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A trigonometric grey prediction approach to forecasting electricity demand
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Energy 31 (2006) 2839–2847 www.elsevier.com/locate/energy
A trigonometric grey prediction approach to forecasting electricity demand P. Zhou, B.W. Ang, K.L. Poh Department of Industrial and Systems Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 4 April 2005
Abstract Electricity demand forecasting plays an important role in electricity systems expansion planning. In this paper, we present a trigonometric grey prediction approach by combining the traditional grey model GM(1,1) with the trigonometric residual modification technique for forecasting electricity demand. Our approach helps to improve the forecasting accuracy of the GM(1,1) and allows a reasonable grey prediction interval to be obtained. Two case studies using the data of China are presented to demonstrate the effectiveness of our approach. r 2005 Elsevier Ltd. All rights reserved. Keywords: Electricity demand; Forecasting; Grey system theory; The GM(1,1) model
1. Introduction Electricity demand forecasting mainly deals with the forecasting of the amount of electricity that should be generated and distributed in a specific region over a specific period by electric utilities. It plays an important role in electricity systems expansion planning. Since electricity has great influence on people’s daily life and the national economy, the accuracy of electricity demand forecasting is important not only for electric utilities themselves but also for the consumers. The forecasting models for electricity demand can be broadly classified into two categories, viz. causal and time-series models [1]. Causal models exploit the relationship between a dependent variable and independent variables, and assume that the variations in dependent variable could be explained by independent variables. Multiple linear regression analysis and econometric models are the most popular causal models [1–4]. A limitation of causal models is that it depends on the availability and reliability of independent variables over the forecasting period which requires further efforts in data collection and estimation. In contrast, time-series models such as the growth curve models and the autoregressive integrated moving average (ARIMA) models, which require only the historical data of the variable of interest to forecast its future evolution behavior, has been equally popular in electricity demand forecasting [5–8]. A large number of historical observations are usually needed to obtain satisfactory forecast accuracy for the ARIMA models. In addition to the models Corresponding author. Tel.: +65 6516 2203; fax: +65 6777 1434.
E-mail address: [email protected] (P. Zhou). 0360-5442/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2005.12.002
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mentioned above, modern machine learning techniques such as artificial neural networks (ANNs), which can be formulated in the form of causal or time-series models, have also gained popularity in electricity demand and load forecasting [1,9–11]. Their performance will depend on the amount of training data and how representative these data are, as well as the experience of practitioners to a large extent. Grey system theory, propounded by Deng [12], is a truly multidisciplinary theory dealing with grey systems that are characterized by both partially known and partially unknown information. It has been widely used in several fields such as agriculture, industry and environmental systems studies [13,14]. As an essential part of grey system theory, grey forecasting models have gained in popularity in time-series forecasting due to their simplicity and ability to characterize an unknown system by using as few as four data points [15–17]. Electricity demand forecasting may be regarded as a grey system problem, because we know some factors such as population, economy conditions and weather have influence, but we do not clearly know how they exactly affect electricity demand [18]. Like the ARIMA models, the grey forecasting models are also a family of forecasting models among which the GM(1,1) model is the most frequently used. Since only two parameters are estimated and used to model a time series in the GM(1,1) model, its prediction accuracy may not be always satisfactory. Some studies have been reported on how to improve the accuracy of the GM(1,1) model [19,20]. More recently, several improved grey forecasting models were developed for electricity demand and load forecasting [21–23]. However, to build a reasonable forecasting model, most of these methods are either rather complex or require a large number of data points. In this paper, we present a simple trigonometric grey prediction approach to forecasting electricity demand by combining the GM(1,1) model with the trigonometric residual modification technique. While keeping the useful characteristic of few data points required, its prediction accuracy is far higher than that of the original GM(1,1) model. In addition, a reasonable prediction interval could be constructed. The rest of this paper is organized as follows. Section 2 introduces the original GM(1,1) model. In Section 3 we propose our trigonometric grey prediction approach. Following the idea of interval prediction in multiple regression analysis, we also present a grey prediction interval based on the proposed approach. In Section 4 two case studies on China electricity demand forecasting are presented. Section 5 concludes this study. 2. GM(1,1) model Among the family of grey forecasting models, the GM(1,1) model is the most frequently used, and is one with a certain degree of accuracy despite its simplicity [13]. Conventional forecasting techniques often deal with the original historical data directly and try to model their evolution behavior approximately. However, the GM(1,1) model begins by converting the original data series into a monotonically increasing data series by a preliminary transformation called accumulated generating operation (AGO). Applying the AGO technique reduces the noise of the original data series efficiently, and the new data series generated will approximately exhibit exponential behavior [19]. Since the solution of first-order differential equations also takes the exponential form, the first-order grey differential equations are then constructed to model the data series from AGO and forecast the future behavior of the system. Assume that X ð0Þ ¼ fxð0Þ ð1Þ; xð0Þ ð2Þ; . . . ; xð0Þ ðnÞjnX4g is the original non-negative data series taken in consecutive order and at equal time interval. The procedures for applying the GM(1,1) model to predict the future value xð0Þ ðn þ kÞ with kX1 can be described as follows [21–23]: Step 1: Form a new data series X ð1Þ ¼ fxð1Þ ð1Þ; xð1Þ ð2Þ; . . . ; xð1Þ ðnÞg by AGO: xð1Þ ðkÞ ¼
k X
xð0Þ ðiÞ.
