B. Deregulated Markets ad Supply Fuctio Biddig I deregulated electricity markets, each geerator tells the ISO the amout of electricity it ca provide give a uit price of electricity. The mappig from the uit price to the supply is called supply fuctio. We adopt the liear supply fuctio from [4], ad defie geerator s supply fuctio as: S p, w = w p, where w R + is geerator s strategic actio, ad p R + is the uit price of electricity. To clear the market, i.e., to fid the price p satisfies the coditio N g S p, w = D, the ISO sets the price p as follows: where w = w Ng pw = R Ng + is the biddig profile. Geerator s payoff is its profit defied as follows: D N g w. u w, w p w, w S [p w, w, w ] c S [pw, w, w ],

4 4 where w is the actio profile of all the geerators other tha geerator. Now we ca formally defie the supply fuctio equilibrium SFE. Defiitio 1: A actio profile w is a supply fuctio equilibrium, if each geerator s actio w satisfies 4 w arg max w s.t. u w, w [ s S p w, w ], w s, [ f [A g ] S pw, w, ] w + m +A l d f. { [Ag ] m S m [ pw, w, wm ]} I a SFE, each geerator s bid maximizes its ow payoff give the others bids. What is special about our SFE is that the set of feasible bids of each geerator depeds o the others bids. Therefore, the SFE is a geeralized Nash equilibrium, which is usually harder to aalyze tha a stadard Nash equilibrium [11]. III. EFFICIENCY OF LINEAR SUPPLY FUNCTION BIDDING I this sectio, we will prove that the SFE exists ad results i a uique equilibrium supply profile. I additio, we will provide a upper boud of the efficiecy loss at SFE, ad discuss how this upper boud depeds o the trasmissio etwork topology. Our first mai result is the characterizatio of SFE through a covex programmig. Specifically, we show that ay SFE results i a uique equilibrium supply profile, which is a optimal solutio to a covex optimizatio problem. Propositio 1: Ay SFE results i a allocatio that is the uique solutio to the followig modified cost miimizatio problem: where ĉ s = Proof: See Appedix A. 1 + mi s s.t. s D 2s ĉ s 4 N g s = D, 5 N g s s s, 6 f 1 A g s + A l d f 2, 7 s D c s 0 D 2x 2 c xdx. 8 Propositio 1 is sigificat i describig the equilibrium outcome. It shows that although there may be multiple equilibrium biddig profiles, the resultig equilibrium supply profile, which is what we care about the most, is always uique. Moreover, Propositio 1 proves that the equilibrium supply profile is the optimal solutio to a covex optimizatio problem, which provides a efficiet way of computig the equilibrium supply profile. 4 We deote the th colum of a matrix A by [A].

5 5 Lower PoA Boud 1 Higher PoA Boud a Network 1 b Network 2 Fig. 1. Impact of etwork topology o the efficiecy loss. Assume that all the lie capacities are the same. I both etworks, the PoA upper boud is determied by ode 3, which has the highest coectivity. By coectig ode 5 to ode 3, the etwork o the right has a higher PoA boud because ode 3 has a higher degree tha ode 3 o the left. The covex optimizatio problem i Propositio 1 is crucial i aalyzig the PoA. Before we go ito detailed aalysis, we formally defie PoA here. Defiitio 2: PoA is defied as N g c s N g c s, where s Ng is the equilibrium supply profile, ad s Ng is the supply profile that miimizes the total cost. The PoA is well defied because the equilibrium supply profile is uique. By defiitio, the PoA is ever smaller tha 1. A larger PoA idicates that the efficiecy loss at the equilibrium is larger. Next, we give a aalytical upper boud of the PoA. Theorem 1: The PoA is upper bouded by 1 + D 2, where max mi s m, d m + Proof: See Appedix B. m, E f m. 9 Theorem 1 gives us a upper boud of the PoA. From the upper boud, we ca get isight o the key factors that ifluece the efficiecy loss. First, the efficiecy loss ca be higher if there is a geerator with large geeratio capacity. This ituitio has bee obtaied i some prior works [3] that do ot cosider flow limit costraits, ad our cotributio here is to show that the same ituitio also applies to the case with flow limit costraits. Secod, aother factor that may affect the efficiecy loss is how coected a geerator is. More specifically, the term d m + m, E f m ca be cosidered as geerator m s weighted degree i the etwork with the weights beig the capacity of trasmissio lies plus its demad. If oe geerator is much better coected tha the others, the efficiecy loss ca be high. This is a ew ituitio that was kow from prior works. Our isights shed some light o plaig the trasmissio etworks ad the locatios of geerators. We illustrate our isights i Fig. 1, where we show two etworks with the same odes but differet etwork topology.node 3 o the left has a degree of 3, while the oe o the right has a higher degree of 4, because ode 5 is added to its set of eighbors. Therefore, the etwork o the right has a higher PoA boud. This example suggests that it may be beeficial to evely distribute the coectios betwee geerators.

