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國立雲林科技大學
National Yunlin University of Science and Technology
ARIMA Models to Predict
Next-Day Electricity Prices
Advisor ：Dr. Hsu
Graduate： Keng-Wei Chang
Author ：Javier Contreras
Rosario Espinola
Francisco J. Nogales
Antonio J. Conejo
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL.18 NO.3, AUGUST 2003
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Outline
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Motivation
Objective
Introduction
ARIMA TIME SERIES ANALYSIS
NUMERICAL RESULTS
Conclusions
Personal Opinion
Review
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Motivation
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There are usually incorporate two instruments for trading
in the electricity markets：
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the pool；
bilateral contracts
For both cases, predicting the prices of electricity for
tomorrow or for the next 12 months is of the foremost
importance.
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Objective
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Price forecasts are developed in bidding strategies or
negotiation skills in order to maximize benefit.
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Introduction
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Therefore, an accurate price forecast for an electricity market
has a definitive impact on the bidding strategies by producers
or consumers.
Auto Regressive Integrated Moving Average (ARIMA)
This paper focuses on the day-ahead price forecast of a daily
electricity market using ARIMA models.
Box & Jenkins 於1976年提出ARIMA模式，認為時間數
列未來的變動會依其過去的資料型態而變動，且運用該
模式進行預測時，時間數列的平均數與共變異數必須是
固定不變的穩定過程，亦即資料達定態，其型態不隨時
間而改變。
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ARIMA TIME SERIES ANALYSIS
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ARIMA processes are a class of stochastic processes
used to analyze time series.
The general scheme is as follows：
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step 0) A class of models is formulated assuming certain
hypotheses.
step 1) A model is identified for the observed data.
step 2) The model parameters are estimated.
step 3) If the hypotheses of the model are validated, go to
step 4, otherwise go to step1 to refine the model.
step 4) The model is ready for forecasting.
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step 0)
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A class of models is formulated assuming certain hypotheses.
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A general ARIMA formulation is selected to model the
price data.
 ( B) pt   ( B) t .....(1)
pt is the price at time t
 ( B) and  ( B) are functions of the backshift operator B : B l pt  pt l
 t is theerror term.
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Example：
 ( B)  (1  1 B1  2 B 2 )(1  24 B 24  48 B 48 )  (1  168B168 )(1  B)(1  B 24 ).....( 2)
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step 1)
A model is identified for the observed data.
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A trial model, as seen in (1), must be identified for the
price data.
In a trial, the observation of the autocorrelation and
partial autocorrelation plots of the price data can help to
make this selection.
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step 2)
The model parameters are estimated.
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After the functions of the model have been specified, the
parameters of these functions must be estimated.
The SCA System is used to estimate the parameters of
the model in the previous step.
The parameter estimation is based on maximizing a
likelihood function for the available data.
Additional information for outlier detection and
adjustment can be found.
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step 3)
If the hypotheses of the model are validated, go to step 4,
otherwise go to step1 to refine the model.
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A diagnosis check is used to validate the model
assumptions of step0.
If the hypotheses made on the residuals are true.
Residuals must satisfy the requirements of a white noise
process：zero mean, constant variance, uncorrelated
process and normal distribution.
If the hypotheses on the residuals are validated by tests
and plots, then, the model can be used to forecast prices.
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step 4)
The model is ready for forecasting.
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The model from step2 can be used to predict future
prices (24 hours ahead).
Due to this requirement, difficulties may arise because
predictions can be less certain as the forecast lead time
becomes larger.
The SCA System is again used to compute.
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Result
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Spanish electricity markets, year 2000
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Californian electricity markets, year 2000
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NUMBERICAL RESULTS
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Spanish, last week of May 2000
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The daily mean errors are around 5%
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NUMBERICAL RESULTS
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Spanish, last week of August 2000
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The daily mean errors are around 8%
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NUMBERICAL RESULTS
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Spanish, third week of November 2000
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The daily mean errors are around 7%
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NUMBERICAL RESULTS
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Californian, third week of April 2000
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The daily mean errors are around 5%
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NUMBERICAL RESULTS
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MWE：Mean Week Error；
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s R ：Standard deviation of the error terms
FMSE：Forecast Mean Square Error
FMSE 
168
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 ( pt  pt )2
i 1
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NUMBERICAL RESULTS
11~12
1
2
Fig. 5. Electricity prices vs. available daily hydro production:
September 1999 to December 2000 in the Spanish market.
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Conclusions
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The Spanish model needs 5 hours to predict future prices,
as opposed to the 2 hours needed by the Californian
model.
These differences may reflect different bidding structures
and ownership.
Average errors in the Spanish market are around 10%,
and 5% in the stable period of the Californian (around
11% considering the three weeks, and without
explanatory variables).
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Personal Opinion
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