The area demands may be related to a 58 number of

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Unformatted text preview: t variable, whereas the other variables would be referred to as the independent variables. The next step is to estimate, statistically, the functional relationship between the dependent variable and the independent variable(s). Having determined this, and also that the relationship is statistically significant, the predicted future values of the independent variables can be used to predict the future values of the dependent variable. In this dissertation, billing data was used as independent variables. The conclusion was that regression and neural networks were not appropriate techniques for area and transmission substation load forecasts. For more details, refer to the dissertation, but some of the major shortcomings were: 1) Only ten data points (annual loads) are available and a twenty-year load forecast is required. 2) The assumptions for the properties of least squares estimators are violated i.e. i) The model error variance is homogeneous (constant variance) with mean zero. ii) The model errors are uncorrelated from observation to observation. iii) The model errors are assumed to be normally distributed. 3) Multiple regression models are short-term forecasting techniques (at least 5 years), not long-term. 4) Some of the independent variables are highly correlated. leads to the problem of multi-collinearity. That In such cases the variance is overestimated, the confidence levels for the coefficients become very wide and sometimes the coefficients have a wrong sign. 5) The model errors (actual minus forecast) are sometimes correlated and called auto-correlation. With auto -correlation the variance is underestimated, the t-tests are high and the confidence levels for the coefficients become very small. 59 6) Heteroschedasticity. The variances are not constant over time, giving forecast results that are either too low or too high (funnel effect). There are techniques such as ridge regression or principal component regression to solve the multi-collinearity problems, but only spreadsheet software is available. Artificial neural networks is another technique to consider for forecasting purposes. In this chapter, a brief overview is given on neural nets, terminology frequently used and appropriate neural net structures for predicting future loads. Artificial neural networks are sometimes considered simplified models of the human brain. Some authors feel that this is misleading, because the human brain is too complex and is not well-understood. There are, nevertheless, a number of similarities that can be examined. Human nerve cells, called neurons, consist of three parts: the cell body, the dendrites and the axon. The body (soma) is the large, relatively round central body in which almost all the logical functions of the neuron are performed. It carries out the biochemical transformations required to synthesise the enzymes and other molecules necessary to the life of the neuron. Each neuron has a hair-like structure of dendrites (inputs) around it. The dendrites are the principal receptors of the neuron and serve to connect its incoming signals. The axon (output) is the outgoing connection for signals emitted by the neuron. Synapses are speed contacts on a neuron, which are the termination 60 points for the axons from other neurons. Synapses play the role of interfaces connecting some axons of the neurons to the spines of the input dendrites. d d Neuron = synapses d = dendrites Axon d o = soma o d d Figure 3.6.1 - Components of a neuron The principle of the single neuron is the following: a number of inputs (xis) are applied, each input is multiplied by a weight (ωji ), and then summated. The output (yj ) of the activation function ( ψj ) is compared with the desired output (dj ). The difference between the actual and the desired outputs are used to adjust to weights. The ideal condition would be to have a difference of zero. x0 x1 x2 x3 ωj0 ωj1 ωj2 ωj3 µj ∑ yj ψ( µj) dj Figure 3.6.2 - Single Layer Perceptron with a neuron in its output layer The summated output uj is given 61 n u j = ∑ ω ji xi (3.2.9) i =0 e j = d j − y j = d j − ψ (u j ) (3.2.10) The following terminology is frequently used in dealing with neural nets: • Supervised training is accomplished by presenting a sequence of training vectors, or patterns, each with an associated target output vector. The weights are adjusted according to a learning algorithm in order to achieve the target output vector as closely as possible. • Unsupervised training is, for example, used with self-organising neural nets. A sequence of input vectors is provided, but no target vectors are specified. The net modifies the weights so that the most similar input vectors are assigned to the same output unit. • Fixed weights are used for constrained optimisation problems. The Boltzmann machine (without learning) and the continuous Hopfield net can be used for these typ es of problems. When these nets are designed, the weights are set to represent t...
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