Short-term Forecasting of Nodal Electricity Demand in New Zealand Erwann Sbaï and Michael Simpson 1 December 2008 1 University of Auckland. This work is part of a Honours dissertation written by Michael Simpson under the supervision of Erwann Sbaï.
Abstract This paper compares the accuracy of several models for short-term electricity load forecasting for nodes in New Zealand. The literature on short-term electricity load forecasting is reviewed, resulting in the selection of the seasonal ARIMA model and the double seasonal Holt-Winters exponential smoothing model for empirical testing. We also estimate a one season ARIMA model with some double seasonal aspects. A multivariate regression model is also developed for comparison, utilising aspects of several multivariate models used in other short-term electricity load forecasting papers. A time series of half-hourly electricity demand in the Hayward node in New Zealand is used for estimation and forecasting. The data clearly indicates a daily and a weekly seasonal pattern. The seasonal ARIMA model outperforms the Holt- Winters and multivariate regression models in forecasting electricity demand from half an hour to three hours ahead. The superiority of the seasonal ARIMA model increases as the forecasting length increases. Significant autocorrelation of residuals in the Holt-Winters and multivariate regression models leads to an error adjustment term being included, which improves the forecasting accuracy of the Holt-Winters models but not the multivariate regression model.
1. Introduction Accurate forecasting of short-term electricity load is important for all participants in the New Zealand electricity spot market. Generators require accurate forecasts in order to determine the most efficient method of producing the load needed in the forthcoming periods, and to submit supply schedules to the spot market for each half-hour period of the day, in order to maximise profits with respect to the other generators bids. Transpower, the systems operator, utilises forecasts of demand in forthcoming periods to determine the lowest cost method of providing the required quantity of electricity for each node in New Zealand, given the supply bids submitted by each generator. Overestimation of the quantity required results in financial losses from generators being on standby that are not needed, while underestimation can result in the costly startup of cold generators or, in the worst case scenario, the frequency dropping from the required 50Hz, with the potential tripping of generators resulting in a blackout (Bunn, 1982). The New Zealand electricity transmission grid is divided into 244 nodes, where generated electricity exits the high-voltage network and is distributed to customers. Electricity generators in New Zealand submit a supply function at least two hours ahead for each node they are willing to supply electricity to for each half hour period of the day. The system allows each generator to submit five price bands, and the quantity they are willing to supply at each price. Transpower forecasts the quantity required in each node for the approaching half hour, and calculates the quantities each generator should supply to each node in order to achieve the lowest cost. Each generator is paid the price of the last quantity required to meet the demand
forecast in each node. This mechanism is illustrated in Figure 1, in which Generators 1, 2 and 3 submit their supply schedules to the systems operator. Transpower determines that 540MW are needed for the approaching half hour. Adding the quantities provided at the lowest cost from the three generators to cover the required quantity results in a price of $75.02 per megawatt generated. Figure 1: Price Determination in Electricity Spot Market Source: Genesis Energy The importance of accurate forecasting and the power of computer algorithms in estimating complex models have encouraged extensive research into forecasting models for short-term electricity demand. Seasonal ARIMA and Holt-Winters exponential smoothing have been widely used as they require only the quantity demanded variable, and are relatively simple and robust in forecasting. A variety of multivariate models have also been empirically tested in other papers. In this paper the forecasting ability of seasonal ARIMA, Holt-Winters exponential smoothing and multivariate regression models will be empirically tested on data from the
Hayward node in New Zealand which, due to its relatively small size, is far more unpredictable than aggregate country demand data used for empirical testing in other papers. The following section discusses the relevant literature on the three short-term forecasting models. We then introduce the data used for estimation and forecasting, and outline the structure of the models to be estimated. Section 4 discusses the estimation of the models and empirical testing, followed by a summary and a conclusion.
