Directional Pricing Theory in Electricity *

Transcription

1 Directional Pricing Theory in Electricity Akira Maeda and Makiko Nagaya : Graduate School of Art and Science, The Univerity of Tokyo -8- Komaa, Meguro, Tokyo 5-89, Japan; maeda(at)gloal.c.u-tokyo.ac.jp Telephone: ; Fax: : Faculty of Gloal Buine, Showa Women Univerity Atract: Thi tudy i a firt attempt of invetigating a theory of directional pricing. Directional pricing i defined a price or rate deign that apply different price to elling and uying the concerned good. A typical example would e rate chedule in the feed-in-tariff (FIT) policy for electricity. Thi tudy dicue how the pricing i ditinctive and how that a new development of the theory i eential for the analyi of uch emerging electricity market. Keyword: directional pricing, feed-in-tariff, rate deign, real option, digital option, inary option. JEL Claification Code: D, L94, Q8.. Introduction Standard economic theory preume that price are non-directional: Auming that there i no tranaction cot, the price of tranferring Good X from Agent A to B mut e equal to that of tranferring the good in the oppoite direction. Thi i owing to a imple no-aritrage argument: If the elling price i greater than the uying price for an agent, the agent could make a profit y uying the good only to immediately ell it ack. However, for ome good in the real world, thi may not e the cae. A typical example i of electricity. It i non-torale and thu mut e conumed at the very moment of production, which make it impoile for any agent to take the opportunity of aritrage on a ingle tranmiion line. Becaue of recent development in electricity production and tranmiion technologie, ipolar trade etween electric utility companie and cutomer have ecome popular. In fact, in ome countrie uch a Germany, Spain, and Japan, the feed-in-tariff (FIT) policy ha een introduced to promote renewale energy generation technologie. Under thi policy, electricity old ack to the utility company i growing in trade volume. With thee recent change in Thi reearch wa upported in part y the Environment Reearch and Technology Development Fund (S-()) of the Minitry of the Environment, Japan and the Grant-in-Aid for Scientific Reearch (No. 54) of the Japan Society for the Promotion of Science. In Japan, a FIT policy ha jut tarted in July,. Couture et al. () provide a comprehenive view of the FIT policy acro the world. Note that ome paper lited there including Agnolucci (7), Butler and Neuhoff (8), Couture and Gagnon (), Fouquet and Johanon (8), Leer and Su (8), Menanteau et al. (), Rader and Norgaard (996) addre topic of electricity market deign. While uch topic are cloely related to electricity pricing policie, none of them identifie iue of our interet in thi paper. For further dicuion of the policy in the context of promotion of renewale energy technologie, ee International Energy Agency (IEA) (8).

2 technology and policy, the purpoe of thi tudy i to develop a theory for pricing policy that allow the market to et directional price. The paper i organized a follow: The next ection addree the model that decrie economic agent ehavior under a directional pricing cheme. Section invetigate the mathematic. Section 4 highlight ditinctive feature of the model. Section 5 conclude the dicuion.. The model Conider a ingle electricity utility firm and a ingle, repreentative cutomer. Aume that the cutomer own hi generator. Electricity generated from the cutomer generator can e either conumed y the cutomer or old to the utility firm. Alo aume that thee two partie are connected to each other with a ingle electric tranmiion line. In a traditional electric ytem, electricity flow from the utility firm to the cutomer. With the FIT policy, electricity may flow in the oppoite direction under a different rate cheme. That i, conumer can ell the electricity they produce uing their own generator ack to the utility firm at a price different from the rate they would pay to the utility firm. The conumer generator may not e a conventional thermal power generator. Rather, it may e photovoltaic (PV), wind powered, or powered y another renewale ource. Thee generator may e more cotly than conventional thermal generator; however, owing to the newer energy policy requirement, they are eing widely promoted throughout the country. To cover uch higher cot, the cutomer electricity elling price to the utility firm may e higher than hi uying price from the firm. Hereafter, we conider the ehavior of the electricity cutomer who face two electricity price: One i the uying price that applie when electricity i purchaed from the utility firm. The other i the elling price that applie when the electricity produced y the cutomer i old ack to the utility firm. We introduce the following notation: U : + welfare function of the cutomer C : + cot function of the cutomer generator x : electricity produced y the cutomer y : electricity purchaed from the utility firm z : electricity old to the utility firm : uying price of electricity : elling price of electricity w : compoite good other than electricity m : upper limit of the udget contraint. Figure depict the ituation that we conider in thi tudy. <Inert Figure here>

