«James W. Taylor Saïd Business School University of Oxford Journal of Operational Research Society, 2003, Vol. 54, pp. 799-805. Address for ...»
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
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.
1 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 Buizza1,2). 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 robustness3. 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.
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 Hill4), general exponential smoothing (e.g. Christiaanse5), artificial neural networks (e.g. Lamedica et al.6), spectral methods (e.g. Laing and Smith7) and seasonal ARIMA models (e.g. Laing and Smith7; Darbellay and Slama8). 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
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, Xt, with just one seasonal pattern can be written as
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-Ls), d and D are the orders of differencing, εt is a white noise error term, and φp, ΦP, θq and ΘQ 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 Ls 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
where s1 and s2 are the number of periods in the different seasonal cycles, and Ω P2 and ΨQ2 are polynomial functions of orders P2 and Q2, respectively. This model can be expressed as ARIMA ( p, d, q ) × ( P1, D1, Q1 ) s1 × ( P2, D2, Q2 ) s2. Applying the model to half-hourly electricity demand, Laing and Smith7 set s1=48 to model the within-day seasonal cycle of 48 half-hours, and s2=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 Slama8 set s1=24 to model the within-day seasonal cycle, and s2=168 to model the within-week cycle.
three or more seasonalities by the introduction of additional polynomial functions of the lag operator and additional difference operators in expression (1). Therefore, the annual seasonal pattern in electricity demand could also be modelled. However, it is usual to assume that it is not significant in the context of lead times up to a day-ahead7.
In this section, we have shown how the multiplicative double seasonal ARIMA model is a straightforward extension of the standard multiplicative seasonal model. Motivated by this, and by the fact that exponential smoothing has been a competitive alternative to ARIMA models with a variety of different types of data11, in the next section, we adapt the standard Holt-Winters method for application to series with two seasonalities.
Double Seasonal Holt-Winters Exponential Smoothing Standard Holt-Winters The standard Holt-Winters method was introduced by Winters12 and is suitable for series with one seasonal pattern. The multiplicative seasonality version of the method is presented in expressions (2)-(5). It assumes an additive trend and estimates the local slope, Tt, by smoothing successive differences, (St - St-1), of the local level, St. The local s-period seasonal index, It, is estimated by smoothing the ratio of observed value, Xt, to local level, St.
seasonality is multiplicative in the sense that the underlying level of the series is multiplied by the seasonal index. Holt-Winters for additive seasonality is an alternative formulation, which involves the addition of seasonal factors to the underlying trend. The multiplicative
does not depend on the current mean level. The multiplicative version is much more widely used and so for simplicity, in this paper, we provide only the multiplicative formulation.
It is worth noting that the use of the word “multiplicative” in the context of seasonal ARIMA models is quite different to its use in Holt-Winters exponential smoothing. By contrast with Holt-Winters for multiplicative seasonality, the seasonal effect for multiplicative seasonal ARIMA models does not depend on the mean level of the series.
There is no equivalence between Holt-Winters for multiplicative seasonality and multiplicative seasonal ARIMA models. This point is, perhaps, emphasised by the fact that, although there is an ARIMA model for which Holt-Winters for additive seasonality is optimal13, there is no ARIMA model for which Holt-Winters for multiplicative seasonality is optimal14.
Double Seasonal Holt-Winters Although standard Holt-Winters is widely used for forecasting seasonal time series, the method is only able to accommodate one seasonal pattern. A formulation that can accommodate more than one seasonal pattern has not been considered in the exponential smoothing literature. This is evident from the recent taxonomies of Hyndman et al.15 and Taylor16. The Holt-Winters method for double multiplicative seasonality is given in expression (6)-(10). The method is suitable when there are two seasonal patterns in the time series. 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, Wt − s2. Similarly, the local s2-period seasonal index, Wt, is estimated by smoothing the ratio of observed value, Xt, to the product of the local level, St, and local s1-period seasonal index, Dt − s1.
where α, γ, δ and ω are smoothing parameters. Applying the method to a series of half-hourly demand, one would set s1=48 and s2=336, as in the multiplicative double seasonal ARIMA model of Laing and Smith7. Dt and Wt would then represent the within-day and within-week seasonalities, respectively. A double additive seasonality method can be developed in a similar way from the standard Holt-Winters method for additive seasonality. The formulation in expressions (6)-(10) can easily be extended for three or more seasonal patterns by introducing an extra seasonal index and smoothing equation for each additional seasonality.
Empirical Comparison of Methods We carried out empirical analysis in order to address two main issues. Firstly, we wished to investigate whether the new double seasonal Holt-Winters method offers an improvement on the standard Holt-Winters method in terms of forecast accuracy. Secondly, we 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. It is shown in Figure 2. We used the first 8 weeks of data to estimate method parameters and the remaining 4 weeks to evaluate post-sample forecasting performance. This amounts to 2,688 half-hourly observations for estimation and 1,344 for evaluation. To simplify our comparison of methods, we chose a period that did not contain any ‘special’ days, such as national holidays. Demand on these days is so very unlike the rest of the year that online univariate methods are generally unable to produce reasonable forecasts. In practice, interactive facilities tend to be used for special
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 process of model identification is impractical in an online demand forecasting system, and so the model is chosen offline. We 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. In view of this, we considered polynomials up to order two, but we also checked the autocorrelation function of the residuals for any remaining higher order autocorrelation.
We compared the Schwartz Bayesian Criterion (SBC) for an extensive range of different ARIMA models. We investigated differencing and a logarithmic transformation for demand but found neither to improve the SBC. The model with lowest SBC and satisfactory residuals was the following ARIMA(2,0,0)×(2,0,2)48×(2,0,2)336 model, which we shall refer to as the
Double Seasonal ARIMA model: