- Email: [email protected]
Applications of improved grey prediction model for power demand forecasting
Energy Conversion and Management 44 (2003) 2241–2249 www.elsevier.com/locate/enconman
Applications of improved grey prediction model for power demand forecasting Che-Chiang Hsu
a,*
, Chia-Yon Chen
b
a
b
Industrial Engineering and Management Department, Nan-Jeon Junior Institute of Technology, 178 Chau-Chin Road, Yen Shui, Tainan Hisen 73701, Taiwan, ROC Institute of Resources Engineering, National Cheng-Kung University, 1 Ta-Hsueh Road, Tainan 70101, Taiwan, ROC Received 10 July 2002; accepted 28 October 2002
Abstract Grey theory is a truly multidisciplinary and generic theory that deals with systems that are characterized by poor information and/or for which information is lacking. In this paper, an improved grey GM(1,1) model, using a technique that combines residual modification with artificial neural network sign estimation, is proposed. We use power demand forecasting of Taiwan as our case study to test the efficiency and accuracy of the proposed method. According to the experimental results, our proposed new method obviously can improve the prediction accuracy of the original grey model. Ó 2003 Published by Elsevier Science Ltd. Keywords: Grey theory; Improved GM(1,1) model; Artificial neural network
1. Introduction Grey theory, developed originally by Deng [1], is a truly multidisciplinary and generic theory that deals with systems that are characterized by poor information and/or for which information is lacking. The fields covered by grey theory include systems analysis, data processing, modeling, prediction, decision making and control. The grey theory mainly works on systems analysis with poor, incomplete or uncertain messages. Grey forecasting models have been extensively used in many applications [2–10]. In contrast to statistical methods, the potency of the original series in the time series grey model, called GM(1,1), has been proven to be more than four [11]. In
*
Corresponding author. Tel.: +886-6-2757575x62826; fax: +886-6-2380421. E-mail address: [email protected] (C.-C. Hsu).
0196-8904/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. doi:10.1016/S0196-8904(02)00248-0
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addition, assumptions regarding the statistical distribution of data are not necessary when applying grey theory. The accumulated generation operation (AGO) is one of the most important characteristics of grey theory, and its main purpose is to reduce the randomness of data. In fact, functions derived from AGO formulations of the original series are always well fitted to exponential functions. In this paper, we introduce a new technique that combines residual modification and residual artificial neural network (ANN) sign estimation to improve the accuracy of the original GM(1,1) model. Furthermore, we use power demand forecasting of Taiwan as our case study to examine the model reliability and accuracy.
2. Original GM(1,1) forecasting model The GM(1,1) is one of the most frequently used grey forecasting model. This model is a time series forecasting model, encompassing a group of differential equations adapted for parameter variance, rather than a first order differential equation. Its difference equations have structures that vary with time rather than being general difference equations. Although it is not necessary to employ all the data from the original series to construct the GM(1,1), the potency of the series must be more than four. In addition, the data must be taken at equal intervals and in consecutive order without bypassing any data [11]. The GM(1,1) model constructing process is described below: Denote the original data sequence by ð1Þ xð0Þ ¼ xð0Þ ð1Þ; xð0Þ ð2Þ; xð0Þ ð3Þ; . . . ; xð0Þ ðnÞ ; where n is the number of years observed. The AGO formation of xð0Þ is defined as: xð1Þ ¼ xð1Þ ð1Þ; xð1Þ ð2Þ; xð1Þ ð3Þ; . . . ; xð1Þ ðnÞ ;
ð2Þ
where xð1Þ ð1Þ ¼ xð0Þ ð1Þ; and xð1Þ ðkÞ ¼
k X
xð0Þ ðmÞ;
k ¼ 2; 3; . . . ; n:
ð3Þ
m¼1
The GM(1,1) model can be constructed by establishing a first order differential equation for xð1Þ ðkÞ as: dxð1Þ ðkÞ=dk þ axð1Þ ðkÞ ¼ b:
ð4Þ
Therefore, the solution of Eq. (4) can be obtained by using the least square method. That is, ! ^ b b^ x^ð1Þ ðkÞ ¼ xð0Þ ð1Þ e^aðk1Þ þ ; ð5Þ a^ a^ where ½^ a; b^T ¼ ðBT BÞ1 BT Xn
ð6Þ
C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 2241–2249
2243
and 2 6 6 B¼6 4
0:5ðxð1Þ ð1Þ þ xð1Þ ð2ÞÞ 0:5ðxð1Þ ð2Þ þ xð1Þ ð3ÞÞ .. .
