Physical and Financial Virtual Power Plants

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1 Physical and Financial Virtual Power Plants by Bert WILLEMS Public Economics Center for Economic Studies Discussions Paer Series (DPS) 05.1 htt:// Aril 005

2 PHYSICAL AND FINANCIAL VIRTUAL POWER PLANTS Bert WILLEMS K.U.Leuven (Belgium) Abstract Regulators in Belgium and the Netherlands use different mechanisms to mitigate generation market ower. In Belgium, antitrust authorities oblige the incumbent to sell financial Virtual Power Plants, while in the Netherlands regulators have been discussing the use of hysical Virtual Power Plants. This aer uses a numerical game theoretic model to simulate the behavior of the generation firms and to comare the effects of both systems on the market ower of the generators. It shows that financial Virtual Power Plants are better for society. Keywords: Futures markets, Otions markets, Cournot, Market ower, Electricity, Arbitrage JEL: C7, D43, G13, L13, L50, L94 1 INTRODUCTION 1.1 Market ower and long run contracts Several countries recently decided to liberalize their electricity markets and to organize cometition in electricity generation. They assumed that economies of scale and entry barriers in the generation sector were sufficiently small to make cometition viable. In ractice, the generation market is not always very cometitive. Generators often succeed in driving u rices significantly above cometitive levels. This also haens in markets with low levels of market concentration. Prices above marginal roduction costs have been shown to exist in several markets. 1 Comaring different electricity markets in the US, Bushnell et al. (004) show that California had a relatively unconcentrated generation market but that the lack of long term contracts led to high rice-cost margins in the summer of 000. With long term contracts, generators sell 1 See Borenstein et al. (00), Joskow and Kahn (00), Wolak and Patrick (001) and Wolfram (1999).

3 art of their electricity ex-ante, at a locked-in rice. As a result, generators will behave more cometitively in the sot market. The intuition is that of the durable goods monoolist in Coase s conjecture (Coase 197). See Figure 1. Grah A shows the rofit maximizing rice M for a monoolist who sells only in the sot market, has roduction costs Cq () and faces an inverse demand function Pq (). The monoolist will set the rice such that marginal revenue equals marginal cost. Grah B shows the same situation for a monoolist who signed long term contracts for k units of electricity. In the sot market (the second stage), k units will therefore disaear both at the demand and the suly side. The rofit maximizing rice then equals M and is lower than M. In the first stage, the contracting stage, consumers will take into account that the rice in the sot market will be equal to M. They will only buy long run contracts for electricity at the rice M, as they would lose money otherwise. Hence, the monoolist will receive the rice M in the roduction as well as in the contracting stage. This fact is called the erfect arbitrage condition. M Grah A q () Grah B q ( + k) c'( q ) M c'( q + k) q q Figure 1 Effect of an ex-ante contract on the behavior of a monoolist The study of Bushnell et al. highlights the imortance of long term contracts in electricity markets. There is, however, no consensus on the role of long term contracts in electricity markets. Historically, olicy makers have been oosed to long term contracting. They feared that long term contracts between incumbent generators and retailers might slow down entry in the generation market. They also assumed that long term contracting would decrease the transarency and the liquidity of the sot markets. Illiquid sot markets would lead to inefficient real time roduction decisions, and would also make entry more difficult. A small entrant will have to rely on the sot market to balance the difference between the energy sold and the energy roduced. Currently, olicy makers are becoming more favorable towards long term contracts. They hoe that long term contracts will ease entry in the generation market by reducing the risk for entrants and will reduce market ower in the sot market. Long term contracts will also hel retailers who sell electricity at fixed regulated rices to hedge their rice risks. Nowadays, the olicy debate is whether one should imose the usage of long term contracts or whether generators and retailers will sign the right amount of long term contracts on their own. See for examle Creti and Fabra (004). k

