A Structural Model for Coupled Electricity Markets Stolberg, 2014 - - PowerPoint PPT Presentation

A Structural Model for Coupled Electricity Markets

Stolberg, 2014

Michael M. Kustermann | Chair for Energy Trading and Finance | University of Duisburg-Essen

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Table of contents

Motivation Electricity Prices in France and Germany in 2010-2011 Central Western Europe Market Coupling Course of Dimensionality A Structural Model for Coupled Markets Assumptions and Requirements Demand and Fuels Market Supply Curve Cross Boarder Physical Flow Spot Prices Futures Options

Michael M. Kustermann | Stolberg | Mai 2014

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CWE Region

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German and French Power Prices from Jan 2010 to Dec 2011

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German vs French Power Prices - 2010 and 2011

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Definitions

◮ A Market Area is a set of nodes and edges in an electric network,

for which a unique energy price is calculated (’spot’ i.e. day-ahead).

◮ Two market areas A and B are interconnected, if there exists an

edge, which connects a node in A with a node in B.

◮ An edge which connects two market areas is called

interconnector.

◮ The sum over the available capacities of all interconnectors

between A and B is called available (cross boarder) transmission capacity (ATC).

◮ ’Market coupling uses implicit auctions in which players do not

actually receive allocations of cross-border capacity themselves but bid for energy on their exchange. The exchanges then use the available cross-border transmission capacity to minimize the price difference between two or more areas.’ (EPEX SPOT)

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Course of Dimensionality

◮ Consider the case of n interconnected market areas. ◮ This network has at most N = n−1 l=1 (n − l) = n(n−1) 2

ATCs.

◮ If 1 ≤ k ≤ N Interconnectors exist with capacities [El min, El max],

then the set of all possible states of the network can be characterized as [E1

min, E1 max] × [E2 min, E2 max] × · · · × [Ek min, Ek max]. ◮ This cube has

• k

l

• 2k−l l-dimensional Volumes.

◮ Thus, the network can be in k

• l=1
• k

l

• 2k−l = 3k

different states.

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Course of Dimensionality - Example

◮ 1 Volume - Markets are coupled ◮ 6 Areas - Markets are coupled ◮ 12 Vertices - 1 Market is decoupled ◮ 8 Edges - Markets are completely decoupled

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Requirements for a Structural Model for Coupled Markets

We focus on the two market case. A structural model for coupled markets should

◮ be simple. ◮ be tractable. ◮ lead to a closed form formula for the cdf of spot prices. ◮ lead to closed form formulae for futures prices.

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Economic Assumptions

Starting point for our model is the following structure of a hybrid model

◮ price independent demand ◮ market supply curve has exponential shape ◮ fuels prices shift market supply curve multiplicatively ◮ market clearing price is given as intersection of supply and

demand

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Model for Demand and Fuel

We assume Demand in Country i ∈ {1, 2} to be given by Di

t = f i t + ˜

Di

t

d ˜ Di

t = −ki ˜

Di

tdt + σidW i t

dW i

t dW j t = ρi,jdt

where f i

t = βi 1 + βi 2 cos(2π t

24 + βi

3)

+ βi

4 cos(2π

t 168 + βi

5) + βi 6 cos(2π

t 8760 + βi

7)

denotes the deterministic seasonal component.

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Model for Demand and Fuel II

Moreover, we assume that only one fuel might be marginal and is given by d ln(St) = kS(θS − ln(St))dt + σSdW S

t

dW S

t dW 1 t = ρS,1

dW S

t dW 2 t = ρS,2.

It follows   D1

t

D2

t

ln(St)   |Fs ∼ N (µ(s, t), Σ(s, t)) The parameters µ(s, t) and Σ(s, t) are explicitely given in terms of the parameters of the SDEs.

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Model for the Market Supply Curve

We assume the Market Supply Curve in Country i ∈ {1, 2}, Ci, to be given as a function of demand D and fuels price S: Ci(D, S) = Seai+biD + c. I.e. we assume

◮ constant production capacities ◮ production costs consist of fuels cost and fuel price independent

costs (labour costs,...).

◮ exponential dependence of the market clearing price on demand.

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Cross Border physical Flows

We denote the physical flow from country 2 to country 1 by Et. The maximum capacity is restricted and depends on the direction of the flow: Et ∈ [Emin, Emax] , Emin ≤ 0, Emax ≥ 0. Note that, if

◮ Emin = Emax = 0, markets are not connected and thus, pricing

might be done independently.

◮ Emax = −Emin → ∞, the interconnector is never congested and

thus, one unique market price for both markets exists at all hours.

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Cross Border physical Flows in case of coupled markets

In interconnected markets, only the electricity which is not imported has to be produced. Thus, the electricity price is determined as P1

t (D1 t , Et, St) = C1(D1 t − Et, St) = Stea1+b1(D1

t −Et) + c.

