Evaluation of global solar radiation models for Konya, Turkey

Energy Conversion and Management 47 (2006) 3149–3173

www.elsevier.com/locate/enconman

Evaluation of global solar radiationmodels for Konya, Turkey

Hakan Okyay Menges a, Can Ertekin b,*, Mehmet Hakan Sonmete a

a Selcuk University, Faculty of Agricultural Engineering, Department of Farm Machinery, Konya, Turkeyb Akdeniz University, Faculty of Agricultural Engineering, Department of Farm Machinery, 07070 Antalya, Turkey

Received 10 August 2005; accepted 15 February 2006Available online 29 March 2006

Abstract

Solar radiation data are required by solar engineers, architects, agriculturists and hydrologists for many applicationssuch as solar heating, cooking, drying and interior illumination of buildings. In order to achieve this, several empiricalmodels have been developed to predicted the solar radiation all over the world.

The main objective of this study is to review the global solar radiation models available in the literature. In order toevaluate the applicability of 50 models for computing the monthly average daily global radiation on a horizontal surface,the geographical and meteorological data of Konya, Turkey (37�52 0N latitude, 32�29 0E longitude) was used. The modelswere compared on the basis of statistical error tests such as the percentage error (e), mean percentage error (MPE), rootmean square error (RMSE), mean bias error (MBE), regression coefficient (R) and Nash–Sutcliffe equation (NSE).According to the results, the Ertekin and Yaldiz model showed the best estimation of the global solar radiation on a hor-izontal surface for Konya, Turkey, by means of the MPE (0.004266%), RMSE (0.022576 MJ/m2), MBE (0.000000 MJ/m2),R (0.999993) and NSE (0.999985) statistical tests: � �

0196-8

doi:10.

* CoE-m

H ¼ 20:296019� 0:096134H 0 þ 0:317593d� 0:146422RHþ 10:705159SS0

� 0:288332T þ 0:021331TS

þ 0:359791C þ 0:207588P � 0:076444E

� 2006 Elsevier Ltd. All rights reserved.

Keywords: Solar energy; Global solar radiation; Solar radiation models; Model comparison; Turkey

1. Introduction

Knowledge of local solar radiation is essential for many applications, including architectural design, solarenergy systems and particularly for design methods, crop growth models and evapotranspiration estimates in

904/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

1016/j.enconman.2006.02.015

rresponding author. Tel.: +90 242 310248/3102411; fax: +90 242 2274564.ail address: [email protected] (C. Ertekin).

mailto:[email protected]

3150 H.O. Menges et al. / Energy Conversion and Management 47 (2006) 3149–3173

the design of irrigation systems [1–7]. Unfortunately, for many developing countries, solar radiation measure-ments are not easily available due to the cost and maintenance and calibration requirements of the measuringequipment. Therefore, it is important to elaborate methods to estimate the solar radiation based on readilyavailable meteorological data.

Several empirical models have been developed to calculate global solar radiation using various parameters.These parameters include extraterrestrial radiation, sunshine hours, mean temperature, maximum tempera-ture, soil temperature, relative humidity, number of rainy days, altitude, latitude, total precipitable water,albedo, cloudiness and evaporation [8–48].

Solar energy technologies offer a clean, renewable and domestic energy source and are essential componentsof a sustainable energy future [49,50]. Turkey is a high insolation country in that the number of sunshine hoursamounts to almost 2640 h per year. Average daily solar energy density is 3.6 kW h/m2. The total gross solarenergy potential of Turkey is about 8.8 mtoe. The yearly total solar radiation varies between 1120 kW h/m2

year in the Black Sea Region with 1971 h of sunshine annually and 1460 kW h/m2 year in the Southeast Ana-tolia with 2993 h of sunshine annually. The most common application of solar energy is solar water heaters inTurkey. The solar water heaters mounted were about 7.5 million m2 and the total energy production related tothis amount was 290 thousand toe in 2001. The solar energy usage is projected to be 431 thousand toe in 2010and 828 thousand toe in 2020 [51–53].

The objective of this study was to validate several models to predict the monthly average daily global radi-ation on a horizontal surface against an independent data set for Konya (Turkey) and, thus, to select the mostaccurate model.

2. Models and data

2.1. Angstrom–Prescott–Page model (model 1)

The Angstrom–Prescott–Page model is the most commonly used model as given by

HH 0

¼ aþ bSS0

� �ð1Þ

where H is the monthly average daily global radiation, H0 is the monthly average daily extraterrestrial radi-ation, S is the day length, S0 is the maximum possible sunshine duration, and a and b are empirical coefficients[54–57].

The monthly average daily extraterrestrial solar radiation on a horizontal surface was calculated from thefollowing equation [57]:

H 0 ¼24

pIgsf cos k cos d sin ws þ

p180

ws sin k sin dh i

ð2Þ

where Igs is the solar constant (=1367 W/m2), f is the eccentricity correction factor, k is the latitude of the site,d is the solar declination and ws is the mean sunrise hour angle for the given month. The eccentricity correctionfactor, solar declination and sunrise hour angle can be computed by Eqs. (3)–(5), respectively [57]:

f ¼ 1þ 0:033 cos360n365

� �� ð3Þ

d ¼ 23:45 sin360 284þ nð Þ

365

� �ð4Þ

ws ¼ cos�1ð� tan k tan dÞ ð5Þ

where n is the number of the day of year starting from the first of January. For a given month, the maximumpossible sunshine duration (S0) can be calculated using the following equation [57]:

S0 ¼2

15ws ð6Þ

H.O. Menges et al. / Energy Conversion and Management 47 (2006) 3149–3173 3151

2.2. Tiris et al. model (model 2)

Tiris et al. gave the a and b empirical coefficients of the Angstrom–Prescott–Page model for Turkey, in gen-eral, as follows [58]:

HH 0

¼ 0:18þ 0:62SS0

� �ð7Þ

2.3. Page model (model 3)

Page has given the coefficients of the Angstrom–Prescott–Page model, which is believed to be applicableanywhere in the world, as the following [56]:

HH 0

¼ 0:23þ 0:48SS0

� �ð8Þ

2.4. Bahel et al. model (model 4)

Bahel et al. suggested the following relationship ([59] from [43]):

HH 0

¼ 0:175þ 0:552SS0

� �ð9Þ

2.5. Louche et al. model (model 5)

Louche et al. presented the model below to predict global solar radiation [60]:

HH 0

¼ 0:206þ 0:546SS0

� �ð10Þ

2.6. Benson et al. model (model 6)

Benson et al. proposed two different formulations for two intervals of a year [61];

HH 0

¼ 0:18þ 0:60SS0

� �ð11Þ

for the interval of January–March and October–December and

HH 0

¼ 0:24þ 0:53SS0

� �ð12Þ

for the interval of April–September.

