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Blind source separation for improved load forecasting in the power system

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Blind source separation for improved load forecasting in the power system
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  Blind Source Separation for Improved Load Forecasting in the Power System Krzysztof Siwek  1 , Stanis ł aw Osowski 1,2 , Ryszard Szupiluk  3,4 , Piotr Wojewnik  4 , Tomasz Z ą   bkowski 4   1  Warsaw University of Technology, Warsaw, Poland, e-mail: [ksiwek, sto]@iem.pw.edu.pl 2  Military University of Technology, Warsaw, Poland 3  Warsaw School of Economics, Warsaw, Poland 4  PTC Era, Warsaw, Poland  Abstract   The paper presents the improved method for 24 hour load forecasting in the power system, integrating different classical neural forecasting results by using blind source separation as an expert system. The numerical results of 24-hour power forecasting for Polish Power System are presented and discussed in the paper. 1 INTRODUCTION The system economy is the motivation for the daily 24 hour forecasting of the load in the electric power system of the country. To the widely used forecasting methods belong the methods applying the artificial neural networks. Multilayer perceptron, Kohonen networks as well as different hybrid neural  based solutions are the most important representatives of this group [2,3,4,5].  This paper will present different approach. It will  be relied on many different methods of load  prediction, integrating them into one expert system, responsible for the final forecasting. In this paper we will integrate the results of partial predictions made  by the multilayer perceptron and two self-organizing  based networks of the competitive type. As the integrating expert system we will apply the blind source separation approach. Thanks to such  processing the accuracy of load forecasting may be significantly improved. The results of the numerical experiments, concerning 24 hour load prediction for the Polish Power System will be presented and discussed in the  paper. They show the significant improvement of the  proposed method in comparison to the individual results of prediction. 2 DESCRIPTION OF THE METHOD 2.1 The general system description The proposed method of load forecasting integrates different methods of prognosis into one forecasting system. Each prognosis algorithm is trained separately and acts independently of each other. The results of prognosis generated by each prognosis network for the period used in training, deliver the time series that are put in parallel to the blind source separation (BSS) system. The number of inputs to BSS is equal to the number  N   of the applied  prognosis networks. The BSS system decomposes the srcinal stream of signals of length  p , forming the matrix  p N  × ⊂ R X  (  p  is the number of prognosis hours used in learning), into independent components using the matrix  N  N  × ⊂ R W.  The independent component signals, generated by BSS, form the matrix Y of  N   rows and  p  columns. This transformation is described by Y = WX . Each row of matrix Y  represents the independent component series. Some of these series represent the essential information and some represent the noise. Reconstructing the srcinal time series back into real  prognosis, on the basis of essential independent components only, will provide the prognosis deprived of noise that is presumably of better quality. The problem is that we don’t know in advance which of the components is the noise and which represents the useful information. We have solved the problem by trying in reconstruction all  possible combinations of independent components and accepting this one, which provides the best results of prediction on the learning data. The reconstruction of the srcinal data matrix X is   done by   using the inverse operation YWX ˆˆ 1 − =  (1) X ˆ  denotes the reconstructed time series matrix and Y ˆ  - the independent component matrix formed from the srcinal matrix Y  by omitting some row or rows. In recovering the signals we try all possible combinations of independent components, substituting the rejected components (appropriate rows of Y ) by zeros. The combination corresponding to the best result of prediction on the learning data is assumed as the final solution. Fig. 1 presents the graphical illustration of the proposed method. The switches in the figure represent the possible elimination of the appropriate independent component at the reconstruction stage of the data. In  practical implementation of this approach we have applied three different neural based prediction methods.   Fig. 1: The general scheme of the proposed solution One is based on the multilayer perceptron (MLP) and the other two on the application of the hybrid approach exploiting the self-organizing networks either in crisp (CSO) or fuzzy (FSO) mode. 2.2 MLP based forecasting The prediction of 24-hour load pattern for the next day using MLP makes use of the universal approximation ability of the MLP networks [1]. To represent the generally unknown function of the next day load it maps the past loads of the system into the  present forecasted load at d  th day and h th hour. Our general MLP based model of the load in the power system may be presented here in the form ( ) ),(),...,,1(,,,),(  H h Dd  P hd  P ur  f  hd  P   −−−= w  (2) where w  represents the weight vector of the network,  H   and  D  - the number of past hours and days, respectively, influencing the prediction process, r   - the type of the day (workday or holiday) and u  - the season of the year (autumn, winter, spring and summer). The neural network architecture (Fig. 2) associated with this mathematical model, possesses certain number of input nodes (one or two to code the type of the day and the season of the year and nodes to represent the loads of some past days) and 24 output linear neurons equal to the number of hours of  prediction (24 hours ahead). Fig. 2: The MLP structure for 24 hour load forecasting The number of hidden layers and neurons of sigmoidal non-linearity is subject to adjustment in an experimental way by training different structures and choosing one of the smallest one, still satisfying the learning conditions. On the basis of some numerical simulations we have found that optimal number of input nodes in our case is 23, which correspond to  D =3 and  H  =4. The number of hidden layers was equal two and each layer contained 20 and 15 sigmoidal neurons, respectively. In this way the optimal structure of the MLP network used in  prediction is described as 23-20-15-24. 