Here, Et is the imported amount and D1

t − Et is the residual demand

which has to be satisfied by local production. Define: A1 = {ω ∈ Ω : P1

t (D1 t , Emax, St) ≥ P2 t (D2 t , −Emax, St)}

A2 = {ω ∈ Ω : P1

t (D1 t , Emin, St) ≤ P2 t (D2 t , −Emin, St)}

A3 = Ω \ (A1 ∪ A2) Then, the cross border flow in case of coupled markets is Et(ω) =    Emax , if ω ∈ A1 Emin , if ω ∈ A2

a1−a2 b1+b2 + b1 b1+b2 D1 t (ω) − b2 b1+b2 D2 t (ω)

, if ω ∈ A3

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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: P1

t (ω) = P1 t (D1 t , Et, St) =

   C1(D1

t (ω) − Emax, St(ω))

, if ω ∈ A1 C1(D1

t (ω) − Emin, St(ω))

, if ω ∈ A2 Cm(D1

t (ω) + D2 t (ω), St(ω))

, if ω ∈ A3 The function Cm can be viewed as the aggregated market supply curve for both countries and is given by Cm(D, S) = Seam+bmD + c with am = a1b2+a2b1

b1+b2

and bm =

b1b2 b1+b2 . Equivalent results hold for P2 t in

country 2.

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Distribution of the market clearing prices - limiting cases

Define the generalized lognormal distribution logN(µ, σ2, c) as the distribution with density f(x) = 1 √ 2πσ(x − c) e− 1

2 ( ln (x−c)−µ σ

)2 , ∀x ∈ (c, ∞).

Then it obviously holds that Pi

t|Fs d

→ logN

• ai + bi

Tµ, bi TΣbi, c

• , if Emax = −Emin → 0+.

Here, bi = (b1, 0, 1)T if i = 1 (0, b2, 1)T if i = 2 . And Pi

t|Fs d

→ logN(am + bmµ, b

T mΣbm, c) , if Emax = −Emin → ∞.

Here, bm = (bm, bm, 1)T.

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Distribution of the market clearing prices

Defining Pt =

• P1

t , P2 t

T we find the distribution function: FPt|Fs (x) = Q(Pt ≤ x|Fs) = Q ({Pt ≤ x} ∩ A1|Fs) + Q ({Pt ≤ x} ∩ A2|Fs) + Q ({Pt ≤ x} ∩ A3|Fs) . We are able to calculate above Probabilities. It turns out Q({P1

t ≤ x1} ∩ {P2 t ≤ x2} ∩ A1|Fs) = Φ3

• d(x); Bµ; BΣBT

where Φ3(y; µ; Σ) denotes the cdf at y of the (degenerate) normal distribution with mean µ and covariance Σ. The parameters are

d(x) =   ln(x1 − c) − a1 + b1Emax ln(x2 − c) − a2 − b2Emax a1 − a2 − (b1 + b2)Emax   , B =   b1 1 b2 1 −b1 b2   .

Similar expressions can be found for the other 2 terms.

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Example of the resulting Copulae

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Futures prices in the structural model

We consider futures with hourly 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 have

F 1(s, t) = EQ

s [P1 t ] =

• Ω

P1

t (ω)Q(dω)

=

• A1

C1 D1

t (ω) − Emax, St(ω)

• Q(dω)

+

• A2

C1 D1

t (ω) − Emin, St(ω)

• Q(dω)

+

• A3

Cm D1

t (ω) + D2 t (ω), St(ω)

• Q(dω)

and equivalent for country 2.

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Futures prices in the structural model II

Similar to the calculation for the cdf, we find

• A1

C1 D1

t (ω) − Emax, St(ω)

• Q(dω) =

c · Φ  a1 − a2 − (b1 + b2)Emax − b

T 3 µ

• b

T 3 Σb3

  + ea1−b1Emax +bT

1 µ+ 1 2 bT 1 Σb1Φ

 a1 − a2 − (b1 + b2)Emax − b

T 3 µ − b T 3 Σb3

• b

T 3 Σb3

 

where b3 = (−b1, b2, 0)T. Futures prices for futures with delivery in a set T of hours are then given by F i (s, T) = 1 |T|

• t∈T

F i(s, t).

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Example of futures prices

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Plain Vanilla Calls

Define the following sets Bmax = {ω ∈ Ω : C1(D1

t − Emax, St) ≥ K}

Bmin = {ω ∈ Ω : C1(D1

t − Emin, St) ≥ K}

Bmid = {ω ∈ Ω : Cm(D1

t + D2 t , St) ≥ K}.

Then, the price of a Call with Strike K is given by

EQ

s

• P1

t − K

+ =

• A1∩Bmax

C1(D1

t − Emax, St)dQ

+

• A2∩Bmin

C1(D1

t − Emin, St)dQ +

• A3∩Bmid

Cm(D1

t + D2 t , St)dQ

− K (Q (A1 ∩ Bmax) + Q (A2 ∩ Bmin) + Q (A3 ∩ Bmid))

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Example of a Plain Vanilla Call

Export: The price of an at the money option increases with increasing futures price, but increasing export capacities further reduces variablility of demand and thus the call loses value.

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Thank you for your attention...

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References

EPEX SPOT SE Data Download Center, www.epexspot.com Rouquia Djabali, Joel Hoeksema, Yves Langer COSMOS description - CWE Market Coupling algorithm, APX Endex, www.apxendex.com 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

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Appendix

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Price Convergence FR - GER (hourly basis)

Figure: Price Convergence between France and Germany

Michael M. Kustermann | Stolberg | Mai 2014