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CERNA, Centre d’économie industrielle

Ecole Nationale Supérieure des Mines de Paris - 60, bld St Michel - 75272 Paris cedex 06 - France Téléphone : (33) 01 40 51 91 26 - E-mail : Nicolas.Rouveyrollis@ensmp.fr,Alain.Galli@ensmp.fr

Using Singular Spectrum Analysis

to Model Electricity Prices

Nicolas Rouveyrollis & Alain Galli

Presented at ETE Workshop Leuven, 15-16 Sept 2005

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OutlineOutline

•Part 1 : Introduction

• Modelling Electricity Prices

• SSA

•Part 2 : Price vs consumption

•Part 3 : A simple model for prices including consumption

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Modelling Electricity PricesModelling Electricity Prices

First Step: Split prices into two components.

a) Part with physical meaning

b) Stochastic noise Ex. Lucia & Schwartz (2000)

like a classical signal/noise decomposition.

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However

• Prices are far from being stationary

• Signal is quasi-periodic

So Fourier methods are not suitable.

But physical part has specific temporal behavior (long range)

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addition

SSA

Component <=> % total information

Singular Spectrum AnalysisSingular Spectrum Analysis

Works directly in the time domain

Ref: Broomhead & King (1986)

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Part 1

X=Trajectory Matrix

1 2 M

i 1 i 2 i M 1

N M 1 N M 2 N

x(t ) x(t ) .. .. x(t )...

x(t ) x(t ) .. .. x(t )X...

x(t ) x(t ) .. .. x(t )

X(t1 ), X(t2 ), …, X(ti+1 ), X(ti+2 ),,…, X(ti+M+1 ),…, X(tN ),

M

SSA - Description of the Method

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Part 1

X=Trajectory Matrix

Univariate SSA - Description of the Method

tC X X11 N M

As in PC compute eigenvalues and eigenvectors of C

But different because the trajectory matrix X

explicitly includes temporal correlations

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Part 1

•Multivariate SSA (Broomhead & King,1986)Multivariates series

•Monte-Carlo SSA (Allen & Smith, 1996)Tests, confidence intervals

•Multiscale SSA (Yiou, Sornette & Ghil,2000)Link with Wavelets

SSA - Extensions

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Part 2 : Prices Vs Consumption

• Day-ahead Powernext prices• Powernext Volumes • RTE Consumption

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Powernext Base Load

LogPrix

-3

-2

-1

0

1

2

3

4

5

6

7

8

Log of Baseload

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Daily Volumes for Powernext

Volume

0

500

1000

1500

2000

2500

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Daily French ConsumptionFrom RTE –web site

RTE CR

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

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SSA on Powernext (prices & volumes) and French consumption

0

10

20

30

40

50

60

0 2 4 6 8 10

LnSpot Volume RTE

First 10 eigenvalues

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SSA on Powernext (prices & volumes) and French consumption

Volume Consumption Ln(Spot)

 Components

% Var % Var % Var

Trend cp1,cp2 56.68 cp1,cp2 60.01 cp3 6.89Weekly periodicity ? ? cp3-cp10 24.95

cp1,cp2,cp4-cp10 42.16

Residual cp3-cp200 43.32 cp11-cp200 15.04 Cp11-cp200 50.94

Components Components

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SSA Illustration

0

500

1000

1500

2000

2500

Volume Trend Volume

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SSA Illustrations

-3

-2

-1

0

1

2

3

4

Residu LnSpot Residual Ln(Spot)

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SSA Illustrations

-1.5

-1

-0.5

0

0.5

1

1.5

Résidu RTE

Residual Consumption

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SSA Illustrations

-1000

-500

0

500

1000

1500

Résidu Volume

Residual Volume

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Rationale behind the use of SSA on market data

The less we have to explain- the easier it will be to predict

Because the long range components generally

have a physical meaning they are easier to:

• Estimate

• Extrapolate

• Correlate

Only the short range data really needs a complex model

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Part 3 : Models

1. Barlow’s Approach2. An Alternative Model

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Volume

=D

-1

1

t

( ) si S

si t t

t

g D D MK D M

Physical

Limit

Sell (Volume))g ( Volume

Barlow’s Approach

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Existence, observability & definition of the physical limit

Correlation Powernext volume / total consumption in France

or Strong Link between Powernext volume & price

Prerequisites for Barlow’s Approach

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y = 0.0784x + 8E-17R2 = 0.0061

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-6

-4

-2

0

2

4

6

8

10

-4 -2 0 2 4 6 8

Log Prices

Volume

The random component for prices is not The random component for prices is not explained by Powernext volumesexplained by Powernext volumes

Drawback of Barlow’s Approach to the French market

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The random component for prices is better The random component for prices is better explained by consumption dataexplained by consumption data

Alternative Approach to the French market

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Residu LnSpot Regression

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Based on consumption data

Advantages• Dynamics of consumption data are closer to

those of prices than the Powernext volumes.• Forecasts are available

(See for example www.rte-France.fr)

Alternative models

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Focusing on the short range components for Comsumption & Log prices

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Residu LnSpot Regression

01-03-2002 to 31-03-2004

Strong correlation

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0 100 200 300 400 500 600 700 800 900-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 réelestimation

Promising Approach

Real

Estimate

An alternative modelAn alternative model

Short range component of logprices

=f(short range component of consumption,Noise)

(stochastic mean reversion)

( ) (pure jump process)

. (short-range consumption factor)

( )

t t t

t Y t Y t

t t

t t t t

dX X dW

dY Y dN

C a SRc b

SRLs C f t X Y

28-11-2001 to 31-03-2004

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Link Prices & Consumption• = Physical nature of Prices. +Noise.• Allows scenario testing. SSA • Natural decomposition « signal/Noise » • Allows us to compare the stochastic

aspects of markets.

ConclusionConclusion

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This work has been done with the support of:– CapGemini,– Electrabel,– EDF,– Gaz de France - Gaselys,– Poweo,– Powernext,– RTE

Aknowledgement Aknowledgement

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