3 1 17 7 ; .. 7 .5
ð7Þ
0:5ðxð1Þ ðn 1Þ þ xð1Þ ðnÞÞ 1
T Xn ¼ xð0Þ ð2Þ; xð0Þ ð3Þ; xð0Þ ð4Þ; . . . ; xð0Þ ðnÞ : We obtained x^ð1Þ from Eq. (5). Let x^ð0Þ be the fitted and predicted series, x^ð0Þ ¼ x^ð0Þ ð1Þ; x^ð0Þ ð2Þ; x^ð0Þ ð3Þ; . . . ; x^ð0Þ ðnÞ; . . . ; where x^ð0Þ ð1Þ ¼ xð0Þ ð1Þ. Applying the inverse AGO, we then have ! b^ ð0Þ ð0Þ ð1 ea^Þe^aðk1Þ ; k ¼ 2; 3; . . . ; x^ ðkÞ ¼ x ð1Þ a^
ð8Þ
ð9Þ
ð10Þ
where x^ð0Þ ð1Þ; x^ð0Þ ð2Þ; . . . ; x^ð0Þ ðnÞ are called the GM(1,1) fitted sequence, while x^ð0Þ ðn þ 1Þ; x^ð0Þ ðn þ 2Þ; . . . ; are called the GM(1,1) forecast values.
3. Improved grey forecasting model Deng [1] also developed a residual modification model, the residual GM(1,1) model. The differences between the real values, xð0Þ ðkÞ, and the model predicted values, x^ð0Þ ðkÞ, are defined as the residual series. We denote the residual series as qð0Þ : qð0Þ ¼ qð0Þ ð2Þ; qð0Þ ð3Þ; qð0Þ ð4Þ; . . . ; qð0Þ ðnÞ ; ð11Þ where qð0Þ ðkÞ ¼ xð0Þ ðkÞ x^ð0Þ ðkÞ:
ð12Þ
The residual GM(1,1) model could be established to improve the predictive accuracy of the original GM(1,1) model. The modified prediction values can be obtained by adding the forecasted values of the residual GM(1,1) model to the original x^ð0Þ ðkÞ. However, the potency of the residual series depends on the number of data points with the same sign, which is usually small when there are few observations. In these cases, the potency of the residual series with the same sign may not be more than four, and a residual GM(1,1) model cannot be established. Here, we present an improved grey model to solve this problem. We establish a modification sub-model that is a combination residual GM(1,1) forecaster that uses the absolute values of the residual series with an ANN for residual sign estimation. The schematic of the improved forecasting system is shown in Fig. 1. The detail process to formulate this improved grey forecast model is described as follows.
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C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 2241–2249 Original Model Data Input
Original GM(1,1) Forecaster
Original Forecast Output
Residual Input Final Forecast
Modification Sub-Model Residual GM(1,1) Residual Forecast Output Forecaster ANN Sign Estimater
Combination Module
Residual Sign Eastimate
Fig. 1. Schematic of the forecasting system.
3.1. Residual forecasting model First, denote the absolute values of the residual series as eð0Þ : eð0Þ ¼ eð0Þ ð2Þ; eð0Þ ð3Þ; eð0Þ ð4Þ; . . . ; eð0Þ ðnÞ ;
ð13Þ
where
eð0Þ ¼ qð0Þ ðkÞ;
k ¼ 2; 3; . . . ; n:
ð14Þ
By using the same methods as Eqs. (1)–(10), a GM(1,1) model of eð0Þ can be established. Denote the forecast residual series as ^eð0Þ ðkÞ, then be ð0Þ ð0Þ ^e ðkÞ ¼ e ð2Þ ð15Þ ð1 eae Þeae ðk1Þ ; k ¼ 2; 3; . . . ae 3.2. ANN residual sign estimation model In recent years, much research has been conducted on the application of artificial intelligence techniques to forecasting problems. However, the model that has received extensive attention is undoubtedly the ANN, cited as among the most powerful computational tools ever developed. Fig. 2 presents an outline of a simple biological neural and an ANNÕs basic elements. ANN models operate like a ‘‘black box’’, requiring no detailed information about the system. Instead, they learn the relationship between the input parameters and the controlled and uncontrolled variables by studying previous data. ANN models could handle large and complex systems with many interrelated parameters. Several types of neural architectures are available, among which the multi-layer back propagation (BP) neural network is the most widely used. As Fig. 3 reveals, a BP network typically employs three or more layers for the architecture: an input layer, an output layer and at least one hidden layer. The computational procedure of this network is described below: ! X ð16Þ Wij Xij ; Yj ¼ f i
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Fig. 2. A simple neural [13] vs. a PE model.