4 1. Virtual Power Plants In this new hilosohy of imosing long term contracts on incumbent generators to mitigate market ower, several Euroean regulators have relied uon a secific tye of contracts: the Virtual Power Plants (VPPs). Such a system is currently used in for examle Belgium and France, and is also being discussed in the Netherlands. With a VPP, the incumbent generator sells art of its roduction caacity to market entrants. This sale of generation caacity remains virtual as no roduction caacity changes hand. Legally, the incumbent generator remains the owner of all its generation lants. Regulators often refer VPPs above a divestiture, because the latter is irreversible, might be more costly and is olitically difficult to imlement. Moreover, as Euroean markets become more integrated, the need for mechanisms to reduce market ower might diminish. VPPs can be imlemented in two different ways. In Belgium, antitrust authorities oblige the incumbent to sell financial VPPs, while in the Netherlands the regulator has been discussing hysical VPPs. The main difference between financial and hysical VPPs is that a hysical VPP is associated with a secific generation lant while a financial VPP is not. If a retailer buys a hysical VPP, he reserves generation caacity of a generation lant i.e. he buys the right roduce one MW of roduction in that generation lant. If a retailer buys a financial VPP, then the retailer receives a ure financial insurance contract. The contract is not linked to any generation lant. If the sot rice increases above a certain level, then the retailer will receive a ayment from the generator. Virtual Power Plants are characterized by a virtual roduction cost S and a maximal roduction caacity k. If a retailer owns a hysical VPP with a roduction cost S and caacity k, then he can decide freely to roduce electricity u to the roduction caacity k, as long as he refunds the generator the roduction costs S. If the retailer owns a financial VPP, he will receive money from the generator when the sot rice is above the virtual roduction costs S. He receives the amount: k max{0, S}. Essentially, hysical and financial VPPs are thus hysical and financial call otions with a strike rice S. Otions might have some advantages comared with futures contracts, which are contracts in which retailers are obliged to use the VPP always at full caacity: Otions allow generators and retailers to hedge quantity risks, while futures can only be used to hedge rice risks. Given that electricity cannot be stored very easily, quantity risks are very imortant in the electricity market, and otions therefore lay an imortant role. Market ower is most ronounced during eriods of eak demand and is characterized by high sot rices. Retailers might sign otion contracts to counter the market ower of generators during these eriods 3. The advantage of hysical VPPs is that because secific roduction lants are assigned to the contracts, the robability that electricity is hysically delivered increases. It might also induce generators to invest more in generation caacity. The disadvantage is that the counterarty risk is larger with hysical otions and that the roduction efficiency decreases when generation lants are scheduled in a more decentralized way.

5 The electricity sector is characterized by a lot of missing markets. Often otions are used to correct these roblems. 1.3 This aer In this aer we look at the strategic effects of hysical and financial VPPs in a Cournot game. We assume that there are only two markets: a VPP market, where retailers buy virtual generation caacity and a sot market. In our set-u, firms decide themselves how many VPPs they sell. There is thus no regulation on the amount of VPPs that has to be sold. In the model there is no uncertainty, so hedging is not an issue. The number of generators is assumed to be fixed. Hence, we do not look at the entry decision of new generation firms. The aer is an extension of Allaz and Vila (1993). They showed that, in a Cournot game, firms have a strategic reason to sell futures contracts, because futures contracts serve as a commitment device for the firms to obtain a larger market share in the sot market. Selling futures leads to a risoners dilemma tye of roblem. All firms sell futures, and as a result the sot rice will decrease. We will use a similar framework as Allaz and Vila to analyze VPPs instead of futures contracts Relation with Chao and Wilson The aer is closely related to recent work of Chao and Wilson (004). They argued that generators should be obliged to sell hysical VPPs to retailers. They see several reasons for this. (1) In the long run electricity markets are contestable and thus more cometitive. () Physical call otions might have better strategic effects than futures contracts. (3) The regulation of market ower might be easier with hysical call otions than with futures. And (4) hysical delivery makes sure that generation is effectively built. Our aer looks at a similar roblem as Chao and Wilson but makes different assumtions. They assume erfect regulation of the number of otions that generators have to sell and free entry in the contracting stage. In our aer we assume a fixed number of firms, and that generators decide themselves about the number of otions they sell. Chao and Wilson assume that generators bid linear suly functions in the sot market while we assume that they behave à la Cournot. BENCHMARK: COURNOT GAME WITH FUTURES CONTRACTS This section exlains the standard Cournot game and the Cournot game with futures contracts (i.e. the Allaz and Vila model). It resents the set u of the model, and the definition of the main variables. The next section then continues with the Cournot game with VPPs. 3 Also the regulator can use otions to aim its regulation more recisely at eriods of high demand, minimizing its intervention in the market. 4 Tthe results of Allaz and Vila deend critically on the assumtions of Cournot cometition, erfect intertemoral arbitrage and observability of the contract ositions. See Willems 004 for references.