2. Literature Review The Holt-Winters exponential smoothing model was pioneered by Winters (1960) in a comparison of forecasting methods for sales, as a modification of existing exponential smoothing models. The basic exponential smoothing model forecast the value of a time series variable in the next period to be a weighted average of the actual value in the current period, and the previous period s forecast for the current period. The Holt-Winters model added a seasonal ratio and linear trend to this model, to allow for seasonal variations and changes in the underlying mean over time. Additional modifications have been suggested in order to improve forecasting accuracy. Chatfield (1978) noted significant autocorrelation of residuals in comparing Holt-Winters models to Box-Jenkins ARIMA models for forecasting ability, and used a method suggested by Reid (1975) for improving the forecasting ability of exponential smoothing. The adjustment involves fitting a linear relationship λe t to the one-step-ahead forecast. The parameter lambda can be estimated after the estimation of the other parameters in the model, or simultaneously, which Chatfield suggests may be more efficient. A second major modification to the Holt-Winters exponential smoothing model is the addition of a second seasonal index, introduced by Taylor (2003) when comparing univariate methods for short-term electricity demand forecasting in England and Wales. The double seasonal model achieves more accurate forecasts than the traditional Holt-Winters model, since electricity demand clearly has both a daily and a weekly seasonal pattern. Taylor found that by combining the double seasonal Holt-Winters model with an adjustment for error autocorrelation, forecasting accuracy was improved. In this paper, one and two season Holt-Winters models will
be estimated with and without the error adjustment term, and a new estimation method will be discussed in order for the model to be estimated with relatively little knowledge of programming. The multiplicative seasonal ARIMA model is well established in short-term load forecasting literature, and is often used as a benchmark to compare alternative methods to. Box, Jenkins, & Reinsel (1994) noted that the model can be extended to include multiple seasons. Darbellay and Slama (2000) found that the double seasonal ARIMA model outperformed the nonlinear artificial neural network model in forecasting hourly electricity demand in the Czech Republic. The suggested reason for the superiority of the linear ARIMA model was the linearity of the autocorrelation in the data. Taylor (2003) compared double seasonal ARIMA with the double seasonal Holt-Winters model and found ARIMA to be less accurate. However, Taylor (2006) used an ARIMA model once again to compare against a variety of nonlinear models, and found ARIMA to be more accurate than all but one. Despite the success of the double seasonal ARIMA model, the lack of an available estimation method will limit this paper to a one season ARIMA model, which will contain certain aspects of a double seasonal model, though not in the orthodox format. 2 Multivariate models for short-term load forecasting are less common than univariate models, due in part to their impracticality for real-time forecasting. Weather variables are the major factors affecting electricity demand, and gathering data for predictions in the short-term would be very demanding (Bunn, 1982). Also, weather variables tend to change in a smooth fashion, which may already be reflected in lagged demand values in univariate models (Taylor, 2003). 2 We restrict our self to the use of one econometric software: Eviews 6.0.
Some models have been developed that achieve accurate results, particularly in longer-term forecasts. Hyde and Hodnett (1997) developed a regression model to predict daily load demand in Ireland, to replace the Electricity Supply Board s method of forecasting based on observed loads from a similar day and adjusting based on weather variables. The proposed model consisted of a normal level for the day type, weather effects, and special events such as power outages or industrial strikes, estimated using regression analysis. An error adjustment term was also included, allowing the model to adjust its forecast based on short-term deviations from previous days forecasts. Favourable results were reported for most times of the year, and it was proposed that periods of significant deviation from the forecast be dealt with using rule-based procedures for handling special days. Cottet and Smith (2003) developed a vector autoregressive model using temperature and humidity variables, and dummy variables for day of the week and public holidays, to estimate separate regressions for each of the 48 half-hour periods of the day. The model is able to predict not only distribution of the daily load, but also the time and quantity of the daily peak load. The resulting accuracy is difficult to judge, as other models were not used for comparison. Our work will combine aspects of these papers to develop a simple multivariate regression model which includes temperature measurements and dummy variables for half-hour of the day instead of separate regressions for each half-hour, to allow for the inclusion of lagged quantity variables.
3. Data and Econometric Models Data The data consists of a time series of half-hourly observations of electricity demand from the Hayward node in New Zealand for the one year and four week period from July 212005 to August 162006. This period was chosen because the univariate models will only use the last twelve weeks from May 252006 to August 162006, which contain only one public holiday and no changes in daylight savings, thus preventing unnecessary challenges to the models. The last four weeks of the sample will be used to test the forecasting accuracy of all of the models, and the previous eight weeks will be used to estimate the seasonal ARIMA and exponential smoothing models. The first year, from July 212005 to July 202006, will be used to estimate the regression model, since the model requires at least one year of data to estimate all of the parameters. Figure 1 shows demand in the Hayward node for each half-hour on July 182006. The variation in demand across time is apparent, with peaks in demand at 8am and 6pm, a trough in the middle of the day, and a deeper trough around 2am.
Figure 1: Electricity Demand in the Hayward Node on 18 July 2006 14000 12000 10000 Demand (kw) 8000 6000 4000 2000 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 Half-hour Figure 2 shows demand in the Hayward node across a two week period in July 2006. The strong daily pattern is reflected in almost all of the twenty-four hour periods shown, with some variations in the magnitude of the peaks in demand. The two weeks have a similar overall pattern, with lower than average demand on Monday, Saturday and Sunday, although even this is variable, with demand on the Saturday of the first week as high as the Wednesday, Thursday and Friday of that week. These variations in the overall pattern will make forecasting more difficult, especially over a longer timeframe than one period ahead. Despite variations in the seasonal pattern, our data clearly has two seasonal patterns, one over 48 periods and one over 336 periods.
Figure 2: Electricity Demand in the Hayward Node 16 July 2006 to 29 July 2006 14000 12000 10000 Demand (kw) 8000 6000 4000 2000 0 1 49 97 145 193 241 289 337 385 433 481 529 577 625 Half-hour Hourly temperature observations for Wellington City were retrieved from the National Institute of Water and Atmospheric Research. Half-hourly observations were not available, resulting in the same temperature observation being used for both periods in each hour. Some smoothing of the data - so that periods between the recorded hourly observations took the average of the two recorded observations may be more accurate, but the sheer scale of observations in the one year and four month period made this approach impractical. Figure 3 shows temperature observations over the same two week period as Figure 2. The variation and lack of a consistent seasonal pattern may explain some of the variation in demand that is not attributed to the time of day or the day of the week, hence allowing for more accurate forecasting.