where Yj is the output of node j, f ðÞ is the transfer function, wij is the connection weight between node j and node i in the lower layer and Xi is the input signal from the node i in the lower layer. BP is a gradient descent algorithm. It tries to improve the performance of the neural network by reducing the total error by changing the weights along its gradient. The BP algorithm minimizes the square errors, which can be calculated by:
Fig. 3. A BP network.
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Hidden Layer
Output Layer
d(n-1) d(n+1) d(n)
Bias
Fig. 4. Structure of ANN sign forecasting system.
E ¼ 1=2
XX p ½Oj Yjp 2 ; p
ð17Þ
j
where E is the square errors, p is the index of the pattern, O is the actual (target) output and Y is the network output. A two state ANN model is used here to predict the signs of the forecast residual series. First, we introduce a dummy variable dðkÞ to indicate the sign of the kth year residual. Assume the sign of the kth year residual is positive, then the value of dðkÞ is 1, otherwise it is 0. Then, we set up an ANN model by using the values of dðn 1Þ and dðnÞ to estimate the values of dðn þ 1Þ. The structure of this ANN sign forecasting system is shown in Fig. 4. Let the sign of the kth year residual, sðkÞ, be þ1; if dðkÞ ¼ 1 sðkÞ ¼ ; k ¼ 1; 2; . . . ; n; . . . ð18Þ 1; if dðkÞ ¼ 0 According to the equations illustrated above, an improved grey model combination residual modification with ANN sign estimation can be further formulated as Eq. (19) b be 0ð0Þ ð0Þ a aðk1Þ ð0Þ ð1 eae Þeae ðk1Þ ; þ sðkÞ e ð2Þ ð1 e Þe x^ ðkÞ ¼ x ð1Þ a ae k ¼ 1; 2; . . . ; n; n þ 1; . . .
ð19Þ
Next, we will proceed to the power demand forecasting of Taiwan for our case study to examine the reliability and accuracy of this improved GM(1,1) model. 4. Results To demonstrate the effectiveness of the proposed method, we use the power demand forecasting of Taiwan as an illustrating example. In this study, we use the historical annual power demand of Taiwan from 1985 to 2000 as our research data. There are 16 observations, where 1985–1998 are used for model fitting and 1999–2000 are reserved for ex post testing. For the purposes of comparison, we also use the same number of observations, 14 (power demand from 1985 to 1998), to formulate an ARIMA (p; d; q) model, where p is the order of the auto-regressive part, d is the order of the differencing, and q is the order of the moving average
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Table 1 Model values and forecast errors (unit: 103 W h) Year
Real value
GM(1,1)
Improved GM(1,1)
ARIMA
Model value Error (%)
Model value Error (%)
Model value Error (%)
1985 1986 1987 1988 1989 1990
47,919,102 53,812,862 59,174,751 65,227,727 69,251,809 74,344,947
47,919,102 56,318,092 60,319,829 64,605,914 69,196,550 74,113,379
0.00 4.66 1.94 )0.95 )0.08 )0.31
47,919,102 53,812,862 59,630,904 65,310,510 69,917,174 74,850,394
0.00 0.00 0.77 0.13 0.96 0.68
47,919,102 52,307,500 53,957,006 60,243,936 65,958,706 72,405,080
0.00 )2.80 )8.82 )7.64 )4.76 )2.61
1991 1992 1993 1994 1995 1996 1997 1998
80,977,405 85,290,354 92,084,684 98,561,004 105,368,193 111,139,816 118,299,046 128,129,801
79,379,577 85,019,971 91,061,148 97,531,587 104,461,790 111,884,424 119,834,482 128,349,438
)1.97 )0.32 )1.11 )1.04 )0.86 0.67 1.30 0.17
80,133,358 85,790,897 91,849,611 98,337,985 105,286,530 111,040,924 118,971,794 127,467,127
)1.04 0.59 )0.26 )0.23 )0.08 )0.09 0.57 )0.52
76,688,020 82,105,943 89,156,925 93,739,526 100,954,923 107,828,630 115,049,615 121,169,150
)5.30 )3.73 )3.18 )4.89 )4.19 )2.98 )2.75 )5.43
MAPEa (1986–1998)
1.54
1999 2000
131,725,892 142,412,887
137,469,433 147,237,458
MAPE (1999–2000) a
MAPE ¼ 1n
4.36 3.39 3.88
Pn
xð0Þ ðkÞ k¼1 ½j^
0.57 133,459,644 144,204,700
1.32 1.26
4.24 128,756,418 139,168,992
1.29
xð0Þ ðkÞj=xð0Þ ðkÞ.