(1)
i¼1
Step 2: Establish the grey differential equation as follows: xð0Þ ðkÞ þ azð1Þ ðkÞ ¼ u;
k ¼ 2; 3; . . . ; n,
(2)
where zð1Þ ðkÞ ¼ 0:5xð1Þ ðkÞ þ 0:5xð1Þ ðk 1Þ.
(3)
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Step 3: Estimate the developing coefficient a and the grey input u in Eq. (2) by the ordinary least-squares (OLS) method: ^ u ^ T ¼ ðU T UÞ1 U T X n , ½a;
(4)
where 3 zð1Þ ð2Þ 1 7 6 ð1Þ 6 z ð3Þ 1 7 7 6 , U¼6 .. 7 .. 6 .7 . 5 4 zð1Þ ðnÞ 1 2
(5)
and X n ¼ ½xð0Þ ð2Þ; xð0Þ ð3Þ; . . . ; xð0Þ ðnÞT .
(6)
Step 4: Establish the following whitened first-order differential equation (or grey reflection equation) for future prediction: dxð1Þ ^ ð1Þ ¼ u, ^ þ ax dt xð1Þ ð1Þ ¼ xð0Þ ð1Þ.
ð7Þ
Step 5: Solve the grey reflection Eq. (7) and obtain the predicted values for the data series X AGO as u^ u^ ^ x^ ð1Þ ðkÞ ¼ xð0Þ ð1Þ eaðk1Þ þ ; k ¼ 0; 1; 2; . . . . a^ a^
ð1Þ
from
(8)
By applying the inverse AGO (IAGO), xð0Þ ðkÞ ¼ xð1Þ ðkÞ xð1Þ ðk 1Þ to Eq. (8), the predictions for the original data series can be obtained: x^ ð0Þ ð1Þ ¼ xð0Þ ð1Þ,
u^ ^ ; x^ ð0Þ ðkÞ ¼ ð1 ea^ Þ xð0Þ ð1Þ eaðk1Þ a^
k ¼ 2; 3; . . . .
ð9Þ
3. Trigonometric grey prediction approach Despite its popularity in real application, the prediction accuracy of the GM(1,1) model may not be always satisfactory. A large number of approaches have been developed to improve the accuracy of the GM(1,1) model [19–23]. Most of them follow the rule that the information in the residual series, given by the differences between the observed and forecasted values, may be used to compensate the initial grey prediction values. Although the prediction accuracy is improved, these methods seem to be a little complex, and a lot of observations might be needed to build a reasonable model. For instance, in the ANN-based grey prediction approach developed by Hsu and Chen [21], the forecasted absolute residual series with the signs estimated by ANN are added to the initial forecasted values of the original series, in order to improve the prediction accuracy of the GM(1,1) model. However, the pattern of residual signs may not be accurately learned from few historical data points. Furthermore, these methods deal with only point prediction, while an interval prediction, especially the upper bound values of predictions, may be very useful for balancing the supply and demand in electric utility operations and management [18]. We, therefore, develop the following trigonometric grey prediction approach which requires few data points and, yet, can provide a relatively accurate point prediction as well as a reasonable prediction interval. Denote the residual series as rð0Þ ¼ frð0Þ ð2Þ; rð0Þ ð3Þ; . . . ; rð0Þ ðnÞg, where rð0Þ ðkÞ ¼ xð0Þ ðkÞ x^ ð0Þ ðkÞ;
k ¼ 2; 3; . . . ; n.
(10)
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Since the GM(1,1) model will always generate an exponentially increasing or decreasing data series, the amplitude of the residual series may become increasingly larger over time. Considering the effects of economic fluctuations or seasonal changes on electricity demand, the cyclic variations may exist in the residual series. The residual series may be treated as a combination of a linear trend with cyclic variations. In time-series analysis the trigonometric models are capable of modeling a seasonal process with a linear trend well [24]. The residual series can then be modeled by the generalized trigonometric model: rð0Þ ðk þ 1Þ ¼ b0 þ b1 k þ b2 sin
2pk 2pk þ b3 cos þ ek ; L L
k ¼ 1; 2; . . . ,
(11)
where L is a user-specified parameter for the period of the main cyclic variations and ek is the random component. For quarterly and monthly electricity demand, L ¼ 4 and 12 are, respectively, recommended because seasonal effects are obviously the main cyclic variations. For annual electricity demand, L should reflect the cyclic period of economic fluctuations and depends on the amount of historical data. When there are annual data for about 20 yr, our recommendation is 15pLp30, because such a choice may well capture the patterns of residual series according to our preliminary experiments. Obviously, the coefficients of Eq. (11) can be estimated by OLS as follows: ½b^0 ; b^1 ; b^2 ; b^3 T ¼ ðBT BÞ1 BT Rn ,
(12)
where 2
1 61 6 6 B¼6. 6 .. 4 1
1 2 .. . n1
sin 2p L sin 4p L .. . sin
2ðn1Þp L
cos cos .. . cos
2p L 4p L
2ðn1Þp L
3 7 7 7 7, 7 5
(13)
and Rn ¼ ½rð0Þ ð2Þ; rð0Þ ð3Þ; . . . ; rð0Þ ðnÞT .
(14)
ð0Þ
The forecasted residual series r^ ðkÞ can be obtained by 2pk ^ 2pk þ b3 cos ; r^ð0Þ ðk þ 1Þ ¼ b^0 þ b^1 k þ b^2 sin L L
k ¼ 1; 2; . . . .