Figure 3: Temperature in Wellington 16 July 2006 to 29 July 2006 14 12 Temperature (degrees Celcius) 10 8 6 4 2 0 1 49 97 145 193 241 289 337 385 433 481 529 577 625 Half-hours Stationarity In order for a reliable econometric model to be estimated, some knowledge of the trends in the data is needed. An important consideration in time series models is whether the data is stationaryif a data series is nonstationary, with the mean or the variance changing over time, then the model may be mis-specified and lead to a spurious regression. The stationarity of the quantity series must be tested, both for the twelve week period for the ARIMA and Holt- Winters models, and for the longer period for the multivariate regression model. If the longer period is found to be nonstationary, and the temperature variable is also nonstationary, the two variables should be tested for cointegration before differencing is initiated.
Multiplicative or Additive Model Selection Both the Holt-Winters exponential smoothing model and the seasonal ARIMA model can be specified in either multiplicative or additive form. The multiplicative form is appropriate in cases where the magnitude of the seasonal variation of the data increases as the mean of the data increases (Taylor, 2003). In other short-term load forecasting papers, multiplicative models have been used, suggesting it is the correct choice. Testing for this condition in our data ensures the correct model is used. A seven-day moving mean is calculated for each observation for a six month period. This is subtracted from each observation to find the deviation from the moving mean. For each week in the six month period, the average of the moving mean and the average of the absolute deviation from the moving mean are determined. The correlation between these averages is over 0.95, which indicates that for our data, seasonal variation is positively related to the mean level, suggesting multiplicative models should be used. One Season Holt-Winters Exponential Smoothing The one season Holt-Winters model is a modification of the exponential smoothing model. It consists of a local level S t which is equivalent to a deseasonalised local mean, a local trend T t which is the difference between local levels (S t S t-1 ), and a seasonal index I t, estimated by dividing the actual observation X t by the local level. The specification of the variables is outlined below. Level S t = α (X t / I t-s ) + (1-α) (S t-1 + T t-1 )
Trend T t = β (S t S t-1 ) + (1-β) T t-1 Index I t = ω(x t / S t ) + (1- ω) I t-s Forecast X t+k = (S t + kt t ) I t+k-s where α, β and ω are smoothing parameters between zero and one, s is the number of periods in the seasonal cycle, and X t+k is the k-step ahead forecast. Since the data on electricity demand has two seasonal components, the most prominent seasonal cycle must be chosen for the one season exponential smoothing model. Choosing a season of 336 periods will incorporate both the effect of the day of the week and the half-hour of the day into the model, since the seasonal index will be taken from the same half-hour and the same day of the week from the previous week, while choosing a season of 48 periods ignores the effect the day of the week has on demand. Therefore, choosing a season of 336 periods is more sensible. Double Seasonal Holt-Winters Exponential Smoothing As described in section 2, the Holt-Winters exponential smoothing method has been adapted for data containing two seasons by Taylor (2003). The major addition is a second seasonal index, resulting in a daily index D t and a weekly index W t. There are also changes in the formulation of the level and the forecast to include the effect of the second season. Level S t = α (X t / (D t-s1 W t-s2 )) + (1-α) (S t-1 + T t-1 ) Trend T t = β (S t S t-1 ) + (1- β) T t-1
Daily Index D t = δ (X t / (S t W t-s2 )) + (1-δ) D t-s1 Weekly Index W t = ω (X t / (S t D t-s1 )) + (1-ω) W t-s2 Forecast X t+k = (S t + kt t ) D t+k-s1 W t+k-s2 where s1 is the number of periods in the first season (48), and s2 the number of periods in the second season (336). Error Adjustment for Holt-Winters Models Holt-Winters estimation may be improved using a simple AR(1) error correction term, of the form e t = λe t-1 +ε t. For forecasting, the forecast equation above will be expanded to as follows: X t+k = (S t + kt t ) D t+k-s1 W t+k-s2 λ k e t Forecasts for more than one period ahead will not have knowledge of the previous period s error term, since the observed demand will not be known. However, the relationship between error terms of k lag is: E[e t ] = E[λe t-1 + ε t ] =E[ λ(λe t-2 + ε t-1 )]= = λ k e t-k Therefore, a forecast for k-periods ahead will subtract λ k times the current period s error term. The Holt-Winters parameters and the error adjustment parameter lambda will be estimated simultaneously to improve efficiency. Seasonal ARIMA
An autoregressive integrated moving average (ARIMA) model explains one variable with respect to its past and the history of its residuals. ARIMA models can be written as ARIMA(p,d,q), where p is the number of autoregressive terms, d is the order of integration, and q is the number of moving average terms in the model. The autoregressive terms explain the variable with respect to previous observations of the variable. For example, an AR(2) model appears as follows: Y t = φ 1 Y t-1 + φ 2 Y t-2 + u t. The moving average terms explain the variable with respect to previous residuals, as the following MA(2) model demonstrates: Y t = u t + φ 1 u t-1 + φ 2 u t-2. The order of integration indicates the number of times the data must be differenced before it becomes stationary. An ARIMA(1,1,1) model is written as (1-φL)(ΔY t c) = (1-θL)ε t, where L is the lag operator, with LY t = Y t-1, and Lε t = ε t-1. This process can also be written linearly as ΔY t = (1-φ)c + φδy t-1 - θε t-1 + ε t. In the case where the data contains not only a relationship between the variable and past observations of the variable and residuals but also a seasonal component, the basic ARIMA