Fig. 5. Real values and model values for power demand of Taiwan from 1985 to 2000.
)2.25 )2.28 2.27
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Fig. 6. Model percentage error distribution from 1985 to 2000.
process [12]. As a result of statistical tests, the ARIMA model with ðp; d; qÞ ¼ ð0; 1; 0Þ is formulated as follows: x^ðkÞ ¼ 2404647:67 þ 1:04^ xðk 1Þ;
k ¼ 2; 3; 4; . . . ; n; . . .
ð20Þ
The predicted results obtained by the original GM(1,1) model, improved GM(1,1) model and ARIMA model are shown in Table 1 and Fig. 5. The model percentage error distribution is also shown in Fig. 6. The mean absolute percentage error (MAPE) of the GM(1,1) model, the ARIMA model and our improved GM(1,1) model from 1999 to 2000 are 3.88%, 2.27% and 1.29%, respectively. According to the results shown above, our improved grey model seems to obtain the lowest post-forecasting errors among these models. It is indicated that the modification of our improved GM(1,1) model can reduce model prediction errors effectively.
5. Conclusions The original GM(1,1) model is a model with a group of differential equations adapted for variance of parameters, and it is a powerful forecasting model, especially when the number of observations is not large. In this paper, we have applied an improved grey GM(1,1) model by using a technique that combines residual modification with ANN sign estimations. Our study results show that this method can yield more accurate results than the original GM(1,1) model and also solve problems resulting from having too few data, which may lead the same sign residuals lower than four and violate the necessary condition of setting up a GM(1,1) model. The improved grey models were then applied to predict the power demand of Taiwan. Finally, through this study, our improved grey model, in this paper, is an appropriate forecasting method to yield more accurate results than the original GM(1,1) model.
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References [1] Deng JL. Grey system fundamental method. Huazhong University of Science and Technology Wuhan, China, 1982. [2] Sun G. Prediction of vegetable yields by grey model GM(1,1). J Grey Syst 1991;2:187–97. [3] Morita H et al. Interval prediction of annual maximum demand using grey dynamic model. Electr Power Energy Syst 1996;18:409–13. [4] Huang YP, Wang SF. The identification of fuzzy grey prediction system by genetic algorithm. Int J Syst Sci 1997;28:15–26. [5] Hsu CI, Wen YH. Improved grey prediction models for the Trans-Pacific air passenger market. Transport Planning Technol 1998;22:87–107. [6] Hsu CC, Chen CY. Application of grey theory to regional electricity demand forecasting. Energy Quart 1999;24:96–108. [7] Hao YH, Wang XM. The study of grey system models of Niangziguan spring. J Syst Eng 2000;16:39–44. [8] Liu SF, Deng JL. The range suitable for GM(1,1). Syst Eng Theor Appl 2000;5:111–24. [9] Xing M. Research on combined grey neural network model of seasonal forecast. Syst Eng Theor Appl 2001;11: 31–5. [10] Yue CL, Wang L. Grey–Markov forecast of the stock price. Syst Eng 2000;16:54–9. [11] Deng JL. Grey prediction and decision. Huazhong University of Science and Technology, Wuhan, China, 1986. [12] Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. NJ: Prentice Hall; 1994. [13] Nelson MM, Illingworth WT. A practical guide to neveal nets. MA: Addison-Wesley; 1993.
