A Structural Model for Interconnected Electricity Markets

Transcription

1 A Structural Model for Interconnected Electricity Markets Toronto, 2013 Michael M. Kustermann Chair for Energy Trading and Finance University of Duisburg-Essen

2 Seite 2/25 A Structural Model for Interconnected Electricity Markets Table of contents General Structural Model Motivation The Model Influence of Connected Markets Multi Market Structural Model Assumptions Basic Objects Electricity Spot Prices Derivatives

3 Seite 3/25 A Structural Model for Interconnected Electricity Markets General Structural Model The Merit Order From: Forschungsstelle fuer Energiewirtschaft e. V. (FfE) Not continuous, piecewise constant, differences in production technology.

4 Seite 4/25 A Structural Model for Interconnected Electricity Markets General Structural Model Simple Example of a Structural Model Barlow (2002) uses a simple parametrization of the merit order: dd t = κ(θ D t )dt + σdw t C t (P t ) = a 0 b 0 (1 + αp t ) β

5 Seite 5/25 A Structural Model for Interconnected Electricity Markets General Structural Model Typical Structural Model - One Market

6 Seite 6/25 A Structural Model for Interconnected Electricity Markets General Structural Model The European Electricity Market

7 Seite 7/25 A Structural Model for Interconnected Electricity Markets General Structural Model Interconnector Capacity - Germany Net transfer capacity (NTC) as reported by ENTSO-E during Winter 2010/11 for peak-hours (in MW). Country FR NL DK SE CH AT Total Import from Export to

8 Seite 8/25 A Structural Model for Interconnected Electricity Markets General Structural Model Price Variablility due to Interconnectors Figure: Shift in Offer due to Interconnectors

9 Seite 9/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Economic Assumptions 1. Marginal demand is normal but correlated over countries (as in Aid, Coulon, Barlow,...) 2. Heat rates are exponential in capacity used (as in Coulon, Elliot,...) 3. Market supply curve is piecewise (per fuel) affine-linear in fuels prices (generalization of Aid, Coulon,...)

10 Seite 10/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Mathematical Assumptions We assume a probability space (Ω, F, P) which supports our model. On this space, we have: Wt D = (Wt 1,..., W t n ) the brownian motion which is driving demand (generating F D = (Ft D ) t ). Wt S = (W S 1 t,..., W Sm t ) the brownian motion driving fuels prices (generating F S = (Ft S ) t ). F t = Ft D Ft S, F = (F t ) t the market filtration which consists of all information contained in fuels prices and demand.

11 Seite 11/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model The Model For the sake of simplicity, we assume that only one fuel is marginal per electricity market. We have country i, j {1,..., n} and a (common) market minimum price of c R. For each country i, we assume a market supply curve of the following form: Demand is given by C i (x, s) = se a i +b i x + c D i t = f i t + D i t, d D i t = k i (θ i D i t)dt + σ i dw i t, dw i t dw j t = ρ ij dt where f i t denotes the seasonal component. We have 1. D is independent from F S 2. D t F D s N(µ(s, t), Σ(s, t)) and we will omit the arguments of µ and Σ in the sequel.

12 Seite 12/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Cross Border physical Flows We denote the physical flow from country j to country i by E ij t E ij t, 1 i < j n. The maximum capacity is restricted and might depend on the direction of the flow: [ ] E ij min, E max ij, E ij min < 0, E max ij > 0.

13 Seite 13/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Cross Border physical Flows - two Market case Note that in the two market case (i.e. only Et 12 exists), if E min = E max = 0, markets are not connected and thus, pricing might be done independently. E max = E min, the interconnector is never congested and thus, one unique market price for both markets exists at all hours. For the rest of the talk, we will consider the two market case only.

14 Seite 14/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Cross Border physical Flows - two Market case II In interconnected markets, only the electricity which is not imported has to be produced. Thus, the electricity price is determined as P i t (D i t, E t, S t ) = C i (D i t E t, S t ). Here, E t is the imported amount and D i t E t is the residual demand which has to be satisfied by local production. Define: A 1 = {ω Ω : P 1 t (D 1 t, E max, S t ) P 2 t (D 2 t, E max, S t )} A 2 = {ω Ω : P 1 t (D 1 t, E min, S t ) P 2 t (D 2 t, E min, S t )} A 3 = Ω \ (A 1 A 2 ) Then, the cross border flow is E 12 t (ω) = E max, if ω A 1 E min, if ω A 2 a 1 a 2 b 1 +b 2 + b 1 b 1 +b 2 D 1 t (ω) b 2 b 1 +b 2 D 2 t (ω), if ω A 3

15 Seite 15/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Market Clearing Prices Given the cross border physical flow which minimizes price differences between countries, the resulting electricity price for country 1 may be stated as follows: P 1 t (ω) = P 1 t (D 1 t, E 12 t, S t ) = C 1 (Dt 1(ω) E max, S t (ω)), if ω A 1 C 1 (Dt 1(ω) E min, S t (ω)), if ω A 2 C m (Dt 1(ω) + D2 t (ω), S t(ω)), if ω A 3 with C m as specified on the next slide. Equivalent results hold for P 2 t in country 2.

