Hybrid auto-regressive neural network model for estimating global solar radiation in Bandar Abbas, Iran

Abstract

In this paper, a neural network auto-regressive model with exogenous inputs (NN-ARX) is utilized for predicting daily horizontal global solar radiation (DHGSR). For this aim, two sets of parameters: (1) sunshine hours (n) and maximum possible sunshine hours (N), and (2) maximum ambient temperature (T _max) and minimum ambient temperature (T _min) collected for Bandar Abbas city of Iran are used as inputs. The efficiency of NN-ARX is compared with that of the adaptive neuro-fuzzy inference system (ANFIS), which is a robust methodology. The attained results reveal the superiority of sunshine hours as input over air temperatures so that the NN-ARX (1) and ANFIS (1) models using n and N as inputs offer higher precision than the NN-ARX (2) and ANFIS (2) models using T _max and T _min as inputs. Statistical results demonstrate that NN-ARX provides favorable precision and outperforms ANFIS. The relative percentage error analysis shows that the capability of the ANN-ARX (1) model in different days of the year is indeed attractive since 89.25 % of the predictions fall within the acceptable range of −10 to +10 %. The influence of introducing extraterrestrial solar radiation (H _o ) as third input on the performance of the NN-ARX models is assessed. It is found that using H _o provides only slight improvements on accuracy for both sunshine duration and temperature-based predictions; thus, considering H _o as the third input may not be really suitable since it also brings further complexity in terms of the required inputs. The survey results prove that NN-ARX would be an efficient alternative approach to predict DHGSR.

Appendices

Appendix 1

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

$$K_{T} = \frac{H}{{H_{o} }}$$

The daily extraterrestrial solar radiation on a horizontal surface, H _o , is expressed as (Duffie and Beckman 2006; Kalogirou 2009):

$$H_{o} = \frac{24 \times 3600}{\pi }G_{\text{sc}} \left( {1 + 0.033\cos \,\frac{{360{\kern 1pt} \,n_{\text{day}} }}{365}} \right) \times \left( {\cos \varphi \,\cos \delta \,\sin \omega_{\text{s}} + \frac{{\pi \omega_{\text{s}} }}{180}\,\sin \varphi \sin \delta } \right)$$

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 (Duffie and Beckman 2006). δ and ω _s are the daily solar declination and sunset hour angles, respectively, as (Duffie and Beckman 2006):

$$\delta = 23.45\,\sin \left( {\frac{{\left( {n_{\text{day}} + 284} \right)360}}{365}} \right)$$

$$\omega_{\text{s}} = \cos^{ - 1} \left( { - \tan \varphi \,\tan \delta } \right)$$

The daily maximum possible sunshine duration is (Duffie and Beckman 2006):$N = \frac{2}{15}\cos^{ - 1} \left( { - \tan \varphi \,\tan \delta } \right)$

Appendix 2

The RPE, which its values ranging between −10 and +10 % is usually considered acceptable (Ertekin and Yaldiz 2000), is defined as:

$${\text{RPE}} = \left( {\frac{{H_{i,c} - H_{i,m} }}{{H_{i,m} }}} \right) \times 100$$

where H _i,c is the ith calculated global solar radiation value by the models and H _i,m is the ith measured global solar radiation value.

The MAPE and the MABE are given, respectively by:

$${\text{MAPE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {\frac{{H_{i,c} - H_{i,m} }}{{H_{i,m} }}} \right| \times 100}$$

$${\text{MABE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {H_{i,c} - H_{i,m} } \right|}$$

where N is the total number of data.

The RMSE is calculated by:

$${\text{RMSE}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {H_{i,c} - H_{i,m} } \right)}^{2} }$$

The RRMSE in percent is achieved by dividing the RMSE to the averaged measured values, which is defined by:

$${\text{RRMSE}} = \frac{{\sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {H_{i,c} - H_{i,m} } \right)}^{2} } }}{{\frac{1}{N}\sum\limits_{i = 1}^{N} {H_{i,m} } }} \times 100$$

Different ranges of RRMSE can be defined to represent the models’ precision as follows (Li et al. 2013):

Excellent for RRMSE < 10 %;

Good for 10 % < RRMSE < 20 %;

Fair for 20 % < RRMSE < 30 %;

Poor for RRMSE > 30 %.

The R is computed by:

$$R = \frac{{\sum\limits_{i = 1}^{N} {\left( {H_{i,c} - H_{{c,{\text{avg}}}} } \right).} \left( {H_{i,m} - H_{{m,{\text{avg}}}} } \right)}}{{\sqrt {\left[ {\sum\limits_{i = 1}^{N} {\left( {H_{i,c} - H_{{c,{\text{avg}}}} } \right)^{2} } } \right]\left[ {\sum\limits_{i = 1}^{N} {\left( {H_{i,m} - H_{{m,{\text{avg}}}} } \right)^{2} } } \right]} }}$$