The Regulation of the Transmission Operation and the E ects on the Wholesale Electricity Market

Transcription

1 The Regulation of the Tansmission Opeation and the E ects on the Wholesale Electicity Maket Daniel A. Benitez Claude Campes y Mach 14, 2005 (Wok in Pogess) Abstact The objective of this aticle is to discuss the ole played by the egulation of the tansmission system in ceating o inceasing competition in the powe industy. In paticula, we focus on the e ects of the egulation of the tansmission chage on the stategy played by geneatos. We conside a two-node netwok with limited capacity of inteconnection in which the wholesale maket is oganized though a nodal pice system. In this setting, we analyze a linea tai and a twopat tansmission tai as a mechanism to emuneate the total cost of poviding inteconnection capacity. We show that the egulation of this chage has two immediate e ects. Fist, it a ects the quantity poduced at each node and moe impotantly, it may a ect the minimum capacity equiement that suppot equilibia with o without congestion. We show that the ow between nodes has an ambiguous impact when inceasing the tansmission chage. Finally, we chaacteize the supply of capacity that it is consistent with a tansmission opeato who behaves competitively and we intoduce some conditions about the optimal tansmission capacity. JEL codes: L94, L51 Keywods: Electicity industy, tansmission, competition, egulation daniel.benitez@univ-tsle1.f, Univesité Toulouse I, GREMAQ. y ccampes@cict.f, Univesité Toulouse I, IDEI and GREMAQ. 1

2 , 1 Intoduction The objective of this aticle is to discuss the ole played by the egulation of the tansmission system in ceating o inceasing competition in the powe industy. In paticula, we focus on the e ects of the egulation of the tansmission tai on the stategy played by geneatos. We conside a twonode netwok with limited capacity of inteconnection in which the wholesale maket is oganized though a nodal pice system. In this setting, we study a linea tai and a two-pat tansmission tai as a mechanism to emuneate the total cost of poviding inteconnection capacity. The egulation of the tansmission system has been analyzed by Léautie [2001], Vogelsang [2001] and Boyce and Hollis [2001]. Léautie consides a meshed netwok in which the TO (Tansmission Opeato) has pivate infomation about the cost of expanding capacity. The social value of an expansion in given by the expected cost eduction in geneation and it is lowe than the maginal cost eduction due to maket powe. In Léautie [2000] the geneation maket is pefect competitive and then, the potential indiect e ects of the TO egulation ove a non-competitive wholesale maket ae not addessed. The main nding shows the similitude between a multipoduct monopolist and the TO allocating the tansmission capacity in a meshed netwok. The e ect of the tansmission tai is not addessed. Vogelsang analyzes a simila setting using a non-bayesian appoach. He focuses on two-pat tai whee the vaiable pat e ects congestion chages (and ancillay sevices) while the xed pat e ects capacity costs. It is mentioned that non-linea pices ae supeio in welfae tems espect to Ramsey pices when economies of scale ae lage. The e ect of this egulation in a non-competitive wholesale electicity ae not analyzed. We should emak that the ole of the tansmission company in the aticle of Vogelsang di es espect to the one analyzed by Léautie and the model that we pesent. In paticula, Vogelsang consides a model in which the tansmission opeato faces the cost of adjusting a dispatch to include the tansmission constaints. This ole may e ect the oganization of the 2

3 POOL and nowadays, the NETA in UK whee a nodal pice system is not used. Boyce and Hollis conside that competition in the wholesale maket is impefect but thee ae no netwok e ects. Geneatos and consumes ae located at the same node and competition is analyzed though a educed fom which is paametized exogenously. They main concen is about the govenance of di eent some oganizational fom of the tansmission system and the minimum cost dispatch. Pepemans and Willems [2003] build a model that eplicates the Belgium electicity system. The model is calibated and seveal scenaios ae simulated. Fo instance, ms can behave competitive o they can compete a la Counot. The monopolist case is also addessed. The cost function of the TO only include the sunk cost which is pefectly knows by the egulato. The main di eences espect to the mentioned efeences is that we explicitly model a non-competitive geneation maket with a tansmission constaint. We speci cally study how di eent policies of tansmission pice a ect competition. It is assumed that geneatos in addition to the cost of the enegy pay a tansmission tai. This tai is used to cove the cost of the tansmission opeato which in fomed by the xed cost of inteconnection plus a vaiable cost that can e ect, depending on the oganization of the industy, the tansmission losses o any vaiable cost associated with the tansmission activity. We show that the egulation of this tai has two immediate e ects. Fist, it a ects the quantity poduced at each node and moe impotantly, it may a ect the minimum capacity equiement that suppot equilibia with o without congestion. We show that the ow between nodes has an ambiguous impact when inceasing the tansmission chage. Finally, we chaacteize the supply of capacity that it is consistent with a tansmission opeato who behaves competitively and we intoduce some conditions about the optimal tansmission capacity. The est of this aticle is oganized as follow. Section 2 pesents competition in the powe maket while in Section 3 we analyze the e ects of the tansmission tai on competition. Finally, in Section 5 we intoduce some nal comments. Most of the poofs ae in a nal appendix. 3

