Short-term electricity demand forecasting using double seasonal exponential smoothing

TL;DR: The forecasts produced by the new double seasonal Holt–Winters method outperform those from traditional Holt-Winters and from a well-specified multiplicative double seasonal ARIMA model.

Abstract: This paper considers univariate online electricity demand forecasting for lead times from a half-hour-ahead to a day-ahead. A time series of demand recorded at half-hourly intervals contains more than one seasonal pattern. A within-day seasonal cycle is apparent from the similarity of the demand profile from one day to the next, and a within-week seasonal cycle is evident when one compares the demand on the corresponding day of adjacent weeks. There is strong appeal in using a forecasting method that is able to capture both seasonalities. The multiplicative seasonal ARIMA model has been adapted for this purpose. In this paper, we adapt the Holt-Winters exponential smoothing formulation so that it can accommodate two seasonalities. We correct for residual autocorrelation using a simple autoregressive model. The forecasts produced by the new double seasonal Holt-Winters method outperform those from traditional Holt-Winters and from a well-specified multiplicative double seasonal ARIMA model.

Summary

Introduction

• Online electricity demand prediction is required for the control and scheduling of power systems.
• The authors consider online, univariate forecasting of half-hourly data.
• A within-week seasonal cycle, of duration 336 half-hour periods, is evident when one compares the demand on the corresponding day of adjacent weeks.
• The authors adapt the Holt-Winters method so that it can accommodate two seasonalities.
• In the next section, the authors describe how ARIMA models have been adapted for online electricity demand forecasting, in order to capture multiple seasonalities in the demand series.

Multiplicative Double Seasonal ARIMA Models

• The one short-term forecasting method that has remained popular over the years, and appears in many papers as a benchmark approach, is multiplicative seasonal ARIMA modelling.
• The model is often expressed as ARIMA(p,d,q)×(P,D,Q)s.
• The multiplicative seasonal ARIMA model can easily be extended to take care of three or more seasonalities by the introduction of additional polynomial functions of the lag operator and additional difference operators in expression (1).

Double Seasonal Holt-Winters

• The method is only able to accommodate one seasonal pattern.
• This is evident from the recent taxonomies of Hyndman et al.15 and Taylor16.
• The formulation involves separate seasonal indices, Dt and Wt, for the two seasonalities.
• The local s1-period seasonal index, Dt, is estimated by smoothing the ratio of observed value, Xt, to the product of the local level, St, and local s2-period seasonal index, .
• A double additive seasonality method can be developed in a similar way from the standard Holt-Winters method for additive seasonality.

Empirical Comparison of Methods

• The authors carried out empirical analysis in order to address two main issues.
• Secondly, the authors wanted to compare forecasting performance of the new formulation with a well-specified multiplicative double seasonal ARIMA model.
• The data used was 12 weeks of half-hourly electricity demand in England and Wales from Monday 5 June 2000 to Sunday 27 August 2000.
• In practice, interactive facilities tend to be used for special days, which allow operator experience to supplement or override the system offline.
• If a forecasting method is unable to tolerate gaps in the historical series, the special days can be smoothed over, leaving the natural periodicities of the data intact7.

Multiplicative Double Seasonal ARIMA

• The authors used the Box-Jenkins modelling methodology to identify the most suitable ARIMA model based on the 2,688 observations in the estimation sample.
• The autocorrelation function and partial autocorrelation function were used to select the order of the model, which was then estimated by maximum likelihood.
• The residuals were inspected for any remaining autocorrelation.
• Laing and Smith7 explain that, in the multiplicative double seasonal ARIMA formulation in expression (1), polynomials of order greater than two are rarely necessary when fitting a model to half-hourly data for England and Wales.
• The authors investigated differencing and a logarithmic transformation for demand but found neither to improve the SBC.

Holt-Winters Exponential Smoothing

• The authors produced forecasts using the following three Holt-Winters methods: Holt-Winters for Within-Day Seasonality -.
• This is the new Holt-Winters for double multiplicative seasonality, given in expressions (6)-(10), using both a 48-period cycle for the within-day seasonality and a 336-period seasonal cycle for the within-week seasonality.
• 9, the authors also calculated the mean absolute error, root mean square error and root mean square percentage error, but they do not report these results here because the relative performances of the methods for these measures were very similar to those for the MAPE.
• Double Seasonal Holt-Winters outperforms Holt-Winters for Within-Week Seasonality for 38 of the 48 lead times, indicating that there is benefit in using a method that is able to pick up both seasonalities.