(15)
Note that the constant term and the trigonometric terms in Eq. (15) can be treated as a part of the Fourier residual model [16] when L is equal to n1. The difference is that we substitute a linear trend term ‘‘b^1 k’’ for other terms in the Fourier residual model, whereby the occupied degree of freedom is highly reduced. Based on the above trigonometric residual modification technique, our trigonometric grey prediction model can be formulated as follows: ð0Þ x^ ð0Þ tr ð1Þ ¼ x ð1Þ,
^ ð0Þ ðkÞ þ r^ð0Þ ðkÞ; x^ ð0Þ tr ðkÞ ¼ x x^ ð0Þ tr ðkÞ
k ¼ 2; 3; . . . ,
ð16Þ
denotes the forecasted value and x^ ð0Þ ðkÞ is the output of the GM(1,1) model. where Our discussion so far is restricted to point prediction. For the purpose of balancing the supply and demand in electric utility operations and management, a reasonable interval prediction of electricity demand is very useful. Morita et al. [18] have provided a grey prediction interval of the annual electricity demand, but their method is only restricted to the original GM(1,1) model. We shall now present a prediction interval for our trigonometric grey prediction model. Note that the trigonometric residual modification model as given by Eq. (11) is essentially a multiple linear regression model. Since the GM(1,1) model and the trigonometric residual modification model have two and four parameters, respectively, the total degrees of freedom occupied by our trigonometric grey prediction approach is six. When the number of historical data points is more than six, the excessive degrees of freedom
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can be used to estimate the variance of residual errors and then the prediction interval. In contrast, in the Fourier grey prediction approach, almost all the degrees of freedom are used for parameter estimation. By following the idea of interval prediction in multiple regression analysis, we present a 100ð1 aÞ% prediction interval of xð0Þ ðkÞ as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þ ^ ð0Þ x^ ð0Þ k ¼ 2; 3; . . . , (17) tr ðkÞ ta=2;n6 s 1 þ hkk px ðkÞpx tr ðkÞ þ ta=2;n6 s 1 þ hkk ; where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 uP h u n ð0Þ t k¼1 x^ tr ðkÞ xð0Þ ðkÞ , s¼ n6 hkk ¼ Y Tk ðBT BÞ1 Y k , and
2pðk 1Þ 2pðk 1Þ ; cos Y k ¼ 1; k 1; sin L L
T .
Although the prediction interval is obtained using the concept of multiple regression analysis, there is no strict deduction or proof. Since the centre point of the prediction interval comes from our improved grey prediction approach, the prediction interval may be taken as a ‘‘grey prediction interval’’. 4. Case studies We first present an empirical illustration on China annual electricity demand forecasting to examine the performance of our trigonometric grey prediction approach. Because of the different approaches in economic development before and after the reform and open policy initiated in the late 1970s, the electricity consumption patterns in China in the two periods are quite different. We therefore use the annual electricity demand data after 1980 in our study. The data were collected from China Statistical Yearbook [25], where the 1981–1998 data are used for model building, while the 1999–2002 data are used as an ex post testing data set. Prediction accuracy is an important criterion for evaluating a forecasting technique [26]. In this study three statistical measures, viz. mean absolute percentage error (MAPE), mean absolute deviation (MAD) and meansquared error (MSE) are used to evaluate the prediction accuracy of our approach. MAPE is a general accepted measure in percent of prediction accuracy. MAD and MSE are two measures of the average magnitude of the forecast errors, but the latter imposes a greater penalty on a large error than several small errors. The three measures are, respectively, defined as follows: MAPEð%Þ ¼
(18)
n 1X ^ yðkÞj, jyðkÞ n k¼1
(19)
n 1X ^ yðkÞÞ2 , ðyðkÞ n k¼1
(20)
MAD ¼
MSE ¼
n ^ yðkÞj 1X jyðkÞ , n k¼1 yðkÞ
^ where yðkÞ and yðkÞ represent the forecasted and observed values, respectively. When the trigonometric grey prediction approach is used to model and forecast China annual electricity demand, we use L ¼ 23, because this is the average of the recommended bounds. Our preliminary experiments show that the forecasted values are quite insensitive to L within the recommended interval in this case study. However, it does not imply that there are no cyclic variations in the residuals for the original GM(1,1) forecasts. From Fig. 1 we can find that the residuals exhibit some cyclic variations, and the amplitude becomes
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increasingly larger over time. One possible reason for the insensitivity of the forecasted values to L in this illustration is that the seasonality of the residuals for annual data case may not be strong. We also apply the GM(1,1) and ARIMA models for comparison purposes. In the case of the ARIMA model, a natural logarithm transformation is first applied to the original time series, and then an ARIMA(0,1,0) model is identified and formulated based on the transformed time series. The resulting ^ ^ 1Þ0:9895 , k ¼ 2; 3; 4; . . .. Table 1 shows the forecasted values as well as ARIMA model is yðkÞ ¼ 1:1837½yðk the relative errors (RE) for the three methods. 600.00 400.00 100 Million KWh
200.00 0.00 -200.00 -400.00 -600.00 -800.00 -1000.00 -1200.00 -1400.00 Fig. 1. GM(1,1) residuals of China annual electricity demand from 1981 to 2002.
