A Structural Model for Coupled Electricity Markets Stolberg, 2014 Michael M. Kustermann | Chair for Energy Trading and Finance | University of Duisburg-Essen
Seite 2 A Structural Model for Coupled Electricity Markets | 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
Seite 3 A Structural Model for Coupled Electricity Markets | Motivation CWE Region Michael M. Kustermann | Stolberg | Mai 2014
Seite 4 A Structural Model for Coupled Electricity Markets | Motivation German and French Power Prices from Jan 2010 to Dec 2011 Michael M. Kustermann | Stolberg | Mai 2014
Seite 5 A Structural Model for Coupled Electricity Markets | Motivation German vs French Power Prices - 2010 and 2011 Michael M. Kustermann | Stolberg | Mai 2014
Seite 6 A Structural Model for Coupled Electricity Markets | Motivation 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) Michael M. Kustermann | Stolberg | Mai 2014
Seite 7 A Structural Model for Coupled Electricity Markets | Motivation 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 ) ATCs. 2 ◮ If 1 ≤ k ≤ N Interconnectors exist with capacities [ E l min , E l max ] , then the set of all possible states of the network can be characterized as [ E 1 min , E 1 max ] × [ E 2 min , E 2 max ] × · · · × [ E k min , E k max ] . � � k 2 k − l l -dimensional Volumes. ◮ This cube has l ◮ Thus, the network can be in k � � k 2 k − l = 3 k � l l = 1 different states. Michael M. Kustermann | Stolberg | Mai 2014
Seite 8 A Structural Model for Coupled Electricity Markets | Motivation 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 Michael M. Kustermann | Stolberg | Mai 2014
Seite 9 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets 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. Michael M. Kustermann | Stolberg | Mai 2014
Seite 10 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets 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 Michael M. Kustermann | Stolberg | Mai 2014
Seite 11 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Model for Demand and Fuel We assume Demand in Country i ∈ { 1 , 2 } to be given by t + ˜ D i t = f i D i t t = − k i ˜ d ˜ D i D i t dt + σ i dW i t t dW j dW i t = ρ i , j dt where 2 cos ( 2 π t f i t = β i 1 + β i 24 + β i 3 ) t t + β i 168 + β i 5 ) + β i 8760 + β i 4 cos ( 2 π 6 cos ( 2 π 7 ) denotes the deterministic seasonal component. Michael M. Kustermann | Stolberg | Mai 2014
Seite 12 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Model for Demand and Fuel II Moreover, we assume that only one fuel might be marginal and is given by d ln ( S t ) = k S ( θ S − ln ( S t )) dt + σ S dW S t dW S t dW 1 t = ρ S , 1 dW S t dW 2 t = ρ S , 2 . It follows D 1 t | F s ∼ N ( µ ( s , t ) , Σ( s , t )) D 2 t ln ( S t ) The parameters µ ( s , t ) and Σ( s , t ) are explicitely given in terms of the parameters of the SDEs. Michael M. Kustermann | Stolberg | Mai 2014
Seite 13 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Model for the Market Supply Curve We assume the Market Supply Curve in Country i ∈ { 1 , 2 } , C i , to be given as a function of demand D and fuels price S : C i ( D , S ) = Se a i + b i D + 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. Michael M. Kustermann | Stolberg | Mai 2014
Seite 14 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Cross Border physical Flows We denote the physical flow from country 2 to country 1 by E t . The maximum capacity is restricted and depends on the direction of the flow: E t ∈ [ E min , E max ] , E min ≤ 0 , E max ≥ 0 . Note that, 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. Michael M. Kustermann | Stolberg | Mai 2014
Seite 15 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets 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 t − E t ) + c . t − E t , S t ) = S t e a 1 + b 1 ( D 1 P 1 t ( D 1 t , E t , S t ) = C 1 ( D 1 Here, E t is the imported amount and D 1 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 in case of coupled markets is E max , if ω ∈ A 1 E min , if ω ∈ A 2 E t ( ω ) = a 1 − a 2 b 1 b 2 b 1 + b 2 D 1 b 1 + b 2 D 2 b 1 + b 2 + t ( ω ) − t ( ω ) , if ω ∈ A 3 Michael M. Kustermann | Stolberg | Mai 2014
Seite 16 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets 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: C 1 ( D 1 t ( ω ) − E max , S t ( ω )) , if ω ∈ A 1 P 1 t ( ω ) = P 1 t ( D 1 C 1 ( D 1 t , E t , S t ) = t ( ω ) − E min , S t ( ω )) , if ω ∈ A 2 C m ( D 1 t ( ω ) + D 2 t ( ω ) , S t ( ω )) , if ω ∈ A 3 The function C m can be viewed as the aggregated market supply curve for both countries and is given by C m ( D , S ) = Se a m + b m D + c with a m = a 1 b 2 + a 2 b 1 b 1 + b 2 . Equivalent results hold for P 2 b 1 b 2 and b m = t in b 1 + b 2 country 2. Michael M. Kustermann | Stolberg | Mai 2014
Seite 17 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Distribution of the market clearing prices - limiting cases Define the generalized lognormal distribution logN ( µ, σ 2 , c ) as the distribution with density 1 ) 2 , ∀ x ∈ ( c , ∞ ) . 2 ( ln ( x − c ) − µ e − 1 f ( x ) = √ σ 2 πσ ( x − c ) Then it obviously holds that � T µ, b i T Σ b i , c � d P i , if E max = − E min → 0 + . t | F s → logN a i + b i � ( b 1 , 0 , 1 ) T if i = 1 Here, b i = if i = 2 . ( 0 , b 2 , 1 ) T And T d P i t | F s → logN ( a m + b m µ, b m Σ b m , c ) , if E max = − E min → ∞ . Here, b m = ( b m , b m , 1 ) T . Michael M. Kustermann | Stolberg | Mai 2014
Seite 18 A Structural Model for Coupled Electricity Markets | A Structural Model for Coupled Markets Distribution of the market clearing prices � T we find the distribution function: � P 1 t , P 2 Defining P t = t F P t | F s ( x ) = Q ( P t ≤ x |F s ) = Q ( { P t ≤ x } ∩ A 1 |F s ) + Q ( { P t ≤ x } ∩ A 2 |F s ) + Q ( { P t ≤ x } ∩ A 3 |F s ) . We are able to calculate above Probabilities. It turns out � d ( x ); B µ ; B Σ B T � Q ( { P 1 t ≤ x 1 } ∩ { P 2 t ≤ x 2 } ∩ A 1 |F s ) = Φ 3 where Φ 3 ( y ; µ ; Σ) denotes the cdf at y of the (degenerate) normal distribution with mean µ and covariance Σ . The parameters are ln ( x 1 − c ) − a 1 + b 1 Emax b 1 0 1 d ( x ) = ln ( x 2 − c ) − a 2 − b 2 Emax B = 0 b 2 1 . , a 1 − a 2 − ( b 1 + b 2 ) Emax − b 1 b 2 0 Similar expressions can be found for the other 2 terms. Michael M. Kustermann | Stolberg | Mai 2014
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