2.3 The self-organizing methods of forecasting The idea of applying self-organizing networks for load forecasting is well known [5].The main task of the self-organizing network is to learn the characteristics of the daily loads of the system. The days of the same type belonging to the same seasons of the year have similar load characteristics and form clusters, grouping the similar data. Each cluster is represented by one neuron acting in the competitive mode. To make the  prediction independent on the general trend, changing from year to year, we transform the input data by cutting out the mean value and dividing the result by the standard deviation of the data for this day. Instead of real data we use in this way the so called profiles defined as follows )()(),( ),( d d  P hd  P  t d  p  m σ   −=  (3) where  P  ( d  , h ) is the real load of d  th day at h th hour,  P  m ( d  ) is the mean value of the load of d  th day and σ   is the standard deviation of the load of this day. The set of profiles for each day of the years taking part in learning process forms the training data of the network. Once the network is trained, each neuron represents the data closest to its weight vector in the chosen metric space. If we want to make prediction for 24-hour load pattern of the particular day we simply take the reversed form of equation (3). The  prediction of the load for d  th day and h th hour may  be expressed in the form )(ˆ),(ˆ)(ˆ),(ˆ d  P hd  pd hd  P  m += σ    (4) where the variables with hat mean predicted values. Two problems have to be solved. One is the accurate  prediction of the profiles ),(ˆ  hd  p  and the second - good estimation of the mean value )(ˆ d  P  m  and standard deviation )(ˆ  d  σ   for the particular day. The best results of mean value and standard deviation prediction have been obtained by applying the MLP network. In our model for mean (variance)  prediction we take the input vector to the network containing 9 nodes, representing the mean or variance of the load for the days of the previous years, the actual season of the year and type of the day. The type of the day (workday or holiday) are coded in a binary way. The destination is associated with the predicted value of the mean or variance for d  th day, respectively. The number of hidden neurons was chosen on the ground of good generalization ability of the network. The profile prediction has been be  solved by us in two different ways. In the classical (crisp) self-organization (CSO) approach [5] we estimate the profiles averaging the winner vectors for this particular day from the past, i.e., ∑∑ == = ni dini idi k k d  11 )(ˆ wp  (5) where k  di  is the number of appearances of i th neuron among the winners in the past for this particular day. The other method is the fuzzy self-organizing approach. It is also associated with one-layer self-organizing network, however in prediction process we take into account not only the winner but also the activity of some losers, closest to the winner. As a result of learning we memorize not only the winner  but also some limited number q  of neurons closest to the winner. At the same time we keep also their relative activities, that represent the membership values. If the activities of the winner and the neighboring neurons are denoted by u w  and u i , respectively, their relative activities are defined by [3]  2)( iuwui  e y  −− =  α   (6) For the winner it is  y w =1 and for all other neurons 0 ≤  y i <1. The coefficient α  is the decaying parameter. The membership value for i th neuron at the  presentation of the input vector x  is given by ∑ = = qi iii  y y 1 µ   (7) The phase of prediction of profile of d  th day takes into account not only the winners but also their neighbors and their relative activities, denoted by  µ i . [3]  ∑ ∑∑ ∑ = == = = niq j jiniq j ji ji  wd  11)(11)()( )(ˆ µ µ  p  (8) The parameter )(  ji µ   denotes the membership value of  j th neuron taking part in prediction, where i  is the notation of the particular day of the previous years. The relations (6)-(7) are of fuzzy nature, thus the method is called fuzzy self-organization (FSO). Observe that both methods (FSO and CSO) are only modifications of the self-organization algorithm. So their results are a bit similar, however not the same. 2.4 Blind source separation system The blind source separation system decomposes the streams of input signals of  N   channels into  N   independent components. The basic assumption is that the input signals are the mixtures of some unknown basic srcinal sources which are to be recovered by the separation algorithm. There are many different solutions to BSS [8]. We have tried many of them installed actually in ICALAB [7] and the best results have been obtained at application of second order blind identification (SOBI) algorithm. SOBI applies the time delayed covariance matrices R  xx (  p i ) ∑ = −=  pk iT i xx  nk k  p p 1 )()( 1)( xxR   (9) for the preselected set of time lags n 1 , n 2 , …, n  L , (  p  is the number of learning samples) determined for orthogonalized vectors x ( k  ), that is x ( k  ):= Qx ( k  ), where Q  is a specially selected orthogonalization matrix [6,7]. The main step is the joint approximate diagonalization (JAD), described by T ii xx  p UUDR   = )( (10) for all values of i =1,2,…,  L . The estimated independent sources for each time step k   are described by the relation (k)(k) T QxUy  =  (11) The demixing matrix is then defined by W=U T Q . The mixed signals can be reconstructed back using the relation (k)(k)ˆ UyQyWx  ++ == , where + means  pseudoinverse (for the number of mixed signals equal the number of independent components it is simple matrix inversion). 