4 2 The Model In pactice the activity of the Tansmission Opeato (heeafte T O) di es among counties. In geneal it is not educe to maintain and maybe to build and opeate the tansmission system. Thee ae electicity systems in which the T O oganizes a scheduled dispatch and it also executes the eal time dispatch. Fo instance, this is the case in UK unde NETA. 1 The National Gid Company eceives fou hous befoe the dispatch all the aangements between geneatos, supplies and distibutos. Since some tades ae not feasible due to the netwok constaints, the T O must nd the best way to accommodate them. It buys o sells enegy in advance to adapt ows. In eal time, the T O also has some eseve of enegy to ensue the equilibium between supply and demand. Since this activity implies a cost, the egulato o es a contact that coves some cost taget. The m eceives any di eence as a po t when the ealized cost is lowe than the taget but it is accountable fo the di eence when it is lage. 2 The egulato socializes this cost among uses. In Agentina, the scheduled and the eal dispatch ae made by di eent agents. The System Opeato (heeafte SO) detemines a scheduled minimum dispatch cost that includes netwok constaints while it is the T O in chage to ensue that this anticipated dispatch is feasible in eal time. The T O is not accountable fo deviation in the scheduled dispatch but it is esponsible when the netwok capacity di es fom the contacted netwok capacity. That is, the T O eceives a given netwok capacity which has been speci ed in the contact o eed by the egulato. Thee is punishment in case of failue which is not diect linked to the geneation cost (o oppotunity cost fo demand) associated of solving fo such a deviation. 3 In this aticle, we do not conside issues like the incentive of the T O to manipulate the optimal dispatch. This poblem has been study by Vogelsang [2001] and Benitez [2005]. We conside a educed fom of T O that executes a scheduled dispatch made by the SO. 1 See Roques et al. [2004]. 2 In any case thee is an uppe and lowe bound in the cost. 3 The TO is also penalized when it executes a non-scheduled maintenance pogam. 4

5 We assume that the cost of this m is compound of two elements. Fist, the xed (sunk) cost elated to all the equipment associated to the netwok tansmission: towes, lines, tansfomes, etc. In addition, we conside that cost is elated to ows. Fo instance, in UK, tansmission losses ae costly fo the T O and they ae paid by all uses independently on thei contibutions. In Agentina, the optimal dispatch includes this cost. Howeve, we assume that pat of the maintenance cost depends on the usage of the system and in paticula, on the ow. That is, the highe of the ow (up to capacity), the highe cost to maintain the system woking coectly. In any case, thee is cost associated to netwok ows. The T O cost function is denoted by: C(z; K) = L(z) + K whee z denotes ows, K is the total equipment associated to the tansmission activity and is the pe-unit capacity cost. 4 The function L(z) is stictly convex and positive with L(0) = 0: If we intepet it as tansmission losses, L(z) = z 2 =2 whee is a positive technical paamete. In the case that L(z) is intepeted as tansmission losses, we ae implicitly assuming that the T O puchases the necessay enegy in a seconday maket and then, the optimal dispatch does not include losses. It is impotant to emak that the xed cost is the most impotant component in the cost function. In this setting, the po t function of a TO is denoted by: T O = T C(z; K) whee T denotes the egulated evenue eceived by the TO. In pactice, the evenue of the TO is collected fom seveal souces. Fo instance, thee is a netwok fee which may depend on the numbe of connection of the uses in the system o diectly to the maximum demand o injection. In addition, thee is a vaiable chage that depends on the injection/demand and it may be also node elated. In geneal the evenue is elated to the speci c ole of the TO in the system 4 This is a model with one line and z epesents the unique physical ow in the system. 5

6 In the analysis we only conside a set of instument that can be used to emuneate the TO. We assume a linea pice pe unit injected o a two-pat tai that includes a linea pat pe unit injected and a xed fee. 5 The netwok is fomed by two nodes denoted with i = 1; 2 connected by a line of capacity K. At each node, thee is a concave demand with negative slope D i (p) and a monopoly geneato with maginal cost c i (q i ): We assume a non-competitive powe maket which is descibed in the next section. 2.1 Competition in the Electicity Maket The liteatue of competition with tansmission constaint was analyzed though auction theoy and Counot competition. Léautie [2001] assumes that geneatos ae bette infomed than the cental dispatche about thei cost. Assuming that capacity of geneation is not binding, a geneato with the lowest maginal cost will satisfy the entie demand, poviding that tansmission capacity is not binding. Othewise, in addition to the lowest cost geneato, it should be anothe geneato which is not constained by tansmission capacity to satisfy demand. Note when the infomation has been evealed and congestion is absent, thee is no ole fo a m having the second lowest maginal cost. That is, thee is no equilibium in which ms with di eent maginal cost ae poducing and the netwok is uncongested. In pactice, we obseve that even without congestion, thee is moe than one m poducing. This fact can be explained by the geneation capacity (inceasing supply function) of each plant o by secuity easons. On the othe hand, Boenstein, Bushnell and Stoft [2000] (heeafte BBS) conside a model of quantity competition in which it is possible to have an equilibium in which two asymmetic geneatos poduce without congestion. We adapt this quantity game to include the e ect of the pice egulation 5 In the NETA only demand pays fo tansmission losses independently of location (unifom chage). See The new electicity tading aangements Volume 1, July 1999, The O ce of Gas and Electicity Makets. It is also ecommended to include geneation. Geneation pays some tansmission chages to cove Tanspot Uplift which does not include tansmission losses and Reactive Powe. In Agentina the geneation pays a tansmission chage that depends on the enegy injected plus some xed pat. 6

7 of the T O. We denote by T i = t i q i + F the tai paid by each geneato. The geneatos compete o eing quantities. The system opeato allocates these quantities in a way that maximizes the goss consume s suplus. The geneatos eceive the nodal pice of thei location independently of whee poduction has been consumed. In the Appendix we descibe the nodal pice dispatch when geneatos bid quantities. Basically, BBS show the existence of thee egimes of demand. That is, an unconstained demand D u (p) = D 1 (p) + D 2 (p); an aggessive demand D a i (p) = D i(p) + K and a passive demand D p i (p) = D i(p) K: These names ae elated to the fact that eithe thee is no congestion, o if thee is congestion, thee is a positive ow fom node i to node j (aggessive) o congestion with a negative ow fom node i to node j (passive). We need to identify egions in the space of quantities q 1 ; q 2 in which the payo associated to each combination leads to an aggessive, passive o unconstained po ts Let s de ne qi a+ (q j ) = K + D i (P j (q j + K)) and q p+ i (q j ) = K + D i (P j (q j K)) with i = 1; 2 and i 6= j: Condition 1 : Fo a given q j and K, if q i > qi a+ (q j ) the geneato i faces an aggessive demand function while geneato j a passive demand function. If q p+ i (q j ) q i qi a+ (q j ); both geneatos compete facing the unconstained demand function. Poof. Fo a given q j and K; let s conside the maximum quantity poduce at node i such that thee is no congestion. Let s denote this quantity by bq i : In this case and since thee is no congestion, it must be tue that P i (bq i K) = P j (q j + K): Fo q i > bq i we have that P i (q i K) < P j (bq j + K) and thee is congestion: Then bq i K = Pi 1 (P j (q j + K)) o bq i = qi a+ (q j ) = K + D i (P j (q j + K)): Suppose that we conside the minimum quantity poduced at node i fo a given q j and K such that thee is no congestion. Let s denote this quantity by eq i : Using the same agument as befoe, fo any q i < eq i it must be tue that P i (q i + K) > P j (bq j K) and then eq i + K = Pi 1 (P j (bq j K)) o eq i = q p+ i (q j ) = K + D i (P j (q j K)): Fo any q i 2 [eq i ; bq i ] thee is no congestion and then qi a+ (q j ) q i q p+ i (q j ): Note that qi a+ (q j ) = q p+ j (q i ): 7

