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Load forecasting for practical power systems by… Y = y y 1 2 y n ,……………..(13) 1 x11 x12 x1 k X = 1 x21 x22 x2k ,……………..(14) 1 x x x 31 32 3k 1 xn 1 xn2 xnk 0 β = and ……………..(15) 1 k 1 € = ……………..(16) 2 n In general, y is an (n x 1) vector of the observations, X is an (n x p) matrix of the levels of the independent variables, β is a (p x 1) vector of the regression coefficients, and € is an (n x 1) vector of random errors. We wish to find the vector of least squares estimators, ˆ that minimizes n L i1 2 i ' ' ( y x ) ( y x ) ……………..(17) Note that L may be expressed as L = Y 1 Y – β 1 X 1 Y – Y 1 X β + β 1 X 1 X β = Y 1 Y - 2β 1 X 1 Y + β 1 X 1 Xβ……………..(18) Because β 1 X 1 Y is a (1 x 1) matrix, and its transpose (β 1 X 1 Y) 1 = Y 1 Xβ is the same matrix. The least squares estimators must satisfy L 1 1 2X Y 2X Xˆ ˆ 0 ……………..(19) 1 This simplifies to X X ˆ 1 X Y equation (19) is the matrix form of the least squares normal equations. It is identical to equation (18). To solve the normal equations, multiply both sides of equation (15) by the inverse of X 1 X. Thus, the least squares estimator of β is ˆ 1 1 1 ( X X ) X Y ……………..(20) It is easy to see that the matrix form of the normal equations is identical to the scalar form. Writing out the above equation in detail we get If the indicated matrix multiplication is performed, the scalar form of the normal equations (i.e equations (3.4)) will result. In this form it is easy to see that X 1 X is a (p x p) symmetric matrix and X 1 Y is a (p x 1) column vector. Note the special structure of the X 1 X matrix. The diagonal elements of X 1 X are the sum of squares of the elements in the columns of X, and the off-diagonal elements are the sums of crossproducts of the elements in the columns of X. Furthermore, note that the elements of X 1 Y are the sums of crossproducts of the columns of X and the observations {Y i }. n n n n n X i1 X i2 X ik Yi ˆ i1 i1 i1 0 i1 n n n n n 2 X i1 X i1 X i1X i2 X i X ˆ 1 ik 1 X i1Yi ……………..(21) i1 i1 i1 i1 i1 n n n n ˆ n 2 k X ik X ik X i1 X ik X i2 X ik X ikYi i1 i1 i1 i1 i1 The fitted regression model is Y ˆ Xˆ .……………..(22) www.<strong>ijcer</strong>online.com ||May ||2013|| Page 59
In scalar notation, the fitted model is Yˆ i k i1 j Load forecasting for practical power systems by… ˆ ˆ X ……………..(23) 0 www.<strong>ijcer</strong>online.com ||May ||2013|| Page 60 ij j = 1, 2… n. The difference between the actual observation Y i and the corresponding fitted value e i Y Yˆi. the (n x 1) vector of residual is denoted by i e Y Yˆ Ŷ i is the residual; say ……………..(24) ANN DESIGN: ANN models work in the manner similar to human brain. In fact the basic ANN element is simulation of biological neuron. A biological neuron is composed of different parts performing different functions. Input information is received by the dendrites and act similar to the input node. These dendrites pass the information to soma the cell body or the processing node. The output of the soma is then transferred through the axon to the synapse. The synapse then gives a weight called synoptic weights to this information. This weighted input serves as the input to the dendrite of the next neuron. Brain consists of large number of such neurons with complex interconnections among them. The processing node in the human brain is a hard limiter. Since it is practically impossible to have such a large number of neurons and interconnection a continuous sigmoid function is used as a processing node in artificial simulation. ANNs works on the basis of mapping of input/output spaces. The relationship between the input variables and the resulting responses there from are determined in an implicit manner. This is done through the training process. Training of the network is accomplished through the correction of the network weights given to each of variables using steepest gradient or other gradient method so as to minimize the sum squared error on responses. The general method of arriving at a neural network design is shown in figure 1. After gathering sample set of inputs and the related output responses (training samples) the network is trained. The learning parameters (learning constant, momentum etc) are adjusted till the network is trained successfully ie, with lowest possible sum squared error. The trained network is tested on the samples other than the training samples. The network parameters ie, number of layers and neurons in each layer are adjusted till successful testing of the network. For any neural network the design procedure involves, (i) feature selection (selection of input variables) and (ii) parameter selection for the network (number of layers, neurons etc). For the current load forecast problem the two procedures go hand in hand. In this case the dependence of the future load could be related to the load at various previous hours based on the past experience and judgment. However the suitability of selected set in modeling the said input/output relationship can only are verified by observing the performance of the network. A network which requires number of layers and neurons in a layer is selected which is fit to capture the input/output relationship. There is no fixed rule to decide the number of layers and neurons in each layer for a given problem. The number of layer depends on the complexity of the problem. It has been observed that a single hidden layer is sufficient for the problem. It is also found that 8-20 neurons in a single hidden layer capture the load pattern closely. For this appropriate number of training and testing patterns are selected from the historical data. It is also important to note that a network with low training error may not always lead to a good prediction, since forecasting not only requires good pattern matching but generalization as well. This balance between training and prediction error can be achieved by training the network of the said training patterns with a few number of epochs (say few thousand) then it is tested on test patterns. This training and testing continues till the prediction error or training error improves. This procedure may be repeated for other combination and one with lowest error is selected. ANN Method :( Applications) Selection of Network Architecture: (Maximum, Minimum, Average load) There are numerous ways to choose the architecture of the models. Here a decision is made to use one network for all day types. It was concluded that networks with only one output node gives better results than forecasting peak-, valley-, and average loads with one multi-output node. Other features to be decided about the architecture of the network are the input variables and the number of hidden layer neuron variables. To forecast the maximum load L max (i) (minimum and average loads) of a certain day, the maximum load (minimum, average loads) of the previous day can be considered as potential input variable. Therefore the network configuration can be summarized as given below: Note: number of neurons in a hidden layer can approximately be determined from the form H = T / {5(N+M)} Where, H= number of hidden layer neurons. N= size of the input layer M= size of the output layer T= size of the training set Training set: The training set for the networks consists of the data over one year, from April 1 st, 1999 to March 31st, 2000.
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