2.7. Monthly specific Dogniaux and Lemoine model (model 7)

Dogniaux and Lemoine obtained the following specific monthly correlations [62]:

JanuaryHH 0

¼ ð�0:00301kþ 0:34507Þ þ ð0:00495kþ 0:34572Þ SS0

� �ð13Þ

FebruaryHH 0

¼ ð�0:00255kþ 0:33459Þ þ ð0:00457kþ 0:35533Þ SS0

� �ð14Þ


MarchHH 0

¼ ð�0:00303kþ 0:36690Þ þ ð0:00466kþ 0:36377Þ SS0

� �ð15Þ

AprilHH 0

¼ ð�0:00334kþ 0:38557Þ þ ð0:00456kþ 0:35802Þ SS0

� �ð16Þ

MayHH 0

¼ ð�0:00245kþ 0:35057Þ þ ð0:00485kþ 0:33550Þ SS0

� �ð17Þ

JuneHH 0

¼ ð�0:00327kþ 0:39890Þ þ ð0:00578kþ 0:27292Þ SS0

� �ð18Þ

JulyHH 0

¼ ð�0:00369kþ 0:41234Þ þ ð0:00568kþ 0:27004Þ SS0

� �ð19Þ

AugustHH 0

¼ ð�0:00269kþ 0:36243Þ þ ð0:00412kþ 0:33162Þ SS0

� �ð20Þ

SeptemberHH 0

¼ ð�0:00338kþ 0:39467Þ þ ð0:00564kþ 0:27125Þ SS0

� �ð21Þ

OctoberHH 0

¼ ð�0:00317kþ 0:36213Þ þ ð0:00504kþ 0:31790Þ SS0

� �ð22Þ

NovemberHH 0

¼ ð�0:00350kþ 0:36680Þ þ ð0:00523kþ 0:31467Þ SS0

� �ð23Þ

DecemberHH 0

¼ ð�0:00350kþ 0:36262Þ þ ð0:00559kþ 0:30675Þ SS0

� �ð24Þ

2.8. Specific monthly Rietveld model (model 8)

Soler applied Rietveld’s model to 100 European stations and gave the following specific monthly correla-tions [63]:

JanuaryHH 0

¼ 0:18þ 0:66SS0

� �ð25Þ

FebruaryHH 0

¼ 0:20þ 0:60SS0

� �ð26Þ

MarchHH 0

¼ 0:22þ 0:58SS0

� �ð27Þ

AprilHH 0

¼ 0:20þ 0:62SS0

� �ð28Þ

MayHH 0

¼ 0:24þ 0:52SS0

� �ð29Þ

JuneHH 0

¼ 0:24þ 0:53SS0

� �ð30Þ

JulyHH 0

¼ 0:23þ 0:53SS0

� �ð31Þ

AugustHH 0

¼ 0:22þ 0:55SS0

� �ð32Þ

SeptemberHH 0

¼ 0:20þ 0:59SS0

� �ð33Þ


OctoberHH 0

¼ 0:19þ 0:60SS0

� �ð34Þ

NovemberHH 0

¼ 0:17þ 0:66SS0

� �ð35Þ

DecemberHH 0

¼ 0:18þ 0:65SS0

� �ð36Þ

2.9. Gopinathan model (model 9)

Gopinathan suggested use of the Angstrom–Prescott–Page coefficients:

a ¼ 0:265þ 0:07Z � 0:135SS0

� �ð37Þ

b ¼ 0:401� 0:108Z þ 0:325SS0

� �ð38Þ

for estimation of the global solar radiation, where Z is the altitude in kilometers [64].

2.10. Zabara model (model 10)

Zabara correlated monthly a and b values of the Angstrom–Prescott–Page model with monthly relativesunshine duration (S/S0) as a third order function and expressed the a and b coefficients as [65]

a ¼ 0:395� 1:247SS0

� �þ 2:680

SS0

� �2

� 1:674SS0

� �3

ð39Þ

b ¼ 0:395þ 1:384SS0

� �� 3:249

SS0

� �2

þ 2:055SS0

� �3

ð40Þ

2.11. Kilic and Ozturk model (model 11)

Kilic and Ozturk calculated the a and b empirical coefficients of the Angstrom–Prescott–Page model forTurkey as follows [66]:

a ¼ 0:103þ 0:000017Z þ 0:198 cosðk� dÞ ð41Þb ¼ 0:533� 0:165 cosðk� dÞ ð42Þ

where Z is the altitude of the site.

2.12. Gariepy’s model (model 12)

Gariepy has reported that the empirical coefficients a and b are dependent on mean air temperature (T) andthe amount of precipitation (P) [67]:

a ¼ 0:3791� 0:0041T � 0:0176P ð43Þb ¼ 0:4810þ 0:0043T þ 0:0097P ð44Þ

2.13. Rietveld model (model 13)

Rietveld examined several published values of the a and b coefficients and noted that a is related linearlyand b hyberbolically to the appropriate mean value of S/S0 such that [68]:


a ¼ 0:10þ 0:24SS0

� �ð45Þ

b ¼ 0:38þ 0:08SS0

� �ð46Þ


Gopinathan has given the following correlation:

HH 0

¼ �0:309þ 0:539 cos k� 0:0693Z þ 0:290SS0

� ��

þ 1:527� 1:027 cos kþ 0:0926Z � 0:359SS0

� �� SS0

� �ð47Þ

where Z is the altitude in kilometers [69].

2.15. Togrul et al. model (model 15)

Togrul et al. proposed the following correlations, where the coefficients of the Angstrom–Prescott–Pagemodel seem to be a function of the sunshine duration ratio [13]:

ðaÞ a ¼ iþ jSS0

� �b ¼ k þ l

SS0

� �ð48Þ

ðbÞ a ¼ i lnSS0

� �þ j b ¼ k ln

SS0

� �þ m ð49Þ

ðcÞ a ¼ iþ jSS0

� �þ k

SS0

� �2

b ¼ lþ mSS0

� �þ n

SS0

� �2

ð50Þ

ðdÞ a ¼ iþ jSS0

� �þ k

SS0

� �2

þ lSS0

� �3

b ¼ mþ nSS0

� �þ p

SS0

� �2

þ rSS0

� �3

ð51Þ

ðeÞ a ¼ iþ jSS0

� �þ k

SS0

� �2

þ lSS0

� �3

þ mSS0

� �4

b ¼ nþ pSS0

� �þ r

SS0

� �2

þ tSS0

� �3

þ vSS0

� �4

ð52Þ

2.16. Almorox and Hontoria model (model 16)

Almorox and Hontoria proposed the following exponential model [2]:

HH 0

¼ aþ b expSS0

� �ð53Þ

2.17. Ampratwum and Dorvlo model (model 17)

Ampratwum and Dorvlo proposed the following model [70]:

HH 0

¼ aþ b logSS0

� �ð54Þ


2.18. Ogelman et al. model (model 18)

Ogelman et al. have correlated (H/H0) with (S/S0) in the form of a second order polynomial equation[33]:

ðaÞ HH 0

¼ aþ bSS0

� �þ c

SS0

� �2

ð55Þ

ðbÞ HH 0

¼ 0:195þ 0:676SS0

� �� 0:142

SS0

� �2

ð56Þ

where a, b and c are empirical coefficients.

2.19. Akinoglu and Ecevit model (model 19)

Akinoglu and Ecevit obtained the correlation below between (H/H0) and (S/S0) in a second order polyno-mial equation for Turkey [42]:

HH 0

¼ 0:145þ 0:845SS0

� �� 0:280

SS0

� �2

ð57Þ

2.20. Samuel model (model 20)

The ratio of global to extraterrestrial radiation was expressed by a function of the ratio of sunshine dura-tion as follows [71]:

ðaÞ HH 0

¼ aþ bSS0

� �þ c

SS0

� �2

þ dSS0

� �3

ð58Þ

ðbÞ HH 0

¼ 0:14þ 2:52SS0

� �� 3:71

SS0

� �2

þ 2:24SS0

� �3

ð59Þ

where a, b, c and d are empirical coefficients.