6 6 a Network 1 b Network 2 Fig. 2. Two radial etworks that have differet local topologies but the same socially optimal ad equilibrium supply profiles. The odes with the same color are homogeeous, while the odes with differet colors ca be differet. IV. SPECIAL RADIAL NETWORKS The results i the previous sectio hold for geeral electricity etworks. I this sectio, we cosider a special class of radial etworks, ad gai deeper isights for this class of etworks. A radial etwork is a etwork with o loop. I a radial etwork, we call each subtree startig from a child of the root a brach. We cosider a special class of radial etworks that have homogeeous odes ad edges withi each brach i.e., the same cost fuctio, the same demad, the same upper ad lower bouds of supply, ad the same lie flow limits. Differet braches ca have differet parameters ad cost fuctios. We proved that at both the socially optimal ad equilibrium supply profiles, the supply levels of odes withi a brach are the same, ad are idepedet of the etwork topology of each idividual brach. We formally state our results for the special radial etworks as follows. Propositio 2: I a radial etwork with idetical odes ad lies withi each brach, the followig statemets hold for both socially optimal ad equilibrium supply profiles: The odes withi a brach have the same supply. The oly possibly cogested lie i.e., a lie with a bidig lie flow costrait is the oe coected to the root. Proof: See Appedix C. Oe implicatio of Propositio 2 is that the local etwork topology i a brach does ot matter. For illustratio, we show i Fig. 2 two etworks with differet local etwork topologies. Based o Propositio 2, these two etworks have the same socially optimal ad equilibrium supply profiles. V. CONCLUSION I this paper, we studied the efficiecy loss of liear supply fuctio biddig i electricity markets. The key feature that sets our work apart from existig works is that we iclude the flow costraits of the trasmissio lies i our model. We show that i the resultig etworked electricity markets, there exists a uique equilibrium supply profile. We provided a aalytical upper boud of the efficiecy loss at the equilibrium. Our upper boud suggests that to reduce the efficiecy loss, we should evely distribute the geeratio capacity ad coectivity amog the geerators. Our upper boud also provides a precise defiitio the coectivity of a geerator as its weighted degree i the etwork weighted by the trasmissio lie capacity plus the demad at its locatio.

7 7 APPENDIX A PROOF OF PROPOSITION 1 Step 1: At ay SFE, there exist at least two geerators with strictly positive bids. Suppose that all the geerators bid zero, amely w = 0 for all. The each geerator supply D, resultig i a total supply of N g D. This supply profile violates the costrait that the supply equals the demad. Hece, at least oe geerator submits strictly positive bids. Suppose that geerator is the oly geerator with a strictly positive bid. The geerator eeds to supply D w w / [ N g 1 D] = N g 2 D < 0, which violates the costrait that s 0. Hece, at least two geerators submit strictly positive bids. Step 2: The payoff fuctio u w, w of each geerator is strictly covex i w at ay equilibrium. We write dow the expressio of the payoff fuctio as where u w, w = pw, w S [pw, w, w ] c S [pw, w, w ] m N = g w m w c S [pw, w, w ], 10 N g 1 S pw, w, w = D w w m N g 1 D, 11 Sice c s is strictly covex ad icreasig i s, ad S pw, w, w is strictly covex i w whe m w m > 0, the payoff fuctio u w, w is strictly covex i w whe m w m > 0. I Step 1, we have proved that m w m > 0 at a equilibrium. Hece, the payoff fuctio u w, w is strictly covex i w at ay equilibrium. Step 3: Sice the payoff fuctio u w, w is strictly covex i w, each geerator s optimizatio problem is a covex optimizatio problem. Therefore, w is a SFE if ad oly if it satisfies the KKT coditios for all. We write the Lagragia multipliers associated with the local costraits?? ad?? as µ SFE R + ad R +, respectively. I additio, we write the Lagragia multipliers associated with the commo costraits µ SFE?? ad?? as λ SFE 1 R E + ad λ SFE 2 R E +, respectively. The the actio profile w is a SFE if ad oly if there exists µ SFE, µsfe, λ SFE 1, ad λ SFE 2 for all such that the followig KKT coditios are satisfied for all : u w, w w + [ s + N g 2 D] µ SFE N g 2 µ SFE 0 µ SFE + λ SFE 1 T b + f 1 λ SFE T b f 2 = 0 12 [ s + N g 2 D] w + s D 0 µ SFE N g 2 w + 2,m,m w m 0 13 w m λ SFE 1 b + f 1 w 0 15 N g 0 λ SFE 2 b f 2 w 0 16 N g

