Best Models Coefficients Determination
Solar radiation data are the best source of information for estimating average incident radiation necessary for the proper design and assessment of solar energy conversion systems. However, many locations lack the necessary equipment for global radiation measurements. By contrast, data on average hours of sunshine or average percentage of possible sunshine hours are widely available from many hundreds of stations in many countries. Several Angström–Black type regression models have been proposed in the literature for estimating solar radiation using data on the duration of sunshine. Newland added a logarithmic term to the original linear regression equation to obtain a better fit. Akinoglu and Ecevit proposed a quadratic regression equation which fitted their data well. Coppolino developed a power function and incorporated a trigonometric term. This power model can also be transformed into the Angström–Black type equation by a logarithmic transformation.
The coefficients a and b, and c where applicable, were computed for each of the models using data for five stations in Oman namely Marmul, Masirah, Salalah, Seeb and Sur. These are the stations that have long-term daily measurements of global radiation and duration of bright sunshine for the years 1986 to 1995. For all the models, the method of least squares was used to determine the coefficients. The clearness index or the natural logarithm of it was the dependent variable, while the relative sunshine or the natural logarithm of it was the independent variable.
The coefficients a, b, c and R2, are reported in Table 3. The other statistics, MBE, RMSE and MPE, are presented in Table 4. The closer these statistics, in Table 4, are to zero the better the estimator. A positive MBE represents an over estimation whereas a negative MBE shows an under estimation.
In this paper, the regression coefficients of all these various models are determined using 10-year (1986?1995) data from five meteorological stations in Oman, which is representative of an area with desert and a sub-tropical climate. The models are compared to determine the best model for each of the stations in Oman.
The coefficients a and b, and c where applicable, were computed for each of the models using data for five stations in Oman namely Marmul, Masirah, Salalah, Seeb and Sur. These are the stations that have long-term daily measurements of global radiation and duration of bright sunshine for the years 1986 to 1995. For all the models, the method of least squares was used to determine the coefficients. The clearness index or the natural logarithm of it was the dependent variable, while the relative sunshine or the natural logarithm of it was the independent variable.
All the models fitted the data adequately. All the coefficients reported are significantly different from zero. The lowest coeffiecient of determination was 70%. The best fits were with data from Salalah and Masirah, where all the models gave coefficients of determination better than 90%. Seeb and Marmul gave the least coefficient of determination. The coefficients are different from station to station. These coefficients are also different from those obtained for the same models in other countries by. This may be due to Oman being largely a desert country. Salalah, which has a climate approaching that of a sub-tropical enviroment, has coefficients close to those obtained by Akinoglu and [7. B.G. Akinoglu and A. Ecevit, Construction of a quadratic model using modified Angström coefficients to estimate global solar radiation. Solar Energy 45 (1990), pp. 85–92. Abstract | View Record in Scopus | Cited By in Scopus (32)7]
For each station, all the models have very similar coefficients of determination. The largest difference between coefficients of determination of the best model and the worst is only 0.075 at Seeb. Considering the RMSE, the results for all the models are very low. This supports the high coefficients of determination mentioned above. All the models slightly under-estimate the solar radiation in Sur, Marmul and Salalah. Slight over-estimation is noted at Seeb and Masirah. It should be noted that the over- and under-estimations are very slight. The absolute percentage bias, MPE, is generally low for all the models for all the stations except Marmul. The power–trigonometric model has the smallest MPE for Masirah, Seeb and Sur. In Salalah and Masirah, the MPEs for the power–trigonometric model were the smallest recorded. It may be noted that Seeb with the largest difference between the best and worst coefficients of determination (0.0746), and Marmul with the largest absolute percentage bias, gave the weakest fits. Generally the linear model gave the largest absolute percent bias. The power–trigonometric model is the best for all the stations. The introduction of the solar altitude-angle has improved the regression equation.
All the models performed very well as estimators of global radiation when the number of relative sunshine hours is used in the regression models studied. The power–trigonometric model is the best overall. This model uses additional information, namely the solar altitude angle of the station. The quadratic and the linear–logarithmic models are the best when only the number of relative sunshine hours is used. All the non-linear models provide more accurate predictions than the basic linear model.