Table 1 Observed and forecasted electricity demands in China, 1981–2002, for three different approaches (unit: 100 million kWh) Year
Observed value
GM(1,1) Forecasted value
TGMa
ARIMA RE (%)
Forecasted value
RE (%)
Forecasted value
RE (%)
Model building stage: 1981–1998 1981 3096 3096 1982 3280 3327.7 1983 3519 3611.5 1984 3778 3919.5 1985 4118 4253.9 1986 4507 4616.7 1987 4985 5010.5 1988 5467 5437.9 1989 5865 5901.7 1990 6230 6405.1 1991 6775 6951.4 1992 7542 7544.3 1993 8426.5 8187.8 1994 9260.4 8886.2 1995 10,023.4 9644.1 1996 10,764.3 10,466.7 1997 11,284.4 11,359.5 1998 11,598.4 12,328.4
0.00 1.45 2.63 3.75 3.30 2.43 0.51 0.53 0.63 2.81 2.60 0.03 2.83 4.04 3.78 2.76 0.67 6.29
3096 3368.7 3662.2 3977.8 4316.8 4680.8 5071.1 5489.3 5937.1 6416.1 6928.1 7475.0 8058.7 8681.1 9344.2 10,050.3 10,801.4 11,599.9
0.00 2.70 4.07 5.29 4.83 3.86 1.73 0.41 1.23 2.99 2.26 0.89 4.36 6.26 6.78 6.63 4.28 0.01
3096 3422.9 3552.1 3756.1 4037.5 4395.1 4824.4 5318.3 5867.3 6461.1 7089.0 7741.4 8410.5 9091.6 9783.3 10,488.4 11,213.7 11,970.6
0.00 4.36 0.94 0.58 1.96 2.48 3.22 2.72 0.04 3.71 4.63 2.64 0.19 1.82 2.40 2.56 0.63 3.21
Ex post testing stage: 1999–2002 1999 12,305.2 13,379.9 2000 13,471.4 14,521.2 2001 14,633.5 15,759.8 2002 16,331.5 17,104.0
8.73 7.79 7.70 4.73
12,448.1 13,348.5 14,303.6 15,315.8
1.16 0.91 2.25 6.22
12,773.9 13,641.8 14,594.9 15,655.6
3.81 1.26 0.26 4.14
a
The proposed trigonometric grey prediction approach.
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Measures of the corresponding forecasting errors are shown in Table 2. Although the GM(1,1) model generates a smaller forecasting error than the ARIMA model in the model building stage, the prediction accuracy of the latter is far better than that of the former in the ex post testing stage. For this particular case, the trigonometric grey prediction approach outperforms the GM(1,1) and ARIMA models in both stages. Fig. 2 shows the upper and lower bounds of the grey prediction intervals at the 95% confidence level, respectively denoted by UPI and LPI, as well as the observed values and point forecasts. Although the width of the prediction intervals is not too large, all the observed values fall between the lower and the upper bound values of the grey prediction intervals. In order to further investigate the applicability of the trigonometric grey prediction approach, we present another case study based on China quarterly gross electricity generation data collected from the Asia Pacific Energy Database [27]. The 12 data points from 2000 to 2002 are used for model building, and the four data points in 2003 are used as an ex post testing data set. The results given by the GM(1,1) model and the trigonometric grey prediction approach (L ¼ 4) as well as the observed values are shown in Fig. 3. It is found that the forecasted values by the trigonometric grey prediction approach fit the observations better than those by the GM(1,1) model. This case study further demonstrates the effectiveness and ability of the trigonometric grey prediction approach in improving the forecasting accuracy of the GM(1,1) model. Fig. 4 shows the model percentage error distributions for the trigonometric grey prediction approach for different L ( ¼ 3, 4 or 5). It is found that the model percentage errors for L ¼ 4 are the smallest in most cases. As a whole, the MAPE for L ¼ 4 is less than half of the MAPE for L ¼ 3 or 5 for a whole experimental period. It indicates that the trigonometric grey prediction approach could capture the quarterly seasonality well when L ¼ 4.
Table 2 Comparative analysis of forecasting errors MAPE (%)
Model building stage: 1981–1998 GM(1,1) ARIMA TGM
2.28 3.25 2.12
170.40 234.21 143.89
60,773.23 104,948.64 32,596.64
Ex post testing stage: 1999–2002 GM(1,1) ARIMA TGM
7.24 2.64 2.37
1005.83 402.86 338.37
1,030,591.53 293,999.26 176,749.80
18000 16000 14000 12000 10000
MAD
MSE
Observed value Forecast value LPI UPI
20 01
19 99
19 97
19 95
19 93
19 91
19 89
19 87
19 85
19 83
8000 6000 4000 2000 0 19 81
100 million KWh
Models
Fig. 2. Grey prediction intervals of China electricity demand from 1982 to 2002.
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550000 Observed value GM forecast value TGM forecast value
500000 GWh
450000 400000 350000 300000 250000
2003-4
2003-3
2003-2
2003-1
2002-4
2002-3
2002-2
2002-1
2001-4
2001-3
2001-2
2001-1
2000-4
2000-3
2000-2
2000-1
200000
Fig. 3. Observed and model values of China quarterly electricity generation from 2000 to 2003.
Percentage error (%)
10 5 0 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
-5 -10
L=4 L=3 L=5
-15 Fig. 4. Percentage errors for the TGM approach for L ¼ 3, 4, 5.
5. Conclusion In this paper, we present a trigonometric grey prediction approach to forecasting electricity demand by combining the GM(1,1) model with the trigonometric residual modification technique. Our approach helps to improve the forecasting accuracy of the GM(1,1) and allows a reasonable grey prediction interval to be obtained. Two cases studies based on China annual and quarterly electricity data are presented to demonstrate the effectiveness and reliability of our approach. Acknowledgements The authors are grateful to the anonymous referees for their helpful comments.