8 In this setting, a geneato located at node i maximizes the e ective po t function e i (q i; q j ; K) taking as given q j and K : 8 q p i >< = agmax f[p i (q i + K) c i (q i )] q i T i g if q i < q p+ i (q j ) q i e i (q i ; q j ; K) = qi c = agmax f[p u (q i + q j ) c i (q i )] q i T i g if q p+ i (q j ) q i qi a+ (q j ) q i >: qi a = agmax f[p i (q i K) c i (q i )] q i T i g if q i > qi a+ (q j ) q i The st ow in e i (q i; q j ; K) is the po t function when demand faced by geneato i is D p i (p):6 Cetainly, if q p i is the optimal quantity fo a passive demand, it must be the case that q p i < qp+ i (q j ): Theefoe, geneato i faces a passive demand while geneato j an aggessive demand. It the case since i (q j ) = qj a+ (q i ) and given that both functions have positive slope, we nd that q j > qj a+ (q i ): Notice that competitos ae isolated fom each othe since they ae local monopolist in its local demand plus o less the esidual q p+ demand given by K. The second ow in e i (q i; q j ; K) is the po t function when demand is P u (); that is the demand deived fom the sum of demands. We efe to this as an unconstained o Counot stategy. The best esponse function in the unconstained egime is qi c(q j) povided that q p+ i (q j ) qi c(q j) qi a+ (q j ): Using the same agument as in the case of a passive stategy, qi a is the best esponse function when q a i > qa+ i (q j ) The E ective Best Response Function We need to de ne the following two thesholds: q sw j is de ned by u i (Minfqc i (qsw j ); q a+ i (qj sw )g; qj sw ) = p i (qp i ; K) and q sw j bq j is de ned by u i (qc i (bq j); bq j ) = u i (qa+ i (bq j ); bq j ) is the quantity played by m j that made m i of being indi eent eithe between the Counot po t o the aggessive po t espect to the po t obtained in a passive stategy. Respect to bq j is the quantity played 6 Fo simplicity, we dop the supescipt a and p in the invese demand functions. 8

9 by m j that made m i of being indi eent eithe between the Counot po t espect to the po t obtained in an aggessive stategy Denote the E ective Best Response Function by e i (q j): The shape e i (q j) depends on whethe qj sw 7 bq j : If qj sw bq j ( Maxfq i e a (q j ) = i ; qi a+ (q j )g fo any q j 2 [0; qj sw ] and, Minfq p i ; qp+ i (q j )g fo any q j qj sw : Othewise 8 >< i e (q j ) = qi c >: (q j) Maxfq a i ; qa+ i (q j )g fo any q j 2 [0; bq j ] and, fo any q j 2 [bq j ; q sw j Minfq p i ; qp+ i (q j )g fo any q j q sw j : ] and, That is, depending on the value of qj sw ; the unconstained best esponse function qi c(q j) can be pat o not of i e(q j): Figue 1 shows 2 e(q 1) in the two cases: q 2 q ( q ) a when q > q ) sw 1 1 Congestion: z 21 >0 No Congestion q ( q ) = q ( q ) a p q ( q ) = q ( q ) Congestion: z 12 >0 p a q a 2 q p 2 ( q ) e 2 1 when q q ) sw 1 1 q ( q ) c 2 1 sw sw q1 q ) 1 q 1 q 1 Figue 1: The E ective Best Response Function in BBS [2000] The best esponse function is not a continuous function. It has a discontinuity when the payo function jumps fom the Counot po t o fom the aggessive po t to the passive po t. 9

10 In the last gue, when q 1 is small, geneato two nds optimal to play an aggessive stategy. As q 1 becomes lage, it is necessay to incease the poduction above q a 2 in ode to maintain an aggessive demand. Howeve, it may be the case that befoe eaching the quantity that belongs to the unconstained best esponse function, it is optimal fo geneato two to play a passive stategy and thee is a jump in the e ective best esponse function. On the contay, this discontinuity may occu afte playing some pat of the unconstained best esponse function. Let s denote by u i (qc i ; q j); a i (qa i ; K) and p i (qp i ; K) the po t function of an unconstained, aggessive and passive stategies espectively when the m i s ival plays q j and the capacity of tansmission is K Nash Equilibium In this section, we successively analyze the case whee demand and technology ae symmetic and the case whee they ae not symmetic. Assume st that c i () = c() and P i () = P () fo all i: Lemma 1 : Let k be de ned by: u i (q c i ; q c j) = p i (qp i (k ); k ) If K k ; thee is an unique symmetic equilibium in pue stategies whee ms play Counot stategies. Othewise, the equilibium is in mixedstategy. Poof. See Theoem 1 in BBS [2000]. When capacity is su ciently lage, it is not a supise that the unique Nash equilibium is a symmetic Counot equilibium. By symmety the line is not used to cay enegy between nodes but geneatos poduce moe enegy than in the case that both nodes wee unconnected. In the theat of deviation that pevents competitos to achieve the monopoly po ts as if K = 0: As the value of the inteconnection capacity diminishes below k ; it is moe po table fo any m to bid a quantity that leads to a passive demand. Howeve the ival will espond with a quantity that necessay leads to an 10

