An adaptive wavelet-network model for forecasting daily total solar-radiation
Introduction
Daily total solar-radiation is considered as the most important parameter in meteorology, solar conversion, and renewable energy applications, particularly for the sizing of stand-alone photovoltaic (PV) systems [1], [2], [3]. This type of data is usually presented as a time series and time series prediction is an important scientific task. Especially challenging are situations where an underlying model, generating observed data, is not known. Modelling time-series includes the area of stochastic prediction and the optimal prediction of a signal sample (in a minimum mean-square sense), given a finite number of past samples. Time series modelling is a conditional expectation [4], but the computation of the conditional expectation requires a knowledge of the joint probability of the current sample and past samples, which is generally not known. Because of this, and the fact that the conditional expectation is generally non-linear, finding the solution is mathematically intractable. Therefore, the methods for designing the non-linear signal predictors are sub-optimal, and they can only attempt to approximate the conditional expectation of the current sample. These sub-optimal methods, like auto-regressive (AR) [5], [6], [7] predictions, Markov chains [8], [9], auto-regressive moving average (ARMA) models [10], and the Markov transitions matrix (MTM) approach [11] use as input only the average monthly clearness index (Kt). All these models are based on simplifying statistical assumptions, about the measured data, which are not always true [12].
A way to solve the problem in such a case can be provided by non-parametric regression methods, such as the artificial neural-network (ANN). Because ANN design is based on training, no statistical assumptions are needed for the source data. Neural networks are widely accepted as a technology offering an alternative way to tackle complex and ill-defined problems [13]. They can be trained to predict results from examples. They are able to deal with non-linear problems, and once trained can perform predictions at very high speed. A review of applications of ANNs in renewable-energy systems is given in [13]. A number of different architectures and learning methods have been applied with varying degrees of success to the problem of predicting future values of time series for total radiation.
Elizondo et al. [14] proposed the use of a feed-forward neural network to estimate the daily solar-radiation. The authors have used as input other meteorological parameters, such as temperature, precipitation, clear-sky radiation, day length and day of the year. Al-Alawi and Al-Hinai [15] have used an ANN for analyzing the relationship between total radiation and climatology variables, this model predicted the total radiation values to a good accuracy of approximately 93%. Guessoum et al. [12] have used the radial basic function (RBF) networks to predict solar-radiation data in Algeria. In their research, the input and the output are the solar radiation data corresponding to a particular day G(j) and those of the next day G(j + 1). The RBF model predicts solar-radiation data, using Kolmokorov–Simernov (KS) statistics, to an accuracy of 1.36%. Kemmoku et al. [16] used single- and multi-stage neural-networks for forecasting the daily insolation; the results show that the mean error reduces from about 30% (by the single-stage) to about 20% (by the multi-stage). Mohandes et al. [17] used data from 41 recording stations in Saudi Arabia to predict the mean monthly solar-radiation. The input values to the network are latitude, longitude, altitude and sunshine duration and the results for testing stations obtained were within 16%. Kalogirou et al. [18] used a recurrent neural-network to predict the maximum solar radiation from relative humidity and temperature. The results indicate that the correlation coefficient obtained varied between 98.58% and 98.75%. Sfetsos and Coonick [19] made a comparative study between the linear methods and proposed a new approach, which combines neural networks and fuzzy logic for forecasting the hourly solar-radiation. However, there still exists a need for satisfactory guidelines when choosing one approach over another for a particular problem. Single layer feed-forward neural-networks, with hidden nodes of adaptive wavelet functions, have been successfully demonstrated to have potential in different applications [20], [21], [22], [23], [24], [25].
The main objective of this work is to investigate the suitability of the wavelet-network architecture with an impulse infinite-response (IIR) filter for modelling and prediction of the daily total solar-radiation. This architecture provides a double local structure which results in an improved speed of learning. In addition, a comparative study has been presented in order to illustrate the importance of this model.
Section snippets
Database of daily total solar-radiation values
The meteorological data, that have been used in this work, are the recorded solar-radiation values during the period extending from 1981 to 2001 from a meteorological station in Algeria (exact geographical position is 36°,43′ North and 3°,2′ East). Fig. 1 shows the variation of total daily solar-radiation (Gi) sequence (total radiation received on a horizontal surface) in each day i.
Wavelet-network structure and algorithm
The combination of wavelet theory and neural networks has led to the development of wavelet-networks. Wavelet-networks are feed-forward networks using wavelets as activation functions. Wavelet-networks have been used in classification and identification problems with some success.
Before beginning the tracking operation, using an adaptive wavelet network model, the unknown non-linear plan must be estimated according to a certain model. In this particular estimation process, the model consists of
Methodology
The objective is to predict the value of daily total solar-radiation from the preceding values in order to solve the problem of missing data: therefore is estimated by the function f(G(n − 1), G(n − 2), …, G(n − k)), which minimizes the mean square error (MSE). For this purpose, we have used a wavelet-network algorithm as described above. This algorithm, however, has been presented for one input and output, therefore it needs to be modified in order to cope with several inputs. The block diagram
Results and discussions
The results obtained by the wavelet-network based on 19 years of training data of total daily solar radiation are presented. Table 1 show the results obtained for the different structures of the wavelet-network. As can be seen, structure 5 gives the best results compared with the other structures. This model can predict the total solar radiation of a day from the preceding 5 values (5 days). This model gives more accurate results, but it is limited to the prediction of one value from five
Conclusions
In this paper, a suitable model for forecasting daily total solar radiation data using an adaptive wavelet-network with IIR filter is described. The prediction of future sequences of solar radiation data is done in a very simple manner. This model can predict the future total solar-radiation values (G(t)) based on previous values (G(t − 1), G(t − 2), …). This model is considered suitable to fill missing data of total solar-radiation values. The validation of the model was performed with previous
Acknowledgements
The first author thanks Prof. A. Guessoum (Faculty of Sciences Engineering, Department of Electronics, Blida University, Algeria) and Dr. A. Hadj Arab (Development Centre of Renewable Energy, Algiers) for their advice.
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