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Energy 31 (2006) 2839–2847 www.elsevier.com/locate/energy
A trigonometric grey prediction approach to forecasting electricity demand P. Zhou, B.W. Ang, K.L. Poh Department of Industrial and Systems Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 4 April 2005
Abstract Electricity demand forecasting plays an important role in electricity systems expansion planning. In this paper, we present a trigonometric grey prediction approach by combining the traditional grey model GM(1,1) with the trigonometric residual modification technique for forecasting electricity demand. Our approach helps to improve the forecasting accuracy of the GM(1,1) and allows a reasonable grey prediction interval to be obtained. Two case studies using the data of China are presented to demonstrate the effectiveness of our approach. r 2005 Elsevier Ltd. All rights reserved. Keywords: Electricity demand; Forecasting; Grey system theory; The GM(1,1) model
1. Introduction Electricity demand forecasting mainly deals with the forecasting of the amount of electricity that should be generated and distributed in a specific region over a specific period by electric utilities. It plays an important role in electricity systems expansion planning. Since electricity has great influence on people’s daily life and the national economy, the accuracy of electricity demand forecasting is important not only for electric utilities themselves but also for the consumers. The forecasting models for electricity demand can be broadly classified into two categories, viz. causal and time-series models [1]. Causal models exploit the relationship between a dependent variable and independent variables, and assume that the variations in dependent variable could be explained by independent variables. Multiple linear regression analysis and econometric models are the most popular causal models [1–4]. A limitation of causal models is that it depends on the availability and reliability of independent variables over the forecasting period which requires further efforts in data collection and estimation. In contrast, time-series models such as the growth curve models and the autoregressive integrated moving average (ARIMA) models, which require only the historical data of the variable of interest to forecast its future evolution behavior, has been equally popular in electricity demand forecasting [5–8]. A large number of historical observations are usually needed to obtain satisfactory forecast accuracy for the ARIMA models. In addition to the models Corresponding author. Tel.: +65 6516 2203; fax: +65 6777 1434.
E-mail address: [email protected] (P. Zhou). 0360-5442/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2005.12.002
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mentioned above, modern machine learning techniques such as artificial neural networks (ANNs), which can be formulated in the form of causal or time-series models, have also gained popularity in electricity demand and load forecasting [1,9–11]. Their performance will depend on the amount of training data and how representative these data are, as well as the experience of practitioners to a large extent. Grey system theory, propounded by Deng [12], is a truly multidisciplinary theory dealing with grey systems that are characterized by both partially known and partially unknown information. It has been widely used in several fields such as agriculture, industry and environmental systems studies [13,14]. As an essential part of grey system theory, grey forecasting models have gained in popularity in time-series forecasting due to their simplicity and ability to characterize an unknown system by using as few as four data points [15–17]. Electricity demand forecasting may be regarded as a grey system problem, because we know some factors such as population, economy conditions and weather have influence, but we do not clearly know how they exactly affect electricity demand [18]. Like the ARIMA models, the grey forecasting models are also a family of forecasting models among which the GM(1,1) model is the most frequently used. Since only two parameters are estimated and used to model a time series in the GM(1,1) model, its prediction accuracy may not be always satisfactory. Some studies have been reported on how to improve the accuracy of the GM(1,1) model [19,20]. More recently, several improved grey forecasting models were developed for electricity demand and load forecasting [21–23]. However, to build a reasonable forecasting model, most of these methods are either rather complex or require a large number of data points. In this paper, we present a simple trigonometric grey prediction approach to forecasting electricity demand by combining the GM(1,1) model with the trigonometric residual modification technique. While keeping the useful characteristic of few data points required, its prediction accuracy is far higher than that of the original GM(1,1) model. In addition, a reasonable prediction interval could be constructed. The rest of this paper is organized as follows. Section 2 introduces the original GM(1,1) model. In Section 3 we propose our trigonometric grey prediction approach. Following the idea of interval prediction in multiple regression analysis, we also present a grey prediction interval based on the proposed approach. In Section 4 two case studies on China electricity demand forecasting are presented. Section 5 concludes this study. 2. GM(1,1) model Among the family of grey forecasting models, the GM(1,1) model is the most frequently used, and is one with a certain degree of accuracy despite its simplicity [13]. Conventional forecasting techniques often deal with the original historical data directly and try to model their evolution behavior approximately. However, the GM(1,1) model begins by converting the original data series into a monotonically increasing data series by a preliminary transformation called accumulated generating operation (AGO). Applying the AGO technique reduces the noise of the original data series efficiently, and the new data series generated will approximately exhibit exponential behavior [19]. Since the solution of first-order differential equations also takes the exponential form, the first-order grey differential equations are then constructed to model the data series from AGO and forecast the future behavior of the system. Assume that X ð0Þ ¼ fxð0Þ ð1Þ; xð0Þ ð2Þ; . . . ; xð0Þ ðnÞjnX4g is the original non-negative data series taken in consecutive order and at equal time interval. The procedures for applying the GM(1,1) model to predict the future value xð0Þ ðn þ kÞ with kX1 can be described as follows [21–23]: Step 1: Form a new data series X ð1Þ ¼ fxð1Þ ð1Þ; xð1Þ ð2Þ; . . . ; xð1Þ ðnÞg by AGO: xð1Þ ðkÞ ¼
k X
xð0Þ ðiÞ.