11 unconstained demand and given the fact that thee is not enough capacity to suppot a Counot equilibium, thee is no equilibium in pue-stategy. The mixed-stategy equilibium of this game has a continuous suppot. That is, a mixed-stategy po le f is a Nash Equilibium if, fo all playes i, i fi ; f i i q i ; f i fo qi 0 whee fi = fi (q) is de ned in the suppot [q i ; q i ] with q i > q i 0: Unfotunately, we could not obtain a close-fom solution and even numeically appoximation of this equilibium is di cult to nd. BBS [2000a] show an algoithm to compute it. They do not use the oiginal one-shot game and it is eplaced by a " ctitious play". This is an in nitely epeated game that may o may not convege to the mixed-stategy equilibium of the oiginal game. Unfotunately, they could not nd the mixed-stategy equilibium in the oiginal game. Suppose now that symmety in the demand functions o in the cost functions does not hold. Fo instance, let s assume w.l.o.g. that m two has a monopoly pice lage than the monopoly pice of m one when K = 0. Let s denote this pice by p m;o i with i = 1; 2: Let s denote by k = k2 the capacity that makes m 2 indi eent between obtaining the Counot po t and the passive po t: u 2(q c 1; q c 2) = p 2 (qp 2 (k ); k ) In addition by b k 2 be the capacity that makes m 2 indi eent between obtaining the Counot po t when its ival plays q1 a (K) and the passive po t. Fomally b k = b k 2 is de ned by: u 2(q a 1( b k); q c 2(q a i ( b k))) = p 2 (qp 2 (b k); b k) that is, q c 2 (qa i (b k)) is the best esponse function of m 2 in the Counot game when m one plays an aggessive stategy. It is impotant to emak that b k 7 k : Lemma 2 : Suppose that p m;o 1 < p m;o 2 : The following equilibia paametized by capacity hold. if K 2 0; minf b k; k g then the Nash equilibium is the pai q1 a; qp 2 : if K > maxf b k; k g then the Nash equilibium is the pai q c 1 ; qc 2 : 11

12 if K 2 minf b k; k g; maxf b k; k g eithe b k < k and thee is no equilibium in pue-stategy o k > b k and thee ae two equilibia: q a 1 ; qp 2 and qc 1 ; qc 2 : Poof. See Theoem 5 in BBS [2000]. The main di eence espect to the symmetic case elies in the existence of an asymmetic equilibia in which, the most e cient povide poduces q a 1 while its ival chooses a passive stategy poducing qp 2 : The second di eence is about the non-existence of a pue-stategy equilibium. It aises when b k < K < k and fo a value of capacity which is not necessaily close to zeo. Finally when k < K < b k thee ae two puestategy equilibia: equilibium. the Aggessive/passive equilibium and the Counot Respect to the latte, since symmety does not hold, thee is a positive ow fom node one to node two. Fom now on, we efe to the symmetic case the model in which demand and the cost function is the same at each node. Othewise, we efe to the asymmetic case. 3 Regulation of the TO We stat assuming that the egulato is not able to impose any disciminatoy tansmission tai. 3.1 Linea Tansmission Tai We assume that T i = tq i and next, we conside a two-pat tansmission fee. The geneatos pay the same pice independently of whethe they ae located. In geneal this is the case in most of the counties and it can be explained mainly by political easons. Howeve, by disciminating the tansmission tai, the govenment may facilitate enty in speci c egions whee the capacity of inteconnection cannot be extended o simply, to facilitate the enty of non-pollutants technologies (i.e. wind and sola geneation, etc.). We conside a game in which competition in the powe maket occus afte the T O; o the egulato, sets the tansmission tai. If we solve 12

13 backwad this game, the po t function of the T O that includes the outcome of the electicity maket is expessed by: T O (t) = tq(t) L(z(t)) K (1) whee Q(t) denotes the sum of q 1 (t) and q 2 (t); that is, the sum of the equilibium quantities in the powe maket. As we descibed befoe, the Nash equilibia depend on k and b k: We show that both thesholds ae a ected by changes in the tansmission tai. To make the analysis less cumbesome, we assume linea demand and constant maginal cost. 7 Let s de ne by t the maximum t such that u 2 (qc 1 (t); qc 2 (t); t) = 0 and by bt the maximum t such that p 2 (qp 2 (b k(t); t); b k(t); t) = 0: That is, the tansmission tai is bounded by the po ts of the geneatos. Lemma 3 : Fo any linea demand and constant maginal cost, povided that the monopoly pice when K = 0 in node one is lowe than the same monopoly pice in node two, the following conditions hold: (i) dk (t)? 0; db k(t) > 0; (ii) k (t ) b k(bt) and t bt, (iii) k (0)? b k(0) and (iv) if b k(0) > k (0); then dk (t) Poof. See the Appendix. > 0 and t = bt: When demand is linea, the equilibium quantities in the unconstained egime ae deceasing function of t as we show below in Poposition 1. It also holds fo the best esponse function in the passive equilibium. The capacity which is needed to make geneato two indi eent of playing the Counot stategy o the passive stategy may incease o decease. Howeve, the equied capacity to make the same geneato indi eent between the passive stategy and to espond to the aggessive stategy chosen by its ival when demand is P u () is always an inceasing function of t. Figue 2 shows the di eent cases and the egions in which the thee types of equilibia occu. Fo instance, let s conside case (c) in Figue 2. Suppose that the existing capacity is K 1 : Then, when the tansmission chage is small, the 7 The analysis also holds when maginal cost is stictly convex. 13