3 Definition: Directional pricing cheme A cutomer cannot imultaneouly purchae and ell the concerned good. When a cutomer purchae the good, the uying price i applied to the trade. On the other hand, when a cutomer ell the good, the elling price i applied to the trade. For the cutomer who face the directional pricing cheme in electricity, the ehavior i decried y the following optimization prolem: max U x+ y zw, () xyzw,,,.t. C( x) + w+ y m+ z, () y z =, () x, y, z, and (4) x+ y z >. (5) The contraint of Equation () ( y z = ) repreent a pecific feature of the model in that electricity cannot flow in two direction at any given moment, and that the direction are alway excluive. Becaue of thi contraint, the prolem need to e treated a mixed-integer programming (MIP). To implify the analyi, we introduce following aumption: Aumption : The welfare function i quai-linear, i.e., U XW, u X + W Alo, u ( ) atifie the Inada condition, i.e., lim u ( X) = and u ( X) Aumption : X + lim =. For the cutomer generator, there i no fixed cot, i.e., C =. Marginal cot i monotonouly increaing from the origin, i.e., MC =. X With thee aumption, the prolem of () (5) reduce to the following: u x+ y z C x y+ z+ m max xyz,,.t. y z =, x, y, z, and x+ y z >. The Lagrangian for thi prolem i written a follow: u x+ y z C x y+ z+ m+ myz+ λ x+ λ y+ λ z+ δ x+ y z The firt order neceary condition (FONC) for the olution et (,,,,,,, ) x y z λ λ λ δ µ to atify are the following: u x + y z MC x + λ + δ =, (6) There are many textook for mathematical programming, among which Hillier and Lieerman (99) are popular..

4 u x + y z + µ z + λ + δ =, (7) u x + y z + + y + =, (8) λ, λ, λ, δ, µ µ λ δ < x, y, λ =, (9) x y λ =, () z, λ z =, () x + y z, δ ( x + y z ) =, () (It can e either poitive or negative). () Conidering the condition provided in Appendix, it turn out that poile cae for thee FONC are limited to the following: Cae I: x >, λ = ; y =, MC x λ > ; z >, λ = ; δ = = (4) ( ) u x z = λ µ z (5) =. (6) Cae II: x >, λ = ; y >, λ = ; z =, λ > ; δ = ( u x + y ) = MC ( x ) (7) = (8) = + λ + µ y. (9) Cae III: x >, λ = ; y =, λ > ; z =, λ > ; δ = u x = MC x () = λ () = + λ. (). Cutomer ehavior In addition to Aumption and, let u introduce pecific form for welfare and cot function. Aumption : u ( ) ha the form of contant relative rik averion (CRRA): X u( X) = A Aumption 4: C ( ) ha the form of quadratic function: = cx C X 4

5 With thee functional form, Cae I III are further implified. Cae I: The cae indicate that x >, λ >, z >. From (4), the following equation hold: ( ) =, or equivalently x z A ( cx ) A x z cx By the condition (6), thi i equivalent to z c A =. =. () Alo, from (6), cx =, and from (5), λ + µ =. z Becaue of the condition z >, we have: z c A = >, or equivalently > c( Ac) +. The welfare for the repreentative cutomer for thi cae i calculated a follow: U = U( x z, w ) = u ( x z ) C ( x ) + z + m = f ( ) + m, (4) where the function f ( ) i defined a follow: f ( X) A X + X c. (5) Cae II: The cae indicate that x >, y >, λ >. Form (7), the following equation hold: ( + ) =, or equivalently x y A ( cx ) A x y cx By the condition (8), thi i equivalent to y A c + =. =. (6) Alo, from (8), cx =, and from (9), λ + µ =. y Becaue of the condition y >, we have y A c = >, or equivalently < c( Ac) +. The welfare for the repreentative cutomer for thi cae i calculated a follow: 5