Adjusting for Error Autocorrelation in the Holt-Winters Methods

• Inspection of the 1-step-ahead errors, in the estimation sample of 2,688 periods, revealed sizeable first-order autocorrelation for all three Holt-Winters methods, indicating that the forecasts were suboptimal.
• The k-step-ahead forecasts from forecast origin τ are then modified by adding the term λkeτ.
• Chatfield22 found that the modification resulted in improvements in accuracy when applied to the autocorrelated errors from Holt-Winters for multiplicative seasonality.
• This led to far greater improvements in post-sample accuracy than were found using the two-stage estimation approach.
• It is less clear why the new method with residual autocorrelation adjustment outperforms the ARIMA model.

Summary and Conclusions

• Online short-term electricity demand forecasting requires a robust, univariate procedure.
• The authors have shown how the method can be adapted for time series with two seasonalities.
• The Holt-Winters methods were improved by the inclusion of an AR(1) model for the residuals.
• We, therefore, conclude that there is strong potential for the use of the new double seasonal HoltWinters formulation in online short-term electricity demand forecasting.
• Rather than recommending the new method in preference to all others, the authors feel that a more useful approach would be to use several different methods.

Figures

Figure 4 Comparison of MAPE results for the 4-week post-sample period. The Holt-Winters methods included an AR(1) model for the residuals.

Table 1 Holt-Winters parameters calculated from the 8-week estimation sample. No adjustment for autocorrelation in the residuals of the methods.

Table 2 Holt-Winters parameters calculated from the 8-week estimation sample. The methods included an AR(1) model for the residuals.

Figure 1 Half-hourly electricity demand in England and Wales from Monday 5 June 2000 to Sunday 18 June 2000.

Figure 2 Half-hourly electricity demand in England and Wales from Monday 5 June 2000 to Sunday 27 August 2000.

Figure 3 Comparison of MAPE results for the 4-week post-sample. No adjustment for autocorrelation in the residuals of the Holt-Winters methods.

Full text

Short-Term Electricity Demand Forecasting Using
Double Seasonal Exponential Smoothing
James W. Taylor
Saïd Business School
University of Oxford
Journal of Operational Research Society, 2003, Vol. 54, pp. 799-805.
Address for Correspondence:
James W. Taylor
Saïd Business School
University of Oxford
Park End Street
Oxford OX1 1HP, UK
Tel: +44 (0)1865 288927
Fax: +44 (0)1865 288805
Email:
james.taylor@sbs.ox.ac.uk

1
Short-Term Electricity Demand Forecasting Using Double Seasonal Exponential
Smoothing
Abstract
This paper considers univariate online electricity demand forecasting for lead times from a
half-hour-ahead to a day-ahead. A time series of demand recorded at half-hourly intervals
contains more than one seasonal pattern. A within-day seasonal cycle is apparent from the
similarity of the demand profile from one day to the next, and a within-week seasonal cycle is
evident when one compares the demand on the corresponding day of adjacent weeks. There is
strong appeal in using a forecasting method that is able to capture both seasonalities. The
multiplicative seasonal ARIMA model has been adapted for this purpose. In this paper, we
adapt the Holt-Winters exponential smoothing formulation so that it can accommodate two
seasonalities. We correct for residual autocorrelation using a simple autoregressive model.
The forecasts produced by the new double seasonal Holt-Winters method outperform those
from traditional Holt-Winters and from a well-specified multiplicative double seasonal
ARIMA model.
Key words: electricity demand forecasting; Holt-Winters exponential smoothing.

2
Introduction
Online electricity demand prediction is required for the control and scheduling of power
systems. The forecasts are required for lead times from a minute-ahead to a day-ahead. At
National Grid, which is responsible for the transmission of electricity in England and Wales,
online prediction is based on half-hourly data. A profiling heuristic is used to produce forecasts
for each minute by interpolating between each half-hourly prediction. The National Grid one
hour-ahead forecasts are a key input to the balancing market, which operates on a rolling one
hour-ahead basis to balance supply and demand after the closure of bi-lateral trading between
generators and suppliers.
Weather is a key influence on the variation in electricity demand (see Taylor and
Buizza
1,2
). However, in a real-time online forecasting environment, multivariate modelling is
usually considered impractical. A multivariate online system would be very demanding in
terms of weather forecast input and would require default procedures in order to ensure
robustness
3
. Univariate methods are considered to be sufficient for the short lead times
involved because the weather variables tend to change in a smooth fashion, which will be
captured in the demand series itself.
In this paper, we consider online, univariate forecasting of half-hourly data. A time
series of electricity demand recorded at half-hourly intervals contains more than one seasonal
pattern. Figure 1 shows half-hourly demand in England and Wales for a fortnight in June
2000. A within-day seasonal cycle, of duration 48 half-hour periods, is apparent from the
similarity of the demand profile from one day to the next, particularly on weekdays. A
within-week seasonal cycle, of duration 336 half-hour periods, is evident when one compares
the demand on the corresponding day of adjacent weeks. There is strong appeal in using a
forecasting method that is able to capture information in both seasonalities.
***** Figure 1 *****