3 THE RESULTS OF NUMERICAL EXPERIMENTS The numerical experiments have been performed using the data of Polish Power System of 5 years. The first 4 years have been used only in learning and the last year has been left for testing the solution. We have made 24 hour prognoses of the load consumption in the country using MLP method and 2 hybrid approaches (CSO and FSO). Method MAPE MSE MAX MLP 1.98% 1.65e5 16.92% CSO 2.35% 2.45e5 18.10% FSO 2.35% 2.43e5 18.08% Table 1: The errors of load forecasting using individual predictors in the testing mode Table 1 presents the obtained individual results of load forecasting for the last (testing) year in the form of the mean absolute percentage error (MAPE), maximum error (MAX) in % and the mean square error (MSE) measured as the mean of squared errors (in Watts). The learning data streams of the 24 hour forecasting corresponding to these three methods have been submitted in the form of 3 parallel time series to the input of the BSS system, performing the independent component analysis. We have applied SOBI algorithm of separation. As a result we have got the separation matrix W and three independent component streams packed in matrix Y , whose graphical forms are presented in Fig. 3. It is evident that only channel 1 depicts the components of essential information and the rest some kind of noise. Using the obtained decomposition we have reconstructed three streams of srcinal data (forecasted  values) by applying the relation (1) at omitting some  basic components (the rows of Y ).   Fig. 3: The independent components of prognosis We have tried all possible combinations (3 combinations of two and 3 of one time series). The  best series on the output of the reconstructing system was assumed as the optimal solution and this one has  been tested on the data of the last year, not taking part in learning. The results of testing the BSS based forecasting system for all 6 combinations of independent components at reconstruction stage are gathered in Table 2. Combination MAPE MSE MAX 1-2 1.99 % 1.67e5 17.18 % 1-3 1.73 % 1.25e5 16.21 % 2-3 99.87 % 2.64e8 110.93 % 1 1.74% 1.26e5 16.42% 2 99.73% 2.63e8 110.05% 3 99.97% 2.65e8 100.87% Table 2: The errors of forecasting the power demand for the testing data using the BSS based system They are presented in the form of MAPE, MSE and MAX errors. As it is seen the best case (bold) corresponds to the case of omitting the component No 2, being the evident noise. Predictor MAPE MSE MAX MLP 13.02 % 23.87 % 4.24 % CSO 26.36 % 48.87 % 8.38 % FSO 25.38 % 47.27 % 8.20 % Table 3: The relative improvement of our forecasting method over individual predictors Table 3 depicts the relative improvements of the best final results (bold) over the individual forecasting methods. There is a significant improvement of forecasting results in all categories of errors. The highest improvement has been observed for MSE error. If we compare this improvement with the best predictor (MLP) we note 13.02% for MAPE, 23.87% of MSE and 4.24% of MAX. The improvements over other  prediction methods are even more impressive. Observe that considering only the basic component  No 1 improves also final results of forecasting, however less than in the previous case. The next experiment has been performed by omitting one of the self-organizing results. We have taken into account the results of MLP and FSO. The results are only slightly worse than in the best case (1.74% of MAPE and 16.36% of MAX errors). This confirms the fact that the highest potential improvement of results may be expected when independent individual predictors are integrated. 4 CONCLUSIONS The paper has presented ensemble machine approach to the prediction of the power consumption for the next 24 hours in the power system. Different neural methods of load forecasting have been combined into one expert system for more accurate prediction. The integration of results has been obtained by applying the blind source separation principle. The obtained reduction of forecasting errors over the individual  predictors was in the range from 4% to 48%. The method is universal, flexible and applicable at any number of individual predictors. The higher is this number the better results are expected. However the  predictors should be independent in their method of forecasting as far as possible. Acknowledgments This work was supported by Ministry of Scientific Research and Information Technology of Poland. References [1]   S. Haykin, “Neural networks, a comprehensive foundation”, Macmillan, 2002, N. Y. [2] A. Bakirtzis, V. Petridis, S. Klartzis, M. Alexiadis, “A neural network short term load forecasting model”, IEEE Tran. Power System, vol. 11, 1996 [3] S. Osowski, K. Siwek, “The self-organizing neural network approach to load forecasting in  power system”, IJCNN, Washington, 2000 [4] S. Osowski, K. Siwek, “Regularization of neural networks for load forecasting in power system”, IEE Proc. GTD, vol. 149, 2002, pp. 340-345 [5] M. Cottrell, B. Girard, Y. Girard, C. Muller, P. Rousset, “Daily electrical power curve:  classification and forecasting using a Kohonen map”, IWANN, Malaga, 1995, pp. 1107-1113 [6] A. Belouchrani, K. Abed-Meraim, J.F. Cardoso and E. Moulines, “A BSS technique using SOS”, IEEE Trans. SP, vol. 45, pp. 434-444, 1997 [7] A. Cichocki, S. Amari, K. Siwek, T. Tanaka et al., ICALAB, www.bsp.brain.riken.jp/ICALAB. [8] A. Cichocki, S. I. Amari, “Adaptive blind signal and image processing”, Wiley, 2003
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