2.21. Bahel model (model 21)

Bahel developed a worldwide correlation based on bright sunshine hours and global radiation data of 48stations around the world, with varied meteorological conditions and a wide distribution of geographic loca-tions [34]:

HH 0

¼ 0:16þ 0:87SS0

� �� 0:16

SS0

� �2

þ 0:34SS0

� �3

ð60Þ

2.22. Glower and McCulloch model (model 22)

Glower and McCulloch proposed the following equation, which depends on the latitude of the site and isvalid for k < 60� [72]:

ðaÞ HH 0

¼ a cos kþ bSS0

� �ð61Þ

ðbÞ H ¼ H 0 0:29 cos kþ 0:52SS0

� �� ð62Þ


2.23. Raja and Twidell model (model 23)

Raja and Twidell offered the following models [39,40]:

ðaÞ H ¼ H 0 0:388 cos kþ 0:367SS0

� �� ð63Þ

ðbÞ H ¼ H 0 0:388 cos kþ 0:407SS0

� �� ð64Þ

2.24. Newland model (model 24)

One of the most interesting models is that of Newland, including a logarithmic term, as follows [73]:

ðaÞ H ¼ H 0 aþ bSS0

� �þ c log

SS0

� �� ð65Þ

ðbÞ H ¼ H 0 0:34þ 0:40SS0

� �þ 0:17 log

SS0

� �� ð66Þ

2.25. Allen model (model 25)

Allen suggested a self calibrating model that is a function of the daily extraterrestrial radiation, meanmonthly maximum and minimum temperatures [74]:

HH 0

¼ a T max � T minð Þ0:5 ð67Þ

2.26. Hargreaves model (model 26)

Hargreaves et al. reported a simple method to estimate the global solar radiation that has an expression likethe Allen model [75]:

HH 0

¼ a T max � T minð Þ0:5 þ b ð68Þ

2.27. Bristow–Champbell model (model 27)

Bristow and Champbell suggested the following relationship for daily global solar radiation as a functionof daily extraterrestrial radiation and the difference between the maximum and minimum air temperatures[76]:

HH 0

¼ a 1� expð�bDT cÞ½ � ð69Þ


2.28. Chen et al. model (model 28)

Chen et al. presented the model below [17]:

ðaÞ HH 0

¼ a lnðT max � T minÞ þ b ð70Þ

ðbÞ HH 0

¼ a lnðT max � T minÞ þ bSS0

� �c

þ d ð71Þ


2.29. Louche et al. model (model 29)

The Angstrom–Prescott–Page model has been modified through the use of the ratio of (S/Snh) instead of(S/S0), where Snh is the sunshine duration, so as to take into account the natural horizon of the site. It canbe calculated as follows [60]:

HH 0

¼ aþ bS

Snh

� �ð72Þ

1

Snh¼ 0:8706

S0

þ 0:0003 ð73Þ

where a and b are empirical coefficients.

2.30. Raja model (model 30)

Based on Bennett’s formula, the following insolation-sunshine relation is proposed [37,77]:

HH 0

¼ 0:368� 0:125 1� ZZa

� �� þ 0:667� 0:018 1� Z

Za

� �� 0:211 cos k

� �S

S04

� �ð74Þ

where Za = 800 m. S04 is the 4� corrected day length used to compensate the finite threshold of the Champ-bell–Stokes sunshine recorder and is calculated as follows:

S04 ¼2

15cos�1 ðsin 4� � sin k sin dÞ

cos k cos d

� �ð75Þ

2.31. Abdalla model (model 31)

Abdalla modified the Gopinathan model for Bahrain as [78]

ðaÞ HH 0

¼ aþ bSS0

� �þ cT þ dRH ð76Þ

ðbÞ HH 0

¼ aþ bSS0

� �þ cT þ dRHþ ePS ð77Þ

where PS is the ratio between mean sea level pressure and mean daily vapour pressure.

2.32. Elagib and Mansell model (model 32)

Elagib and Mansell suggested the following models [25]:

ðaÞ HH 0

¼ a exp bSS0

� �� ð78Þ

ðbÞ HH 0

¼ aþ bSS0

� �n

ð79Þ

ðcÞ HH 0

¼ aþ bkþ cZ þ dSS0

� �ð80Þ

ðdÞ HH 0

¼ aþ bkþ cSS0

� �ð81Þ

ðeÞ HH 0

¼ aþ bZ þ cSS0

� �ð82Þ


2.33. Trabe and Shaltout model (model 33)

Trabe and Shaltout suggested the following model [48]:

HH 0

¼ aþ bSS0

� �þ cT þ dV þ eRHþ fP ð83Þ

where V is the water vapor pressure (in h Pa).


Gopinathan introduced a multiple linear regression equation of the form [69]:

HH 0

¼ aþ b cos kþ cZ þ dSS0

� �þ eT þ f RH ð84Þ

2.35. Dogniaux and Lemoine model (model 35)

Dogniaux and Lemoine proposed the following correlation, where the coefficients of the Angtrom–Pres-cott–Page model seem to be a function of the latitude of the site [62]:

HH 0

¼ 0:37022þ 0:00506SS0

� �� 0:00313

� �kþ 0:32029

SS0

� �ð85Þ

2.36. Ojosu and Komolafe model (model 36)

Ojosu and Komolafe proposed the following equation:

HH 0

¼ aþ bSS0

� �þ c

T min

T max

� �þ d

RH

RHmax

� �ð86Þ

where Tmin is the mean minimum air temperature, Tmax is the mean maximum air temperature, RHmax is themaximum relative humidity and a–d are empirical coefficients [79].

2.37. Ododo et al. model (model 37)

Ododo et al. proposed two new models as follows:

ðaÞ HH 0

¼ aSS0

� �b

T cmaxRHd ð87Þ

ðbÞ HH 0

¼ eþ fSS0

� �þ gT max þ hRHþ iT max

SS0

� �ð88Þ

where a–i are empirical coefficients [41].

2.38. Garg and Garg model (model 38)

Garg and Garg proposed the model below:

HH 0

¼ aþ bdþ cW ð89Þ

W ¼ 0:0049RHexp 26:23� 5416

T K

� �T K

24

35 ð90Þ


where W is the atmospheric precipitable water vapor per unit volume of air (cm), a, b and c are coefficients tobe determined and both RH and TK are the monthly daily means of both humidity (%) and air temperature(K) [80].

2.39. Jin et al. model (model 39)

Jin et al. proposed the following models:

ðaÞ HH 0

¼ aþ bCoskþ cZ þ dSS0

� �ð91Þ

ðbÞ HH 0

¼ ðaþ bkþ cZÞ þ ðd þ ekþ fZÞ SS0

� �ð92Þ

ðcÞ HH 0

¼ ðaþ b cos kþ cZÞ þ ðd þ e cos kþ fZÞ SS0

� �ð93Þ

ðdÞ HH 0

¼ ðaþ bkþ cZÞ þ ðd þ ekþ fZÞ SS0

� �þ ðg þ hkþ jZÞ S

S0

� �2

ð94Þ

ðeÞ HH 0

¼ ðaþ b cos kþ cZÞ þ ðd þ e cos kþ fZÞ SS0

� �þ ðg þ h cos kþ jZÞ S

S0

� �2

ð95Þ

where a–j are empirical constants [12].

2.40. Klabzuba et al. model (model 40)

Klabzuba et al. suggested the model below to predict global radiation:

HH 0

¼ aþ bSS0

� �þ c d þ S

S0

� �� n� eð Þ2 ð96Þ

where n is the day of the year starting from the first of January [81].

2.41. El-Metwally model (model 41)

A non-linear equation has been used as the following [18]:
HH 0
¼ að1=SÞ ð97Þ

2.42. Badescu model (model 42)

Badescu suggested the following models [82]:

ðaÞ HH 0

¼ aþ bC ð98Þ

ðbÞ HH 0

¼ aþ bC þ cC2 ð99Þ

ðcÞ HH 0

¼ aþ bC þ cC2 þ dC3 ð100Þ

2.43. Black model (model 43)

Black, using data from many parts of the world, proposed the following quadratic equation:
HH 0
¼ 0:803� 0:340C � 0:458C2 ðC 6 0:8Þ ð101Þ

where C is the monthly average fraction of the daytime sky obscured by clouds [83].


2.44. Supit and Kappel model (model 44)

Supit and Kappel proposed the following model:

H ¼ H 0 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT max � T minð Þ

pþ b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� C

8

� �s" #þ c ð102Þ

where C is the mean total cloud cover during the daytime observations [45].