Applications of improved grey prediction model for power demand forecasting Che-Chiang Hsu
a,*
, Chia-Yon Chen
b
a
b
Industrial Engineering and Management Department, Nan-Jeon Junior Institute of Technology, 178 Chau-Chin Road, Yen Shui, Tainan Hisen 73701, Taiwan, ROC Institute of Resources Engineering, National Cheng-Kung University, 1 Ta-Hsueh Road, Tainan 70101, Taiwan, ROC Received 10 July 2002; accepted 28 October 2002
Abstract Grey theory is a truly multidisciplinary and generic theory that deals with systems that are characterized by poor information and/or for which information is lacking. In this paper, an improved grey GM(1,1) model, using a technique that combines residual modification with artificial neural network sign estimation, is proposed. We use power demand forecasting of Taiwan as our case study to test the efficiency and accuracy of the proposed method. According to the experimental results, our proposed new method obviously can improve the prediction accuracy of the original grey model. Ó 2003 Published by Elsevier Science Ltd. Keywords: Grey theory; Improved GM(1,1) model; Artificial neural network
1. Introduction Grey theory, developed originally by Deng [1], is a truly multidisciplinary and generic theory that deals with systems that are characterized by poor information and/or for which information is lacking. The fields covered by grey theory include systems analysis, data processing, modeling, prediction, decision making and control. The grey theory mainly works on systems analysis with poor, incomplete or uncertain messages. Grey forecasting models have been extensively used in many applications [2–10]. In contrast to statistical methods, the potency of the original series in the time series grey model, called GM(1,1), has been proven to be more than four [11]. In
*
Corresponding author. Tel.: +886-6-2757575x62826; fax: +886-6-2380421. E-mail address: [email protected] (C.-C. Hsu).
0196-8904/03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. doi:10.1016/S0196-8904(02)00248-0
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addition, assumptions regarding the statistical distribution of data are not necessary when applying grey theory. The accumulated generation operation (AGO) is one of the most important characteristics of grey theory, and its main purpose is to reduce the randomness of data. In fact, functions derived from AGO formulations of the original series are always well fitted to exponential functions. In this paper, we introduce a new technique that combines residual modification and residual artificial neural network (ANN) sign estimation to improve the accuracy of the original GM(1,1) model. Furthermore, we use power demand forecasting of Taiwan as our case study to examine the model reliability and accuracy.
2. Original GM(1,1) forecasting model The GM(1,1) is one of the most frequently used grey forecasting model. This model is a time series forecasting model, encompassing a group of differential equations adapted for parameter variance, rather than a first order differential equation. Its difference equations have structures that vary with time rather than being general difference equations. Although it is not necessary to employ all the data from the original series to construct the GM(1,1), the potency of the series must be more than four. In addition, the data must be taken at equal intervals and in consecutive order without bypassing any data [11]. The GM(1,1) model constructing process is described below: Denote the original data sequence by ð1Þ xð0Þ ¼ xð0Þ ð1Þ; xð0Þ ð2Þ; xð0Þ ð3Þ; . . . ; xð0Þ ðnÞ ; where n is the number of years observed. The AGO formation of xð0Þ is defined as: xð1Þ ¼ xð1Þ ð1Þ; xð1Þ ð2Þ; xð1Þ ð3Þ; . . . ; xð1Þ ðnÞ ;
ð2Þ
where xð1Þ ð1Þ ¼ xð0Þ ð1Þ; and xð1Þ ðkÞ ¼
k X
xð0Þ ðmÞ;
k ¼ 2; 3; . . . ; n:
ð3Þ
m¼1
The GM(1,1) model can be constructed by establishing a first order differential equation for xð1Þ ðkÞ as: dxð1Þ ðkÞ=dk þ axð1Þ ðkÞ ¼ b:
ð4Þ
Therefore, the solution of Eq. (4) can be obtained by using the least square method. That is, ! ^ b b^ x^ð1Þ ðkÞ ¼ xð0Þ ð1Þ e^aðk1Þ þ ; ð5Þ a^ a^ where ½^ a; b^T ¼ ðBT BÞ1 BT Xn
ð6Þ
C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 2241–2249
2243
and 2 6 6 B¼6 4
0:5ðxð1Þ ð1Þ þ xð1Þ ð2ÞÞ 0:5ðxð1Þ ð2Þ þ xð1Þ ð3ÞÞ .. .