6 = ( +, ) U U x y w = + + u x y C x y m = f ( ) + m, (7) where the function f ( ) i defined y Equation (5). Cae III: Thi cae indicate that Ax = cx. Thi i equivalent to + x Ac =. x >, λ >, λ >. From (), the following equation hold: From (), (), and (), cx < c Ac + <. < <, or equivalently The welfare for the repreentative cutomer for thi cae i calculated a follow: U = U x, w ( ) ( ) = u x C x + m = f c Ac + + m, (8) where the function f ( ) i defined y Equation (5). It i noted that function f ( X ) ha it minimum value at + + f( X) f c( Ac) + = A c Thi i proven a elow: f ( X) = A X + X, c f X < and thu, Alo, we have ( ) + + X c Ac. < if and only if + f X = A X + > X. c Thi prove that f ( X ) ha it minimum value at + X= c Ac, a follow:. (9) X= c Ac. 6

7 On the ai of the invetigation of FONC, we can contruct demand or upply curve and the welfare gain for the repreentative cutomer. Becaue we have two price for electricity, i.e., the uying price and the elling price, the cutomer ehavior i claified on the - plane. Region : c( Ac) + min {, } In thi region, among the three cae mentioned aove, only Cae I can apply: Only elling to the utility firm can occur and it invere upply curve i given y Equation (). The welfare i given y Equation (4). Region : < c Ac + < In thi region, among the aovementioned three cae, oth Cae I and II can apply. The choice i up to the cutomer and will e determined y the comparion of poile welfare gain a follow: The cutomer gain i f ( ) + m firm. In thi cae, the invere demand curve i given y Equation (6). On the other hand, the gain i f ( ) + m firm. In thi cae, the invere upply curve i given y Equation (). if he/he chooe to uy electricity from the utility if he/he chooe to ell electricity to the utility Note that there exit an indifference curve that divide the choice of elling and uying. The curve i determined y the following condition: = f ( ) where f < c Ac + <. () Region : { } + max, c Ac a In thi region, only Cae II can apply: Only uying from the utility firm can occur and it invere demand curve i given y Equation (6). The welfare i given y Equation (7). Region 4: < c Ac + < Thi region i equivalent to Cae III where there i neither elling nor uying. In other word, the cutomer chooe to ue electricity in the form of tand-alone. The welfare i given y Equation (8). Figure ummarize thee reult. 7

8 <Inert Figure here> Let u introduce the following function of uying and elling price: Y(, ) : the net demand of electricity purchaed y the cutomer from the utility firm, i.e., y z. Z(, ) : the net upply of electricity old y the cutomer to the utility firm, i.e., z y. υ (, ) : the welfare for the cutomer. By comining all cae of the four region, thee function are written a follow: (, ) where { } Z(, ) Y(, ) υ(, ) = max { φ( ), φ( ) } + m Y = A q qc, () { } q arg max φ, φ J, + c Ac +a J,, =, () a, () X f X I X + f c A c ( I X ) and (4) ( ) X f X I X + f c A c I( X). (5) where f + f + The function I ( ) and (, ) J are index function a follow: if X c ( A c) + I X, otherwie (6) { } if max f, B f S f c A c +a = J( BS, ). otherwie (7) Note that function φ ( X ) of (4) and ( X ) φ of (5) are depicted a in Figure. <Inert Figure here> 4. Ditinctive feature The ehavior of the cutomer under a directional pricing cheme i very complicated in a mathematical form, a wa dicued in the previou ection. For undertanding our feature, let u make ome modification on our original prolem decried y Equation () (5). Conider an alternative pricing cheme that allow the cutomer to make imultaneou trade of uying and elling of good. It doe not uually occur in electricity while it uually happen for other ordinary good. Even in electricity, however, when the cutomer ha two eparate 8