2.45. Lewis model (model 45)

The global radiation on a horizontal surface can be calculated by the following equations [84]:

ðaÞ H ¼ eRHf ; log H ¼ eþ f log RH ð103Þ

ðbÞ H ¼ gShRHi; log H ¼ g þ h log S þ i log RH ð104ÞðcÞ H ¼ j expðkSÞ; ln H ¼ jþ kS ð105ÞðdÞ H ¼ l expðmRHÞ; ln H ¼ lþ mRH ð106ÞðeÞ H ¼ n exp rðS �RHÞ; ln H ¼ nþ rðS �RHÞ ð107Þ

where RH is the mean relative humidity and e–r are empirical coefficients.

2.46. El-Metwally model (model 46)

The proposed models are the following:

ðaÞ H ¼ aH 0 þ bT max þ cT min þ dC þ e ð108ÞðbÞ H ¼ aH 0 þ bT max þ cT min þ dC ð109ÞðcÞ H ¼ expðaH 0 þ bT max þ cT min þ dC þ eÞ ð110Þ

where C is the daily mean of cloud cover [85].

2.47. Swartman and Ogunlade model (model 47)

Swartman and Ogunlade stated that the global radiation can be expressed as a function of (S/S0) and meanrelative humidity (RH) [86];

ðaÞ H ¼ aSS0

� �b

RHc ð111Þ

ðbÞ H ¼ aþ bSS0

� �þ cRH ð112Þ


2.48. Ertekin and Yaldiz model (model 48)

Ertekin and Yaldiz estimated the monthly average daily global radiation by a multiple linear regression modelbased on nine variables: the extraterrestrial radiation, solar declination, relative humidity, ratio of sunshineduration, mean air temperature, soil temperature, cloudiness, precipitation and evaporation as follows [14]:

H ¼ aþ bH 0 þ cdþ dRHþ eSS0

� �þ fT þ gTS þ hC þ iP þ jE ð113Þ

where a–j are empirical coefficients.


2.49. Togrul and Onat model (model 49)

Togrul and Onat investigated the effect of geographical and meteorological parameters on global radiationand gave the following models [27]:

ðaÞ H ¼ aþ bSS0

� �þ c sin dþ dT ð114Þ

ðbÞ H ¼ aþ b sin dþ cSS0

� �þ dRHþ eT ð115Þ

ðcÞ H ¼ aþ bH 0 þ cSS0

� �þ dRHþ eTS ð116Þ

ðdÞ H ¼ aþ bH 0 þ cSS0

� �þ d sin dþ eRHþ fT ð117Þ

ðeÞ H ¼ aþ bH 0 þ cSS0

� �þ dRHþ eTS þ fT ð118Þ

ðfÞ H ¼ aþ bH 0 þ cSS0

� �þ d sin dþ eRHþ fTS þ gT ð119Þ

2.50. Chen et al. model (model 50)

Chen et al. presented the following models [27]:

ðaÞ H ¼ aþ bSS0

� �þ c sin dþ dT max ð120Þ

ðbÞ H ¼ aþ bH 0 þ cSS0

� �þ d sin dþ f RHþ gT max ð121Þ

ðcÞ H ¼ aþ bH 0 þ cSS0

� �þ dRHþ fTS þ gT max ð122Þ

ðdÞ H ¼ aþ bH 0 þ cSS0

� �þ d sin dþ eRHþ fTS þ gT max ð123Þ

2.51. Data and methods of comparison

The location where the measurements have been performed (Konya Meteorology Station) is situated1031 m above sea level at a latitude of 35�52 0N and longitude of 32�29 0E. The global radiation and meteoro-logical data reported in the paper are part of the data measured at the meteorological station of Konya. Therecord times of the parameters used in this study are given in Table 1. The performance of the models wasevaluated on the basis of the following statistical error tests: the mean percentage error (MPE), mean biaserror (MBE) and root mean square error (RMSE). These tests are the ones that are applied most commonlyin comparing the models of solar radiation estimations [4,16,25,26,29,43–45,60,63,85,87–97]. MPE, MBE andRMSE are defined as below:

MPE ¼Xn

i¼1

Hi;m � H i;cð Þ=H i;m½ �100

Nð124Þ

MBE ¼PN

i¼1H i;c � H i;m

Nð125Þ

RMSE ¼PN

i¼1H i;c � H i;m

N

!1=2

ð126Þ

Table 1Record time of some collected parameters in the Meteorological Station of Konya, Turkey

Parameters Record time (years)

Global solar radiation 31Day length 31Mean air temperature 73Mean precipitation 73Mean relative humidity 53Mean minimum air temperature 73Mean maximum air temperature 73Mean maximum relative humidity 53Mean soil temperature 31Mean cloudiness 31Mean evaporation 31


where Hi,m is the ith measured value, Hi,c is the ith calculated value and N is the total number of observations.Also, the percentage error (e) is the calculated-measured variation of an individual month and is defined asfollows [3,26,30,31,36,38,88,96,98–100]:

e ¼ H i;m � Hi;cð Þ=H i;m½ �100 ð127Þ

A relative percentage error between �10% and +10% is considered acceptable. The mean percentage errorcan be defined as the percentage deviation of the calculated and measured monthly average daily globalsolar radiation. The mean bias error test provides information on the long term performance. A lowMBE is desired. A positive value gives the average amount of over estimation of an individual observation,which will cancel an under-estimation in a separate observation. The root mean square error gives informa-tion on the short term performance of the correlations by allowing a term by term comparison of the actualdeviation between the calculated and measured values. The smaller the value, the better is the model’s per-formance. However, a few large errors in the sum can produce a significant increase in the RMSE[9,28,34,101].

We used the correlation coefficient, R, to test the linear relationship between calculated and measured val-ues, which is defined by

R ¼PN

i¼1ðH i;c � HcÞðH i;m � HmÞPNi¼1ðH i;c � HcÞ2

h i PNi¼1ðH i;m � HmcÞ2

h in o1=2ð128Þ

where Hc is the mean calculated global radiation and H m is the mean measured global radiation [102].The Nash–Sutcliffe equation is also selected as an evaluation criterion:

NSE ¼ 1�PN

i¼1ðH i;m � Hi;cÞ2PNi¼1ðH i;m � HmÞ

ð129Þ

A model is more efficient when NSE is closer to 1 [17].

3. Results and discussion

Solar radiation data are essential in the design and study of solar energy conversion devices. Empiricalmodels to estimate global solar radiation are a suitable tool if the parameters can be calibrated for differ-ent locations. These models have the advantage of using meteorological data that are commonlyavailable.

Validation of these 50 models has been performed by using the percentage error, MPE, MBE, RMSE, R

and also the NSE and the results are given in Table 2, where all the empirical coefficients are presented foreach model at the end of the column for that model. According to the results, Model 48, the Ertekin and

Table 2Statistical results of different models

e

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

January 0.033909 13.983532 12.965400 1.356288 13.542023 15.448856February 2.782433 7.379568 9.666723 �10.490326 8.676562 9.174127March 2.969852 1.630769 6.077081 �20.761700 4.064961 3.665928April �0.521403 �2.999175 1.998779 �27.128505 �0.271066 �5.397859May �1.701278 �9.822328 �2.714185 �39.746933 �5.981526 �10.689450June 0.121548 �14.079579 �4.825275 �51.234904 �9.109980 �13.202898July 0.202561 �18.111753 �7.360726 �61.680442 �12.353801 �16.092732August �0.634880 �20.425192 �9.102045 �66.694932 �14.365068 �18.023096September 0.861512 �16.919276 �6.391058 �59.493301 �11.279313 �15.029211October 0.632165 �9.467077 �1.728074 �41.187186 �5.296182 �7.023527November �0.369586 �0.893685 3.388637 �23.331269 1.457297 1.175889December �6.069855 9.452051 8.109675 �3.543991 8.845527 10.977236