(1)
i¼1
Step 2: Establish the grey differential equation as follows: xð0Þ ðkÞ þ azð1Þ ðkÞ ¼ u;
k ¼ 2; 3; . . . ; n,
(2)
where zð1Þ ðkÞ ¼ 0:5xð1Þ ðkÞ þ 0:5xð1Þ ðk 1Þ.
(3)
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Step 3: Estimate the developing coefficient a and the grey input u in Eq. (2) by the ordinary least-squares (OLS) method: ^ u ^ T ¼ ðU T UÞ1 U T X n , ½a;
(4)
where 3 zð1Þ ð2Þ 1 7 6 ð1Þ 6 z ð3Þ 1 7 7 6 , U¼6 .. 7 .. 6 .7 . 5 4 zð1Þ ðnÞ 1 2
(5)
and X n ¼ ½xð0Þ ð2Þ; xð0Þ ð3Þ; . . . ; xð0Þ ðnÞT .
(6)
Step 4: Establish the following whitened first-order differential equation (or grey reflection equation) for future prediction: dxð1Þ ^ ð1Þ ¼ u, ^ þ ax dt xð1Þ ð1Þ ¼ xð0Þ ð1Þ.
ð7Þ
Step 5: Solve the grey reflection Eq. (7) and obtain the predicted values for the data series X AGO as u^ u^ ^ x^ ð1Þ ðkÞ ¼ xð0Þ ð1Þ eaðk1Þ þ ; k ¼ 0; 1; 2; . . . . a^ a^
ð1Þ
from
(8)
By applying the inverse AGO (IAGO), xð0Þ ðkÞ ¼ xð1Þ ðkÞ xð1Þ ðk 1Þ to Eq. (8), the predictions for the original data series can be obtained: x^ ð0Þ ð1Þ ¼ xð0Þ ð1Þ,
u^ ^ ; x^ ð0Þ ðkÞ ¼ ð1 ea^ Þ xð0Þ ð1Þ eaðk1Þ a^
k ¼ 2; 3; . . . .
ð9Þ
3. Trigonometric grey prediction approach Despite its popularity in real application, the prediction accuracy of the GM(1,1) model may not be always satisfactory. A large number of approaches have been developed to improve the accuracy of the GM(1,1) model [19–23]. Most of them follow the rule that the information in the residual series, given by the differences between the observed and forecasted values, may be used to compensate the initial grey prediction values. Although the prediction accuracy is improved, these methods seem to be a little complex, and a lot of observations might be needed to build a reasonable model. For instance, in the ANN-based grey prediction approach developed by Hsu and Chen [21], the forecasted absolute residual series with the signs estimated by ANN are added to the initial forecasted values of the original series, in order to improve the prediction accuracy of the GM(1,1) model. However, the pattern of residual signs may not be accurately learned from few historical data points. Furthermore, these methods deal with only point prediction, while an interval prediction, especially the upper bound values of predictions, may be very useful for balancing the supply and demand in electric utility operations and management [18]. We, therefore, develop the following trigonometric grey prediction approach which requires few data points and, yet, can provide a relatively accurate point prediction as well as a reasonable prediction interval. Denote the residual series as rð0Þ ¼ frð0Þ ð2Þ; rð0Þ ð3Þ; . . . ; rð0Þ ðnÞg, where rð0Þ ðkÞ ¼ xð0Þ ðkÞ x^ ð0Þ ðkÞ;
k ¼ 2; 3; . . . ; n.
(10)
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Since the GM(1,1) model will always generate an exponentially increasing or decreasing data series, the amplitude of the residual series may become increasingly larger over time. Considering the effects of economic fluctuations or seasonal changes on electricity demand, the cyclic variations may exist in the residual series. The residual series may be treated as a combination of a linear trend with cyclic variations. In time-series analysis the trigonometric models are capable of modeling a seasonal process with a linear trend well [24]. The residual series can then be modeled by the generalized trigonometric model: rð0Þ ðk þ 1Þ ¼ b0 þ b1 k þ b2 sin
2pk 2pk þ b3 cos þ ek ; L L
k ¼ 1; 2; . . . ,
(11)
where L is a user-specified parameter for the period of the main cyclic variations and ek is the random component. For quarterly and monthly electricity demand, L ¼ 4 and 12 are, respectively, recommended because seasonal effects are obviously the main cyclic variations. For annual electricity demand, L should reflect the cyclic period of economic fluctuations and depends on the amount of historical data. When there are annual data for about 20 yr, our recommendation is 15pLp30, because such a choice may well capture the patterns of residual series according to our preliminary experiments. Obviously, the coefficients of Eq. (11) can be estimated by OLS as follows: ½b^0 ; b^1 ; b^2 ; b^3 T ¼ ðBT BÞ1 BT Rn ,
(12)
where 2
1 61 6 6 B¼6. 6 .. 4 1
1 2 .. . n1
sin 2p L sin 4p L .. . sin
2ðn1Þp L
cos cos .. . cos
2p L 4p L
2ðn1Þp L
3 7 7 7 7, 7 5
(13)
and Rn ¼ ½rð0Þ ð2Þ; rð0Þ ð3Þ; . . . ; rð0Þ ðnÞT .
(14)
ð0Þ
The forecasted residual series r^ ðkÞ can be obtained by 2pk ^ 2pk þ b3 cos ; r^ð0Þ ðk þ 1Þ ¼ b^0 þ b^1 k þ b^2 sin L L
k ¼ 1; 2; . . . .