9 line, one of which i ued for uying and the other i ued for elling, then the ituation can occur. Figure 4 depict that ituation where two eparate tranmiion line are intalled. <Inert Figure 4 here> With uch two-line etting, the prolem of () (5) i modified a follow: Typical ilateral trade prolem: max U x+ y zw, () xyzw,,,.t. C( x) + w+ y m+ z, () x, y, z, and (4) x+ y z >. (5) Note that the only difference i the removal of Equation (). With Aumption and, FONC for thi prolem are tated a follow: FONC for the inner olution et (,,,,,, ) x y z λ λ λ δ to atify are the following: u x + y z MC x + λ + δ =, (6) u x + y z + λ + δ =, (7) u x + y z + + λ δ =, (8) λ, x, x λ, y, y z, z δ, λ, From Equation (7) and (8), λ =, (9) λ =, () λ =, () x y z δ x + y z =. () +, = λ + λ or equivalently mut hold for an inner olution to exit. It can e replaced y a imple aritrage dicuion: From (), we have the following: C( x) + w+ m+ ( ) z, where y z: the net purchae. If < hold, the cutomer can make the udget contraint virtually limitle y etting z and uch that z and <. Thu, the price cheme mut e uch that. Figure 5 depict an aritrage opportunity in the two-line etting, which illutrate the aove dicuion. <Inert Figure 5 here> With Aumption and 4, the olution of the prolem i decried a follow: y = and y A c z c A = and c Ac, = if + z = if c Ac +, and 9

10 y = and > c Ac >. z = if + Figure 6 depict the olution on the - plane. <Inert Figure 6 here> It i eay to oerve that thi olution i a part of the olution for our original directional pricing prolem. The difference i that no olution exit for the region of <. Thi fact i a critical weakne of the model when it i applied to the analyi of the FIT policie. Becaue the aic idea of the FIT policy i to promote renewale energy ource that have relatively higher unit cot, < i the mot proale cheme to et. It i concluded that no typical market tranaction and/or conumer ehavior model tudied o far in economic theory can e applied to the analyi of the FIT policie. A i dicued aove, the exitence of olution for the cae of < make the directional pricing cheme very ditinctive. To further clarify thi point, conider the cae of + + c A. From Equation () and (), the net demand and upply curve are depicted a in Figure 7 and 8. Note that ( ) and in thee figure are the olution for the following equation: f = f and (8) ( ) f = f, (9) where + + c A. <Inert Figure 7 here> <Inert Figure 8 here> The unique feature that appear on thee figure i that net demand and upply curve can e dicontinuou. A we have dicued in the previou ection, there exit an indifference curve that divide the choice of elling and uying in the third region of < c Ac + < on the - plane. The dicontinuou feature of net demand and upply curve i due to the indifference curve that i determined y (). It i eay to ee that () i conitent with (8) and (9). 5. Concluion The model analytical olution depict a complicated feature of directional pricing cheme. Thi indicate that the rate deign ha a higher degree of freedom than that of traditional electric utility pricing. The analyi preented here i preliminary in a ene that it only focue on cutomer ehavior; however, it i ufficient to how that typical market tranaction and/or conumer ehavior