MBE �0.004022 1.322669 0.218388 6.239890 0.726538 1.148664RMSE 0.223073 2.295267 1.025719 8.057533 1.583779 2.119578MPE �0.141085 �5.022679 0.840411 �33.661433 �1.839214 �3.751395R 0.999270 0.993053 0.996128 0.982793 0.994701 0.995058NSE 0.998537 0.845061 0.969058 �0.909407 0.926229 0.867873

a = 0.356782b = 0.266239

Model 7 Model 8 Model 9 Model 10 Model 11 Model 12

January 5.733395 11.052884 4.898502 9.518246 16.902388 �8.793244February 2.344864 5.064225 4.103426 4.922616 11.054478 �12.403594March �1.421773 �2.138755 1.008832 0.342323 5.343297 �14.080721April �5.805931 �7.028055 �3.262491 �4.057069 �1.105673 �16.046492May �10.791205 �9.484720 �8.603548 �8.911173 �5.468135 �17.554967June �12.958100 �13.202898 �12.367909 �9.829925 �5.780933 �23.754499July �15.620128 �14.336502 �16.811429 �11.211310 �6.787546 �29.670059August �17.473494 �17.389733 �19.384391 �12.634558 �7.555717 �32.197490September �14.582788 �16.289254 �15.556461 �10.333389 �4.227649 �28.487456October �9.689029 �8.896696 �7.951969 �7.538535 0.856654 �20.544275November �4.340911 �2.991894 �1.868730 �2.432586 7.460910 �16.459209December 0.459455 7.164274 �0.703237 4.485270 13.363408 �12.162291

MBE 1.430319 1.328325 1.345815 0.956074 0.147124 3.273969RMSE 2.038540 2.034350 2.085290 1.522666 1.050355 3.920387MPE �7.012137 �5.706427 �6.374950 �3.973341 2.004624 �19.346191R 0.996274 0.996412 0.993166 0.997519 0.998461 0.993611NSE 0.877783 0.878284 0.872113 0.931813 0.967554 0.547986

Model 13 Model 14 Model 15-A Model 15-B Model 15-C Model 15-D

January 30.119869 17.662895 0.695200 2.029326 1.718475 2.172734February 20.684853 5.949912 2.887497 1.369010 2.056772 �0.756684March 13.083969 �1.973617 2.864464 2.150666 2.294478 2.256811April 8.514682 �7.019993 �0.654767 �1.180959 �1.122965 �0.743461May �0.158833 �14.821771 �1.890015 �1.401467 �1.611935 �0.554363June �7.067396 �18.772381 0.037030 0.747765 0.671585 �0.163566July �12.928452 �21.828506 0.291283 0.186274 0.240511 �0.140045August �15.816550 �23.708075 �0.470219 �1.048893 �0.999922 �0.011756September �11.578155 �20.742673 0.927920 0.950237 0.998796 0.354279October �0.856949 �14.435037 0.462507 1.175856 0.974160 1.514529November 11.236605 �4.365733 �0.455822 �1.361238 �1.132480 �1.557127December 26.746446 13.835179 �5.315044 �3.279999 �3.936049 �2.298208

MBE �0.025551 1.829481 �0.008066 �0.008144 �0.008580 �0.005410RMSE 1.960409 2.878217 0.221267 0.193876 0.205837 0.144454

(continued on next page)


Table 2 (continued)

Model 13 Model 14 Model 15-A Model 15-B Model 15-C Model 15-D

MPE 5.165007 �7.518317 �0.051664 0.028048 0.012619 0.006095R 0.986639 0.994029 0.999287 0.999455 0.999385 0.999696NSE 0.886972 0.756364 0.998560 0.998895 0.998754 0.999386

i = 0.340269 i = 1.322794 i = 0.169759 i = �1.416495j = 0.161510 j = 3.827013 j = 0.650596 j = 6.745476k = 0.161510 k = 2.400207 k = �0.907525 k = �17.710444l = �0.046110 m = �3.166297 l = 0.650596 l = 20.349687

m = �0.907525 m = 6.745477n = 1.017439 n = �17.710444

p = 20.349686r = �17.026728

Model 15-E Model 16 Model 17 Model 18-A Model 18-B Model 19

January 2.280092 �1.885097 3.708218 0.695200 9.878138 12.056467February �0.596763 2.387768 2.862078 2.887497 4.835802 5.353245March 2.221519 3.198320 2.111622 2.864464 0.494395 0.387791April �0.775592 �0.226803 �1.485792 �0.654767 �3.880942 �4.062699May �0.469907 �1.185515 �2.675206 �1.890015 �8.936005 �9.275102June �0.165740 0.404105 �0.134896 0.037030 �10.744269 �10.769033July �0.202784 �0.003169 0.765218 0.291283 �12.820870 �12.323078August 0.075151 �1.065781 0.249518 �0.470219 �14.401841 �13.673420September 0.272288 0.716237 1.325620 0.927920 �11.874240 �11.445145October 1.615400 1.129701 �0.153013 0.462507 �7.802748 �8.087638November �1.588668 �0.212715 �1.176811 �0.455822 �2.299833 �2.342831December �2.324101 �8.242215 �1.758658 �5.315044 5.011408 7.480969

MBE �0.007789 0.008625 �0.022943 �0.008066 1.051908 1.005911RMSE 0.144689 0.245108 0.247888 0.221267 1.687093 1.660885MPE 0.028408 �0.415430 0.303158 �0.051664 �4.378417 �3.891706R 0.999697 0.999125 0.999155 0.999287 0.996781 0.997156NSE 0.999384 0.998233 0.998193 0.998560 0.916291 0.918871

i = �0.894570 a = 0.255516 a = 0.600627 a = 0.340269j = 4.238461 b = 0.141838 b = 0.154448 b = 0.323019k = �8.373374 c = �0.046110l = 3.433554m = 6.435634n = 4.238461p = �8.373374r = 3.433554t = 6.435634v = �10.333290

Model 20-A Model 20-B Model 21 Model 22-A Model 22-B Model 23-A

January 1.718475 �45.213483 20.093172 0.033909 10.269574 4.031243February 2.056772 �47.849667 14.508543 2.782433 6.291588 4.120342March 2.294478 �49.307263 9.530291 2.969852 2.210853 2.615164April �1.122965 �54.913433 5.327835 �0.521403 �2.097216 �1.213867May �1.611935 �57.841950 �0.654357 �1.701278 �7.329883 �3.997602June 0.671585 �57.930230 �4.251988 0.121548 �9.881041 �3.906209July 0.240511 �62.201874 �7.746583 0.202561 �12.758471 �5.000937August �0.999922 �65.623485 �9.798202 �0.634880 �14.655987 �6.260737September 0.998796 �60.561050 �6.677337 0.861512 �11.718355 �4.192670October 0.974160 �54.891382 �0.222424 0.632165 �6.420120 �2.220094November �1.132480 �54.278844 7.163335 �0.369586 �0.537987 �0.496951December �3.936049 �53.260761 15.839547 �6.069855 5.309606 �1.620962

MBE �0.008580 8.673581 �0.063684 �0.004022 0.918302 0.371177RMSE 0.205837 9.527592 1.242022 0.223073 1.630909 0.686352MPE 0.012619 �55.322785 3.592653 �0.141085 �3.443120 �1.511940


Table 2 (continued)