3 1 17 7 ; .. 7 .5
ð7Þ
0:5ðxð1Þ ðn 1Þ þ xð1Þ ðnÞÞ 1
T Xn ¼ xð0Þ ð2Þ; xð0Þ ð3Þ; xð0Þ ð4Þ; . . . ; xð0Þ ðnÞ : We obtained x^ð1Þ from Eq. (5). Let x^ð0Þ be the fitted and predicted series, x^ð0Þ ¼ x^ð0Þ ð1Þ; x^ð0Þ ð2Þ; x^ð0Þ ð3Þ; . . . ; x^ð0Þ ðnÞ; . . . ; where x^ð0Þ ð1Þ ¼ xð0Þ ð1Þ. Applying the inverse AGO, we then have ! b^ ð0Þ ð0Þ ð1 ea^Þe^aðk1Þ ; k ¼ 2; 3; . . . ; x^ ðkÞ ¼ x ð1Þ a^
ð8Þ
ð9Þ
ð10Þ
where x^ð0Þ ð1Þ; x^ð0Þ ð2Þ; . . . ; x^ð0Þ ðnÞ are called the GM(1,1) fitted sequence, while x^ð0Þ ðn þ 1Þ; x^ð0Þ ðn þ 2Þ; . . . ; are called the GM(1,1) forecast values.
3. Improved grey forecasting model Deng [1] also developed a residual modification model, the residual GM(1,1) model. The differences between the real values, xð0Þ ðkÞ, and the model predicted values, x^ð0Þ ðkÞ, are defined as the residual series. We denote the residual series as qð0Þ : qð0Þ ¼ qð0Þ ð2Þ; qð0Þ ð3Þ; qð0Þ ð4Þ; . . . ; qð0Þ ðnÞ ; ð11Þ where qð0Þ ðkÞ ¼ xð0Þ ðkÞ x^ð0Þ ðkÞ:
ð12Þ
The residual GM(1,1) model could be established to improve the predictive accuracy of the original GM(1,1) model. The modified prediction values can be obtained by adding the forecasted values of the residual GM(1,1) model to the original x^ð0Þ ðkÞ. However, the potency of the residual series depends on the number of data points with the same sign, which is usually small when there are few observations. In these cases, the potency of the residual series with the same sign may not be more than four, and a residual GM(1,1) model cannot be established. Here, we present an improved grey model to solve this problem. We establish a modification sub-model that is a combination residual GM(1,1) forecaster that uses the absolute values of the residual series with an ANN for residual sign estimation. The schematic of the improved forecasting system is shown in Fig. 1. The detail process to formulate this improved grey forecast model is described as follows.
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C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 2241–2249 Original Model Data Input
Original GM(1,1) Forecaster
Original Forecast Output
Residual Input Final Forecast
Modification Sub-Model Residual GM(1,1) Residual Forecast Output Forecaster ANN Sign Estimater
Combination Module
Residual Sign Eastimate
Fig. 1. Schematic of the forecasting system.
3.1. Residual forecasting model First, denote the absolute values of the residual series as eð0Þ : eð0Þ ¼ eð0Þ ð2Þ; eð0Þ ð3Þ; eð0Þ ð4Þ; . . . ; eð0Þ ðnÞ ;
ð13Þ
where
eð0Þ ¼ qð0Þ ðkÞ;
k ¼ 2; 3; . . . ; n:
ð14Þ
By using the same methods as Eqs. (1)–(10), a GM(1,1) model of eð0Þ can be established. Denote the forecast residual series as ^eð0Þ ðkÞ, then be ð0Þ ð0Þ ^e ðkÞ ¼ e ð2Þ ð15Þ ð1 eae Þeae ðk1Þ ; k ¼ 2; 3; . . . ae 3.2. ANN residual sign estimation model In recent years, much research has been conducted on the application of artificial intelligence techniques to forecasting problems. However, the model that has received extensive attention is undoubtedly the ANN, cited as among the most powerful computational tools ever developed. Fig. 2 presents an outline of a simple biological neural and an ANNÕs basic elements. ANN models operate like a ‘‘black box’’, requiring no detailed information about the system. Instead, they learn the relationship between the input parameters and the controlled and uncontrolled variables by studying previous data. ANN models could handle large and complex systems with many interrelated parameters. Several types of neural architectures are available, among which the multi-layer back propagation (BP) neural network is the most widely used. As Fig. 3 reveals, a BP network typically employs three or more layers for the architecture: an input layer, an output layer and at least one hidden layer. The computational procedure of this network is described below: ! X ð16Þ Wij Xij ; Yj ¼ f i
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Fig. 2. A simple neural [13] vs. a PE model.