model outlined above can be extended to a seasonal ARIMA model, containing seasonal autoregressive and moving average terms, as well as seasonal differencing. The seasonal ARIMA model can be classified as ARIMA(p,d,q)х(P,D,Q,s), where P is the number of seasonal autoregressive terms, D is the order of seasonal differencing, Q the number of seasonal moving average terms included in the model and s the number of periods in the season. For example, ARIMA(1,0,1) х(1,0,1,336) is written as (1-φL)(1-φL 336 )(Y t -c) = (1-θL)(1-ωL 336 )ε t. The requirement for seasonal AR and MA terms can be found by analysing the correllelogram of residuals after the basic ARIMA model has been estimated. Seasonal ARIMA models with more than one seasonal period can be developed. Darbellay and Slama (2000), and also Taylor (2003), used seasonal periods of one day and one week for their double seasonal ARIMA models. An example that would be a natural starting-point for the data in this paper would be an ARIMA(1,0,1)х(1,0,1,48)х(1,0,1,336) model. This can be written as (1- φl)( 1-φ 1 L 48 )(1-φ 2 L 336 )(Y t -c) = (1-θL)(1-ω 1 L 48 )(1-ω 2 L 336 )ε t. A season of 336 periods will be used in the estimation of a seasonal ARIMA model for this data. Some seasonal autoregressive or moving average terms may be added for the 48-period season, though not in the form outlined in the double seasonal model above. The estimation techniques used prevent a second season being added in a separate bracket as written above, but lags of 48 periods may be included in the 336-period bracket; i.e. (1-φL)(1-φ 1 L 48 -φ 2 L 336 )(Y t - c) = (1-θL)(1-ω 1 L 48 -ω 2 L 336 )ε t.
Multivariate Regression A simple multivariate regression model can be used to compare the univariate ARIMA and exponential smoothing models against. Although there are few time series variables to regress electricity demand against, with the one obvious choice being temperature, dummy variables can be used for the half-hour period of the day, the day of the week, the month of the year and also the year itself. Other dummy variables can be tested, and adding lagged dependent variables is another way to improve the accuracy of the regression model. Bunn (1982) discussed some special effects that may be taken into account when forecasting short-term demand in electricity. These include weather variables, television audience behaviour, holidays, on-set of darkness and daylight-savings time. Limitations in the availability of data prevent some of these variables being tested. The general formula for such a model is as follows: X t = c + Σ i=2 48 φ i h t + Σ j=2 7 θ j d t +Σ k=2 12 ρ k m t + ωy t + α*temp t + {other dummy variables} + {lagged dependent variables} + e t, where h is the half-hourly dummy variable, d is the day of the week dummy variable, m the month dummy variable, and y the year dummy variable, and φ i, θ j, ρ k, and ω are the parameters for the 47 half-hours, 6 days, 11 months and the one year requiring dummy variables. Dummy variables need to be estimated for all but one of the half-hours, days, months and years, with the half-hour, day month and year not allocated a dummy variable used as a base. Hyde and Hodnett (1997) found that adding an error adjustment term improved forecasting accuracy. A simple error adjustment term will be added, as for Holt-Winters, in the form of e t = λe t-1 + ε t.
4. Results and Interpretation Empirical analysis is performed to compare the forecasting accuracy of the models. Demand is forecast for one to six periods ahead since the generators and the Systems Operator for the spot market require different lengths of foresight in demand for their different roles. The possibility exists that some models that accurately forecast one period ahead may be less accurate in forecasting six periods ahead. As mentioned previously, the univariate models have estimation periods of eight weeks, the multivariate regression model have an estimation period of one year, and the same four week period is used as a forecast sample for all models. Stationarity The twelve-week period of quantity demanded observations used for univariate estimation and forecasting is tested for nonstationarity. There is very strong evidence against quantity containing a unit root. Therefore, quantity does not need to be differenced before estimating Holt-Winters and ARMA models. Testing the one year and four week period of quantity data provides no evidence against nonstationarity in the data. In the absence of other time series variables in the data, this would require quantity to be differenced to remove the nonstationarity. However, since temperature is a time series variable, there may exist cointegration between quantity and temperature. A linear regression with an intercept is run on quantity and temperature, and the residuals are tested for nonstationarity. There is very strong evidence against nonstationarity, so Engle-
Granger cointegration exists between quantity and temperature, which means the data is not required to be differenced before the regression is run. One Season Holt-Winters Exponential Smoothing The initial trend value is calculated by taking the average of: (i) the difference between the mean of the first and the second 336 observations divided by 336; and (ii) the mean of the first differences from the first two weeks of the sample. The initial level value is calculated by taking the mean of the first 672 observations, and adding 336.5 times the initial trend. The model also requires initial seasonal indices for each of the 336 periods in the season. These are calculated as the average of the ratio of observation to 336-point moving means for each corresponding half-hour period, taken from the first two weeks of the sample. The parameters are estimated in Excel to minimise the mean absolute percentage error (MAPE) of one-step-ahead forecasts. MAPE is the most commonly used forecasting error summary in short-term electricity demand forecasting (Taylor, 2003). Random numbers between zero and one are generated for the parameter values, and the combinations of parameters that resulted in the lowest MAPEs over time are selected and used to narrow the range of random number generation for each parameter. This process is followed until the specific parameters that minimised the MAPE for the estimation sample were identified. Double Seasonal Holt-Winters Exponential Smoothing