14 When k * (0) k ) (0) K C N A/P k * () t ) kt () K N C A/P k * () t ) kt () (a) t (b) t K K 1 C ) kt () k * () t C = Counot Equilibium N = No Pue Stategy Equilibium A/P =Aggessive/Passive Equilibium Othewise A/P (c) A/P and C t Figue 2: Paametization of Equilibia in Tem of K and t: installed capacity is lage than b k(t); k (t) and the geneatos compete in a lage maket obtaining the Counot po ts. As t incease, thee is a egion in which b k(t) < K 1 < k (t): In this case, thee ae two Nash Equilibia: the Counot and the Aggessive/Passive Equilibium. Finally, when the tansmission chage is lage enough, capacity is lowe than b k(t); k (t) and thee is a unique Aggessive/Passive Equilibium. As a coollay of this lemma, we can establish that when demand and cost function is the same at both nodes, dk (t) < 0: Theefoe, Q(t) in equation (1) will be a function of K; b k(t) and k (t): Fo the symmetic case we have that: Q(t) = ( Q c (t) = 2q c (t) if K k (t) Q e (t) = q1 e(t) + qe 2 (t) if othewise 14

15 whee q1 e(t); qe 2 (t) epesent the expected quantities played in a mixed-stategy equilibium. Respect to the asymmetic case: 8 Q a=p (t) = q1 a(t) + qp 2 0; (t) if K 2 minf b k(t); k (t)g >< Q c (t) = q1 c(t) + qc 2 (t) if K > maxfb k(t); k (t)g Q(t) = Q a=p (t) = q1 a(t) + qp 2 (t) and Q c (t) = q1 c >: (t) + qc 2 (t) if b k(t) > K > k (t) Q e (t) = q1 e(t) + qe 2 (t) if k (t) > K > b k(t) As consequence of Lemma 2 and 3 we can establish the following poposition Poposition 1 : Inceasing the linea tansmission tai has the following e ects on the wholesale electicity maket: i) when demand is linea, it deceases the quantity poduced at each node, ii) it inceases o deceases the ow between nodes, ii) it may a ect the equilibium in the geneation maket. Poof. See the Appendix. The st pat in this poposition even when it seems to be logical, it does not necessay holds when demand is not linea. It is possible to daw cases in which the poduction at one node inceases while deceases at the opposite node. By Lemma 1, we know that in the symmetic case thee is not ow in the Counot equilibium. It may hold in a symmetic mixed-stategy equilibium 8. Theefoe, the second pat in this poposition efes to the asymmetic case. The intuition fo this esult is explained by the elative impact of t in the poduction at each node. That is, the poduction will be educed at each node but the impact may di e in such a way that ow incease o decease. Finally, thee is a second e ect when setting the tansmission chage. It may a ect the type of equilibia fo a given capacity. This is explained by the e ect on the thesholds that suppot the di eent types of equilibia. Note that this e ect is missing if we conside an inelastic demand. Fo instance, conside that ms compete bidding pices like in game. 8 We do not know if thee is an asymmetic mixed-stategy equilibium in the symmetic 15

16 Léautie [2001]. In this case, the equilibium is not a ected by changes in the tansmission fee. 9 Suppose that capacity of tansmission is K and conside the minimum tansmission chage that makes the T Os po t non-negative. Denote this pice by t pc : Note that the xed cost K pevents t pc to be equal to the maginal cost of poviding inteconnection. Fomally, t pc is de ned by: t pc = K + L(z(tpc )) Q(t pc ) Natually, fo any t > t pc ; the tansmission opeato gets positive po ts. Conside the tansmission chage chosen by a monopolist tansmission opeato. It is the solution of: Max t T O (t) = tq(t) L(z(t)) K whee this chage denoted by t m satis T O = tq(t) + Q(t) L(z(t))z(t) = 0 with Q(t) ; L(z(t)) and : We assume that T O (t m ) is non-negative. Thee is pivate inteest to povide inteconnection, even without any fom of subsidy. In that case, t m > t pc : In addition, since z(t) 7 0 it means that when z(t) > 0; t m is lage than the same monopoly pice when z(t) < 0: Suppose that in the long-tem, the govenment can pefectly egulates the activity of the T O while the wholesale maket is still non-competitive. That is, the tansmission opeato gets zeo po t. This condition should hold even when thee is a netwok expansion. The linea tansmission tai denoted by t 0 that satis es the zeo po t condition is: t 0 = K + L(z(t0 )) Q(t 0 ) fo K K: Implicitly K 0 = K 0 (t) fo any t t 0 (K): 9 The geneatos include the tansmission fee in thei bids but the type of equilibium is not a ected. When maginal cost is pivate infomation and demand is inelastic, it can be shown that geneatos bid the maginal cost plus the tansmission fee. 16

17 It is impotant to emak that we ae not the detemining the optimal long-tem capacity of inteconnection. We only set a condition in which the capacity is piced at the minimum cost without violating the budget balance of the tansmission company. Let s call K 0 (t) the minimum cost long-tem supply of capacity that it is compatible with a competitive tansmission opeato. This function includes the e ect of a non-competitive geneation maket. Let s de ne t 0 and bt 0 by the value of t that satis es K 0 (t) = k (t) when Q(t) = Q c (t) and K 0 (t) = b k(t) when Q(t) = Q a=p (t) espectively. Both thesholds detemine the value of the tansmission chage that equal the long-tem supply with the minimum capacity to suppot the di eent equilibia in the geneation maket. The next poposition descibes the shape of K 0 (t). Poposition 2 : The minimum cost long-tem supply of capacity that it is compatible with a competitive tansmission opeato is chaacteized by: In the symmetic case: ( tq e (t)+l(z(t)) K 0 fo any t 2 [0; t 0 ) (t) = tq c (t)+l(z(t)) fo any t 2 [t 0 ; t ] In the asymmetic case when k (t) > b k(t) : 8 tq >< a=p (t)+l(z(t)) fo any t 2 [0; bt 0 ) K 0 tq (t) = e (t)+l(z(t)) fo any t 2 [bt 0 ; t 0 ) >: tq c (t)+l(z(t)) fo any t 2 [t 0 ; t ] Finally, the asymmetic case when k (t) < b k(t) : 8 >< K 0 (t) = >: tq a=p (t)+l(z(t)) fo any t 2 [0; bt 0 ) tq a=p (t)+l(z(t)) o tqc (t)+l(z(t)) fo any t 2 [bt 0 ; t 0 ) tq c (t)+l(z(t)) fo any t 2 [t 0 ; t ] Poof. See the Appendix. Figue 3 shows the shape of K 0 (t) in the symmetic case. When the inteconnection capacity is lowe than k (t); the geneation maket has an equilibium in mixed-stategy and then K 0 (t) is computed 17

18 K Counot Equilibium K 0 () t k * (0) Π ( t ) < 0 TO Π ( t ) > 0 TO c tq ( t ) + Lzt ( ( )) Mixed stategy Equilibium k * () t e tq () t + Lzt ( ()) t 0* t Figue 3: The Shape of K 0 (t) in the Symmetic Case. using Q e (t): Recall that K 0 (t) is an inceasing function that minimize the total cost of poviding inteconnection and any capacity level that maximize the net consume s suplus has to satisfy this condition. Then, at t = t 0 ; K 0 (t) shifts to k (t 0 ) and K 0 (t) is computed using Q c (t); the Counot equilibium in the powe maket. Respect to the optimal long-tem capacity, note that fo any K k (t 0 ) thee is no exta welfae impovement due to moe competition in the wholesale electicity maket. Suppose that the net social welfae is de ned by the sum of the net consume s suplus plus the po t of the geneatos and tansmission company. Simple computation shows that the welfae function can be witten by: " 2X Z qi (t;k) W (K) = P i (v)dv i=1 0 c i (q i (t; K)) # K L(z(t)) whee t is the implicit solution of t = K+L(z(t)) Q(t) : We denote the welfae by W c (K) when thee is a Counot equilibium and so on fo the othe two types of equilibia. Let s de ne by K e and K a=p the value of inteconnection that maximize W e (K) and W a=p (K) espectively Poposition 3 : The optimal long-tem inteconnection capacity that maximizes the net social welfae is given by: 18

19 i) In the symmetic case by K e o K c depending on whethe W e (K e ) 7 W c (K c ) with K e 2 (0; k t 0 ) and K c = k (t 0 ): ii) In the asymmetic case by K a=p ; K e o K c depending on the max W a=p (K a=p ); W e (K e ); W c (K c ) with K a=p 2 0; b k bt 0 ; K e 2 bk bt 0 ; k (t 0 ) and K c = k (t 0 ): Poof. The poof of this poposition is staightfowad. The welfae function is not a continuous function of capacity due to the disceet change in q 1 ; q 2 when switching fom one equilibium to the othe. As mentioned befoe, fo any K k (t 0 ) thee is no exta welfae impovement due to moe competition in the wholesale electicity maket and then, it is not socially desiable to invest in capacity beyond this theshold. In the symmetic case, the capacity that maximizes welfae in the mixed-stategy equilibium can lage than k (t 0 ) and then it is bounded. Note that having a capacity K e > k (t 0 ) cannot be optimal since the equilibium will be Counot. Compaing the welfae in the two egimes of equilibium lead to the optimal tansmission capacity. The same agument applies in the asymmetic case, independently whethe k (t) 7 b k(t): This poposition chaacteizes the optimal inteconnection capacity when the tansmission company is pefectly egulated and the wholesale maket is non-competitive. This capacity is not a continuous function of the tansmission chage as Poposition 2 states since thee is a discontinuity in the poduction of electicity when switching fom one equilibium to the othe. It also includes the e ect of changes in the tansmission fee on the thesholds. We should emak that ou analysis captues some aspect of the long-tem capacity expansion but we ae ignoing the possibility of new enty geneation units. Thee is some empiical evidence (Roques et al. 2004) that egions with fequently high pices ae those in which invest in geneation capacity occus moe often. In this case, the optimal tansmission capacity should be lowe as the di eence in the nodal pices between egions. We will include this possibility a futue eseach. This analysis is consistent with the esult of Léautie [2001] that shows the po-competitive e ect of a tansmission expansion. We let as a futue extension the analysis of the incentives of the geneatos and the tansmission 19

20 company to decide on futue expansion. 3.2 Two-pat Tansmission Tai Conside now the use of two pat tai T i = tq i +F: Suppose that t is equal to the shot-tem maginal cost of poviding inteconnection. Since the ow is also a function of t; the pe unit tansmission chage is the implicit solution of : t = L(z(t)) whee t is necessay not lage than L(K): In the symmetic case since thee is no ow between nodes, this chages has to be set equal to zeo while the xed pat is used to cove the total cost of poviding inteconnection. Lemma 4 : In the symmetic case, when the tansmission company use a two-pat tai in the fom of T i = tq i + F; if t = L(z(t)) we have that k (t) = k : Poof. The poof is staightfowad. Since in the Counot equilibium each geneato poduces the same amount of enegy and demand is symmetic, thee is no ow and then t = L(0) = 0: In this case k is not a function of t. In the asymmetic case, thee is a positive ow in the Aggessive/passive equilibium. Thee is a positive ow in the Counot equilibium as well. Fo instance, let s conside that the unique di eence between nodes is the amount of demand, while maginal cost is the same. Natually both ms poduce at equilibium the same amount of enegy, but the consumption is di eent at each node and then, thee is a ow between the nodes with low demand towad to node with high demand. Theefoe, Lemma 3 is also valid in the asymmetic case when a two-pat tai is used and the vaiable chage epesents the maginal losses. Note that since L() > 0 and L 00 () > 0, the sum of the xed pat of the tai is necessaily lowe than K if the zeo-po t condition of the tansmission company holds. That is, when t = L(z(t)), the maginal losses ae lage than the aveage losses and it ceates some bene ts that can be 20

