Predicting the direct, diffuse, and global solar radiation on a horizontal surface and comparing with real data

Abstract

This paper deals with the computation of solar radiation flux at the surface of earth in locations without solar radiation measurements, but with known climatological data. Simple analytical models from literature are calibrated, and linear regression relations are developed for diffuse, and global solar radiation. The measured data for the average monthly global and diffuse irradiation in Kerman, Iran are compared to the calculated results from the existing models. The data are further compared to values calculated with a linear model using seven relevant parameters. The results show that the linear model is favoured to predict irradiation data in various parts of Iran.

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Abbreviations

a:

Climatological constant in Eq. 1

a w :

Water vapor absorptance, Eq. 33

B a :

The percentage of diffuse radiation to the ground due to particles, Eq. 45

b:

Climatological constant in Eq. 1

c:

Climatological constant in Eq. 2

d:

Climatological constant in Eq. 2

e:

Relative error, Eq. 51

E0 :

Eccentricity correction factor for the earth’s orbit, Eq. 26

H :

Daily global irradiation at earth’s surface, Eq. 1

H o :

Extraterrestrial daily irradiation, Eq. 1

H DF :

Daily diffuse irradiation at earth’s surface, Eq. 2

H B :

Direct daily global irradiation at earth’s surface, Eq. 3

\( \bar{H} \) :

Mean monthly global radiation flux

h :

Elevation from sea level, Eq. 31

I B :

Direct beam irradiance for clear sky, Eq. 4

I DF :

Diffuse irradiance for clear sky, Eq. 5

I:

Global Irradiance for clear sky, Eq. 6

I o :

Solar constant, Eq. 8

K * :

Constant used in Eq. 5

m :

Air mass, Eq. 8

MAPE:

Mean average percentage error, Eq. 53

MBD:

Mean bias differences

n:

Hours of measured sunshine, Eq. 1

n* :

Day of year, Eq. 26

N:

Potential astronomical sunshine hours, Eq. 29

P :

Local absolute pressure, Eq. 31

P o :

Reference pressure at sea level, Eq. 31

q :

Coefficient in Eq. 8

r g :

Ground albedo, Eq. 44

r s :

Sky albedo, Eq. 42

R h :

Relative humidity

RMSE:

Root mean square error, Eq. 52

T A :

Transmissivity due to absorption and scattering by particles, Eq. 32

T AA :

Transmissivity due to absorption by particles, Eq. 38

T AS :

Ratio of T A to T AA, Eq. 40

T as :

Function defined in Eq. 43

T o :

Transmissivity due to ozone, Eq. 41

T M :

Transmissivity of atmospheric gases except water vapor, Eq. 30

T R :

Transmissivity due to Rayleigh Scattering, Eq. 39

t :

Test statistic

t c :

Critical value

T UM :

Transmissivity due to oxygen and carbon dioxide, Eq. 37

T W :

Transmissivity of water vapor, Eq. 34

U W :

Precipitable water in a vertical column, Eq. 33

X o :

The amount of ozone times the air mass, Eq. 49

\( \alpha_{s} \) :

Solar altitude angle, Eq. 3

δ:

Declination angle, Eq. 28

θ :

Zenith angle, Eq. 5

τA :

Turbidity coefficient, Eq. 32

φ:

Lattitude, Eq. 25

\( \omega \) :

Hour angle, Eq. 50

\( \omega_{s} \) :

Sunset hour angle, Eq. 25

References

  1. 1.

    Moon P (1940) Proposed standard solar-radiation curves for engineering use. J Frankl Inst 230:583–617

    Google Scholar 

  2. 2.

    Mahaptra AK (1973) An evaluation of a spectro-radiometer for the visible-ultraviolet and near-ultraviolet. Ph.D. dissertation; University of Missouri; Columbia, Mo, p 121 (University Microfilms 74-9964)

  3. 3.

    Watt D (1978) On the nature and distribution of solar radiation. HCP/T2552-01. U.S. Department of Energy

  4. 4.

    Atwater MA, Ball JT (1978) A numerical solar radiation model based on standard meteorological observation. Sol Energy 21:163–170

    Article  Google Scholar 

  5. 5.

    Atwater MA, Ball JT (1979) Sol Energy 23:725

    Google Scholar 

  6. 6.