Initial values for the level and the trend are calculated in the same manner as for Holt-Winters with one season. The initial values for the 48 day indices are estimated by taking the average of the ratios of observations to 48-point moving mean for the corresponding periods from the first week of the sample. The initial values for the 336 week indices are estimated by taking the average of the ratios of observations to 336-point moving mean from the first two weeks of the sample, divided by the corresponding day indices. The same process for estimation of the parameters is used for double seasonal Holt-Winters as is used for one season Holt-Winters. Error Adjustment for Holt-Winters Models After estimating the Holt-Winters model for one and two seasons, significant positive autocorrelation of residuals was found. This is unsurprising, given the prevalence of autocorrelation in other literature on Holt-Winters forecasting. Our estimation of both the one season and double seasonal Holt-Winters models with error correction estimates the error correction parameter lambda in conjunction with estimating the other parameters, and using the same method as the estimation of the other parameters. Random numbers between 0 and 1 are generated for lambda since the autocorrelation coefficient is significantly positive, and the range is narrowed down to find the value of lambda that minimises the MAPE of the estimation sample. The estimated coefficients for the Holt-Winters exponential smoothing models without and with error adjustment terms are reported in Tables 1 and 2 respectively. Table 1: Estimated Coefficients of Holt-Winters Exponential Smoothing Models
Level α Trend β Within-day seasonality δ Within-week seasonality ω One Season Holt-Winters 0.94 0-1 Double Seasonal Holt-Winters 0.95 0 0.8 1 Table 2: Estimated Coefficients of Holt-Winters Exponential Smoothing Models with Error Adjustment Level α Trend β Within-day seasonality δ Within-week seasonality ω Error adjustment λ One Season Holt-Winters 0.86 0-1 0.4 Double Seasonal Holt-Winters 0.91 0 0.14 0.14 0.34 Seasonal ARIMA Several statistics are considered important for identifying the correct specification of the ARIMA model. In keeping with the estimation of the Holt-Winters model, the MAPE of onestep-ahead forecasts is one of the statistics considered. Other criteria used to select the specification of the model include the Akaike Information Criterion, the adjusted R-squared, and the Box-Jenkins methodology to minimise or eliminate the autocorrelation of residuals. Due to the difficulty in estimating a true double seasonal ARIMA model, the hybrid model which performed the best overall with regards to the criteria above is difficult to write in compact form. In expanded form, the model is written as: (1-1.36L + 0.41L 2 )(1-0.32L 48-0.24L 336-0.15L 672-0.12L 1008-0.12L 1344 )(Y t - 8421) = (1 + 0.16L 48 )(1 + 0.04L 336 + 0.04L 672 )ε t This specification has two autoregressive terms, seasonal autoregressive terms of lags 48, 336, 672, 1008 and 1344, a moving-average term with a lag of 48, and seasonal moving-average terms of lags 336 and 672. Of the many combinations of autoregressive and moving-average terms tested, this was the only specification that did not have statistically significant
autocorrelation of residuals. It also has the second-lowest Akaike Information Criterion and MAPE, and the highest adjusted R-squared, which all indicate a well-specified model. Multivariate Regression Initially, a model with only dummy variables for the half-hour, day, month and year is estimated, with an MAPE for the period of estimation of 16.4. Adding temperature to the model decreases the MAPE to 15.9, though of the other variables added, only a dummy variable for public holidays that fall on weekdays was significant. Variables found to be insignificant include dummy variables for daylight savings and school holidays, and a time variable that increases by one for each subsequent period. Adding lagged dependent variables has a significant effect on decreasing the MAPE of the forecasts. For example, adding a dependent variable of one lag decreases the MAPE from 15.8 to 4.32. Adding lags from 1 to 5 periods behind, 46 to 49 periods behind, and 329 and 334 to 339 periods behind all added to the forecasting accuracy of the model. The Breusch-Godfrey serial correlation test provides extremely strong evidence against no autocorrelation of residuals. Autocorrelation is particularly severe with residuals from 48 periods behind. However, autocorrelation only affects the standard errors of the coefficients, not the estimates of the coefficients themselves. In testing the significance of variables this would be a considerable problem. Since we are only interested in estimating a model to provide accurate forecasts, autocorrelation is far less important. Forecasting accuracy may be improved by including a lagged error term as was used in the Holt-Winters models. In this case, the results when including a lagged error term for one period behind and for 48 periods behind did
