A Comparison of the Forecast Performance of. Double Seasonal ARIMA and Double Seasonal. ARFIMA Models of Electricity Load Demand

Transcription

1 Applied Mathematical Sciences, Vol. 6, 0, no. 35, A Comparison of the Forecast Performance of Double Seasonal ARIMA and Double Seasonal ARFIMA Models of Electricity Load Demand Siti Normah Hassan Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia, 830, UTM Johor Bahru Johor Darul Takzim, Malaysia Maizah Hura Ahmad Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia, 830, UTM Johor Bahru Johor Darul Takzim, Malaysia maizah@utm.my Suhartono Department of Statistics, Institut Teknologi Sepuluh Nopember, Indonesia Norizan Mohamed 4 Mathematics Department, Faculty of Science and Technology Universiti Malaysia Terengganu (UMT) 030 Kuala Terengganu, Terengganu, Malaysia Abstract A half-hourly electricity load demand of Malaysia for 6 months, from January 00 to 30 June 00 is used in thisstudy with the purpose of improving theaccuracy of short term electricity load demand forecasts. The results of the identification step show that the load data have daily and weekly seasonal periods. Thus, the aim of this paper is to develop a forecasting model by studying the long-term characteristics of time series based on double seasonal ARFIMA model. The best order of the model which is based on fractional differencing is

2 6706 Siti Normah Hassan et al determined and the results are compared with the established seasonal ARIMA model. Using the mean absolute percentage error (MAPE) as the forecast accuracy measure, the current study shows that double seasonal ARFIMA model performs better than double seasonal ARIMA in forecasting electricity load demand in Malaysia where the mean absolute percentage error of the forecast value is reduced by about 0.04%. Keywords: Seasonal ARIMA, ARFIMA, Fractional Differencing, Electricity Load Demand Introduction In many countries, electricity load demand is mainly influenced by meteorological conditions, seasonal effects such as daily and weekly cycles, calendar holiday and special events []. Just like in other countries, load demand of electricity in Malaysia, need to be determined to prevent energy wasting and system failure. Many forecasting methods have been developed and implemented to forecast electricity load demand. These methods can be classified into two categories: time series forecasting and intelligent system forecasting. In some cases, methods from both categories are combined to get better forecasting performances. Methods that are used to forecast electricity load demand include ARMA and back-propagation neural network by El-Telbany and El-Karmi [], double seasonal exponential smoothing, double seasonal ARIMA and regression by Taylor [6] and Taylor et al. [7], multi-layer perception by Gonzalez-Romera et al. [] and neural networks by Hippert et al. [4]. The current study focuses on the special characteristic of electricity load demand which contains double seasonal pattern for daily and weekly cycles. In such a case, Gould et al. [3] suggested the use of multiple seasonal method. In the current study, the established seasonal ARIMA method is used and due to the long-term characteristics of electricity load data in Malaysia, we also study the ARFIMA method which is based on fractional differencing. Since the data contains both double seasonal pattern of daily and weekly cycles, double seasonal method for both models are developed. This paper will be organized as follows. We will start by presenting the methodology on double seasonal ARIMA and double seasonal ARFIMA. We then present the results when both models are used to forecast electricity load demand in Malaysia. Finally we give our conclusions based on the evaluation method presented in this study.

3 Comparison of forecast performance 6707 Methodology ARIMA and ARFIMA Models Time series methods use historical data as the basis of estimating future outcomes. Some notable methods are moving average, exponential smoothing, autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA). The method that will be used in this paper are double seasonal ARIMA and double seasonal ARFIMA. The Box- Jenkins seasonal ARIMA method can be written as: φ d D s s s ( )( ) ( )( ) θ ( ) ( ) B B Φ B B z = B Θ B a () p P t q Q t where φ B = φ B φ B φ B p( )... Φ B = Φ B Φ B Φ B S P( )... θ B = θ B θ B θ B q( )... Θ B = Θ B Θ B Θ B S Q( )... p q p q P Q PS QS and where z t is the data set after appropriate transformation of time t while ( B) d D s and ( B ) are the seasonal and non-seasonal differencing operators respectively. B is the backshift operator and {z t } is white noise with zero mean and variance σ. Due to the existence of double seasonal pattern of daily and weekly in the load demand data, we analyze the data by using double seasonal ARIMA method. The model of double seasonal ARIMA can be written as: φ d s s D s s D ( B)( B) Φ ( B )( B ) Φ ( B )( B ) z s s = θ ( ) Θ ( ) Θ ( ) p P P t B B B a, q Q Q t () where z t is the data set after appropriate transformation of time t while and s ( B ) D are the seasonal and non-seasonal differencing operators respectively. B is the backshift operator and {z t } is white noise with zero mean σ B θ B are regular autoregressive and moving and variance. φ ( ) and ( ) p average polynomials of orders p and q. Φ s P ( B ), s Φ P ( B ), Θ s Q ( B ) q and ( B ) d

4 6708 Siti Normah Hassan et al ( B s ) Θ are the autoregressive and moving average of orders Q,,, P P Q Q. The double seasonal ARFIMA model is the same as the double seasonal ARIMA model in equation () except for the operator ( B) d which is defined by the following binomial expansion { ψ j } is square summable where function for d < 0.5, d 0,,,... ( B) d ψ B j = j= 0 j Γ ( j+ d) ψ j = and Γ () is the gamma Γ ( j + ) Γ ( d) In general, when we define ψ ( z) = ( z) d, this function is analytic in the open z: z for negative d. disk { : } z z < and analytic in the closed unit disk { } Forecasting Evaluation Method To evaluate the accuracy of the ARIMA and ARFIMA models, we calculate the mean absolute percentage error (MAPE) in the forecasting result. The MAPE is determined by using the following formula, n x $ i xi i= xi MAPE = 00% n where x i is the actual observed value, x $ i is the predicted value and n is the number of predicted value. 3. Results The data used in this study were obtained from TenagaNasionalBerhad (TNB), Malaysia. The time series data correspond to half-hourly six months Malaysia electricity load demand, measured in Megawatt (MW), for the time period from st January 00 to 30 th June 00. For the purpose of time series modeling, observations from st January 00 to 3 rd June 00 were used to fit the ARIMA and ARFIMA models while observations from 4 th June 00 to 30 th June 00 were kept for the purpose of sample forecast accuracy checking. The time series plot of the load demand observed from st January 00 to 3 rd June 00 are shown in Figure.