Model 20-A Model 20-B Model 21 Model 22-A Model 22-B Model 23-A

R 0.999385 0.997995 0.993609 0.999270 0.995638 0.998610NSE 0.998754 �1.669690 0.954632 0.998537 0.921774 0.986146

a = 0.169759 a = 0.451908b = 1.301193 b = 0.266239c = �1.815050d = 1.017439

Model 23-B Model 24-A Model 24-B Model 25 Model 26 Model 27

January 1.100595 1.118225 12.738862 2.674955 4.139156 4.332420February 0.531225 2.861474 6.812389 4.120410 5.001279 5.156154March �1.455154 2.773253 2.143463 1.770776 2.182693 2.276615April �5.519625 �0.768777 �2.228134 �5.866354 �5.754113 �5.715396May �8.816522 �1.961308 �7.593287 �4.495154 �4.507925 �4.496929June �9.151586 0.042644 �9.951146 1.088311 0.987193 0.976564July �10.581558 0.331668 �12.533650 3.596648 3.435385 3.409452August �11.996983 �0.424069 �14.308657 2.574709 2.330268 2.282880September �9.702309 0.961276 �11.530566 0.923366 0.575820 0.500422October �7.107193 0.429057 �6.648964 �2.837173 �3.009194 �3.036861November �4.636098 �0.556571 �0.548813 �3.549928 �3.201252 �3.116926December �4.671332 �4.801090 8.143622 �4.324847 �2.800770 �2.595031

MBE 1.101314 �0.010239 0.881639 0.016966 �0.000382 �0.003567RMSE 1.443224 0.219254 1.629363 0.544782 0.541482 0.540632MPE �6.000545 0.000482 �2.958740 �0.360357 �0.051788 �0.002220R 0.998270 0.999305 0.996004 0.995660 0.995679 0.995694NSE 0.938742 0.998586 0.921922 0.991271 0.991377 0.991404

a = 0.423073 a = 0.144663 a = 0.153796 a = 14.350191b = 0.194102 b = �0.033739 b = 0.009035c = 0.042385 c = 0.549104

Model 28-A Model 28-B Model 29 Model 30 Model 31-A Model 31-B

January 4.776340 1.391091 0.032126 2.989927 2.337217 0.821846February 4.757251 1.043506 2.788407 0.135854 �0.151270 1.202188March 1.746138 1.805422 2.974479 �3.135601 1.386846 0.622664April �6.039724 0.246376 �0.527344 �7.280208 5.033801 �0.809559May �4.656538 �1.823160 �1.712788 �12.167263 �0.312505 �0.241726June 0.965603 �0.417292 0.109387 �14.264064 1.017303 0.905490July 3.510461 �0.110917 0.196759 �16.853379 1.814854 0.541785August 2.546471 �0.355661 �0.627752 �18.698527 �1.856632 �0.781224September 0.989645 1.888374 0.881866 �16.033668 �3.388660 �0.773506October �2.929115 1.197017 0.656873 �11.766251 �4.772957 �0.112883November �3.637199 �0.566955 �0.348881 �7.137666 �5.472730 0.083726December �2.213670 �3.543445 �6.071094 �2.647995 �2.625059 �2.811493

MBE �0.001825 �0.013361 �0.004320 1.682009 0.033796 0.005359RMSE 0.558405 0.189710 0.223756 2.237689 0.428492 0.124552MPE �0.015361 0.062863 �0.137330 �8.904903 �0.582483 �0.112724R 0.995405 0.999491 0.999266 0.996327 0.997411 0.999775NSE 0.990829 0.998942 0.998528 0.852737 0.994600 0.999544

a = 0.264341 a = �0.125214 a = 0.357005 a = �0.107838 a = 0.441916b = �0.155612 b = 42.871362 b = 0.304034 b = 0.704958 b = 0.366762

c = 0.005054 c = �0.002523 c = �0.004300d = �41.917482 d = 0.004041 d = �0.001648

e = 0.001169

Model 32-A Model 32-B Model 32-C Model 32-D Model 32-E Model 33

January �0.993576 1.000631 0.033909 0.110379 0.362329 �0.114249February 2.568395 2.879739 2.782433 2.844733 3.004707 0.068977



Table 2 (continued)


March 3.082091 2.796388 2.969852 3.024066 3.127585 0.178003April �0.365234 �0.727756 �0.521403 �0.466674 �0.369570 �0.177000May �1.437707 �1.947929 �1.701278 �1.653463 �1.608460 �0.220229June 0.261764 0.031791 0.121548 0.160129 0.145328 0.329249July 0.100836 0.323018 0.202561 0.235581 0.181772 0.625329August �0.846260 �0.428085 �0.634880 �0.603357 �0.670111 �0.926908September 0.791537 0.956006 0.861512 0.894868 0.845366 �0.144735October 0.878921 0.421561 0.632165 0.675957 0.699298 0.064626November �0.290245 �0.522626 �0.369586 �0.312344 �0.197026 1.544126December �7.235846 �4.952055 �6.069855 �5.987753 �5.713615 �1.572216

MBE �0.290444 �0.009641 �0.004022 �0.011024 �0.016696 0.000674RMSE 0.231533 0.219899 0.223073 0.223189 0.223750 0.089950MPE 0.002982 �0.014110 �0.141085 �0.089823 �0.016033 �0.028752R 0.999212 0.999299 0.999270 0.999270 0.999272 0.999881NSE 0.998423 0.998578 0.998537 0.998535 0.998528 0.999762

a = 0.379518 a = 0.301048 a = 32.889918 a = 0.061467 a = 0.385102 a = 0.544776b = 0.509485 b = 0.315804 b = �1.781862 b = 0.007788 b = �0.029939 b = 0.406154

n = 0.736718 c = 33.877931 c = 0.266561 c = 0.269593 c = �0.015904d = 0.266239 d = 0.021165

e = �0.004521f = 0.003558

Model 34 Model 35 Model 36 Model 37-A Model 37-B Model 38

January 0.457878 8.822358 �1.847975 �0.270305 1.164275 �2.966894February 1.073633 3.674741 1.747844 �0.442727 0.744494 3.239578March 0.743348 �4.396521 1.914128 0.950472 0.190785 2.844234April �0.728902 �9.320202 �0.533067 0.569853 �0.628260 �3.740912May �0.041516 �12.855237 �0.195742 �0.593836 �0.156522 0.370703June 0.936242 �14.578079 1.067737 0.459903 0.747774 1.440766July 0.417221 �15.558100 �0.295118 0.245111 0.453417 �0.714872August �0.892156 �15.547319 �1.099040 �0.773237 �0.655259 �0.891476September �0.708096 �13.476627 0.258593 0.033777 �0.683697 1.064151October �0.001662 �7.505446 0.151863 0.132771 0.193171 1.876744November �0.120891 �0.868381 �0.550720 �0.529905 0.113796 1.048732December �3.351355 5.155839 �4.923913 �1.830165 �2.263013 �9.156908

MBE 0.008621 1.428880 0.012824 0.007478 0.003311 0.010246RMSE 0.126641 2.082323 0.172743 0.092503 0.100863 0.337697MPE �0.184688 �6.371081 �0.358784 �0.170691 �0.064920 �0.465513R 0.999773 0.998752 0.999580 0.999881 0.999852 0.998334NSE 0.999528 0.872476 0.999122 0.999748 0.999701 0.996646

a = 0.095971 a = 0.484227 a = 1.013087 e = 0.335073 a = 0.771912b = 0.372242 b = 0.274823 b = 0.462180 f = 0.368064 b = �0.000926c = 0.042876 c = �0.024595 c = �0.100556 g = �0.000365 c = �0.866601d = 0.353497 d = �0.159206 d = �0.035145 h = 0.000054e = �0.003863 i = �0.001565f = �0.001411


January 0.033909 �0.206186 �0.308020 0.695200 0.886538 �0.252620February 2.782433 2.649346 2.570513 2.887497 2.953338 2.511421March 2.969852 2.903303 2.837959 2.864464 2.879490 2.729345April �0.521403 �0.578370 �0.643659 �0.654767 �0.645636 �0.662335May �1.701278 �1.696064 �1.749476 �1.890015 �1.900030 �1.809725June 0.121548 0.196312 0.157881 0.037030 0.037605 0.016356July 0.202561 0.323331 0.294196 0.291283 0.319808 0.160192August �0.634880 �0.498340 �0.524753 �0.470219 �0.428601 �0.590901