8 8 By pluggig i the expressio of the payoff fuctio u w, w, we ca rewrite 12 as 1 + S [pw, w, w ] N g 2 D 0 = dc S [pw, w, w ] ds + + s m D m N s pw µsfe g,m m D m N s g m D s m D,m s m D λ SFE T pw 1 b + c 1 pw 17 pw µ SFE s m D,m s m D λ SFE T pw 2 b c 2 Step 4: Sice the optimizatio problem MSW is a covex optimizatio problem, s is a solutio if ad oly if it satisfies the KKT coditios. We write the Lagragia multipliers associated with the costraits 0 s ad s s i 6 as µ MWS = µ MWS ad µ MWS = µ MWS, respectively. Similarly, we write the Lagragia multiplier associated with the costrait N g s = D i 5 as p MSW. Fially, we write the Lagragia multipliers associated with the costraits c 1 A g s + A l d ad A g s + A l d c 2 i 7 as λ MWS 1 ad λ MWS 2, respectively. The KKT coditios are dc s s 1 + ds m N s g,m m D p MSW µ MSW + µ MSW λ MSW 1 T [Ag ] + λ MSW T [Ag ] = N g s = D 19 0 µ MSW s µ MSW s s 21 0 λ MSW 1 A g s + A l d + c λ MSW 2 A g s + A l d c Step 5: Give ay SFE w, the allocatio s, where s = s w pw,, is a solutio to MSW. For ay SFE w, there exists µ SFE, µsfe, λ SFE 1, ad λ SFE 2 such that the KKT coditios are satisfied. Defie s = s w pw, ad the Lagragia multipliers of the problem MSW as follows p MSW = pw 24 { m N 1 g s m D [ λ m N s g,m m D SFE T Ag s m + A l d + c 1 + λ SFE ] } T Ag s m + A l d c 2 µ MSW = µ MSW = 1 s m Ng m D pw µ SFE 25 s m D m N s pw µsfe 26 g,m m D m Ng s m D 2 2 λ MSW 1 =,m s m D pw λsfe 1 27 m Ng s m D 2 λ MSW 2 =,m s m D pw λsfe 2 28

9 9 It is ot difficult to check that the KKT coditios of MSW are satisfied by s, p MSW, µ MSW Step 6: Give a solutio s to MSW, we ca costruct a SFE w., µmsw This ca be proved similarly as i Step 5, usig the mappig betwee the Lagragia multipliers., λ MSW 1, ad λ MSW 2. First, we show that for ay s [s, s ], we have We rewrite ĉ s here: It is clear that ĉ s 1 + s c s ĉ s D 2s To see c s ĉ s, we look at ĉ s c s = ĉ s = APPENDIX B PROOF OF THEOREM s s D 2s s c s. c s D 2s D D 2x 2 c xdx. c s because the itegral i the expressio of ĉ s is oegative. s s D c s D 2s 0 D 2x 2 c xdx. Note that ĉ 0 c 0 = 0, ad that the derivative of ĉ s c s is s D 2s dc s ds 0, because c is icreasig ad 0 s s < D 2. Therefore, we have ĉ s c s 0 for all s [s, s ]. Give the above iequality, we have c s ĉ s N g N g N g ĉ s N g s 1 + D 2s c s N g max m Ng s m D 2 max m Ng s m max m Ng s m D 2 max m Ng s m where the first ad third iequalities come from c s ĉ s the secod iequality follows from the fact that s Ng problem. We ca see that a upper boud of the PoA is max m Ng s m 1 + D 2 max m Ng s, m 1 + s c s N g c s, D 2s c s for ay feasible s, ad is the optimal solutio to the modified cost miimizatio

