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 : galli@cerna.ensmp.fr

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

Review Review of Modelsof Models

•Fundamental modelling

•Cost based modelling

•Economic equilibrium

•Agent based modelling

•Quantitative modelling

- Based on stochastic models ( finance )

- Finance & « physical »

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

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

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= =

∂ ∂∂ = − + + ∂ ∂ ∂ ∑ ∑∫ ∫

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)

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

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

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

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

Market Market DataData

Daily average of 24 hourly spot prices

Characteristics of weekly seasonality

then Spot after normalisation

PowernextPowernext EEXEEX Spot Spot

EEX-Powernext +80

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

PowernextPowernext, , EEXEEX: Variograms: Variograms

Before normalisation

After normalisation

Before

After

APX SpotAPX Spot

APX SpotAPX Spot

Variogram before

normalisation

Variogram after

normalisation

Powernext PricePowernext Price & & TemperatureTemperature

T+50°

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

Simulating price knowing TemperatureSimulating price knowing Temperature

Price

Price | | Temp

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

CopulasCopulas

Two bivariate distributions with Gaussian margins

and correlation =0.6

Bigaussian Copula

A Copula based co-simulation.

Copula Gaussian

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!