New models to compute short-run forecasts of
electricity prices: application to the Spanish
market case
Carolina García Martos, Julio Rodríguez, María Jesús Sánchez 1
Laboratorio de Estadística. Escuela Técnica Superior de Ingenieros
Industriales. Universidad Politécnica de Madrid.
Summary. Due to the last changes in the electricity markets, not only producers but also
consumers need one-day-ahead accurate forecasts of electricity prices. In this article two alternatives to the models used by Contreras et al. (2003) are proposed to predict next day
electricity prices based on ARIMA methodology. Both of them use and model the 24
hourly time series, instead of the complete time series of the prices. This allows taking advantage of the homogeneity of these 24 time series. A design of experiments has been carried out to reach the aim of this work: select the model that leads to smaller prediction errors as well as the preferred length of the time series to build the ARIMA models used to
forecast. A mixed model which combines the advantages of the two new models discussed
is proposed. Some numerical results among the ones obtained (forecasts for each and every
hourly price in the period 1998-2003) for the Spanish market are shown.
Keywords: Time series analysis, electricity markets, forecasting, design of experiments, marginal price.
1. Introduction
Nowadays, in competitive markets, there are two ways to trade with electricity: bilateral contracts and the pool. Referred to bilateral contracts, which is interesting
is to reduce the risk they imply. In the pool both the generating companies and the
consumers submit to the market operator their respective generation and consumption bids for each hour of the forthcoming day. The marginal price is defined as
1
The authors would like to thank financial support of the Project MTM2005-08897 (Ministerio de
Educación y Ciencia, Spain).
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the bidding one submitted by the last generation unit needed to satisfy the whole
demand.
Having short-term accurate forecasts let the producers schedule the production of their units in order to maximize its benefits. The disposal of adequate models to predict electricity prices can be considered of great interest.
Just a few years ago only demand was predicted in centralized markets. For
forecasting electricity prices the techniques applied can be divided into two main
groups: Neural networks and time series models. Neural networks had been used
by Ramsay et al. (1998), Szkuta et al. (1999) and Rodríguez et al. (2004). Besides,
Nicholaisen et al. (2000) had made forecasts for electricity prices using neural
networks filtering the non-linearity of the prices. Models based on time series
have also been used to model electricity prices. Contreras et al. (2003) forecasts
electricity prices of the spanish market and the californian applying ARIMA models. Troncoso et al. (2003) made a comparison between kWNN technique and dynamic regression. Crespo-Cuaresma et al. (2003) propose a group of univariate
models to predict electricity prices in the Leipzig market. Conejo et al. (2005)
compare several methods including wavelets approximation, ARIMA models and
neural networks. Nogales et al. (2005) forecast the prices in the PJM interconexion
through transfer function, showing that the inclusion of explanatory variables does
not reduce significantly the prediction errors.
In this work, two new models to predict short term electricity prices are proposed. Both of them build ARIMA models separately for the 24 hourly time series.
This paper is organized as follows. In section 2 the methodology and the
computational implementation are described. In section 3 the design of experiments carried out its explained. Section 4 presents numerical results for the Spanish market. In section 5 some conclusions are provided.
2. Methodology and computational implementation
Making a brief descriptive analysis of the prices corresponding to the period 19982003, a conclusion is reached: Not only the level of the prices but also the variability of them depend on the corresponding hour of the day.
Two new different models are proposed. The first one, which we will refer to,
from now on as Model 24, forecasts the electricity prices for each of the 24 hours
of the next day using the ARIMA models built for each of the 24 time series. The
second one compute the forecasts for working days using the 24 time series of
these days and the forecasts of the prices in weekends with the data corresponding
to this kind of days. This second model will be hereafter referred as Model 48, be-
1511
cause of the number of series it works with (24 for the working days and another
24 for weekends). In order to determine which model leads to more accurate forecasts a design of experiments has been carried out. This has also let us determine
the preferred length of the time series used to build the models to forecast with.
Till now a default length between two and three months (10 -12 weeks) had been
used to build the models, but a rigorous study to determine the convenience of this
or other length had not been ever made.
The factors which influence is interesting and their respective levels are:
1.
Model: It takes in account if different models have been built (Model
48) or not (Model 24) for working days and weekends.
2.
Length of the time series used to build the model to forecast with.
Levels of this factor:
8 weeks (Level 3),12 weeks (Level 5), 16 weeks (Level 7), 20
weeks (Level 9), 24 weeks (Level 11).
Something remarkable is the great number of models that have been identified
and estimated. The objective is ambitious (computing forecasts of the prices for
the six years considered (1998-2003) with ten possible combinations of the levels
of the two factors under study). The number of models that we have identified is
over 500000, and the automatization of the procedure turned necessary. Identification of all these models had been made using TRAMO. Models are selected using
BIC model selection criteria.
3. Design of experiments
The main objective of this work is to determine whether is interesting or not to
identify, estimate and forecast with different models for working days and weekends, as well as the number of observations in the time series used to build the
models. The day forecasts are being computed for is included as a block. Doing
this, the effect of the day is eliminated. This is because if the price in one day is
rather unexpected, the forecast will not be accurate whatever combination of the
levels of the factors were considered. Also the possible correlation between forecasting errors is eliminated when including the day as a block.
The variable under study, yijt, is the logarithm of the average daily prediction error, chosen to be able to compare our results with previous ones (Contreras et al.