11 model tudied o far in economic theory are not feaile for the analyi of emerging market and policy iue with regard to electricity. We elieve that our reult contriute to a etter undertanding of rate deign for newly defined electricity market. Poile future reearch i recommended in two direction. One i further invetigation of the feed-in-tariff (FIT) policy formulation. The current tudy identifie the net demand (or net upply) curve for electricity cutomer a a function of two price elling and uying price. It i uggeted that uing the function a a uilding lock, one can explore market deign iue of the FIT policy. It would alo e intereting to conider an electricity market where n cutomer and m electric utility firm complete with each other. Another direction i to explore the inary feature of directional pricing. We define directional a eing ynonymou with a inary choice of either uying or elling. In fact, there i a parallel intrument in the finance ector financial derivative called inary option or digital option. Therefore, directional pricing propoed here can e interpreted a a inary real option or a digital real option. Note that while there i a long lit of tudie on the concept of real option, virtually none invetigate uch type of real option. Oviouly, more reearch need to e done in thi direction. Reference Agnolucci, P. (7), The Effect of Financial Contraint, Technological Progre and Long-Term Contract on Tradale Green Certificate, Energy Policy 5(6): Butler, L. and K. Neuhoff (8), Comparion of Feed-in Tariff, Quota and Auction Mechanim to Support Wind Power Development, Renewale Energy (8): Couture, T.D., K. Cory, C. Kreycik, and E. William (), A Policymaker Guide to Feed-in Tariff Policy Deign, Technical Report NREL/TP-6A-44849, National Renewale Energy Laoratory, The US Department of Energy. Couture, T. and Y. Gagnon (), An Analyi of Feed-in Tariff Remuneration Model: Implication for Renewale Energy Invetment, Energy Policy 8(): Fouquet, D. and T.B. Johanon (8), European Renewale Energy Policy at Croroad: Focu on Electricity Support Mechanim, Energy Policy 6(): Hillier, F.S. and G.J. Lieerman (99), Introduction to Mathematical Programming, Mcgraw-Hill (TX). International Energy Agency (8), Deploying Renewale: Principle for Effective Policie, ISBN , IEA/OECD. Leer, J.A. and X. Su (8), Deign of an Economically Efficient Feed-In Tariff Structure for Renewale Energy Development, Energy Policy 6(): Menanteau, P., D. Finon, and M. Lamy (), Price Veru Quantitie: Chooing Policie for Promoting the Development of Renewale Energy, Energy Policy (8): Rader, N.A. and R.B. Norgaard (996), Efficiency and Sutainaility in Retructured Electricity Market: The Renewale Portfolio Standard, The Electricity Journal 9(6): 7 49.

12 Appendix : The arrangement of FONC (i) Suppoe x =, and thu, λ >. Then, Equation (6) (8) are reduced to the following: u y z = MC λ δ = λ δ <. Thi contradict to the Inada condition of Aumption. Thu, x > and λ = mut hold true. (ii) With the condition of x >, uppoe y =. Thi mean λ >. Equation (6) (8) and () are reduced to the following equalitie: ( x z ) + = u x z MC x δ u x z + z + + = µ λ δ u x z λ + δ = δ =. Now uppoe x = z >. By the aumption of the Inada, it indicate that u x z = u, which i impoile for the aove equalitie to hold. Thu, δ = mut hold true. x > z and (iii) With the condition of x >, uppoe z =. Thi mean λ >. By Equation (6) (8) and (), the following equalitie mut hold: ( x y ) + + = u x y MC x δ u x + y + λ + δ = u x + y y + = µ λ δ δ + =. Now uppoe x + y =. By the aumption of the Inada, it indicate that u x + y = u, which i impoile for the aove equalitie to hold. Thu, x + y > and δ = mut hold true. Further arranging thee condition, we otain Cae I III.

13 Figure : Analytical frame The curve of = f ( ) f Sell region U increae = + a U f m + c Ac a Buy region No uy, no ell region U = min f ( X) + U increae m a c( Ac) + a = + a U f m Figure : Solution on the - plane φ ( X ) φ ( X) φ ( X ) φ ( X) φ c( Ac) + φ c( Ac) + + c Ac X + c Ac X Figure : Function φ ( X ) and φ ( X )

14 Figure 4: Two-line etting Figure 5: Aritrage opportunity = + c Ac Buy region + c Ac Sell region No uy, no ell region Figure 6: Solution on the - plane for typical ilateral trade 4

15 + c Ac ( ) Y(,) A c Figure 7: Net demand curve a a function of uying price for the cae of + + c A + c Ac ( ) Z(,) c A Figure 8: Net upply curve a a function of elling price for the cae of + + c A 5