    Kondratyev KY (1969) Radiation in the atmosphere. Academic press, New York

    Google Scholar 

  7. 7.

    McDonald JE (1960) Direct absorption of solar radiation by atmospheric water vapor. J Metrol 17:319–328

    Article  Google Scholar 

  8. 8.

    Lacis AL, Hansen JE (1974) A parameterization for absorption of solar radiation in the earth’s atmosphere. J Atmospheric Sci 31:118–133

    Article  Google Scholar 

  9. 9.

    Davies JA, Hay JE (1979) Calculation of the solar radiation incident on a horizontal surface. In: Proceedings, first Canadian solar radiation data workshop, 17–19 April 1978. Canadian Atmospheric Environment Service

  10. 10.

    Hoyt DV (1978) A model for the calculation of solar global insolation. Sol Energy 21:27–35

    Article  Google Scholar 

  11. 11.

    Bemporad A (1904) Zur Theorie der Extinktion des Lichtes in der Erd-atmosphare Mitteilungen der Grossherzoglichen Sternmwarte Zu Heidel-berg; No. 4

  12. 12.

    Kasten F (1964) A new table and approximation formula for the relative optical air mass. Thechnical Report 136, Hanover, New Hampshire: U. S. Army Material Command, Cold Region Research and Engineering Laboratory

  13. 13.

    Page JK (1964) The estimation of monthly mean values of daily total short—wave radiation on vertical and inclined surfaces from sunshine records for latitude 40°N–40°S. Sol Energy 10:119

    Google Scholar 

  14. 14.

    Bird RE, Hulstrom RE (1980) Direct insolation models. SERI/TR-335–344. Solar Energy Research Institute, Golden

    Google Scholar 

  15. 15.

    Bird RE, Hulstrom RE (1981) Review evaluation and improvement of direct irradiance models. J Sol Energy Eng 103:183

    Article  Google Scholar 

  16. 16.

    Bird RE, Hulstrom RE (1981) A simplified clear sky model for direct and diffuse insolation on horizontal surface, U. S. Solar Energy Research Institute (SERI), Technical Report TR-642-761, Golden, Colorado

  17. 17.

    Ashjaee M, Roomina MR, Ghafouri-Azar R (1993) Estimating direct, diffuse, and global solar radiation for various cities in Iran by two methods and their comparison with the measured data. Sol Energy 50(5):441–446

    Article  Google Scholar 

  18. 18.

    Barbaro S, Coppolino S, Leone C, Sinagra E (1979) An atmospheric model for computing direct and diffuse solar radiation. Sol Energy 22:225–228

    Article  Google Scholar 

  19. 19.

    Saffaripour MH (2009) Study of influencing parameters and developing meteorological models for solar energy gain in a dry and hot region of Iran including five provinces. Ph. D. dissertation, Department of Mechanicl Engineering, Shahid Bahonar University of Kerman, May 2009

  20. 20.

    Pisimanis D, Notaridou V (1987) Estimating direct, diffuse and global solar radiation on an arbitrary inclined plane in Greece. Sol Energy 39:159–172

    Article  Google Scholar 

  21. 21.

    Daneshyar M (1978) Solar radiation statistics for Iran. Sol Energy 21:345–349

    Article  Google Scholar 

  22. 22.

    Sabziparvar AA (2008) A simple formula for estimating global solar radiation in central arid deserts of Iran. Renew Energy 33(5):1002–1010

    Article  Google Scholar 

  23. 23.

    Bahadorinejad M, Mirhosseini SA (2004) Clearness index data for various cities in Iran. Presented at the third conference on optimization of fuel consumption in building, persion volume, pp 603–619

Download references

Acknowledgments

The authors would like to express their gratitude to the Iranian Meteorological Organization (IMO) for their sincere cooperation in providing the files and documents available in their archive containing meteorological information regarding Kerman airport station. Without this information this research would have not been successful. The authors also extend their appreciation to Mr. Pouyan Talbizadeh, the graduate student at SBUK, for his sincere support and encouragement.

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Correspondence to M. A. Mehrabian.