10 10 which is icreasig i max m Ng s m. Hece, our ext step is to fid a upper boud for max m Ng s m. Sice max m Ng s m is the maximum of all geerators socially optimal supplies, it must be bouded by the maximum of all geerators feasible supplies. I other words, it is bouded by the optimal value of the followig optimizatio problem: max s s.t. max s m 29 N g s = D, s s s, N g, f A g s + A l d f. The optimal value of 29 is upper bouded by the optimal value of aother optimizatio problem with some costraits removed. Specifically, it is upper bouded by the optimal value of the followig optimizatio problem: max s s.t. max s m 30 N g s = D, s s s, N g. We ca see that the optimal value of 30 is max s m. This ca be achieved by lettig the geerator with the largest capacity provides its maximum supply, ad the other geerators provide ay supply profile that satisfies the supply ad demad balace costrait. The optimal value of 29 is also upper bouded by the optimal value of the followig optimizatio problem: max s max s m 31 s.t. f A g s + A l d f. Sice it is hard to solve 31 directly, we work o a equivalet formulatio where the variables are the supply profile ad the flows o each lie p = p ij {i,j} E. max s max s m 32 s.t. f ij p ij f ij, p ij = 0, for ay loop l, B ij i j l s i d i = j:i j p ij k:k i p ki, i N. Note that the costraits are flow limit costraits the first costrait, voltage agle costraits the secod oe, ad eergy coservatio at each ode the third oe. We further relax the problem by removig the voltage agle

11 11 costrait, ad get: max s max s m 33 s.t. f ij p ij f ij, s i d i = p ij p ki, i N. j:i j Now we ca aalytically calculate the optimal value of 33 as max d m + m, E k:k i f m. I summary, we have proved that the optimal value of 29 is upper bouded by mi max s m, max d m + f m m, E = max mi s m, d m + f m, which yields the upper boud of the PoA 1 + D 2 m, E i Theorem 1. APPENDIX C PROOF OF PROPOSITION 2 For a special radial etwork, the social welfare maximizatio problem ca be simplified as mi s s.t. K k=0 K k=0 N k c k s 34 N k s = D, 35 s k s s k, k, N k, 36 f k s d k f k, k, l E k. 37 :l E Before provig the propositio, we first prove a useful mootoicity property of the optimal supply profile. Lemma 1: At the optimal supply profile, o ay path from the root to a leaf ode, the supplies of the odes o this path must satisfy either oe of the two coditios: all the supplies are o smaller tha the demad d k, ad are o-icreasig i the ode s distace from the root, amely s i s j d k for ay i j; or all the supplies are o larger tha the demad d k, ad are o-decreasig i the ode s distace from the root, amely s i s j d k for ay i j. Proof: We choose a arbitrary path i brach k. If all the odes i this path have the optimal supply equal to the demad i.e., s = d k, N k, the mootoicity property is immediately satisfied. I the rest of the proof, we cosider the cases where at least oe ode has a optimal supply that is differet from the demad d k.