21 used to educe the oveall cost of poviding inteconnection (Schweppe at al. 1988). 4 Final Comments The objective of this aticle is to discuss the ole played by the egulation of the tansmission system in ceating o inceasing competition in the powe industy. In paticula, we focus on the e ects of the egulation of the tansmission tai on the stategy played by geneatos. We conside a twonode netwok with limited capacity of inteconnection in which the wholesale maket is oganized though a nodal pice system. In this setting, we study a linea tai and a two-pat tansmission tai as a mechanism to emuneate the total cost of poviding inteconnection capacity. It is assumed that geneatos in addition to the cost of the enegy, they pay a tansmission chage. This tai is used to pay the cost of the tansmission opeato which in fomed by the xed cost of inteconnection plus a vaiable cost that can e ect, depending on the oganization of the industy, the tansmission losses o any vaiable cost associated with the tansmission activity. We show that the egulation of this chage has two immediate e ects. Fist, it a ects the quantity poduced at each node and moe impotantly, it may a ect the minimum capacity equiement that suppot equilibia with o without congestion. We show that the ow between nodes has an ambiguous impact when inceasing the tansmission chage. Finally, we chaacteize the supply of capacity that it is consistent with a tansmission opeato who behaves competitively and we intoduce some conditions about the optimal tansmission capacity. Appendix Nodal Pice with Quantity Bids Let s assume that geneatos bid quantities. The SO pefoms the ole of Tansmission Opeato eceiving the quantity bids q 1 ; q 2 : Let s denote the goss consumes suplus at nodes i by S i (d i ) whee d i epesents the quantities consumed. 21

22 The ole of the SO is just to allocate the poduction chosen by the geneatos among consumes in the way that maximizes the goss consumes suplus subject to the netwok constaint: 10 Max d 1 ;d 2 S 1 (d 1 ) + S 2 (d 2 ) st: jq 1 d 1 j K () q 1 + q 2 d 1 + d 2 This is not the standad dispatch in which ms bid pice o supply function. Since quantities ae given, the ole is just to allocate them, egading the cost of geneation. Howeve, the geneatos know the ule of the game an implicitly, when they bid quantities, they ecognize the cost of poduction. () Using the fact that S 0 i () = P i(); the FOCs ae: P 1 (d 1 ) = ; P 2 (d 2 ) = jq 1 d 1 j K; (q 1 d 1 K) = 0; 0 q 1 + q 2 d 1 + d 2 ; 0 (q 1 + q 2 d 1 d 2 ) = 0 Conside the case whee q 1 d 1 < K. It implies = 0 and P 1 (d 1 ) = P 2 (d 2 ). The line is not congested and the SO allocates the quantities to each node. A single pice fo both egions is enough to implement this optimal dispatch. Theefoe only one pice is equied to integate both egions in a lage unconstained maket and the supplies face a total demand equal to D u (p) = D 1 (p) + D 2 (p). As a consequence of the pool mechanism and since o es ae pefect substitutes, geneatos do not cae if thei enegy is sold in egion one and two. 10 The tansmission constaint jq 1 d 1j K can be ewitten by these two inequalities: q 1 d 1 K q 1 d 1 K The st inequality if it is binding means that thee ae some expots fom node one two node two. If it is not binding, it is not necessay that congestion is absent, since thee can be expots fom node two to node one. It is captued by the second inequality. 22

23 Suppose now that > 0. The st consequence is that egion two is constained to impot at most K units (and in egion one; d 1 = q 1 Then, ationing using di eent pices is optimal when the line is congested. The pice at each egion is P 2 (q 2 +K) and P 1 (q 1 K) = P 2 (q 2 +K) : The nodal pice is used to emuneate geneatos fo thei bids. The constained demand faced by geneato one can be witten as D a 1 (p) = D 1(p) + K while by geneato two is D p 2 (p) = D 2(p) K): K: The supescipt a and p mean aggessive and passive demand. The same agument holds when congestion is due to a ow fom node two to node one. Poof of Lemma 3. Conside a linea demand at both nodes: D i (p i ) = a i b i p i and constant maginal cost C i (q i ) = c i q i fo i = 1; 2 whee a i ; b i > 0 and c i 0: We assume that a j b j > a i b i (c j c i ) and it implies a positive ows fom node one towad node two. Note that it is equivalent to have a monopoly in node j lage than the monopoly pice in node i when the link has no capacity. Then. Respect to b k(t) : dk (t) 0 if b j + 2p b j p bi + b j 0 3 d b k(t) > 0 since 6b i b j 4b 2 j + 4(b i + b j ) 3 2 p bj > 0 By compaing t and bt we can show that the di eence depends on whethe maginal costs ae di eent. The same applied fo k (t ) b k(bt): Respect to k (0)? b k(0); thee ae two examples in BBS[2000]. Finally since db k(t) 0: > 0 and k (t ) b k(bt); if b k(0) > k (0) it is necessay that dk (t) > Poof of Poposition 1. Pat i) Let s conside the st ode condition of each geneato when the link is unconstained: P u (q 1 + q 2 )q i + P u (q 1 + q 2 ) c(q i ) t = 0 fo i = 1; 2 23

24 Taking total deivatives with espect to t; we obtain: dq 1 dq 2 = = P u () c 2 "() (q 1 q 2 )P u" () [P u () c 2 "()] soc 1 + [P u () c 1 "()] [P u () + q 2 P u" ()] 7 0; P u () c 1 "() + (q 1 q 2 )P u" () [P u () c 2 "()] soc 1 + [P u () c 1 "()] [P u () + q 2 P u" ()] 7 0 whee soc 1 denotes the second ode condition fo m one in the po tmaximization poblem. Note that denominatos ae positive while the sign of the numeato is ambiguous. When demand is linea (o no to much convex), both deivatives ae negative. Using the same agument we can show that the quantity poduced in an aggessive o in a passive equilibium ae deceasing function of t. As mentioned befoe, since we cannot detemine a close-fom solution of the mixed-stategy equilibium, we assume that it also holds in such a case. Pat ii) Note that by assumption, node one expots enegy towad node two. This ow can be expessed by: z = q 1 D 1 (P u (q 1 + q 2 )) whee z denotes ows. The capacity constaint implies that z 2 [0; K]: Let s di eentiate the ow espect to t: dz = (1 h)dq 1 h dq 2 whee h = D 1 P u (): Note that h 2 (0; 1): To show this claim, be D 1 = F (p) and D 2 (p) = G(p) with negative deivatives f(p) and g(p): In this case, Q u = D(P u ) + G(P u ): Di eentiating espect to Q u leads to 1 = P u (f + g): Then, D 1 P u is equal to f=(f + g) which is positive and lowe than one. In the case whee dq 1 is positive, it is clea that ow inceases when t inceases. Suppose that dq 1 is negative. Then, when h tends to zeo, in the limits, the ow decease and on the contay, when h tends to one, the ow incease. Pat iii) Fo simplicity conside the symmetic case. By Lemma 3, we know that dk (t) is negative. Suppose that capacity K 2 [ k (t ); k (0)]. It means that fo any t 2 [0; t ]; thee is no equilibium in pue-stategy as it is expessed in Lemma 2. Theefoe, by inceasing the tansmission chage thee exists et de ned by K = k (et) such that fo any t et; thee is an 24