where Yj is the output of node j, f ðÞ is the transfer function, wij is the connection weight between node j and node i in the lower layer and Xi is the input signal from the node i in the lower layer. BP is a gradient descent algorithm. It tries to improve the performance of the neural network by reducing the total error by changing the weights along its gradient. The BP algorithm minimizes the square errors, which can be calculated by:
Fig. 3. A BP network.
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C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 2241–2249 Input Layer
Hidden Layer
Output Layer
d(n-1) d(n+1) d(n)
Bias
Fig. 4. Structure of ANN sign forecasting system.
E ¼ 1=2
XX p ½Oj Yjp 2 ; p
ð17Þ
j
where E is the square errors, p is the index of the pattern, O is the actual (target) output and Y is the network output. A two state ANN model is used here to predict the signs of the forecast residual series. First, we introduce a dummy variable dðkÞ to indicate the sign of the kth year residual. Assume the sign of the kth year residual is positive, then the value of dðkÞ is 1, otherwise it is 0. Then, we set up an ANN model by using the values of dðn 1Þ and dðnÞ to estimate the values of dðn þ 1Þ. The structure of this ANN sign forecasting system is shown in Fig. 4. Let the sign of the kth year residual, sðkÞ, be þ1; if dðkÞ ¼ 1 sðkÞ ¼ ; k ¼ 1; 2; . . . ; n; . . . ð18Þ 1; if dðkÞ ¼ 0 According to the equations illustrated above, an improved grey model combination residual modification with ANN sign estimation can be further formulated as Eq. (19) b be 0ð0Þ ð0Þ a aðk1Þ ð0Þ ð1 eae Þeae ðk1Þ ; þ sðkÞ e ð2Þ ð1 e Þe x^ ðkÞ ¼ x ð1Þ a ae k ¼ 1; 2; . . . ; n; n þ 1; . . .
ð19Þ
Next, we will proceed to the power demand forecasting of Taiwan for our case study to examine the reliability and accuracy of this improved GM(1,1) model. 4. Results To demonstrate the effectiveness of the proposed method, we use the power demand forecasting of Taiwan as an illustrating example. In this study, we use the historical annual power demand of Taiwan from 1985 to 2000 as our research data. There are 16 observations, where 1985–1998 are used for model fitting and 1999–2000 are reserved for ex post testing. For the purposes of comparison, we also use the same number of observations, 14 (power demand from 1985 to 1998), to formulate an ARIMA (p; d; q) model, where p is the order of the auto-regressive part, d is the order of the differencing, and q is the order of the moving average
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Table 1 Model values and forecast errors (unit: 103 W h) Year
Real value
GM(1,1)
Improved GM(1,1)
ARIMA
Model value Error (%)
Model value Error (%)
Model value Error (%)
1985 1986 1987 1988 1989 1990
47,919,102 53,812,862 59,174,751 65,227,727 69,251,809 74,344,947
47,919,102 56,318,092 60,319,829 64,605,914 69,196,550 74,113,379
0.00 4.66 1.94 )0.95 )0.08 )0.31
47,919,102 53,812,862 59,630,904 65,310,510 69,917,174 74,850,394
0.00 0.00 0.77 0.13 0.96 0.68
47,919,102 52,307,500 53,957,006 60,243,936 65,958,706 72,405,080
0.00 )2.80 )8.82 )7.64 )4.76 )2.61
1991 1992 1993 1994 1995 1996 1997 1998
80,977,405 85,290,354 92,084,684 98,561,004 105,368,193 111,139,816 118,299,046 128,129,801
79,379,577 85,019,971 91,061,148 97,531,587 104,461,790 111,884,424 119,834,482 128,349,438
)1.97 )0.32 )1.11 )1.04 )0.86 0.67 1.30 0.17
80,133,358 85,790,897 91,849,611 98,337,985 105,286,530 111,040,924 118,971,794 127,467,127
)1.04 0.59 )0.26 )0.23 )0.08 )0.09 0.57 )0.52
76,688,020 82,105,943 89,156,925 93,739,526 100,954,923 107,828,630 115,049,615 121,169,150
)5.30 )3.73 )3.18 )4.89 )4.19 )2.98 )2.75 )5.43
MAPEa (1986–1998)
1.54
1999 2000
131,725,892 142,412,887
137,469,433 147,237,458
MAPE (1999–2000) a
MAPE ¼ 1n
4.36 3.39 3.88
Pn
xð0Þ ðkÞ k¼1 ½j^
0.57 133,459,644 144,204,700
1.32 1.26
4.24 128,756,418 139,168,992
1.29
xð0Þ ðkÞj=xð0Þ ðkÞ.