6 Our aer considers an oligooly with two identical firms ij, {1,}. Firm i roduces q i units at a roduction cost Cq ( i ) = cq i. Total roduction of both firms is equal to q 1 + q, and the sot rice is given by the inverse demand function:.1 Standard Cournot game = P( q + q ). (1) 1 We start with the standard Cournot game without futures contracts, for which we will use the suerscrit C. The rofit of a firm i is equal to its revenue minus roduction costs: C π i = ( c) qi. () In a Cournot game, firm i maximizes its rofit (), by setting its roduction quantity q i, taking into account that the sot rice deends on the joint roduction of the firms (1). All firms set their roduction level q i simultaneously. The Nash equilibrium of this game is the intersection of the best resonse functions of the layers. To illustrate our aer, we will use a numerical examle in which the inverse demand function is linear, and normalized to Pq () = 1 q. We will write all solutions as function of the cost arameter c. C C The equilibrium roduction quantities and sot rice q () c and P () c are given by C 1 c q i = (3) 3 1 C + c = (4) 3 where q is shorthand for the vector ( q1, q ).. Cournot game with futures contracts If there are futures contracts, then we need to model the game with two stages: a contracting stage and a roduction stage. Figure shows the timing of the game. Contracting Stage Production Stage TIME Generators sell k i Futures at a rice F Generators learn each other s contracting osition Generators sell q i electricity on sot market Figure Timing of the Cournot game with futures In the first stage, the contracting stage, generators decide simultaneously about the number of futures contracts k i they sell to retailers. Each futures contract is a two-sided insurance contract which insures the rice of one unit of electricity. If the sot rice is above the futures rice f, then the generator will refund the

7 retailer the difference of the sot rice and the futures rice f. If the sot rice is below the futures rice, then the retailer will ay the generator the difference between the futures rice and the sot rice. The total ayment of generator i is thus k ( f). 5 After the first stage and before the second stage, each firm learns the contract osition of the other firms. In Figure this haens at time = 1.5. There is therefore erfect information at the beginning of the second stage. In the second stage, the roduction stage, the firms simultaneously set their roduction level q i. Each firm will take the contracting ositions k as given. Firm i s rofit is equal to revenue in the sot market, minus roduction costs and ayments related to the futures contracts. The suerscrit F denotes the game with futures contracts. F i i i π = ( c) q k ( f) (5) We will solve the game by backward induction and derive first the Nash Equilibrium in the second stage of the game as a function of the number of futures sold in the first stage qii F ( k, c). After deriving the second stage equilibrium, we will solve the equilibrium of the first F stage k () c. I.3 Second Stage Firm i maximizes its rofit (5) by setting its roduction q i, taking the contracting osition k as given and taking into account that the sot rice is determined by (1). In our small numerical examle, the equilibrium quantity of the generators in the second stage is: q F II, i 1 c ki + k ( k, c) = 3 j i. (6) The fact that firm i owns futures contracts changes its incentives to roduce in the second stage. Firm i needs to refund buyers of the futures contract for high strike rices. It has therefore less interest in high sot rices, and roduces more in the second stage of the game. Hence, owning futures contracts makes a firm more aggressive in the second stage, i.e. it roduces more, and its reaction function moves outwards. This effect is based on exactly the same intuition as Coase s conjecture as exlained in Figure 1..4 Perfect arbitrage Allaz and Vila, assume erfect arbitrage between the contracting and the roduction stage. This means that there is no rofit to be made by arbitraging between the sot market and the futures market, i.e. F F f = Pq ( ( kc, ) + q ( kc, )). (7) II,1 II, 5 This discussion exlains the futures contract as a financial insurance contract. An alternative exlanation considers the futures contract as a hysical contract. It can be shown that both aroaches are equivalent.

8 Note that (7) imlies that arbitrageurs correctly anticiate the strategic effects of the futures contracts on the sot rice..5 First Stage In the first stage the firms maximize their rofit (5), taking into account that q i is determined by the second stage behavior of the firms (6), the rice by the inverse demand function (1) and the forward rice f by arbitrage condition (7). By selling more futures in the first stage, a generator can change the second stage equilibrium. By selling futures, total roduction in the second stage is increased, leading to a lower rice. This influences firm 1 s rofit negatively. However, selling futures increases the market share of firm 1, which increases rofit. At the otimal number of futures both effects are balanced. This trade-off defines the first stage reaction functions of both firms. In equilibrium, the generators will sell F 1 c ki,1() c = (8) 5 futures contracts in the first stage. The equilibrium rice will be equal to F 1+ 4c I () c =. (9) 5 3 VIRTUAL POWER PLANTS VPPs will not be modeled as ower lants with a constant virtual marginal roduction cost S. Instead we assume that the virtual roduction costs are quadratic. q Virtual Cost = (10) γ The arameter γ is a measure for the size of the Virtual Power Plant. As the VPP becomes larger (growing γ ) the virtual marginal roduction costs will have a lower sloe. The unit of γ is MW /$ We choose this aroach in order to reduce roblems which are related with corner solutions in the model. (See Willems 004 for constant virtual marginal costs.) Another interretation of VPPs with increasing marginal costs is that of a linear bundle of call otions with different strike rices. The generators sell γ bundles of call otions to retailers. Each bundle contains ds otions with a strike rice between S and S + ds. 4 FINANCIAL VIRTUAL POWER PLANTS A Financial VPP is a linear bundle of financial call otions. A financial call otion is a onesided insurance contract which insures retailers against rice increases above the strike rice S '. If the sot rice is above the strike rice, then the generator will refund the retailer the difference between the sot rice and the strike rice: S'. When the sot rice