25 equilibium in pue-stategy. The same analysis holds fo the asymmetic case. Poof of Poposition 2. We only pove the poposition fo the symmetic case. The othe two cases follow the same agument. Fist, note that K 0 (t) is positive fo any non-negative tansmission chage. It is an inceasing function fo any t 2 [0; t m ]: Let s conside the deivative of K 0 (t) espect to 0 = tq(t) + Q(t) L(z(t))z(t) whee Q(t) 2 fq(t) e ; Q(t) c g: Note that the numeato is equal to the F OC of a monopolist tansmission opeato and it is necessay non-negative fo t 2 [0; t m ]: Howeve, we know that a pefect competitive T O sets t pc < t m and is positive in the elevant ange. Let s conside any t 2 [0; t 0 ): By constuction K 0 (t) < k (t) and then, the equilibium in the wholesale maket of electicity is Q e (t): At t = t 0 ; the following inequality holds, povided that Q e (t 0 ) < Q c (t 0 ) : t 0 Q c (t 0 ) + L(z(t 0 )) > t0 Q e (t 0 ) + L(z(t 0 )) that is, K 0 (t 0 ; Q c (t 0 )) > K 0 (t 0 ; Q e (t 0 )) By de nition K 0 (t) is the supply of capacity that it is compatible with a competitive tansmission opeato. Theefoe, at t = t 0 ; it is possible to supply a lage capacity of tansmission since even that capacity incease, the aveage cost of the tansmission opeatos is constant due to the incease of the electicity poduction. Then, since by de nition K 0 (t) is the minimum cost long-tem supply of capacity that it is compatible with a competitive tansmission opeato, fo t t 0 ; we have that : K 0 (t) = tqc (t) + L(z(t)) Note that the consumes s suplus is inceasing in the quantities, it means that fo the same tansmission chage, it is possible to o e a lage capacity inceasing the total suplus. Finally, we pove that Q c (t) > Q a=p (t) since it is equied fo the two asymmetic cases. 25

26 Conside K = k > b k: In that case the Counot equilibium is feasible. Howeve, we show that Q c is lage than Q a=p when K = b k. W.l.o.g. assume q1 c > qc 2 : Necessay, fo some q 1 q1 c ; thee is a jump in 2 e(q 1) fom 2 u(q 1) to q p 2 since node two is now congested by expots fom node one. In that case q p 2 (k ) < q2 c: Since fo some q 2 < q2 c ; it implies that 1 e(q 2) = q1 a+ (q 2) > q1 a and then, necessay qa 1 < qc 1 : Conside now K = b k: Note that q1 a(k ) > q1 a(b k) but q p 2 (k ) < q p 2 (b k): Note that u 2 (q 1) intesects q p 2 (b k) when q 1 > q c 1 : Since u 2 (q 1) has negative slope, then q p 2 (b k) < q c 2 : To conclude, Qc > Q a=p when k > b k: Conside K = b k > k : Thee ae two equilibia. BBS[2000] show that it occus when q1 a > qc 1 but necessay, qp 2 < qc 2 : Conside u 1 (qp 2 ) which is necessay lage than q1 a: By assumption u i (q j) has slope in absolute value lowe than one and given that q p 2 < qc 2 ; then qc 1 + qc 2 > u 1 (qp 2 ) + qp 2 > qa 1 + qp 2 : Refeences [1] Benitez, D. 2005, Decentalization, Incentives and the Optimal Oganizational Stuctue, GREMAQ, Univesité Toulouse I. [2] Boenstein, S., Bushnell, J. and Stoft, S., 2000, The Competitive Effects of Tansmission Capacity in the Deegulated Electicity Industy, Rand Jounal of Economics, Vol. 31, pp [3] Boenstein, S., Bushnell, J. and Stoft, S., 2000a, Supplement to The Competitive E ects of Tansmission Capacity in the Deegulated Electicity Industy, available at [4] Boyce, J. ana Hollis, A. 2001, Govenance of Electicity Tansmission System,Univesity of Calgay, (fothcoming in the Enegy Economics). [5] Kleindofe, P. 2004, Economic Regulation unde Distibuted Owneship: The Case of Electicity Powe Tansmission, mimeo, The Whaton School, Univesity of Pennsylvania. 26

27 [6] Léautie, T.O. 2000, Regulation of an Electic Powe Tansmission Company, The Enegy Jounal Vol. 21, pp [7] Léautie, T.O. 2001, Tansmission Constaints and Impefect Makets fo Powe, Jounal of Regulatoy Economics Vol. 19, pp [8] Roques, F., Newbey, D., and Nuttall W., 2004, Geneation Adecuacy and Investment Incentives: fom the Pool to NETA, CMI Electicity Poject Woking Pape 58, Octobe [9] Pepemans, G. and Willems, B. 2003, Regulating Tansmission in a Spatial Oligopoly: a Numeical Illustation fo Belgium, WP , Cente fo Economics Studies, Katholiek Univesiteit Leuven. [10] Schweppe, F., Caamanis, M., Tabos, R. and Bohn, R., 1988, Spot Picing of Electicity, Kluwe Academic Publishes. [11] Vogelsang, I. 2001, Pice Regulation fo Independent Tansmission Companies, Jounal of Regulatoy Economics, Vol. 20-2, pp