5 Comparison of forecast performance Time Series Plot of Y t Yt In d e x Figure The time series plot of half-hourly electricity load demand observed from st January 00 to 3 rd June 00 Figure illustrates the ACF and PACF plots of the electricity load demand. It can be clearly seen that the data is non-stationary and contains double seasonal patterns which are daily seasonal and weekly seasonal. The ACF plot also shows that this data contains long memory characteristic which can be analyzed by using fractional integration method namely, ARFIMA. Autocorrelation Function for Yt (with 5%significance limits for the autocorrelations) Partial Autocorrelation Function for Yt (with 5%significance limits for the partial autocorrelations) Autocorrelation Partial Autocorrelation Lag Lag Figure ACF and PACF plots of half-hourly electricity load demand observed from st January 00 to 3 rd June 00 In this study, to analyze the data using double seasonal ARIMA and double seasonal ARFIMA models, R program and SAS program are developed. The parameter coefficients of the model are estimated by using maximum likelihood method [5].

6 670 Siti Normah Hassan et al Fitting an ARIMA model The best model that we get for double seasonal ARIMA after differencing at lag (non-seasonal), lag 48 (daily seasonal) and lag 336 (weekly seasonal) is given as follows: ARIMA ([7,8,9,0,,4,47],,)(0,,) (0,,) We then fit an ARIMA model by using maximum likelihood method. Based on AIC criterion, we get the best model fitted by ( B B B B B B B )( B )( B ) zt = ( 0. B)( B )( B ) at Fitting an ARFIMA model Since the load data contains long memory characteristic, we need to find the fractional differencing value first. By using GPH method, we get the value for fractional d, which is After doing fractional differencing to the load data, we get a new set of data series. From the data plot, we can see that the data is not yet stationary and contains double seasonal pattern. The data became stationary after differencing at lag 48 and 336. After taking fractional differencing at d=0.4757, the best ARFIMA model is given as follows: ARFIMA ([,,3,8,9,3,4,6,47],,)(0,,) (0,,) We then fit an ARFIMA model by using maximum likelihood method. We get the best model fitted by ( 0.45B+ 0.69B 0.048B B B 0.035B B B 0.09 B ) ( B) ( B )( B ) zt = ( 0.69 B)( B )( B ) at Forecasting Using Double Seasonal ARIMA and Double Seasonal ARFIMA Models Figure 3 plots the actual electricity load demand and the forecasts using double seasonal ARIMA and ARFIMA models. The plot in Figure 3 shows that the forecasts produced using both methods follow the actual values closely. However, based on the MAPE values, the double seasonal ARFIMA model appears to have a better forecasting performance when compared to the double seasonal ARIMA model.

7 Comparison of forecast performance 67 Axis Title FORECASTING ARIMA ARFIMA Figure 3 Forecasting plot for double seasonal ARIMA and double seasonal ARFIMA from 4 th June 00 to 30 th June 00 In Table, the MAPE of out-sample forecasts using both models are presented. Table The MAPE values of the model for double seasonal ARIMA and double seasonal ARFIMA Model MAPE Double Seasonal ARIMA Double Seasonal ARFIMA Conclusion In the electricity load demand forecasting literature, a number of researchers only used single SARIMA and in some cases, they used double SARIMA when double seasonal patterns exist. While double SARIMA outperformed single SARIMA in those studies, the current study showed that forecasting performance can be improved by studying the long term characteristic of the time series. In the case of forecasting Malaysia electricity load demand, using an ARFIMA model instead of an ARIMA model, the mean absolute percentage error of the forecast value is reduced by about 0.04%. References [] M. El-Telbany and F. El-Karmi, Short-term Forecast of Jordanian Electricity Demand Using Particle Swarm Optimization,Electricity Power Systems Research, 78 (007),

8 67 Siti Normah Hassan et al [] E. Gonzalez-Romera, M. A. Jaramillo-Moran and D. Carmono-Fernadez, Forecasting of the Electric Energy Demand Trend and Monthly Fluctuation with Neural Networks, Computer & Industrial Engineering, 5 (007), [3] P. G. Gould, A. B. Koehler, J. K. Ord, R. D.Snyder, R. J. Hyndman and F. V. Araghi, Forecasting Time Series with Multiple Seasonal Patterns, European Journal of Operational Research,9 (008), 07-. [4] H. S. Hippert, D. W. Bunn and R. C. Souza, Large Neural Networks for Electricity Load Forecasting. Are They Overfitted, International Journal of Forecasting, (005), [5] G. O. L. C. Marcques, Empirical Aspects of the Whittle-based Maximum Likelihood Method in Jointly Estimating Seasonal and Non-seasonal Fractional Integration Parameter, Physic A, 390 (0), 8-7. [6] J. W. Taylor, Triple Seasonal Methods for Short-term Electricity Demand Forecasting,European Journal of Operational Research, 04 (00), [7] J. W. Taylor, L. M. de Menezes and P. E. McSharry, A Comparison of Univariate for Forecasting Demand up to a Day Ahead, International Journal of Forecasting, (006), -6. Received: September, 0