Power Market and Models Convergence ?
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CERNA, Centre d’économie industrielle
Ecole Nationale Supérieure des Mines de Paris - 60, bld St Michel - 75272 Paris cedex 06 - FranceTéléphone : (33) 01 40 51 9314 - Télécopie : (33) 01 44 07 10 46 - E-mail : [email protected]
Power Power MarketsMarkets and Models: and Models:
Convergence ? Convergence ?Alain Galli, Nicolas Rouveyrollis
& Margaret Armstrong
ENSMP
Presented at Le printemps de la recherche -EDF, 20 May 2003
Web Site: www.cerna.ensmp.fr
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Review Review of Modelsof Models
•Fundamental modelling
•Cost based modelling
•Economic equilibrium
•Agent based modelling
•Quantitative modelling
- Based on stochastic models ( finance )
- Finance & « physical »
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Models Models derived from derived from financefinance
•Black & Scholes
•Mean reverting (OU) exp (OU)
•Multifactor type models
•Jumps models
•Stochastic volatility models
•Levy processes
• HJM type models
•Garch
•Switching models
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Multifactor Multifactor modelsmodels
Variants of Brennan’s model (for interest rates)
or Gibson-Schwartz extended by Schwartz (for commodity)
( )
( )
S S
C C
S C
dS C dt dWSdC C dt dWdW dW dt
µ σ
κ α σρ
= − +
= − +
=
Drawback:
• C non observable
• 6 parameters
Pilipovic
S ~ OU
C ~ GBM
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HJM type (HJM type (multifactormultifactor))
1
( , ) ( , )( , )
ni
i ti
dF t T t T dWF t T
σ=
= ∑
Clewlow &Strikland (1999)
0 01 1
( , ) ( , )( ) ( (0, ) ( , ) ( , )( )
n nt t i ii ii u i t
i i
u t u tdS t Log F t u t du dW dt t t dWS t t t t
σ σσ σ= =
∂ ∂∂ = − + + ∂ ∂ ∂ ∑ ∑∫ ∫
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Jump Jump modelsmodels
Electricity spot prices show strong variations
Strong variations = Jumps
•Jumps « mean reverting »
•Positive and negative Jumps
Examples
•OU +Jumps (Villaplana - 2003)
•GS two factors +Jumps
•Jump +switching (Roncoroni - 2002)
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Stochastic volatilityStochastic volatility
Example
( )
( ) ( ( )) ( )
S
S
dS dt t dWSt t dt t dW
dW dW dtν
ν
µ ν
ν κ θ ν ξ νρ
= +
= − +
=
Heston
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Switching Switching ModelsModels
( )
~ (0, )t
t t
t
t
r
Ln S
N
rµ ε
ε σ
= +
rt is a Markov Chain
Example (Elliott, Sick & Stein, 2003)
Markov chain = the number of active generators at time t
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Bid based Stochastic Bid based Stochastic ModelsModels
Skantze, P., Gubina, A., & Ilic, M. (2000)
(( )) ()aL tS e b tt +=
L(t) = Stochastic Load
b(t) = Stochastic shift with jumps due to outage
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Comments Comments on Modelson Models
•Most models (except the last ones) are transposed directly fromfinance
•Seasonality is considered not a problem
•From practical point of view similar results can be obtained from
Jumps, Switching and Volatility -If Jump amplitude ~Vol-
•Still few models consider external variables
(eg Temperature,Capacity, Outage,..)
• Many practical studies on markets but few proposals for marketdriven diffusion models
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Market Market DataData
Daily average of 24 hourly spot prices
Characteristics of weekly seasonality
then Spot after normalisation
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PowernextPowernext EEXEEX Spot Spot
EEX-Powernext +80
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PowernextPowernext & & EEXEEX
Average Average Spot Spot Price Price on on Different DaysDifferent Days
Daily average Daily variance
Mon
day Su nday
Mon
day
Su nday
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PowernextPowernext, , EEXEEX: Variograms: Variograms
Before normalisation
After normalisation
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Before
After
APX SpotAPX Spot
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APX SpotAPX Spot
Variogram before
normalisation
Variogram after
normalisation
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Powernext PricePowernext Price & & TemperatureTemperature
T+50°
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PowernextPowernext PricePrice & & TemperatureTemperature
ρ=0.52
ρ = 0.43
Price Skew (1% >2 0% <-2)
25 % in [-2,-0.5] 12% in [0.5 2]
exp(-Temp)
Normalised
Price
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Simulating price knowing TemperatureSimulating price knowing Temperature
Price
Price | | Temp
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Price Price & & TemperatureTemperature: :
Is correlation enough Is correlation enough ??
Cor(P,T) = 0.43
but visually high peaks of Temperature
are strongly correlated to high prices.
•Switching models
•Copulas
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CopulasCopulas
Two bivariate distributions with Gaussian margins
and correlation =0.6
Bigaussian Copula
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A Copula based co-simulation.
Copula Gaussian
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ConclusionConclusion
Initially models were taken directly from finance.
Studies have demonstrated the complexity of thesemarkets and the similarities and differences between them.
Better suited models are starting to be developed, forexample, by incorporating the impact of temperature.
But much work still remains to be done!