12 12 Amog all the odes with a optimal supply differet from the demad d k, we pick oe that has the logest distace from the root if there are multiple odes with the same logest distaces, we pick a arbitrary oe. We suppose that this ode is ode, amely arg max N m. 38 m N k :s m d k There are two possibilities, s > d k ad s < d k. We first cosider the case of s > d k. First, all the odes i have s i = d k. Hece, we have s i s j d k for ay i, j that are descedats of. Secod, we claim that ay ode m must have s m s. Assume that the cotrary is true: amely, there exists odes that are acestors of ad have supply smaller tha s. Suppose that amog these odes, the oe furthest away from the root is ode m. We defie a ew supply profile s, where s = s ε, s m = s m + ε, ad s i = s i, i m,. Here ε > 0 is a positive umber small eough such that s m < s. Clearly, this ew supply profile satisfies the geerator capacity costraits. We ow check the trasmissio lie flow costraits. For ay lie l above ode m, we have i:l N i s i d i = i:l N i s i d i, because l N m ad l N. For ay lie l below ode, we have i:l N i s i d i = i:l N i s i d i, because we did ot chage the supply of those odes. For ay lie l betwee ode m ad ode, we have s i d i = s i d i ε < s i d i, 39 i:l N i i:l N i i:l N i because l N ad l / N m. Sice ode m is the furthest ode with supply lower tha d k, we must have i:l N i s i d i > 0. Therefore, we ca fid a ε small eough such that 0 i:l N i s i d i < i:l N i s i d i f k. I summary, we ca fid a ε small eough such that s is feasible. Sice all the odes i brach k have the same covex cost fuctio, the supply profile s results i a lower total cost tha s. This is cotradictory to the fact that s is the optimal supply. This proves our claim that ay ode m must have s m s. We ca repeat the above procedure for the eighborig acestor of ode, for the eighborig acestor of that ode, ad so o. As a result, we have s i s j d k for ay i j. Usig the same argumet, we ca prove that if ode has a optimal supply s < d k, the we have s i s j d k for ay i j. Now we prove Propositio 2 based o the mootoicity of the optimal supply profile. First, i oe brach, there caot be two paths with supplies lower tha the demad o oe path ad supplies larger tha the demad o the other path. This is because ay two paths i same brach must share some odes close to the root two disjoit paths would be i two differet braches. Ay ode shared by such two paths must have s < d k ad s > d k, which is impossible. Hece, i oe brach, the supplies must be o greater tha the demad or o smaller tha the demad for all the odes. We prove the propositio for the case where all the odes have supplies o smaller tha the demad. The same logic applies to the proof for the case where all the odes have supplies o larger tha the demad. Sice s d k, we have :l E s d 0 > f k for ay lie l, ad hece λ 1,l = 0 for ay lie l. Cotrary to the claim i the propositio, we assume that there is a lie l that is ot coected to the root ad has a bidig flow limit costrait. To esure that the flow costraits for the lies above lie l are satisfied, we must have s d k for ay that is above lie l. Hece, we have s = d k for ay that is above lie l. Due to

13 13 the mootoicity property, we must have s = d k for all the odes i this brach. However, i this case, the flow costrait for lie l caot be bidig. This leads to a cotradictio. Hece, the oly possible cogested lie is the oe that coects to the root. To prove that all the odes have the same optimal supply, we look at the KKT coditios. Defie p R as the Lagragia multiplier associated with 35, µ, µ R + as the Lagragia multipliers associated with 36, ad λ 1,l, λ 2,l R + as the Lagragia multipliers associated with 37. The we ca write dow the KKT coditios as follows: dc k s ds p µ + µ λ 1,l + λ 2,l = 0, k, N k, 40 l E l E K k=0 N k s = D 41 0 µ s s k,, 42 0 µ s s k,, 43 0 λ 1,l s d :l E f k, k, N k, 44 0 λ 2,l s d :l E f k, k, N k. 45 We have show that all the lies, except the oe coectig to the root, have o-bidig flow limit costraits. We idex the lie coectig to the root by l. The we have λ 2,l = 0 for all l l. Sice we are i the case where Hece, the first-order coditio ca be simplified to dc k s ds = p + µ µ λ 2,l, N k. 46 We look at the root ode m of brach k. If s m = s k, due to the mootoicity property, we must have s = s k for all m. The the propositio is proved. If s m s k, s k, we have µ m = µ m = 0, ad dck s m ds m = p λ 2,l. For ay m, sice s s m, we have µ = 0, ad dck s ds = p + µ λ 2,l p λ 2,l = dck s m ds m. Sice c k is strictly covex ad s s m, the oly possible solutio is that µ = 0, which results i s = s m. If s m = s k, we have µ m = 0, ad dck s m = p µ m λ 2,l. For ay m, if s < s k, we have µ = 0, ad ds m dc k s ds = p + µ λ 2,l dck s m ds m = p µ m λ 2,l = dck s m ds m. But sice c k is strictly covex ad s < s m, we must have dck s ds < dck s m ds m. This leads to a cotradictio. Hece, we must have s = s k for ay m. I coclusio, all the odes must have the same optimal supply. REFERENCES [1] J. D. Glover, M. S. Sarma, ad T. J. Overbye, Power System Aalysis ad Desig. Cegage Learig, [2] R. Johari ad J. N. Tsitsiklis, Parameterized supply fuctio biddig: Equilibrium ad efficiecy, Oper. Res., vol. 59, o. 5, pp , [3] Y. Xu, N. Li, ad S. H. Low, Demad respose with capacity costraied supply fuctio biddig, IEEE Tras. Power Syst., forthcomig. [4] N. Li ad S. H. Low, Demad respose usig liear supply fuctio biddig, IEEE Tras. Smart Grid, Jul [5] E. J. Aderso, O the existece of supply fuctio equilibria, Math. Program. Ser. B, vol. 140, o. 2, pp , Sep

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