Fig. 5. Real values and model values for power demand of Taiwan from 1985 to 2000.
)2.25 )2.28 2.27
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Fig. 6. Model percentage error distribution from 1985 to 2000.
process [12]. As a result of statistical tests, the ARIMA model with ðp; d; qÞ ¼ ð0; 1; 0Þ is formulated as follows: x^ðkÞ ¼ 2404647:67 þ 1:04^ xðk 1Þ;
k ¼ 2; 3; 4; . . . ; n; . . .
ð20Þ
The predicted results obtained by the original GM(1,1) model, improved GM(1,1) model and ARIMA model are shown in Table 1 and Fig. 5. The model percentage error distribution is also shown in Fig. 6. The mean absolute percentage error (MAPE) of the GM(1,1) model, the ARIMA model and our improved GM(1,1) model from 1999 to 2000 are 3.88%, 2.27% and 1.29%, respectively. According to the results shown above, our improved grey model seems to obtain the lowest post-forecasting errors among these models. It is indicated that the modification of our improved GM(1,1) model can reduce model prediction errors effectively.
5. Conclusions The original GM(1,1) model is a model with a group of differential equations adapted for variance of parameters, and it is a powerful forecasting model, especially when the number of observations is not large. In this paper, we have applied an improved grey GM(1,1) model by using a technique that combines residual modification with ANN sign estimations. Our study results show that this method can yield more accurate results than the original GM(1,1) model and also solve problems resulting from having too few data, which may lead the same sign residuals lower than four and violate the necessary condition of setting up a GM(1,1) model. The improved grey models were then applied to predict the power demand of Taiwan. Finally, through this study, our improved grey model, in this paper, is an appropriate forecasting method to yield more accurate results than the original GM(1,1) model.
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References [1] Deng JL. Grey system fundamental method. Huazhong University of Science and Technology Wuhan, China, 1982. [2] Sun G. Prediction of vegetable yields by grey model GM(1,1). J Grey Syst 1991;2:187–97. [3] Morita H et al. Interval prediction of annual maximum demand using grey dynamic model. Electr Power Energy Syst 1996;18:409–13. [4] Huang YP, Wang SF. The identification of fuzzy grey prediction system by genetic algorithm. Int J Syst Sci 1997;28:15–26. [5] Hsu CI, Wen YH. Improved grey prediction models for the Trans-Pacific air passenger market. Transport Planning Technol 1998;22:87–107. [6] Hsu CC, Chen CY. Application of grey theory to regional electricity demand forecasting. Energy Quart 1999;24:96–108. [7] Hao YH, Wang XM. The study of grey system models of Niangziguan spring. J Syst Eng 2000;16:39–44. [8] Liu SF, Deng JL. The range suitable for GM(1,1). Syst Eng Theor Appl 2000;5:111–24. [9] Xing M. Research on combined grey neural network model of seasonal forecast. Syst Eng Theor Appl 2001;11: 31–5. [10] Yue CL, Wang L. Grey–Markov forecast of the stock price. Syst Eng 2000;16:54–9. [11] Deng JL. Grey prediction and decision. Huazhong University of Science and Technology, Wuhan, China, 1986. [12] Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. NJ: Prentice Hall; 1994. [13] Nelson MM, Illingworth WT. A practical guide to neveal nets. MA: Addison-Wesley; 1993.