3
Holt-Winters exponential smoothing is a popular approach to forecasting seasonal
time series. The robustness and accuracy of exponential smoothing methods has led to their
widespread use in applications where a large number of series necessitates an automated
procedure, such as inventory control. This suggests that Holt-Winters might be a reasonable
candidate for the automated application of online electricity demand forecasting. However,
the method is only able to accommodate one seasonal pattern. The multiplicative seasonal
ARIMA model has been extended in order to model the within-day and within-week
seasonalities in electricity demand. In this paper, we adapt the Holt-Winters method so that it
can accommodate two seasonalities. This involves the introduction of an additional seasonal
index and an extra smoothing equation for the new seasonal index.
In the next section, we describe how ARIMA models have been adapted for online
electricity demand forecasting, in order to capture multiple seasonalities in the demand series.
We then show how the Holt-Winters method can be adapted for series with more than one
seasonality. The section that follows presents an empirical forecast comparison of the new
formulation with the standard Holt-Winters method and with a multiplicative double seasonal
ARIMA model. In the final section, we provide a summary and conclusion.
Multiplicative Double Seasonal ARIMA Models
The literature on short-term load forecasting contains a variety of univariate methods
that could be implemented in an online prediction system. The range of different approaches
includes state space methods with the Kalman filter (e.g. Infield and Hill
4
), general
exponential smoothing (e.g. Christiaanse
5
), artificial neural networks (e.g. Lamedica et al.
6
),
spectral methods (e.g. Laing and Smith
7
) and seasonal ARIMA models (e.g. Laing and
Smith
7
; Darbellay and Slama
8
). The most noticeable development in demand forecasting over
the last decade has been the increasing interest shown by researchers and practitioners in
artificial neural networks (see Hippert et al.
9
). Although there is obvious appeal to using this

modelling approach to find the non-linear relationship between demand and weather
variables, its appeal for univariate modelling is far less clear. The one short-term forecasting
method that has remained popular over the years, and appears in many papers as a benchmark
approach, is multiplicative seasonal ARIMA modelling.
The multiplicative seasonal ARIMA model, for a series, X
t
, with just one seasonal pattern
can be written as
()
(
)
(
)
(
)
t
s
Qqt
D
s
ds
Pp
LLXLL
εθφ
Θ=∇∇Φ
where L is the lag operator, s is the number of periods in a seasonal cycle, ∇ is the difference
operator, (1-L), ∇
s
is the seasonal difference operator, (1-L
s
), d and D are the orders of
differencing,
ε
t
is a white noise error term, and
φ
p
, Φ
P
,
θ
q
and Θ
Q
are polynomial functions of
orders p, P, q and Q, respectively. The model is often expressed as ARIMA(p,d,q)×(P,D,Q)
s
.
It is multiplicative in the sense that the polynomial functions of L and L
s
are multiplied on
each side of the equation to give a rich function of the lag operator. Box et al.
10
(p 333)
comment that the model can be extended for the case of multiple seasonalities. The
multiplicative double seasonal ARIMA model can be written as
()
(
)
(
)
(
)
(
)
(
)
t
s
Q
s
Qqt
D
s
D
s
d
s
P
s
Pp
LLLXLLL
εθφ
2
2
1
1
2
2
1
1
2
2
1
1
ΨΘ=∇∇∇ΩΦ
(1)
where s
1
and s
2
are the number of periods in the different seasonal cycles, and and
2
P
Ω
2
Q
Ψ
are polynomial functions of orders P
2
and Q
2
, respectively. This model can be expressed as
ARIMA
. Applying the model to half-hourly electricity
demand, Laing and Smith
21
),,(),,(),,(
222111 ss
QDPQDPqdp ××
7
set s
1
=48 to model the within-day seasonal cycle of 48 half-hours,
and s
2
=336 to model the within-week cycle of 336 half-hours. The forecasts from ARIMA
models of this type are currently used at National Grid. In an application to hourly demand in
the Czech Republic, Darbellay and Slama
8
set s
1
=24 to model the within-day seasonal cycle,
and s
2
=168 to model the within-week cycle.
4