Table 2 (continued)


September 0.861512 0.976815 0.946941 0.927920 0.952601 1.046943October 0.632165 0.661606 0.614319 0.462507 0.452301 1.084482November �0.369586 �0.448148 �0.517696 �0.455822 �0.434436 0.408886December �6.069855 �6.332638 �6.442305 �5.315044 �5.100406 �4.782759

MBE �0.004022 �0.006054 0.001684 �0.008066 �0.012784 �0.010874RMSE 0.223073 0.223698 0.223829 0.221267 0.221377 0.211776MPE �0.141085 �0.170753 �0.230342 �0.051664 �0.002286 �0.011726R 0.999270 0.999265 0.999263 0.999287 0.999289 0.999354NSE 0.998537 0.998528 0.998527 0.998560 0.998559 0.998681

a = 0.067455 a = �0.223014 a = 0.318190 a = 39.891292 a = 0.194207 a = �2.738119b = 0.348575 b = �0.000157 b = 0.225004 b = �2.163631 b = 0.360266 b = �0.006176c = 0.013698 c = 0.570335 c = �0.132033 c = 41.090266 c = �0.137780 c = �0.000002d = 0.266239 d = 0.386045 d = 0.180104 d = 18.186846 d = 0.102822 d = 11.118269

e = �0.013619 e = 0.196924 e = �0.981943 e = 0.486514 e = �397.719922f = 0.380282 f = �0.071465 f = 18.731831 f = �0.147660

g = 9.483900 g = �0.030064h = �0.517682 h = 0.142622j = 9.766651 j = �0.133630

Model 41 Model 42-A Model 42-B Model 42-C Model 43 Model 44

January 55.347749 �4.669382 �1.450581 0.229014 40.372600 3.241498February 31.549429 1.716631 2.958987 2.827766 30.012481 5.836140March 15.062622 3.040606 2.846146 2.179395 17.107690 4.046834April 3.421607 0.838000 0.806633 0.131710 17.086904 �1.085002May �8.353899 0.260854 �0.767475 �1.077983 2.758088 �1.434717June �12.016718 0.528637 �0.407616 0.492038 �15.824141 0.695315July �12.212925 �0.467367 0.060484 �0.034756 �25.865443 1.090713August �10.097257 �0.972650 �0.058955 �0.661663 �27.712107 �0.227650September �3.031246 0.651423 0.829577 1.123143 �23.157546 �1.183893October 8.035900 0.301870 �0.955773 �0.385366 �9.546371 �3.475854November 25.824621 �3.352217 �4.235731 �4.728664 2.554355 �5.002896December 56.354058 �11.008162 �7.200027 �4.930958 39.613764 �3.845298

MBE �0.584958 0.045153 0.023569 0.014284 0.489360 0.000000RMSE 2.493241 0.290591 0.242940 0.219204 3.436213 0.350986MPE 12.490328 �1.094313 �0.631194 �0.403027 3.950023 �0.112067R 0.996609 0.999002 0.999197 0.999317 0.961065 0.998187NSE 0.817180 0.997517 0.998264 0.998587 0.652740 0.996377

a = 0.006088 a = 0.609649 a = 0.582627 a = 0.630831 a = 0.105717b = �0.187758 b = �0.031630 b = �0.433562 b = 0.165090

c = �0.180229 c = 0.774294 c = 0.584622d = �0.682010

Model 45-A Model 45-B Model 45-C Model 45-D Model 45-E Model 46-A

January �30.094422 11.745490 �17.381723 �20.804707 �20.073817 �0.353933February 0.544386 4.066545 2.172099 4.068915 3.762725 0.187953March 13.562771 7.840435 14.306042 12.743144 12.990462 0.096060April 14.917176 16.256658 21.067892 12.289108 13.851587 �0.000605May 25.293050 5.931074 16.263130 22.885482 21.628905 �0.431472June 16.382918 2.920635 6.253526 14.388355 12.791451 0.359642July �4.257762 3.721014 �3.477853 �2.488198 �2.947574 0.140892August �10.580956 �2.778034 �8.339560 �9.592899 �9.652842 �0.436063September �12.295001 �11.544875 �9.389242 �14.554584 �13.843334 0.054663October �11.783530 �23.348731 �12.284198 �14.782961 �14.433069 1.033850November �26.319319 �27.575662 �25.275321 �23.469297 �23.774586 �1.697465December �49.080407 2.754782 �34.395659 �37.280446 �36.621956 1.027968



Table 2 (continued)

Model 45-A Model 45-B Model 45-C Model 45-D Model 45-E Model 46-A

MBE 0.074742 �0.018297 0.055081 0.044924 0.063870 0.000000RMSE 2.642737 1.629652 2.060333 2.368265 2.294985 0.075486MPE �6.142591 �0.834222 �4.206739 �4.716507 �4.693504 �0.001543R 0.892366 0.960195 0.935941 0.914490 0.919927 0.999916NSE 0.794599 0.921894 0.875156 0.835049 0.845099 0.999832

e = 5793.487645 g = 0.095252 j = 5.812594 l = 73.087813 n = 46.513347 a = 0.566283f = �1.475592 h = 1.274525 k = 0.124428 m = �0.027415 r = 0.022547 b = �0.231323

i = 0.634376 c = 0.132530d = �1.160484e = 6.814798

Model 46-B Model 46-C Model 47-A Model 47-B Model 48 Model 49-A

January 4.713904 �2.036625 2.607777 �14.516400 �0.123475 3.868765February 3.455733 2.437876 2.970467 13.319581 0.048820 0.669949March 1.391815 0.676332 2.465974 8.240181 �0.070888 1.329117April �3.230859 2.288609 �1.146622 �6.519238 �0.003172 �0.657715May �2.716051 0.298570 �2.347331 16.963108 �0.032304 �2.283886June 0.741435 �2.528344 �0.086511 10.615758 0.135533 0.051973July 1.711598 0.000988 0.518279 �3.621141 �0.115766 1.384918August 0.809451 2.354636 �0.075128 �6.488272 �0.085584 0.895671September 0.438731 �0.865387 1.163731 �9.785705 0.288630 0.395703October �2.021867 1.806610 0.138078 �6.406272 �0.205547 �2.304617November �2.777384 �5.938437 �0.795869 �0.402082 �0.130431 �4.400397December 1.025909 �3.550147 �3.069868 �24.268551 0.345379 1.852243

MBE �0.014675 0.006450 �0.019015 0.000000 0.000000 0.000000RMSE 0.325123 0.320758 0.229481 1.639536 0.022576 0.248521MPE 0.295201 �0.421276 0.195248 �1.905753 0.004266 0.066810R 0.998469 0.998510 0.999266 0.959658 0.999993 0.999091NSE 0.996891 0.996974 0.998451 0.920944 0.999985 0.998184

a = 0.528313 a = 0.044932 a = 0.616723 a = 81.431683 a = 20.296019 a = 9.656848b = 0.126389 b = 0.033206 b = 0.300470 b = �29.813096 b = �0.096134 b = 11.399876c = �0.114461 c = �0.068358 c = �0.004258 c = �0.822776 c = 0.317593 c = 17.773486d = �0.454928 d = �0.122164 d = �0.146422 d = �0.097917

e = 1.576552 e = 10.705159f = �0.288332g = 0.021331h = 0.359791i = 0.207588j = �0.076444