16 Seite 16/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Supply curve in the case of market convergence In the case of market convergence, aggregated demand has to be met by the cheapest production units in both countries. For S t fixed, we thus define (C i ) 1 (y, S t ) = inf{x R : C i (x, S t ) y} and find the aggregated supply curve as C m (x, S t ) = inf{y R : (C 1 ) 1 (y, S t ) + (C 2 ) 1 (y, S t ) x} which has the form C m (x, s) = se am+bmx + c with a m = a 1b 2 +a 2 b 1 b 1 +b 2 and b m = b 1b 2 b 1 +b 2.

17 Seite 17/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Distribution of the market clearing price - limiting cases Assuming lognormal fuels prices, i.e. log(s t ) F S s N(µ S, σs 2 ), we define the generalized lognormal distribution logn(µ, σ 2, c) as the distribution of a absolutely continous random variable X with density f X (x) = 1 e 1 ln (x c) µ 2 ( σ ) 2, x (c, ) 2πσ(x c) then, it obviously holds that Pt i d Fs logn(µ S + a i + b i µ i, σs 2 + b2 i σi 2, c) as E max = E min 0 + Pt i d Fs logn(µ S + a m + b m (µ 1 + µ 2 ), σs 2 + b2 i (σ ρσ 1 σ 2 + σ2), 2 c) as E max = E min

18 Seite 18/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Distribution of the market clearing price In the case of known fuels prices (realistic over short time horizon), we calculate: F P 1 t Fs (x) = Q(P 1 t x F s ) = Q({P 1 t x} A 1 F s ) + Q({P 1 t x} A 2 F s ) + Q({P 1 t x} A 3 F s ). We are able to calculate above Probabilities. It turns out Q({Pt 1 x} A 1 Fs ) = ln(x c) a 1 Φ µ 2 b 1 + Emax 1 a 1 a 2 (b 1 + b 2 )Emax + b 2 µ 2 b 1 µ 1 [ ; σ 2 1 b 2 ρσ 1 σ 2 b 1 σ 1 2 b 2 ρσ 1 σ 2 b 1 σ 1 2 b 2 2 σ2 2 2b 1 b 2 ρσ 1 σ 2 + b2 1 σ2 1 ] and similar (i.e. the differences of bivariate normal expressions with constant covariance matrix evaluated at affine-linear transformations of a vector which depends on the shifted logarithm of x) for the other 2 terms.

19 Seite 19/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Distribution of the market clearing prices in GER and NL

20 Seite 20/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Futures prices in the structural model We first consider futures with immediate delivery. Denote by F i (s, t) the futures price of electricity in country i at time s for delivery in t. Under the risk-neutral measure we should have F 1 (s, t) = E Q s [Pt 1 ] = P t(ω)q(dω) Ω = P t(ω)q(dω) + A 1 P t(ω)q(dω) + A 2 P t(ω)q(dω) A 3 and again, assuming deterministic fuels prices, we find P t(ω)q(dω) = cφ a 1 a 2 E max(b 1 + b 2 ) + b 1 µ 1 b 2 µ 2 + A 1 b1 2σ2 1 2b 1b 2 ρσ 1 σ 2 + b2 2σ2 2 e a 1+b 1 (µ 1 E max )+ b 2 1 σ2 1 2 Φ a 1 a 2 E max(b 1 + b 2 ) + b 1 µ 1 b 2 µ 2 + b 2 1σ 2 1 b 1 b 2 ρσ 1 σ 2 b 2 1 σ2 1 2b 1b 2 ρσ 1 σ 2 + b 2 2 σ2 2

21 Seite 21/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Futures prices depending on Interconnector capacity

22 Seite 22/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Thank you for your attention...

23 Seite 23/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model References EPEX SPOT SE Data Download Center, Rouquia Djabali, Joel Hoeksema, Yves Langer COSMOS description - CWE Market Coupling algorithm, APX Endex, Rene Carmona, Michael Coulon, Daniel Schwarz Electricity Price Modeling and Asset Valuation: A Multi-Fuel Structural Approach Rene Carmona, Michael Coulon A Survey of Commodity Markets and Structural Models for Electricity Prices

24 Seite 24/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Appendix

25 Seite 25/25 A Structural Model for Interconnected Electricity Markets Multi Market Structural Model Price Convergence FR - GER (hourly basis)