(2003), Conejo et al. (2005)). The factors considered are the model (24 or 48) and
the length of the time series (8, 12, 16, 20 and 24 weeks). The day is included as a
block. The equation for the design of experiments is:
y ijt = μ + α i + β j + γ t + (αβ )ij + uijt
(3.1)
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where μ is a global effect that takes in account the average level of the response (logarithm of the prediction error), αi is the main effect related to the
model, βj is the main effect associated with the length of the time series and an effect like γt takes in account the increasing effect of the block (the day in this particular case). The term (αβ)ij measures the difference between the expected value
of the response and the one computed using a model that does not includes the interactions. An aleatory effect, uijt, includes the effect of all the causes not considered in the rest of the sources of variability in the experiment. The response yijt is
calculated as it follows.
mt
t
1 24 ph − ph
)
yijt = log( ∑
24 h =1
pht
(3.2)
mh
where the forecast of the price pt for the day t in the hour h has been calculated using the model i and estimating the multiplicative ARIMA (p,d,q) x (P,D,Q)
using the previous j observations. The models, that have been identified and estimated for the hourly time series are different depending on the activity (and consequently consumption) of the corresponding hour. The analysis of the residuals
let us check the assumption verification. In some cases ARCH-effects have been
detected and volatility has been modelled. (García-Martos, C. (2006)).
The forecasts,
(
)
m
pth
have been calculated minimizing the expression
2
⎡ h
⎤
E⎢ m
pt − pth ⎥ .
⎣
⎦
Working days and weekends are studied separately. Table 1 shows the
ANOVA table for the design of experiments of working days. Interactions are not
significant, but main effects are.
SOURCE
A: day
B: length
C: model
BC
Residual
TOTAL
Sum of
squares
5026.73
3.89
9.98
0.03
216.54
5285.03
D. f.
903
4
1
4
8043
8955
Mean
square
5.56
0.97
9.98
0.01
0.03
F-ratio
p-value
206.76
36.14
370.95
0.29
0.0
0.0
0.0
0.88
Table 1. ANOVA for working days
Figure 1 shows the convenience of model 48 for working days.
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From Figure 2 the preferred length for the time series used to build the models
is obtained, and is the one corresponding to level 9, which means using the previous 20 weeks to forecast one-day-ahead.
Fig. 1. Main effects (Model, working days)
Fig. 2. Main effects (Length, working days)
For weekends the results that arise from the Analysis of variance are shown in
Table 2.
SOURCE
A: day
B: length
C: model
BC
Residual
TOTAL
Sum of
squares
585.63
0.56
327.64
0.27
279.30
1189.66
D. f.
353
4
1
4
3159
3521
Mean
square
1.65
0.14
327.64
0.06
0.08
F-ratio
p-value
18.76
1.59
3705.73
0.77
0.0
0.0
0.0
0.74
Table 2. ANOVA for weekends
From Figure 3 and 4 it can be stated for weekends that the main effects of the
prediction error is significantly smaller using model 24. Also the convenience of
building the models using the previous 16, 20 or 24 weeks arises from Figure 4,
without significant difference between the corresponding levels.
Fig. 3. Main effects (Model, weekends)
Fig. 4. Main effects (Model, weekends)
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The main conclusions that can be stated from the design of experiments are the
convenience of using the previous 20 weeks to the day we try to compute the forecast, as well as the proposal of a mixed model that forecasts for working days with
model 48 and for weekends using model 24.
4. Numerical results
The average daily prediction error for the full period considered (1998-2003) is
sligthly above 13%. Bearing in mind that forecasts have been calculated for all the
hourly prices in a really representative and large period of time (6 years), the results, in terms of prediction error, reflect the accuracy of the new mixed model
proposed.
Only as an example (since we have calculated forecasts for all the weeks in the period under study), we show here the results for two weeks selected among the
whole period. The first one has been chosen to illustrate the improvement, in
terms of prediction errors, in comparison with the model proposed by Contreras et
al. (2003). It corresponds to the last week of May 2000 (25th-31st). Figure 5 shows
the real values and the forecasts computed. Daily mean errors for this week appear
in Table 3, as well as the prediction errors obtained with the model proposed by
Contreras et al. (2003).
Week 25th−31st May 2000
45
Forecasts
Real prices
40
day 1
day 2
day 3
day 4
day 5
day 6
day 7
price (euro/MWh)
35
30
25
20
15
10
0
20
40
60
80
100
Hours in the week
120
140
160
180
Mixed
model
4.6 %
2.8 %
6.0 %
14.4 %
4.3 %
5.4 %
2.9 %
Contreras
(2003)
4.73 %
4.13 %
3.71 %
6.84 %
6.03 %
6.96 %
3.41 %
Table 3. Daily prediction errors
(25th – 31st May 2000)
Fig. 5. Real prices and forecasts 25th –
31st May 2000
The second week selected which results we show is the third one in April 2002.
Results are shown in Fig 6 and Table 4.
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15th−21st April 2002
50
Forecasts
Real prices
45
day 1
day 2
day 3
day 4
day 5
day 6
day 7
price (euro/MWh)
40
35
30
25
20
0
20
40
60
80
100
Hours in the week
120
140
160
180
Fig. 6. Prices and forecasts (15th – 21st april 2002)
Mixed model
proposed
3.06 %
2.69 %
2.13 %
2.36 %
3.02 %
11.62 %
7.23 %
Table 4. Daily prediction
errors (15th – 21st April
2000)
5. Conclusions
This paper proposes several methods to forecast one-day-ahead electricity
prices, corresponding to the different combinations of the levels of the factors under study.
A complete study to determine whether forecasting with model 24 or 48 has
been carried out as well as the length of the time series. The exhaustive analysis
made. A mixed model to forecast next-day electricity prices is proposed. We recommend computing forecasts for working days with model 48 and with model 24
for weekends. We have also determined the preferred length for the time series to
build the models: 20 weeks. Numerical results provided in section 5 let us check
the performance of the alternative mixed model proposed.
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