9 is below strike rice, then there is no ayment. In short, the generator ays the retailer the amount max{ P S ', 0}. If a retailer buys γ bundles, it owns γ ds ' financial call otions with a strike rice S '. The generator ays the retailer the amount ( γds ') max{ S ',0}. The total ayment by the generator to the retailer is equal to ' ' γmax{0, S } ds = γ. (11) 0 The rofit of the generation firm is the sum of the rofit in the sot market and rofit in the financial market. In the sot market the generator sells q 1 units of electricity at a rice. The roduction cost is c. In the financial market, the firm sells γ VPPs at a rice f but it will need to refund the retailers the amount (11). The rofit of the generator is equal to fo πi = ( c) qi + γ( f ) (1) where we use the suerscrit fo for financial otions. 4.1 Second Stage In the second stage generators set their roduction quantity q 1 and maximize their rofit (1), taking into account that the sot rice is determined by (1), and assuming that f and γ are fixed. In our examle, the Nash equilibrium of second stage is 4. Perfect arbitrage q fo II, i 1+ γi + c ( γi γj 1) ( γ, c) = 3 + γi + γj. (13) We assume erfect arbitrage between the first stage and the second stage of the game. The rice for buying a financial VPP needs to be equal to the exected ay-out (11). 4.3 First Stage f = (14) In the first stage of the game, generators will sell bundles of financial VPPs, maximizing their rofit function (1), taking into account that the sot rice, the roduction quantities q and the rice of the bundle f are determined by equations (1) (13) and (14). The Nash Equilibrium in the first stage of the game is equal to: fo c 5 γi () c =. (15) 4 4 The equilibrium rice for electricity is then equal to fo I = c(1+ c). (16) c c

10 5 PHYSICAL VIRTUAL POWER PLANTS Physical otions are more difficult to model than financial otions. The reason for this is that we now also have to model the roduction decisions of the retailers who reserved roduction caacity. If a retailer bought γ bundles of VPPs, then it reserved γ ds ' MW of a virtual lant with a virtual roduction cost equal to S '. This bundle of infinitesimal small roduction lants with increasing roduction costs is equivalent to a virtual roduction lant with a virtual roduction cost V q VC( qv ) =. (17) γ In the second stage the retailer will minimize his rocurement costs for electricity. It can buy electricity on the sot market at a rice, or can use its virtual ower lant to roduce electricity q V at a cost q V. In the otimum, the retailer will use its virtual ower lant u to γ the oint where the virtual marginal roduction costs is equal to the sot rice VC '( qv ) =, or: qv = γ. (18) Equation (18) describes the behavior of the retailers. We can now derive the sot rice when generators roduce q = ( q1, q) with the generation lants which were not reserved, and sold γ = ( γ1, γ) VPPs in the first eriod of the game. The equilibrium rice deends on γ and q, and is determined by the following two equations: Pq ( 1 + q + qv ) =. (19) qv = ( γ1 + γ) The rofit of a generator firm is equal to O πi = q i + γif + γi ( qi + γic ). (0) The first term reflects the revenue from selling roduction with unreserved roduction caacity in the sot market, the second term is the revenue from selling hysical VPPs in the contracting stage of the game. The third term is the revenue received from retailers when they use their VPPs, and the last term reflects the total roduction costs of the firm. Total roduction of firm i is equal to q i + γ i. 5.1 Second Stage Equilibrium In the second stage, generators will simultaneously set their roduction quantities q i in order to maximize their rofit function (0), taking into account that the amount γ of VPPs sold in the first stage are fixed and that the sot rice is determined by equations (19). The second stage Nash Equilibrium in the numerical alication is:

11 O (1 + γj)(1 c ( 1 + γi + γj ) qii, i( γ, c) = 3 + γi + γj. (1) The equation describes how the generators will roduce in the second stage of the game. 5. Arbitrage As in the Allaz and Vila model we assume that there is erfect arbitrage between the first eriod and the second eriod of the game. The erfect arbitrage condition requires that γf + γ = γ. () The left side is the cost of buying γ bundles of hysical otions for a rice f and aying γ for a total roduction of γ units of electricity. The right hand side is the cost of buying γ units of electricity on the sot market at a rice. Hence we obtain that the rice f of a hysical VPP is 5.3 First Stage Equilibrium f =. (3) In the first stage of the game, generators will sell a bundle of hysical otions, maximizing their rofit function (0), taking into account that roduction quantities q, the sot rice and the rice of VPPs f are given by (1), (19) and (3). Generators can only sell a ositive amount of hysical VPPs γ 0. The generators will sell their VPPs in a non-cooerative way. In our examle, the equilibrium in the first stage is γ O II where γ * () c is the root of a olynomial A of the third order: () c = max(0, γ * (c)) (4) ( ) A = 1+ 5c - 4c -(4-10c + 1 c ) γ 3-8 c( c) γ - 16c γ. (5) According to equation (4) generators will sell no hysical VPPs when the roduction cost is below 1/4. For roduction costs above 1/4 generators will sell a ositive amount. The equilibrium rice is given by 6 COMPARISON O I O 1+ c(1 + γi ( c)) () c =. (6) O γ ( c) Figure 3 shows the equilibrium amount of VPPs as function of the cost c. The dotted and continuous line resent the financial and hysical VPPs resectively. Note that generators do not sell hysical VPPs if the roduction cost is below 1/4. I

12 g 1 Amount of VPPs sold c Figure 3 Amount of VPPs sold in equilibrium. Figure 4 shows the final rice as function of the roduction cost of the generators for four different cases. The thick dashed and thick continuous lines are the equilibrium rices of the Cournot game with financial and hysical VPPs resectively. The thin dotted and thin continuous lines are equilibrium rices of the Cournot game with forward contracts, and the standard Cournot game. Price 1 Equilibrium Price c Figure 4 Equilibrium rices Figure 4 shows that rices with financial VPPs are lower than those with hysical VPPs. Prices with forward contracts lie somewhere in between. In the examle total welfare is uniquely determined by the equilibrium rice. A lower rice corresonds with a higher welfare. Financial VPPs are thus referable from a societal viewoint. 7 CONCLUSION This aer shows that there are large differences in the strategic effects of hysical and financial VPPs. It also shows that financial Virtual Power Plants might be referred to hysical Virtual Power Plants. 8 REFERENCES Allaz, B. and Vila, J. L. (1993). " Cournot Cometition, Forward Markets and Efficiency." Journal of Economic Theory, 59, Borenstein, S., Bushnell, J. B., and Wolak, F. A. (00). "Measuring Market Inefficiencies in California's Restructured Wholesale Electricity Market." American Economic Review, 9(5), Bushnell, J., Mansur, E., and Saravia, C. (004). "Market Structure and Cometition: A Cross-Market

13 Analysis of U.S. Electricity Deregulation." Re. No. CSEM WP-16, University of California Energy Institute. Chao, H.-P. and Wilson, R. (004). "Resource Adequacy and Market Power Mitigation via Otion Contracts." University of California Energy Institute Power Conference. Coase, R. H. (197). "Durability and Monooly." Journal of Law & Economics, 15(1), Creti, A. and Fabra, N. (004). "Caacity Markets for Electricity." Re. No. CSEM WP-14, University of California Energy Institute. Joskow, P. L. and Kahn, E. (00). "A Quantitative Analysis of Pricing Behavior in California's Wholesale Electricity Market During Summer 000." Energy Journal, 3(4), Willems, B. (004). "Cournot cometition, financial otion markets and efficiency" CES- Discussion aer 04.14, K.U.Leuven. Wolak, F. A. and Patrick, R. (001). "The imact of Market Rules and Market Structure on the Price Determination Process in England and Wales Electricity Market." Re. No. Working aer 848, NBER. Wolfram, C. D. (1999). "Measuring Duooly Power in British Electricity Sot Market." American Economic Review, 89(4),

14 Coyright the author(s). Discussion aers are in draft form. This discussion aer is distributed for uroses of comment and discussion only. It may not be reroduced without ermission of the coyright holder. Coies of working aers are available from the author.