not improve the forecasting accuracy of the model. The White Test provided extremely strong evidence against no heteroskedasticity in the model. This implies that the magnitude of the residuals in the model vary as at least one of the variables in the model varies. Once again, heteroskedasticity affects the standard errors of the coefficients and not the estimation of the coefficients themselves, so for building a forecasting model this will not have an impact. The estimated coefficients and standard errors for the multivariate regression model are reported in Table 7 of Appendix 1. Discussion Figures 6 and 7 compare the accuracy of the models for forecasting one to six periods ahead. The seasonal ARIMA model performed best in forecasting for all time periods except one period ahead, where it was second best with an MAPE of 2.02, slightly behind the double seasonal Holt-Winters exponential smoothing model s MAPE of 2.00. The one season Holt-Winters, double seasonal Holt-Winters and seasonal ARIMA were very similar in accuracy for one period ahead, and even the multivariate regression model was not far behind. From two to six periods ahead, the errors for the Holt-Winters and multivariate regression models were fairly similar in accuracy, highlighting the relative supremacy of the seasonal ARIMA. Seasonal ARIMA forecasting errors for six periods ahead were 6.21, well below the other models MAPEs of between 6.93 and 7.37. One reason for this could be the superior error adjustment structure of the seasonal ARIMA model. Three moving average terms including two seasonal moving average terms with one and two-week lags were included, and each had the same power for any number of forecasting periods ahead. However, for the Holt-Winters models with error
adjustment terms, the coefficient for the error adjustment decreases greatly in magnitude as the forecasting period ahead is extended, so that when forecasting six periods ahead, the error adjustment would have had very little impact on the actual forecast. The multivariate regression model had no error adjustment term as it did not improve forecasting ability in estimation. The inaccuracy in forecasting from two to six periods ahead for the multivariate model is not surprising given the relative inaccuracy in forecasting one period ahead. It must be noted that there are many multivariate models in literature for short-term electricity load forecasting, so we cannot conclude from these results that univariate models are superior. These results are surprising given that Taylor found the double seasonal Holt-Winters model with an error adjustment to be clearly more accurate than the seasonal ARIMA model. Equally surprising is that Taylor s paper estimated a true double seasonal ARIMA model, while our seasonal ARIMA model was effectively a one season model with a crude adjustment. A properly formatted double seasonal ARIMA model would be expected to yield even more accurate results. Figure 6: MAPE s of forecasts from one to six periods ahead for data from July 21 to August 162006. Forecast Periods Ahead Forecasting Method 1 2 3 4 5 6 One Season Holt-Winters 2.29 3.73 4.90 5.83 6.65 7.37 One Season Error Adjusted Holt-Winters 2.04 3.53 4.76 5.73 6.57 7.27 Double Seasonal Holt-Winters 2.18 3.54 4.64 5.55 6.31 7.00 Double Seasonal Error Adjusted Holt-Winters 2.00 3.41 4.55 5.46 6.24 6.93 Multivariate Regression 2.10 3.63 4.85 5.83 6.56 7.13 Seasonal ARIMA 2.02 3.26 4.27 5.08 5.71 6.21
Figure 7: MAPE s of forecasts from one to six periods ahead for data from July 21 to August 16 2006. 8.00 7.00 6.00 5.00 One Season Holt-Winters One Season Error Adjusted Holt-Winters MAPE 4.00 3.00 2.00 Double Seasonal Holt-Winters Double Seasonal Error Adjusted Holt- Winters Multivariate Regression Seasonal ARIMA 1.00 0.00 1 2 3 4 5 6 Forecast Periods Ahead The results when comparing the four Holt-Winters models are unsurprising, with the double seasonal Holt-Winters performing better than the one season version. The data clearly contained two seasonal trends, so the one season Holt-Winters would not have adjusted for the seasonal variation as successfully as the double seasonal model. The error adjustment versions also performed better than the models without them, especially for very short-term forecasting, for reasons explained above.
5. Conclusion The importance of short-term electricity load forecasting and the significant financial gains for players in the spot market that can be achieved through increased accuracy has encouraged a substantial amount of research into various univariate and multivariate models. Seasonal ARIMA, Holt-Winters exponential smoothing and variations on weather-related multivariate models have proven most popular in recent years. In this study, one season and two season Holt-Winters exponential smoothing models with and without error adjustment terms, a hybrid of a single and double seasonal ARIMA model, and a dummy variable model with temperature and lagged dependent variables were compared for forecasting electricity load in the Hayward node of New Zealand for one to six periods ahead. The seasonal ARIMA model performed the best in minimising the mean absolute percentage errors, especially for longer-term forecasting. The error adjustment term in the Holt-Winters model significantly improved very short-term forecasting, but the improvements were reduced for longer-term forecasts. Given the relative superiority of the seasonal ARIMA model in forecasting for the Hayward node, the estimation and performance of a pure double seasonal ARIMA model for nodal electricity demand should be investigated. Also, other multivariate models should be empirically tested before it can be concluded that univariate models are superior for short-term load forecasting, as the multivariate model used in this study is only one of a variety of multivariate models introduced in other literature. Another extension would be to use the models to forecast demand for other nodes in New Zealand. The distribution of demand for electricity over a day varies greatly across nodes, given the characteristics of
electricity consumers, especially non-residential consumers, also varies greatly across nodes. A combination of models may be a sensible way to deal with the relative strengths and weaknesses of models, with weights varying for different periods of time (Smith, 1989) or different forecast periods ahead.
