Applications of improved grey prediction model for power demand forecasting

Transcription

1 Energy Conversion and Management 44 (2003) Applications of improved grey prediction model for power demand forecasting Che-Chiang Hsu a, *, Chia-Yon Chen b a Industrial Engineering and Management Department, Nan-Jeon Junior Institute of Technology, 178 Chau-Chin Road, Yen Shui, Tainan Hisen 73701, Taiwan, ROC b 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.: x62826; fax: address: [email protected] (C.-C. Hsu) /03/$ - see front matter Ó 2003 Published by Elsevier Science Ltd. doi: /s (02)

2 2242 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) 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 x ð0þ ¼ x ð0þ ð1þ; x ð0þ ð2þ; x ð0þ ð3þ;...; x ð0þ ðnþ ; ð1þ where n is the number of years observed. The AGO formation of x ð0þ is defined as: x ð1þ ¼ x ð1þ ð1þ; x ð1þ ð2þ; x ð1þ ð3þ;...; x ð1þ ðnþ ; ð2þ where x ð1þ ð1þ ¼x ð0þ ð1þ; and x ð1þ ðkþ ¼ Xk m¼1 x ð0þ ðmþ; k ¼ 2; 3;...; n: ð3þ The GM(1,1) model can be constructed by establishing a first order differential equation for x ð1þ ðkþ as: dx ð1þ ðkþ=dk þ ax ð1þ ðkþ ¼b: ð4þ Therefore, the solution of Eq. (4) can be obtained by using the least square method. That is, where ^x ð1þ ðkþ ¼ x ð0þ ð1þ ^b ^a! e ^aðk 1Þ þ ^b ^a ; ð5þ ½^a; ^bš T ¼ðB T BÞ 1 B T X n ð6þ

3 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) and 2 3 0:5ðx ð1þ ð1þþx ð1þ ð2þþ 1 0:5ðx ð1þ ð2þþx ð1þ ð3þþ 1 B ¼ ; ð7þ 0:5ðx ð1þ ðn 1Þþx ð1þ ðnþþ 1 T: X n ¼ x ð0þ ð2þ; x ð0þ ð3þ; x ð0þ ð4þ;...; x ð0þ ðnþ ð8þ 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;... ; ð9þ where ^x ð0þ ð1þ ¼x ð0þ ð1þ. Applying the inverse AGO, we then have ^x ð0þ ðkþ ¼ x ð0þ ð1þ ^b! ð1 e^a Þe ^aðk 1Þ ; k ¼ 2; 3;...; ð10þ ^a 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, x ð0þ ðkþ, and the model predicted values, ^x ð0þ ðkþ, are defined as the residual series. We denote the residual series as q ð0þ : q ð0þ ¼ q ð0þ ð2þ; q ð0þ ð3þ; q ð0þ ð4þ;...; q ð0þ ðnþ ; ð11þ where q ð0þ ðkþ ¼x ð0þ ðkþ ^x ð0þ ðkþ: 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. ð12þ

4 2244 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) Data Input Original Model Original GM(1,1) Forecaster Original Forecast Output Modification Sub-Model Residual Input Residual GM(1,1) Forecaster Residual Forecast Output Combination Module Final Forecast ANN Sign Estimater Residual Sign Eastimate Fig. 1. Schematic of the forecasting system Residual forecasting model First, denote the absolute values of the residual series as e ð0þ : e ð0þ ¼ e ð0þ ð2þ; e ð0þ ð3þ; e ð0þ ð4þ;...; e ð0þ ðnþ ; ð13þ where e ð0þ ¼ q ð0þ ðkþ ; k ¼ 2; 3;...; n: ð14þ By using the same methods as Eqs. (1) (10), a GM(1,1) model of e ð0þ can be established. Denote the forecast residual series as ^e ð0þ ðkþ, then ^e ð0þ ðkþ ¼ e ð0þ ð2þ b e ð1 e a e Þe aeðk 1Þ ; k ¼ 2; 3;... ð15þ a e 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: Y j ¼ f X W ij X ij!; ð16þ i

5 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) Fig. 2. A simple neural [13] vs. a PE model. where Y j is the output of node j, f ðþ is the transfer function, w ij is the connection weight between node j and node i in the lower layer and X i 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.