(15)
Note that the constant term and the trigonometric terms in Eq. (15) can be treated as a part of the Fourier residual model [16] when L is equal to n1. The difference is that we substitute a linear trend term ‘‘b^1 k’’ for other terms in the Fourier residual model, whereby the occupied degree of freedom is highly reduced. Based on the above trigonometric residual modification technique, our trigonometric grey prediction model can be formulated as follows: ð0Þ x^ ð0Þ tr ð1Þ ¼ x ð1Þ,
^ ð0Þ ðkÞ þ r^ð0Þ ðkÞ; x^ ð0Þ tr ðkÞ ¼ x x^ ð0Þ tr ðkÞ
k ¼ 2; 3; . . . ,
ð16Þ
denotes the forecasted value and x^ ð0Þ ðkÞ is the output of the GM(1,1) model. where Our discussion so far is restricted to point prediction. For the purpose of balancing the supply and demand in electric utility operations and management, a reasonable interval prediction of electricity demand is very useful. Morita et al. [18] have provided a grey prediction interval of the annual electricity demand, but their method is only restricted to the original GM(1,1) model. We shall now present a prediction interval for our trigonometric grey prediction model. Note that the trigonometric residual modification model as given by Eq. (11) is essentially a multiple linear regression model. Since the GM(1,1) model and the trigonometric residual modification model have two and four parameters, respectively, the total degrees of freedom occupied by our trigonometric grey prediction approach is six. When the number of historical data points is more than six, the excessive degrees of freedom
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can be used to estimate the variance of residual errors and then the prediction interval. In contrast, in the Fourier grey prediction approach, almost all the degrees of freedom are used for parameter estimation. By following the idea of interval prediction in multiple regression analysis, we present a 100ð1 aÞ% prediction interval of xð0Þ ðkÞ as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þ ^ ð0Þ x^ ð0Þ k ¼ 2; 3; . . . , (17) tr ðkÞ ta=2;n6 s 1 þ hkk px ðkÞpx tr ðkÞ þ ta=2;n6 s 1 þ hkk ; where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 uP h u n ð0Þ t k¼1 x^ tr ðkÞ xð0Þ ðkÞ , s¼ n6 hkk ¼ Y Tk ðBT BÞ1 Y k , and
2pðk 1Þ 2pðk 1Þ ; cos Y k ¼ 1; k 1; sin L L
T .
Although the prediction interval is obtained using the concept of multiple regression analysis, there is no strict deduction or proof. Since the centre point of the prediction interval comes from our improved grey prediction approach, the prediction interval may be taken as a ‘‘grey prediction interval’’. 4. Case studies We first present an empirical illustration on China annual electricity demand forecasting to examine the performance of our trigonometric grey prediction approach. Because of the different approaches in economic development before and after the reform and open policy initiated in the late 1970s, the electricity consumption patterns in China in the two periods are quite different. We therefore use the annual electricity demand data after 1980 in our study. The data were collected from China Statistical Yearbook [25], where the 1981–1998 data are used for model building, while the 1999–2002 data are used as an ex post testing data set. Prediction accuracy is an important criterion for evaluating a forecasting technique [26]. In this study three statistical measures, viz. mean absolute percentage error (MAPE), mean absolute deviation (MAD) and meansquared error (MSE) are used to evaluate the prediction accuracy of our approach. MAPE is a general accepted measure in percent of prediction accuracy. MAD and MSE are two measures of the average magnitude of the forecast errors, but the latter imposes a greater penalty on a large error than several small errors. The three measures are, respectively, defined as follows: MAPEð%Þ ¼
(18)
n 1X ^ yðkÞj, jyðkÞ n k¼1
(19)
n 1X ^ yðkÞÞ2 , ðyðkÞ n k¼1
(20)
MAD ¼
MSE ¼
n ^ yðkÞj 1X jyðkÞ , n k¼1 yðkÞ
^ where yðkÞ and yðkÞ represent the forecasted and observed values, respectively. When the trigonometric grey prediction approach is used to model and forecast China annual electricity demand, we use L ¼ 23, because this is the average of the recommended bounds. Our preliminary experiments show that the forecasted values are quite insensitive to L within the recommended interval in this case study. However, it does not imply that there are no cyclic variations in the residuals for the original GM(1,1) forecasts. From Fig. 1 we can find that the residuals exhibit some cyclic variations, and the amplitude becomes
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increasingly larger over time. One possible reason for the insensitivity of the forecasted values to L in this illustration is that the seasonality of the residuals for annual data case may not be strong. We also apply the GM(1,1) and ARIMA models for comparison purposes. In the case of the ARIMA model, a natural logarithm transformation is first applied to the original time series, and then an ARIMA(0,1,0) model is identified and formulated based on the transformed time series. The resulting ^ ^ 1Þ0:9895 , k ¼ 2; 3; 4; . . .. Table 1 shows the forecasted values as well as ARIMA model is yðkÞ ¼ 1:1837½yðk the relative errors (RE) for the three methods. 600.00 400.00 100 Million KWh
200.00 0.00 -200.00 -400.00 -600.00 -800.00 -1000.00 -1200.00 -1400.00 Fig. 1. GM(1,1) residuals of China annual electricity demand from 1981 to 2002.
