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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 : [email protected],[email protected]
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
-8
-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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