Appendices

Appendix A

$$ H_{0} = \frac{{24I_{0} E_{0} }}{\pi }(\cos \varphi \cos \delta \sin \omega_{s} + \frac{{\pi \omega_{s} }}{180}\sin \varphi \sin \delta ) $$
(25)
$$ E_{0} = \left( {1 + 0.033\cos \frac{{360n^{ * } }}{365}} \right) $$
(26)
$$ \omega_{s} = \cos^{ - 1} ( - \tan \varphi \tan \delta ) $$
(27)
$$ \delta = 23.45\sin \left( {360\frac{{284 + n^{*} }}{365}} \right) $$
(28)
$$ N = \frac{{2\omega_{s} }}{15} $$
(29)

Appendix B

$$ T_{M} = 1.041 - 0.15[m(9.368*10^{ - 4} P + 0.051)]^{\frac{1}{2}} $$
(30)
$$ \frac{P}{{P_{0} }} = \exp \left( {\frac{h}{1000( - 0.174 - 0.0000017h)}} \right) $$
(31)
$$ T_{A} = \exp ( - \tau_{A}^{0.873} (1 + \tau_{A} - \tau_{A}^{0.7088} )m^{0.9108} ) $$
(32)
$$ a_{\text{w}} = 2.4959mU_{\text{w}} [(1.0 + 79.03mU_{\text{w}} )^{0.6824} + 6.385mU_{\text{w}} ]^{ - 1} $$
(33)
$$ T_{\text{w}} = 1 - a_{\text{w}} $$
(34)
$$ m = 1/[\cos \theta + 0.15(93.885 - \theta )^{ - 1.253} ] $$
(35)
$$ \tau_{\text{A}} = 0.2758\tau_{{{\text{A}}(0.38)}} + 0.35\tau_{{{\text{A}}(0.50)}} $$
(36)
$$ T_{\text{UM}} = \exp ( - 0.127m^{0.26} ) $$
(37)
$$ T_{\text{AA}} = 1 - 0.1(1 - T_{\text{A}} )(1 - m + m^{1.06} )) $$
(38)
$$ T_{R} = \exp ( - 0.093m^{0.84} ) $$
(39)
$$ T_{\text{AS}} = T_{\text{A}} /T_{\text{AA}} $$
(40)
$$ T_{0} = 1 - 0.161X_{0} (1 + 139.48X_{0} )^{ - 0.3035} - \frac{{0.00271X_{0} }}{{1.0 + 0.0044X_{0} + 0.0003X_{0}^{2} }} $$
(41)
$$ r_{\text{s}} = 0.0685 + (1 - B_{a} )(1 - T_{\text{as}} ) $$
(42)
$$ T_{\text{as}} = 10^{{ - 0.045[(P/P_{0} )m]^{0.7} }} $$
(43)
$$ r_{\text{g}} = 0.2 $$
(44)
$$ B_{\text{a}} = 0.84 $$
(45)
$$ K^{*} = 0.32 $$
(46)
$$ \tau_{{{\text{A}}(0.38\mu {\text{m}})}} = 0.35 $$
(47)
$$ \tau_{{{\text{A}}(0.50\mu {\text{m}})}} = 0.27 $$
(48)
$$ X_{0} = 0.3m $$
(49)
$$ \cos \theta = \cos \phi \cos \delta \cos \omega + \sin \phi \sin \delta $$
(50)

Appendix C

$$ e = \frac{{H_{i,m} - H_{i,c} }}{{H_{i,m} }} \times 100 $$
(51)
$$ {\text{RMSE}} = \left[ {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {H_{i,m} - H_{i,c} } \right)^{2} } } \right]^{1/2} $$
(52)
$$ {\text{MAPE}} = \frac{100}{n}\sum\limits_{i = 1}^{n} {\left| {\frac{{H_{i,m} - H_{i,c} }}{{H_{i,m} }}} \right|} $$
(53)

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Safaripour, M.H., Mehrabian, M.A. Predicting the direct, diffuse, and global solar radiation on a horizontal surface and comparing with real data. Heat Mass Transfer 47, 1537 (2011). https://doi.org/10.1007/s00231-011-0814-8

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Keywords

  • Diffuse Radiation
  • Global Solar Radiation
  • Solar Radiation Intensity
  • Page Model
  • Diffuse Solar Radiation