Table 1 Observed and forecasted electricity demands in China, 1981–2002, for three different approaches (unit: 100 million kWh) Year
Observed value
GM(1,1) Forecasted value
TGMa
ARIMA RE (%)
Forecasted value
RE (%)
Forecasted value
RE (%)
Model building stage: 1981–1998 1981 3096 3096 1982 3280 3327.7 1983 3519 3611.5 1984 3778 3919.5 1985 4118 4253.9 1986 4507 4616.7 1987 4985 5010.5 1988 5467 5437.9 1989 5865 5901.7 1990 6230 6405.1 1991 6775 6951.4 1992 7542 7544.3 1993 8426.5 8187.8 1994 9260.4 8886.2 1995 10,023.4 9644.1 1996 10,764.3 10,466.7 1997 11,284.4 11,359.5 1998 11,598.4 12,328.4
0.00 1.45 2.63 3.75 3.30 2.43 0.51 0.53 0.63 2.81 2.60 0.03 2.83 4.04 3.78 2.76 0.67 6.29
3096 3368.7 3662.2 3977.8 4316.8 4680.8 5071.1 5489.3 5937.1 6416.1 6928.1 7475.0 8058.7 8681.1 9344.2 10,050.3 10,801.4 11,599.9
0.00 2.70 4.07 5.29 4.83 3.86 1.73 0.41 1.23 2.99 2.26 0.89 4.36 6.26 6.78 6.63 4.28 0.01
3096 3422.9 3552.1 3756.1 4037.5 4395.1 4824.4 5318.3 5867.3 6461.1 7089.0 7741.4 8410.5 9091.6 9783.3 10,488.4 11,213.7 11,970.6
0.00 4.36 0.94 0.58 1.96 2.48 3.22 2.72 0.04 3.71 4.63 2.64 0.19 1.82 2.40 2.56 0.63 3.21
Ex post testing stage: 1999–2002 1999 12,305.2 13,379.9 2000 13,471.4 14,521.2 2001 14,633.5 15,759.8 2002 16,331.5 17,104.0
8.73 7.79 7.70 4.73
12,448.1 13,348.5 14,303.6 15,315.8
1.16 0.91 2.25 6.22
12,773.9 13,641.8 14,594.9 15,655.6
3.81 1.26 0.26 4.14
a
The proposed trigonometric grey prediction approach.
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Measures of the corresponding forecasting errors are shown in Table 2. Although the GM(1,1) model generates a smaller forecasting error than the ARIMA model in the model building stage, the prediction accuracy of the latter is far better than that of the former in the ex post testing stage. For this particular case, the trigonometric grey prediction approach outperforms the GM(1,1) and ARIMA models in both stages. Fig. 2 shows the upper and lower bounds of the grey prediction intervals at the 95% confidence level, respectively denoted by UPI and LPI, as well as the observed values and point forecasts. Although the width of the prediction intervals is not too large, all the observed values fall between the lower and the upper bound values of the grey prediction intervals. In order to further investigate the applicability of the trigonometric grey prediction approach, we present another case study based on China quarterly gross electricity generation data collected from the Asia Pacific Energy Database [27]. The 12 data points from 2000 to 2002 are used for model building, and the four data points in 2003 are used as an ex post testing data set. The results given by the GM(1,1) model and the trigonometric grey prediction approach (L ¼ 4) as well as the observed values are shown in Fig. 3. It is found that the forecasted values by the trigonometric grey prediction approach fit the observations better than those by the GM(1,1) model. This case study further demonstrates the effectiveness and ability of the trigonometric grey prediction approach in improving the forecasting accuracy of the GM(1,1) model. Fig. 4 shows the model percentage error distributions for the trigonometric grey prediction approach for different L ( ¼ 3, 4 or 5). It is found that the model percentage errors for L ¼ 4 are the smallest in most cases. As a whole, the MAPE for L ¼ 4 is less than half of the MAPE for L ¼ 3 or 5 for a whole experimental period. It indicates that the trigonometric grey prediction approach could capture the quarterly seasonality well when L ¼ 4.
Table 2 Comparative analysis of forecasting errors MAPE (%)
Model building stage: 1981–1998 GM(1,1) ARIMA TGM
2.28 3.25 2.12
170.40 234.21 143.89
60,773.23 104,948.64 32,596.64
Ex post testing stage: 1999–2002 GM(1,1) ARIMA TGM
7.24 2.64 2.37
1005.83 402.86 338.37
1,030,591.53 293,999.26 176,749.80
18000 16000 14000 12000 10000
MAD
MSE
Observed value Forecast value LPI UPI
20 01
19 99
19 97
19 95
19 93
19 91
19 89
19 87
19 85
19 83
8000 6000 4000 2000 0 19 81
100 million KWh
Models
Fig. 2. Grey prediction intervals of China electricity demand from 1982 to 2002.
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550000 Observed value GM forecast value TGM forecast value
500000 GWh
450000 400000 350000 300000 250000
2003-4
2003-3
2003-2
2003-1
2002-4
2002-3
2002-2
2002-1
2001-4
2001-3
2001-2
2001-1
2000-4
2000-3
2000-2
2000-1
200000
Fig. 3. Observed and model values of China quarterly electricity generation from 2000 to 2003.
Percentage error (%)
10 5 0 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
-5 -10
L=4 L=3 L=5
-15 Fig. 4. Percentage errors for the TGM approach for L ¼ 3, 4, 5.
5. Conclusion In this paper, we present a trigonometric grey prediction approach to forecasting electricity demand by combining the GM(1,1) model with the trigonometric residual modification technique. Our approach helps to improve the forecasting accuracy of the GM(1,1) and allows a reasonable grey prediction interval to be obtained. Two cases studies based on China annual and quarterly electricity data are presented to demonstrate the effectiveness and reliability of our approach. Acknowledgements The authors are grateful to the anonymous referees for their helpful comments.
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