6. Acknowledgements I would like to sincerely thank Dr Erwann Sbai for his constant guidance and assistance on all aspects of this paper. I would also like to thank Dr James W. Taylor for his helpful response to queries about estimation of the Holt-Winters exponential smoothing model.
Appendix 1 Table 1 Dickey-Fuller Unit Root Test for Quantity for One Year and Four Week Period Null Hypothesis: QUANTITY has a unit root Exogenous: Constant Lag Length: 44 (Automatic based on SIC, MAXLAG=44) t-statistic Prob.* Augmented Dickey-Fuller test statistic -1.884183 0.3401 Test critical values: 1% level -3.430527 5% level -2.861502 10% level -2.566791 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(QUANTITY) Method: Least Squares Date: 11/16/08 Time: 13:53 Sample (adjusted): 46 18816 Included observations: 18771 after adjustments Table 2 Dickey-Fuller Unit Root Test for Temperature for One Year and Four Week Period Null Hypothesis: TEMPHH has a unit root Exogenous: Constant
Lag Length: 40 (Automatic based on SIC, MAXLAG=44) t-statistic Prob.* Augmented Dickey-Fuller test statistic -1.111129 0.2426 Test critical values: 1% level -2.565085 5% level -1.940841 10% level -1.616688 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(TEMPHH) Method: Least Squares Date: 11/16/08 Time: 13:57 Sample (adjusted): 42 18816 Included observations: 18775 after adjustments Table 3 Dickey-Fuller Unit Root Test for Residuals of Regression for One Year and Four Week Period Null Hypothesis: RESID01 has a unit root Exogenous: Constant Lag Length: 44 (Automatic based on SIC, MAXLAG=44)
t-statistic Prob.* Augmented Dickey-Fuller test statistic -3.624805 0.0053 Test critical values: 1% level -3.430527 5% level -2.861502 10% level -2.566791 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RESID01) Method: Least Squares Date: 11/16/08 Time: 13:59 Sample (adjusted): 46 18816 Included observations: 18771 after adjustments Table 4 Dickey-Fuller Unit Root Test for Quantity for Twelve Week Period Null Hypothesis: Q has a unit root Exogenous: Constant Lag Length: 30 (Automatic based on SIC, MAXLAG=31)
t-statistic Prob.* Augmented Dickey-Fuller test statistic -12.10734 0.0000 Test critical values: 1% level -3.431567 5% level -2.861963 10% level -2.567038 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(Q) Method: Least Squares Date: 11/11/08 Time: 14:16 Sample (adjusted): 32 4704 Included observations: 4673 after adjustments Table 5 Estimated Coefficients and Standard Errors from ARIMA Estimation Dependent Variable: Q Method: Least Squares Date: 10/28/08 Time: 18:35 Sample (adjusted): 1347 2688
Included observations: 1342 after adjustments Convergence achieved after 11 iterations Backcast: OFF (Roots of MA process too large) Variable Coefficient Std. Error t-statistic Prob. C 8420.591 2630.438 3.201212 0.0014 AR(1) 1.366646 0.025380 53.84793 0.0000 AR(2) -0.406294 0.025366-16.01724 0.0000 SAR(48) 0.318523 0.033220 9.588285 0.0000 SAR(336) 0.244005 0.042613 5.726087 0.0000 SAR(672) 0.151092 0.032099 4.707000 0.0000 SAR(1008) 0.120835 0.025237 4.787964 0.0000 SAR(1344) 0.120494 0.024087 5.002441 0.0000 MA(48) -0.159622 0.041761-3.822280 0.0001 SMA(336) -0.039639 0.053064-0.747005 0.4552 SMA(672) -0.043258 0.051023-0.847815 0.3967 R-squared 0.991933 Mean dependent var 8151.314 Adjusted R-squared 0.991872 S.D. dependent var 2359.418 S.E. of regression 212.7091 Akaike info criterion 13.56589 Sum squared resid 60221297 Schwarz criterion 13.60853 Log likelihood -9091.713 F-statistic 16366.24 Durbin-Watson stat 1.995216 Prob(F-statistic) 0.000000 Table 6 Autocorrelation Test for Preferred ARIMA Model Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.198490 Probability 0.096106 Obs*R-squared 119.0366 Probability 0.094168
Test Equation: Dependent Variable: RESID Method: Least Squares Date: 11/11/08 Time: 14:25 Presample missing value lagged residuals set to zero. Table 7 Estimated Coefficients and Standard Errors of Multivariate Regression Model Dependent Variable: QUANTITY Method: Least Squares Date: 10/30/08 Time: 15:26 Sample (adjusted): 340 17472 Included observations: 17133 after adjustments
