Solar radiation estimation using artificial neural networks

Solar radiation estimation using artificial neural networks

Applied Energy 71 (2002) 307–319 www.elsevier.com/locate/apenergy Solar radiation estimation using artificial neural networks Atsu S.S. Dorvloa,*, Jos...

216KB Sizes 2 Downloads 174 Views

Applied Energy 71 (2002) 307–319 www.elsevier.com/locate/apenergy

Solar radiation estimation using artificial neural networks Atsu S.S. Dorvloa,*, Joseph A. Jervaseb, Ali Al-Lawatib a

Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod, Muscat, Oman 123 b Information Engineering Department, College of Engineering, Sultan Qaboos University, PO Box 33 Al-Khod, Muscat, Oman 123 Received 30 May 2001; accepted 9 February 2002

Abstract Artificial Neural Network Methods are discussed for estimating solar radiation by first estimating the clearness index. Radial Basis Functions, RBF, and Multilayer Perceptron, MLP, models have been investigated using long-term data from eight stations in Oman. It is shown that both the RBF and MLP models performed well based on the root-mean-square error between the observed and estimated solar radiations. However, the RBF models are preferred since they require less computing power. The RBF model, obtained by training with data from the meteorological stations at Masirah, Salalah, Seeb, Sur, Fahud and Sohar, and testing with those from Buraimi and Marmul, was the best. This model can be used to estimate the solar radiation at any location in Oman. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Solar radiation; Radial basis functions; Artificial neural networks; Clearness index

1. Introduction Solar energy is the portion of the sun’s energy available at the earth’s surface for useful applications, such as raising the temperature of water or exciting electrons in a photovoltaic cell, in addition to supplying energy to natural processes like photosynthesis. This energy is free, clean and abundant in most places throughout the year. Its effective harnessing and use are of importance to the world, especially at the * Corresponding author. Tel.: +1-968-515-400; fax: +968-513-415. E-mail addresses: [email protected] (A.S.S. Dorvlo), [email protected] (J.A. Jervase), [email protected] squ.edu.om (A. Al-Lawati). 0306-2619/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(02)00016-8

308

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

Nomenclature ANN bo b1 ðjÞ b2 ðjÞ H Hˆ Ho M MLP RBF S So W1 ðk; jÞ W2 ðjÞ   j ðxÞ

Artificial Neural Network data independent bias term variable at output node bias term applied to jth neuron of hidden layer bias term applied to output layer neuron observed solar radiation estimated solar radiation maximum solar radiation number of hidden layer neurons Multilayer Perceptron Radial Basis Function sunshine hours maximum sunshine hours weight between kth input and jth summation node of hidden layer weight between jth hidden layer neuron and output summation node latitude of location sun’s declination at location output of jth node in hidden layer

time of high fossil fuel costs and the degradation of the atmosphere by the use of these fossil fuels. Solar radiation data provide information on how much of the sun’s energy strikes a surface at a location on earth during a particular time period. These data are needed for effective research into solar-energy utilization. Due to the cost and difficulty in measurement, these data are not readily available [1,2]. Therefore, there is the need to develop alternative ways of generating these data. The aim of the present work is to develop artificial neural network (ANN) models that can be used to estimate solar radiation at any given location in Oman based on its latitude, longitude, altitude, sunshine hours and the month of the year. This can be viewed as a multivariable interpolation problem in which it is required to estimate the function relating the input to the output using a set of input-output data. This kind of problem is referred to in the literature by different names such as nonparametric regression, function approximation and supervised learning in neural network terminology [3]. Artificial neural networks and in particular Radial Basis Function (RBF) networks have been successfully used in tackling such problems [4]. Several other models have been developed for estimating solar radiation for particular stations. Regression-type models were first applied to this problem by Angstro¨m [5] based on sunshine duration. This model was developed for other locations by Akinoglu and Ecevit [6], Newland [7], Gopinathan [8] and Rietveld [9]. Some regression models incorporating trigonometric functions have been proposed by Dorvlo and Ampratwum [10] and Coppolino [2]. Radiation data being time dependent have also been modeled using harmonic analysis methods by Dorvlo and

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

309

Ampratwum [11], Herrero [12], Philips [13]. Mohandes et al. [14] have compared radial basis function methods with the regression models for Saudi stations and found the radial basis method models to be better than the regression models. Hokoi et al. [15] used a stationary autoregressive moving average model (ARMA) of order three to model hourly solar radiation while Mora-Lopez and Sidrach-de-Cardona [16] also used an ARMA model of order unity to model monthly solar radiation data. Sfetsos and Coonick [17] used artificial intelligence techniques for forecasting hourly solar radiations. All these models are however location dependent. We aim to develop neural network based models for estimating solar radiation in those places in Oman that do not have measuring instruments.

2. Artificial neural networks for solar radiation estimation Neurons are the basic elements of the human brain. The basic function of these neurons is to provide us with the ability to apply our previous experiences to our actions [18–20]. Artificial Neural Networks (ANNs) are computing algorithms that mimic the four basic functions of these biological neurons. These functions receive inputs from other neurons or sources, combine them, perform operations on the result and output the final result [20]. What makes ANNs exciting is the fact that once a network has been set up, it can learn in a self-organizing way that emulates the brain functions such as pattern recognition, classification, and optimization [18– 22]. An ANN is characterized by its architecture, training or learning algorithm and activation function. The architecture describes the connections between the neurons. It consists of an input layer, an output layer and generally, one or more hidden layers in-between as depicted in Fig. 1. Fig. 1 shows one of the commonly used networks, namely, the layered feed-forward neural network with one hidden layer.

Fig. 1. A typical layered feed-forward neural network.

310

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

The layers in these networks are interconnected by communication links that are associated with weights that dictate the effect on the information passing through them. These weights are determined by the learning algorithms, which lead to the categorization of the ANNs as [21]:  Fixed weight ANNs: these do not need any kind of learning.  Unsupervised ANNs: these networks are trained (weights are adjusted) based on input data only. The networks learn to adapt using experience gained from previous inputs.  Supervised ANNs: These are the most commonly used ANNs. In these networks, the system makes use of both input and output data. The weights are updated for every set of input/output data. The multilayer perceptron and radial basis function artificial neural networks fall into this category. The activation function, on the other hand, relates the output of a neuron to its input based on the neuron’s input activity level. Some of the commonly used functions include the threshold function, piece-wise linear function, sigmoid function [18] and the Gaussian function. Each activation function in an ANN is usually associated with a bias. This bias may be determined using the training algorithm by considering it as a weight with a fixed input. A brief introduction to RBF and MLP networks for solar radiation estimation follows. 2.1. A. RBF Network Implementation The radial basis function network consists of an input layer, an output layer and usually but not necessarily one hidden layer [3]. The architecture used is shown in Fig. 2. The activation function in the hidden layer of the RBF network is Gaussian, that is characterized by its response that decreases monotonically with distance from a central point.

Fig. 2. A radial basis function (RBF) neural network.

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

311

The problem of estimating solar radiation at a given location based on its latitude, longitude, altitude, sunshine ratio and the month of the year may be considered as a multivariable interpolation one. This entails finding an approximating function H(x) representing solar radiation in which x is an N-dimensional vector. In this case, the x components are the independent variables, namely, latitude, longitude, altitude, sunshine ratio and the month of the year (i.e. N=5). In RBF networks, H(x) is expressed as a linear combination of multivariate Gaussian basis functions [18,23]. Thus HðxÞ ¼

M X Wj j ðxÞ þ bo

ð1Þ

j¼1

where "

5 X 1 j ðxÞ ¼ exp  ðxj  ckj Þ2 2 2 j j¼1

j ðxÞ ckj j Wj bo M

is is is is is is

# ð2Þ

the output of the jth node in the hidden layer (see Fig. 2), the center of the jth RBF node for the kth input variable xk , the width of the Gaussian function, the weight between jth RBF unit and output layer neuron, the bias term i.e. data independent variable at output node, the number of hidden layer neurons.

The learning process of a radial basis function network involves using the inputoutput data to determine the parameters ckj , j and Wj . One of the techniques used to obtain these parameters is based on assuming fixed radial-basis functions. In this method, the centers are randomly selected from the training data set. On the other hand, the width of the Gaussian radial basis function is expressed in terms of the maximum distance between the chosen center d and the number of centers M as d j ¼ pffiffiffiffiffiffiffiffi 2M

ð3Þ

With ckj and j specified for all the hidden nodes, it remains to determine the weights.These can be computed using multiple linear regression techniques [23]. This involves processing the P training patterns through M hidden nodes to generate an MP matrix, Q, say. The aim is to find the weights vector W (1M vector) such that the error between the computed output vector S (1P vector) and the target output vector, T, is minimum. In matrix form, this translates to minimizing kT  Sk ¼ kT  WQk where kk

represents the Euclidean norm.

ð4Þ

312

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

Fig. 3. A multilayer perceptron (MLP) feed-forward neural network.

The least-squares solution of (4) for the vector W can be found using the pseudoinverse of Q as follows W ¼ TQT ðQQT Þ1

ð5Þ

The Matlab Neural Network Toolbox was used for the implementation of the radial basis function network [24]. The Matlab function ‘newrb’ was used [25]. Based on a given width (spread), this function iteratively adds one neuron at a time to the network until the sum-squared error falls below a specified error goal or a maximum number of neurons is attained. 2.2. MLP Network Implementation The multilayer perceptron network consists of an input layer, an output layer and usually one or more hidden layers. The architecture used in this work, shown in Fig. 3, has an input layer of five inputs, one hidden layer with a sigmoidal activation function, , defined by the logistic function ¼

1 1 þ expðyÞ

ð6Þ

where y is the corresponding input. For the output layer, a linear activation function was used in the implementation. The approximating function H(x), representing solar radiation is defined as " HðxÞ ¼

M X Aj ðxÞW2 ð jÞ þ b2 ð1Þ

# ð7Þ

j¼1

where

"

# 5 X xðkÞW1 ðk; jÞ þ b1 ð jÞ Aj ðxÞ ¼  k¼1

ð8Þ

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

W1(k,j) W2( j) b1( j) b2(1) x(k) M

313

is the linear activation function at the output layer (see Fig. 3), is the weight between kth input and jth summation hidden layer node, is the weight between jth hidden layer neuron and output summation node, is the bias term applied to the jth hidden layer neuron, is the bias term applied to the output layer neuron, is the kth element of the input vector x, is the number of hidden layer neurons.

The learning process of the MLP network involves using the input-output data to determine the weights and biases. One of the techniques used to obtain these parameters is the backpropagation algorithm [18]. In this method, the weights and biases are adjusted iteratively to achieve a minimum specified mean square error between the network output and target value. The Matlab Neural Network Toolbox was again used for the implementation of the MLP network. For the training of the neural network, the Bayesian regulation backpropagation algorithm ‘trainbr’ was used. This training function updates the weights and bias values according to Levenberg-Marquardt optimization [26]. It minimizes a linear combination of squared errors and weights, and then uses Bayesian regularization to determine the correct combination that results in a network that generalizes satisfactorily.

3. Data The Sultanate of Oman lies between latitude 16 400 N and 26 200 N and longitudes 51 500 E and 59 400 E which is in the solar belt. The maximum radiation, Ho, levels expected in Oman is between 23.32 MJ/m2/day and 40.31 MJ/m2/day. The theoretical sunshine duration, So, is given by So ¼ ð2=15Þcos1 ðtantanÞ, where a and d represent respectively the latitude and declination of a location. For Oman, So, ranges between 10.5 hours and 13.5 hours a day. The climatic conditions are mainly desert in the north and subtropical in the far south. The average solar radiation in Oman is 18.71 MJ/m2/day with a standard deviation of 4.0 MJ/m2/day, and the daily sunshine duration is between 8.0 and 20.5 hours [27]. Oman has a great potential for solar energy harnessing because of the long daily duration of sunshine hours and high levels of solar radiation. There are at the moment 25 weather stations in Oman that routinely measure climatic parameters like solar radiation, sunshine hours, temperature, rainfall, atmospheric pressure, temperature and humidity. However, not all the stations measure all these parameters, especially solar radiation. Most stations are relatively new and therefore have not accumulated any long-term data yet. In addition, there are large areas of Oman that do not have any weather stations. The raw data for this study were obtained through personal communication from the Directorate General of Civil Aviation and Meteorology. However, monthly summaries are published annually by the Oman Ministry of Transportation and Housing [28]. Table 1 gives the geographical details of the location of the stations

314

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

Table 1 Average daily sunshine duration, solar radiation and location of meteorological stations in Oman considered in the study Station no.

Location

Latitude (N)

Longitude (E)

Altitude (m)

Sunshine durationa (hours)

Solar radiationa (MJ/m2/day)

1 2 3 4 5 6 7 8

Sohar Fahud Sur Seeb Salalah Masirah Marmul Buraimi

24 280 22 210 22 320 23 350 17 020 20 400 18 080 24 140

56 380 56 290 59 280 58 170 54 050 58 540 55 110 55 470

3.63 170.00 13.77 8.40 20.00 18.80 269.00 296.89

9.28 9.98 9.63 9.75 7.96 9.49 10.11 10.30

21.01 20.08 15.18 19.59 16.37 17.32 22.21 19.25

a

(2.31) (1.93) (2.12) (2.21) (3.84) (2.04) (1.38) (1.91)

(5.30) (4.35) (4.19) (4.75) (4.04) (3.71) (2.23) (4.60)

Values between brackets represent the standard deviation.

Table 2 Summary of observed clearness index (H/Ho) and sunshine ratios (S/So) used as part of the input for training and validation of the neural networks Month

S/So

H/Ho

Buraimi 1 2 3 4 5 6 7 8 9 10 11 12

0.8077 0.7734 0.7838 0.8392 0.8980 0.9098 0.9064 0.9014 0.9097 0.9154 0.9056 0.8523

0.8401 0.8448 0.8553 0.8536 0.8679 0.8007 0.7621 0.8206 0.8757 0.8902 0.8613 0.8499

H/Ho

Seeb 0.5832 0.5796 0.5695 0.5890 0.6197 0.5902 0.5346 0.5721 0.6181 0.6358 0.6031 0.5783

Marmul 1 2 3 4 5 6 7 8 9 10 11 12

S/So

0.7814 0.7867 0.7178 0.7745 0.8695 0.8372 0.7394 0.7809 0.8481 0.8965 0.8978 0.8336

0.7907 0.8167 0.7226 0.8025 0.8529 0.8213 0.6639 0.7390 0.8214 0.8831 0.9008 0.8278

H/Ho

Masirah 0.5594 0.5852 0.5713 0.5957 0.6162 0.6128 0.5560 0.5661 0.6047 0.6204 0.6192 0.5830

Sur 0.6862 0.6971 0.6773 0.6660 0.6467 0.5766 0.5424 0.5923 0.6644 0.7007 0.6954 0.6698

S/So

0.8597 0.8447 0.7875 0.8003 0.8762 0.7523 0.5985 0.6492 0.7490 0.8769 0.9028 0.8541

0.8614 0.8108 0.7860 0.8265 0.8635 0.5085 0.1447 0.1074 0.4676 0.8809 0.8897 0.8680

H/Ho

Fahud 0.5282 0.5231 0.5412 0.5358 0.5462 0.4995 0.4312 0.4707 0.4997 0.5431 0.5458 0.5245

Salalah 0.4654 0.4531 0.4521 0.4632 0.4684 0.4411 0.3998 0.4293 0.4539 0.4917 0.4947 0.4608

S/So

0.7927 0.7733 0.7850 0.8263 0.8785 0.8864 0.8490 0.8499 0.8734 0.8899 0.8777 0.7420

0.6194 0.5965 0.5901 0.6106 0.6304 0.6109 0.5663 0.5786 0.6169 0.6265 0.6134 0.6015

Sohar 0.5466 0.5223 0.5319 0.5334 0.5382 0.4629 0.3172 0.2776 0.4201 0.5476 0.5541 0.5432

0.7697 0.7702 0.6830 0.7492 0.8377 0.7975 0.6673 0.6925 0.7843 0.8620 0.8871 0.8126

0.5869 0.6412 0.5874 0.6595 0.6946 0.6763 0.6177 0.6305 0.6657 0.6596 0.6356 0.6294

315

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319 Table 3 RMSE of solar radiation H in MJ/m2/day for all the models developed Model no.

Training set

Validating set

MLP-1a

MLP-2a

MLP-3a

RBF

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

1,2,3,4,5,6 1,2,3,4,5,7 1,2,3,4,5,8 1,2,3,4,6,7 1,2,3,4,6,8 1,2,3,4,7,8 1,2,3,5,6,7 1,2,3,5,6,8 1,2,3,5,7,8 1,2,3,6,7,8 1,2,4,5,6,7 1,2,4,5,6,8 1,2,4,5,7,8 1,2,4,6,7,8 1,2,5,6,7,8 1,3,4,5,6,7 1,3,4,5,6,8 1,3,4,5,7,8 1,3,4,6,7,8 1,3,5,6,7,8 1,4,5,6,7,8 2,3,4,5,6,7 2,3,4,5,6,8 2,3,4,5,7,8 2,3,4,6,7,8 2,3,5,6,7,8 2,4,5,6,7,8 3,4,5,6,7,8

7,8 6,8 6,7 5,8 5,7 5,6 4,8 4,7 4,6 4,5 3,8 3,7 3,6 3,5 3,4 2,8 2,7 2,6 2,5 2,4 2,3 1,8 1,7 1,6 1,5 1,4 1,3 1,2

3.30 3.44 5.25 2.75 3.00 4.54 2.72 7.15 5.08 3.67 4.76 4.65 2.38 5.18 3.40 2.12 3.52 3.66 5.69 5.07 5.29 7.16 2.76 1.01 9.41 2.58 2.64 1.66

5.71 4.70 6.88 1.90 2.96 2.27 2.66 3.58 4.47 5.66 3.78 2.18 5.21 8.08 6.25 3.00 6.09 4.92 3.94 7.84 6.64 6.69 2.19 2.57 8.21 3.19 2.32 1.78

1.51 1.58 5.95 1.60 3.25 5.16 1.46 3.34 4.57 3.10 4.16 1.88 3.15 5.69 1.35 2.72 3.66 2.74 4.03 2.76 6.06 8.12 2.01 2.06 3.81 2.63 3.22 3.52

4.04 2.86 3.28 1.71 2.36 2.13 9.58 3.67 3.96 3.80 8.96 10.08 6.47 3.12 1.75 1.67 1.62 2.27 3.07 4.02 2.90 3.58 1.97 1.57 5.01 1.89 3.66 0.83

a

MLP-i represents a multilayer perceptron network with i hidden layers.

used in this study. These eight stations, Buraimi, Fahud, Marmul, Masirah, Salalah, Seeb, Sohar and Sur have long term data of ten years or more. There is a concentration of stations in the north and south of the country. The central portion of the country which is very sparsely populated has only three stations. Table 2 gives the clearness index which is the ratio of the average daily solar radiation, H, and the daily maximum radiation, Ho, and sunshine ratio, i.e. the ratio of the average daily sunshine hours, S, and the theoretical sunshine duration, So, by month for all the eight stations in the study.

4. Implementation and results Data from the eight stations averaged over at least ten years (1986–1998) were used to train and validate the MLP neural networks and an RBF network. The

316

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

input parameters were latitude, longitude, altitude, sunshine ratio, (S/So) and month of year. The output parameter is the clearness index (H/Ho). The estimated solar radiation was obtained by multiplying the estimated clearness index by Ho. The data used to train and validate are provided in Tables 1 and 2. Data from six stations were used to train the networks and the data from the remaining 2 stations were used for the validation of the models. To arrive at the best possible model, all possible combinations of 6 out of 8 stations data, a total of 28 combinations, were used in training the networks and the remaining two for validation. MLP networks with

Fig. 4. Solar radiation as measured and estimated using the Radial Basis Function network model 28.

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

317

1, 2 and 3 hidden layers were implemented. The size of the root mean square error, RMSE, was used to determine the best models. The RMSE, is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðH ^ i Hi Þ2 ^ i is the estiRMSE ¼ , where Hi is the measured solar radiation and H n mated solar radiation. The computed RMSEs are shown in Table 3. All the four networks performed very well. The range of errors for the RBF networks was 0.83 to 10.08 MJ/m2/day, while the range of errors for MLP networks was 1.01 to 9.41 MJ/ m2/day. A Kruskal–Wallis k-sample test showed no significant difference between

Fig. 5. Solar radiation as measured and estimated using the Multilayer Perceptron network model 15.

318

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

the networks (p-value=0.159). However, a least significant difference comparison showed that the MLP networks with 2 and 3 hidden layers are different (p-value=0.030). Mohandes et al. [14] also reported mixed results with none of the networks having an edge in terms of the mean absolute percentage error. The performances of the networks are similar but we would recommend the use of the RBF network because it does not need as much computing power as the MLP networks. Model #28 which results from training with stations Buraimi, Marmul, Masirah, Salalah, Seeb and Sur and validating with stations Fahud and Sohar gave the smallest RMSE (0.83 MJ/m2/day) and is judged the best model using the RBF network. The results were obtained using Matlab Neural Toolbox function ‘newrb’ [25] with the spread constant of the Gaussian function set to 0.75 and training goal of sum-squared errors set to 0.008. However, among the MLP networks, the one which utilizes 3 hidden layers is judged the best because its mean and standard deviation of the root mean square errors were the lowest. For the MLP network, model #15 which results from training using stations Sohar, Fahud, Salalah, Masirah and Marmul and validating with stations Seeb and Sur was the best (RMSE=1.35 MJ/ m2/day). The results were obtained using Matlab Neural Toolbox function ‘trainbr’ [25] with the training goal of sum-squared errors set to 0.008. Fig. 4 shows the measured and estimated solar radiation using the model that resulted in the minimum overall RMSE value of 0.83 MJ/m2/day, namely, model #28 for the RBF network. Fig. 5 shows results obtained for model #15 which resulted in the minimum overall RMSE of 1.01 MJ/m2/day for the MLP network. It is apparent from Figs. 4 and 5 that the training sets show almost a perfect fit as expected, since the data were used to train the networks.

5. Conclusion We have demonstrated the use of neural network methods in modeling solar radiation. Both the Radial Basis Function (RBF) and Multilayer Perceptron (MLP) networks are good in modeling this set of data. However, the RBF is to be preferred because it requires less computing power and time. The models presented here can be used to predict solar radiation in those locations in the Sultanate of Oman where measurements of sunshine hours are available in addition to the latitude, longitude and altitude measurements of the location. References [1] Duffie JA, Beckman WA. Solar engineering of thermal processes. New York:1991 John Wiley and Sons. [2] Coppolino S. A new correlation between clearness index and relative sunshine. Renewable Energy 1994;4(4):417–23. [3] Orr MJL. Introduction to radial basis function networks. Technical Report, Institute for Adaptive and Neural Computation, Division of Informatics, Edinburgh University, Edinburgh, Scotland, UK. http:// www.anc.ed.ac.uk/ mjo/rbf.html. 1996. [4] Hagan MT, Demuth HB, Beale M. Neural network design. PWS Publishing Company, 1996.

A.S.S. Dorvlo et al. / Applied Energy 71 (2002) 307–319

319

[5] Angstro¨m A. On the computation of global solar radiation from records of sunshine. Arkiv Geophysik 1956;3(23):551. [6] Akinoglu BG, Ecevit A. Construction of a quadratic model using modified Angstro¨m coefficients to estimate global solar radiation. Solar Energy 1990;45:85–92. [7] Newland FJ. A Study of Solar Radiation models for the Coastal Region of South China. Solar Energy 1986;43(4):227–35. [8] Gopinathan KK. Computing the monthly mean daily diffuse radiation from clearness index and percent possible sunshine. Solar Energy 1988;41(4):379–85. [9] Rietveld MR. A new method for estimating the regression coefficients in the formula relating solar radiation to sunshine. Agric Meteorology 1978;19:243–52. [10] Dorvlo ASS, Ampratwum DB. Modeling of weather data for Oman. Renewable Energy 1999;17: 421–8. [11] Dorvlo ASS, Ampratwum DB. Harmonic analysis of global irradiation. Renewable Energy 2000;20: 435–43. [12] Herrero AC. Harmonic analysis of monthly solar radiation data in Spain. Ambient Energy 1993; 14(1):35–40. [13] Philips WF. Harmonic Analysis of Climatic data. Solar Energy 1984;32(3):319–28. [14] Mohandes M, Balghonaim A, Kassa M, Rehman S, Halawani TO. Use of radial basis functions for estimation monthly mean daily solar radiation. Solar Energy 2000;68(2):161–8. [15] Hokoi S, Matsumoto M, Kagawa M. Stochastic models of Solar radiation and outdoor Temperature. ASHRAE Trans 1990;2:245–52. [16] Mora-Lopez LL, Sidrach-de-Cardona M. Multiplicative ARMA models to generate hourly series of Global Irradiation. Solar Energy 1998;63:283–91. [17] Sfetsos A, Coonick AH. Univariate and Multivariate Forecasting of Hourly Solar Radiation with Artificial Intelligence Techniques. Solar Energy 2000;68(2):169–78. [18] Haykin S. Neural networks, MacMillan College Publishing Company, 1994. [19] Fausett L. Fundamentals of Neural networks, Prentice Hall International, 1994. [20] Klerfors D. Artificial neural networks, Project MISB-420–0, Saint Louis University, 1998. [21] Kung SY. Digital neural networks, PTR Prentice-Hall, 1998. [22] Tarassenko L. A guide to neural computing applications, John Wiley & Sons, 1998. [23] El-Sharkawi M, Niebur D, editors. Artificial neural networks with applications to power systems, IEEE Catalog Number 96 TP 112–0, 1996. [24] The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760–2098,USA, www.mathworks.com. [25] Demuth H, Beale M. (1998). Neural network toolbox. User’s Guide Version 3.0, The MathWorks, Inc., 1998. [26] Nocedal J, Wright SJ. Numerical optimization. Springer-Verlag, New York, 1999. [27] Dorvlo ASS, Ampratwum DB. Summary climatic data for solar technology development in Oman. Renewable Energy 1998;14(1–4):255–62. [28] Oman Ministry of Transportation and Housing (1986–1998). Annual Climatic Summaries. Directorate General of Civil Aviation and Meteorology, Department of Meteorology, Sultanate of Oman.