RETRACTED ARTICLE: Temperature-based estimation of global solar radiation using soft computing methodologies

Abstract

Precise knowledge of solar radiation is indeed essential in different technological and scientific applications of solar energy. Temperature-based estimation of global solar radiation would be appealing owing to broad availability of measured air temperatures. In this study, the potentials of soft computing techniques are evaluated to estimate daily horizontal global solar radiation (DHGSR) from measured maximum, minimum, and average air temperatures (T _max, T _min, and T _avg) in an Iranian city. For this purpose, a comparative evaluation between three methodologies of adaptive neuro-fuzzy inference system (ANFIS), radial basis function support vector regression (SVR-rbf), and polynomial basis function support vector regression (SVR-poly) is performed. Five combinations of T _max, T _min, and T _avg are served as inputs to develop ANFIS, SVR-rbf, and SVR-poly models. The attained results show that all ANFIS, SVR-rbf, and SVR-poly models provide favorable accuracy. Based upon all techniques, the higher accuracies are achieved by models (5) using T _max–T _min and T _max as inputs. According to the statistical results, SVR-rbf outperforms SVR-poly and ANFIS. For SVR-rbf (5), the mean absolute bias error, root mean square error, and correlation coefficient are 1.1931 MJ/m², 2.0716 MJ/m², and 0.9380, respectively. The survey results approve that SVR-rbf can be used efficiently to estimate DHGSR from air temperatures.

Appendices

Appendix A

The daily clearness index, K_T, can be computed by (Duffie and Beckman 2006; Kalogirou 2009)

$$ {K}_T=\frac{H}{H_o} $$

The H_o, is expressed as (Duffie and Beckman 2006; Kalogirou 2009)

$$ \begin{array}{c}\hfill {H}_o=\frac{24\times 3600}{\pi }{G}_{\mathrm{s}\mathrm{c}}\left(1+0.033 \cos\;\frac{360\kern0.1em {n}_{\mathrm{day}}}{365}\right)\hfill \\ {}\hfill \times \left( \cos \varphi\;\cos \delta\;\sin {\omega}_{\mathrm{s}}+\frac{\pi {\omega}_{\mathrm{s}}}{180}\; \sin \varphi \sin \delta \right)\hfill \end{array} $$

where G _sc is the solar constant, assumed equal to 1367 W/m², and n _day is the day number of the year, counted from the first of January. δ and ω _s are the daily solar declination and sunset hour angles, respectively, as (Duffie and Beckman 2006)

$$ \begin{array}{c}\hfill \delta =23.45\; \sin \left(\frac{\left({n}_{\mathrm{day}}+284\right)360}{365}\right)\hfill \\ {}\hfill {\omega}_{\mathrm{s}}={ \cos}^{-1}\left(- \tan \varphi\;\tan \delta \right)\hfill \end{array} $$

Appendix B

The MABE, as an accuracy level estimator, gives the mean absolute value of bias error. The MABE is obtained by

$$ \mathrm{MABE}=\frac{1}{N}{\displaystyle \sum_{i=1}^N\left|{H}_{i,c}-{H}_{i,m}\right|} $$

where H _i,c is the ith calculated solar radiation value by models, and H_i,m is the ith measured solar radiation value. Also, N is the total number of observations.

The RMSE, as a widely used indicator, determines the precision of the model by comparing the deviation between the estimated and real data. The RMSE always has a positive value and is calculated by

$$ \mathrm{RMSE}=\sqrt{\frac{1}{N}{{\displaystyle \sum_{i=1}^N\left({H}_{i,c}-{H}_{i,m}\right)}}^2} $$

The R indicates the strength of a linear relationship between the measured and the estimated values and is obtained by

$$ R=\frac{{\displaystyle \sum_{i=1}^N\left({H}_{i,c}-{H}_{c,\mathrm{a}\kern0.1em \mathrm{v}\kern0.1em \mathrm{g}}\right).}\left({H}_{i,m}-{H}_{m,\mathrm{a}\kern0.1em \mathrm{v}\kern0.1em \mathrm{g}}\right)}{\sqrt{\left[{\displaystyle \sum_{i=1}^N{\left({H}_{i,c}-{H}_{c,\mathrm{a}\kern0.1em \mathrm{v}\kern0.1em \mathrm{g}}\right)}^2}\right]\left[{\displaystyle \sum_{i=1}^N{\left({H}_{i,m}-{H}_{m,\mathrm{a}\kern0.1em \mathrm{v}\kern0.1em \mathrm{g}}\right)}^2}\right]}} $$