Variable Coefficient Std. Error t-statistic Prob. C 194.7551 29.96662 6.499069 0.0000 QUANTITY(-1) 1.227663 0.007560 162.3895 0.0000 QUANTITY(-2) -0.321918 0.012022-26.77658 0.0000 QUANTITY(-3) 0.022960 0.012195 1.882772 0.0597 QUANTITY(-4) -0.029077 0.011742-2.476353 0.0133 QUANTITY(-5) 0.021643 0.006714 3.223700 0.0013 QUANTITY(-46) 0.023362 0.007015 3.330206 0.0009 QUANTITY(-47) 0.023487 0.011872 1.978360 0.0479 QUANTITY(-48) 0.135758 0.011926 11.38327 0.0000 QUANTITY(-49) -0.150994 0.007183-21.02119 0.0000 QUANTITY(-329) -0.021161 0.002086-10.14464 0.0000 QUANTITY(-334) 0.044443 0.007351 6.046013 0.0000 QUANTITY(-335) 0.078885 0.011893 6.632837 0.0000 QUANTITY(-336) 0.095609 0.012219 7.824341 0.0000 QUANTITY(-337) -0.147344 0.012198-12.07947 0.0000 QUANTITY(-338) -0.014878 0.011944-1.245573 0.2129 QUANTITY(-339) -0.026491 0.007308-3.625099 0.0003 NUM2006 27.46326 19.29675 1.423207 0.1547 FEBRUARY 12.65403 9.012376 1.404073 0.1603 MARCH 13.73540 9.330467 1.472102 0.1410 APRIL 19.48600 9.386951 2.075861 0.0379 MAY 95.99835 13.45055 7.137132 0.0000 JUNE 102.5430 15.79644 6.491528 0.0000 JULY 94.52589 16.41292 5.759238 0.0000 AUGUST 51.76024 23.89098 2.166518 0.0303 SEPTEMBER 37.93423 22.91510 1.655425 0.0979 OCTOBER 21.43332 22.39648 0.956995 0.3386 NOVEMBER 22.70637 21.82012 1.040616 0.2981 DECEMBER 21.08347 21.39019 0.985661 0.3243 TUESDAY 51.81510 7.029219 7.371388 0.0000 WEDNESDAY 34.98138 7.041428 4.967939 0.0000 THURSDAY 31.24492 7.050731 4.431444 0.0000 FRIDAY 24.48236 7.011854 3.491568 0.0005 SATURDAY 20.24399 6.946393 2.914317 0.0036 SUNDAY -12.13907 6.900171-1.759242 0.0786 H2-19.15437 18.43024-1.039290 0.2987 H3 17.84026 18.17636 0.981509 0.3264 H4-49.21197 18.30412-2.688573 0.0072 H5 1.915839 18.47415 0.103704 0.9174 H6-24.22863 19.06027-1.271159 0.2037 H7 24.23929 19.46469 1.245296 0.2130 H8 22.28745 20.12121 1.107660 0.2680 H9 31.54093 20.39318 1.546641 0.1220 H10-4.991769 20.55144-0.242891 0.8081 H11 55.69474 20.74062 2.685298 0.0073
H12-6.815370 21.08962-0.323162 0.7466 H13 211.5525 21.29253 9.935525 0.0000 H14 88.29046 21.76058 4.057359 0.0000 H15 199.9586 21.56358 9.272978 0.0000 H16 66.28875 21.78833 3.042397 0.0024 H17 73.84176 21.59334 3.419655 0.0006 H18-6.354603 21.24608-0.299095 0.7649 H19 155.9929 20.74055 7.521152 0.0000 H20 73.40924 20.27106 3.621381 0.0003 H21 105.6186 19.55145 5.402089 0.0000 H22 80.32300 19.40940 4.138355 0.0000 H23 91.14822 19.30067 4.722543 0.0000 H24 119.2945 19.18206 6.219067 0.0000 H25 74.02396 19.27193 3.841025 0.0001 H26 65.27203 19.28548 3.384517 0.0007 H27 200.3019 19.48236 10.28119 0.0000 H28 39.66476 19.98651 1.984577 0.0472 H29 141.8016 20.09571 7.056310 0.0000 H30 146.0759 20.53381 7.113922 0.0000 H31 67.96664 20.44214 3.324829 0.0009 H32 150.5739 20.26742 7.429360 0.0000 H33 135.5470 20.76032 6.529141 0.0000 H34 113.6618 21.08011 5.391897 0.0000 H35 242.5684 21.22070 11.43075 0.0000 H36 274.8353 21.52374 12.76894 0.0000 H37 42.16129 21.28147 1.981127 0.0476 H38 83.16012 20.82901 3.992514 0.0001 H39 14.90985 20.91307 0.712944 0.4759 H40 87.30350 20.45918 4.267204 0.0000 H41 121.8005 19.87114 6.129516 0.0000 H42 116.4392 19.47258 5.979650 0.0000 H43 38.31945 19.16478 1.999472 0.0456 H44 43.00934 18.90118 2.275485 0.0229 H45-46.20064 18.80425-2.456925 0.0140 H46-92.62841 18.65968-4.964094 0.0000 H47 177.9301 18.38378 9.678647 0.0000 H48-141.9948 19.04131-7.457196 0.0000 HOLIDAYWKD 69.38390 45.04178 1.540434 0.1235 TEMPHH -10.41107 0.931714-11.17410 0.0000 R-squared 0.990608 Mean dependent var 5195.558 Adjusted R-squared 0.990563 S.D. dependent var 2435.223 S.E. of regression 236.5724 Akaike info criterion 13.77528 Sum squared resid 9.54E+08 Schwarz criterion 13.81327 Log likelihood -117921.9 F-statistic 21666.12 Durbin-Watson stat 2.000209 Prob(F-statistic) 0.000000
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