6 2246 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) d(n-1) d(n) Input Layer Hidden Layer Output Layer d(n+1) Bias Fig. 4. Structure of ANN sign forecasting system. E ¼ 1=2 X X ½O p j Y p j Š 2 ; ð17þ p 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 ¼1 sðkþ ¼ ; k ¼ 1; 2;...; n;... ð18þ 1; if dðkþ ¼0 According to the equations illustrated above, an improved grey model combination residual modification with ANN sign estimation can be further formulated as Eq. (19) ^x 0ð0Þ ðkþ ¼ x ð0þ ð1þ b ð1 e a Þe aðk 1Þ þ sðkþ e ð0þ ð2þ b e ð1 e a e Þe aeðk 1Þ ; a a e 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 are used for model fitting and 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

7 Table 1 Model values and forecast errors (unit: 10 3 Wh) Year Real value GM(1,1) Improved GM(1,1) ARIMA Model value Error (%) Model value Error (%) Model value Error (%) ,919,102 47,919, ,919, ,919, ,812,862 56,318, ,812, ,307,500 ) ,174,751 60,319, ,630, ,957,006 ) ,227,727 64,605,914 ) ,310, ,243,936 ) ,251,809 69,196,550 ) ,917, ,958,706 ) ,344,947 74,113,379 ) ,850, ,405,080 ) ,977,405 79,379,577 ) ,133,358 ) ,688,020 ) ,290,354 85,019,971 ) ,790, ,105,943 ) ,084,684 91,061,148 ) ,849,611 ) ,156,925 ) ,561,004 97,531,587 ) ,337,985 ) ,739,526 ) ,368, ,461,790 ) ,286,530 ) ,954,923 ) ,139, ,884, ,040,924 ) ,828,630 ) ,299, ,834, ,971, ,049,615 ) ,129, ,349, ,467,127 ) ,169,150 )5.43 MAPE a ( ) C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) ,725, ,469, ,459, ,756,418 ) ,412, ,237, ,204, ,168,992 )2.28 MAPE ( ) a P MAPE ¼ 1 n n k¼1 ½j^x ð0þ ðkþ x ð0þ ðkþj=x ð0þ ðkþš Fig. 5. Real values and model values for power demand of Taiwan from 1985 to 2000.

8 2248 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) Fig. 6. Model percentage error distribution from 1985 to process [12]. As a result of statistical tests, the ARIMA model with ðp; d; qþ ¼ð0; 1; 0Þ is formulated as follows: ^xðkþ ¼ :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.

9 C.-C. Hsu, C.-Y. Chen / Energy Conversion and Management 44 (2003) References [1] Deng JL. Grey system fundamental method. Huazhong University of Science and Technology Wuhan, China, [2] Sun G. Prediction of vegetable yields by grey model GM(1,1). JGrey Syst 1991;2: [3] Morita H et al. Interval prediction of annual maximum demand using grey dynamic model. Electr Power Energy Syst 1996;18: [4] Huang YP, Wang SF. The identification of fuzzy grey prediction system by genetic algorithm. Int JSyst Sci 1997;28: [5] Hsu CI, Wen YH. Improved grey prediction models for the Trans-Pacific air passenger market. Transport Planning Technol 1998;22: [6] Hsu CC, Chen CY. Application of grey theory to regional electricity demand forecasting. Energy Quart 1999;24: [7] Hao YH, Wang XM. The study of grey system models of Niangziguan spring. JSyst Eng 2000;16: [8] Liu SF, Deng JL. The range suitable for GM(1,1). Syst Eng Theor Appl 2000;5: [9] Xing M. Research on combined grey neural network model of seasonal forecast. Syst Eng Theor Appl 2001;11: [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, [12] Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. NJ: Prentice Hall; [13] Nelson MM, Illingworth WT. A practical guide to neveal nets. MA: Addison-Wesley; 1993.