Model 49-B Model 49-C Model 49-D Model 49-E Model 49-F Model 50-A

January 2.107013 3.675026 �0.010508 0.436658 �1.236625 3.896023February 1.118371 2.749265 �0.010812 1.128706 �0.354531 1.408224March 0.797859 1.304672 0.923973 1.100985 1.046592 1.809833April �2.373064 �3.412453 �1.040376 �1.857892 �0.455993 �0.856161May �0.678964 �1.454449 0.223269 �1.701492 �0.133745 �2.405562June 0.807603 1.591487 �0.052094 1.541353 0.046109 0.078752July 0.748667 1.507358 �0.236453 1.468528 �0.104507 1.380770August 0.428724 0.111170 0.624989 �0.064418 0.472277 0.921007September �0.221119 �0.522221 0.129670 �1.247059 �0.233817 0.378537October �1.443526 �2.634541 �0.224020 �2.594931 �0.456131 �2.489773November �1.845015 �1.648062 �2.276876 2.604712 �0.244980 �4.635292December 1.551635 �0.411385 2.269001 0.032573 1.873409 0.991739

MBE 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000RMSE 0.179024 0.293261 0.105766 0.243935 0.076158 0.265495MPE 0.083182 0.071322 0.026647 0.070644 0.018172 0.039841


Table 2 (continued)

Model 49-B Model 49-C Model 49-D Model 49-E Model 49-F Model 50-A

R 0.999529 0.998735 0.999835 0.999125 0.999915 0.998963NSE 0.999057 0.997471 0.999671 0.998250 0.999829 0.997927

a = 19.030535 a = 2.653112 a = 37.505080 a = �3.055218 a = 31.062486 a = 9.592015b = 16.634146 b = 0.473598 b = �0.534338 b = 0.494609 b = �0.425730 b = 10.284013c = 7.000115 c = 3.888313 c = 9.627841 c = 5.129711 c = 9.560000 c = 17.469063d = �0.106934 d = �0.060921 d = 35.107491 d = 0.001039 d = 31.662296 d = �0.078593e = �0.138048 e = 0.011078 e = �0.156145 e = 0.472454 e = �0.112867

f = �0.277318 f = �0.491177 f = 0.226608g = �0.481640

Model 50-B Model 50-C Model 50-D

January 0.097697 4.326212 �0.485062February 0.853504 2.733082 0.870327March 1.322759 1.229791 1.444299April �1.534383 �3.469376 �1.412334May 0.090690 �1.265048 �0.031706June �0.094182 1.521521 �0.103811July �0.162044 1.383337 �0.125716August 1.072272 �0.020069 1.184217September �0.098800 �0.160241 �0.330143October �0.630892 �2.349474 �0.786342November �2.968333 �2.576324 �2.353846December 2.362670 �0.578607 2.428311

MBE 0.000000 0.000000 0.000000RMSE 0.148663 0.290721 0.146189MPE 0.025913 0.064567 0.024849R 0.999675 0.998756 0.999686NSE 0.999350 0.997514 0.999371

a = 41.467572 a = 4.195501 a = 41.275673b = �0.670176 b = 0.475886 b = �0.702315c = 8.238759 c = 4.407817 c = 7.950240d = 39.186699 d = �0.075486 d = 40.229185f = �0.170072 f = �0.116407 e = �0.160818g = �0.303878 g = 0.136067 f = 0.093686

g = �0.409071


Yaldiz model (Eq. (113)) was found as the most accurate model for the prediction of global solar radiation ona horizontal surface for Konya. While the percentage error changed between �0.205547% and 0.345379%, theMPE, MBE, RMSE, R and NSE were 0.004266%, 0.000000 MJ/m2, 0.022576 MJ/m2, 0.999993 and 0.999985,respectively. This model can be described for Konya with its coefficients as follows:

H ¼ 20:296019� 0:096134H 0 þ 0:317593d� 0:146422RHþ 10:705159SS0

� �� 0:288332T þ 0:021331TS

þ 0:359791C þ 0:207588P � 0:076444E

Fig. 1 shows the variations between the measured and calculated values of global solar radiation accordingto the months of the year. After comparing the measured global radiation values with the predicted valuesat any particular month for validation of the established model, these values laid around the straight line(Fig. 2). This means that the generalised model is valid for the geographical and meteorological data ofKonya.

0

5

10

15

20

25

0 5 10 15 20 25

Hmeasured

Hca

lcul

ated

Fig. 2. The measured and calculated values of montly average daily global radiation on horizontal surface (MJ/m2).

0

5

10

15

20

25

yraunaJ

yraurbeF

hcraM

lirpA

yaM

en uJ

yluJ

tsuguA

rebmet peS

rebotcO

r ebmev o

N

rebmece

D

Months

onoitaidarlabolg

yliadegareva

yl htnoM

n

m/JM(

ecafruslatnoziroh2 )

Measured values

Calculated values

Fig. 1. The measured and calculated values of global radiation with the Ertekin and Yaldiz model (48) (MPE = 0.004266%,MBE = 0.000000 MJ/m2, RMSE = 0.022576 MJ/m2, R = 0.999993 and NSE = 0.999985).


Acknowledgement

The corresponding author thanks the Scientific Research Administration Unit of Akdeniz University,Antalya, Turkey.

References

[1] Wong LT, Chow WK. Solar radiation model. Appl Energy 2001;69:191–224.[2] Almorox J, Hontoria C. Global solar radiation estimation using sunshine duration in Spain. Energy Convers Manage

2004;45:1529–35.[3] Al-Mohamad A. Global, direct and diffuse solar radiation in Syria. Appl Energy 2004;79(2):191–200.[4] Driesse A, Thevenard D. A test of Suehrcke’s sunshine radiation relationship using a global data set. Solar Energy 2002;72:167–75.


[5] Li DHW, Lam JC. Solar heat gain factors and the implications for building designs in subtropical regions. Energy Build2000;32:47–55.

[6] Lu Z, Piedrahita RH, Neto CDS. Generation of daily and hourly solar radiation values for modeling water quality in aquacultureponds. Trans ASAE 1998;41:1853–9.

[7] Kumar R, Umanand L. Estimation of global radiation using clearness index model for sizing photovoltaic system. Renew Energy2005;30(15):2221–33.

[8] Tarhan S, Sari A. Model selection for global and diffuse radiation over the Central Black Sea (CBS) region of Turkey. EnergyConvers Manage 2005;46:605–14.

[9] Ulgen K, Hepbasli A. Comparison of solar radiation correlations for Izmir, Turkey. Int J Energy Res 2002;26:413–30.[10] Ulgen K, Hepbasli A. Prediction of solar radiation parameters through clearness index for Izmir, Turkey. Energy Source

2002;24:773–85.[11] Ulgen K, Hepbasli A. Estimation of solar radiation parameters for Izmir, Turkey. Int J Energy Res 2002;26:807–23.[12] Jin Z, Yezheng W, Gang Y. General formula for estimation of monthly average daily global solar radiation in China. Energy

Convers Manage 2005;46:257–68.[13] Togrul IT, Togrul H, Evin D. Estimation of monthly global solar radiation from sunshine duration measurements in Elazig. Renew

Energy 2000;19:587–95.[14] Ertekin C, Yaldiz O. Estimation of monthly average daily global radiation on horizontal surface for Antalya, Turkey. Renew Energy

1999;17:95–102.[15] Ertekin C, Yaldiz O. Comparison of some existing models for estimating global solar radiation for Antalya (Turkey). Energy

Convers Manage 2000;41:311–30.[16] Tadros MTY. Uses of sunshine duration to estimate the global solar radiation over eight meteorological stations in Egypt. Renew

Energy 2000;21:231–46.[17] Chen R, Ersi K, Yang J, Lu S, Zhao W. Validation of five global radiation models with measured daily data in China. Energy

Convers Manage 2004;45:1759–69.[18] El-Metwally M. Sunshine and global solar radiation estimation at different sites in Egypt. J Atmos Solar-Terrestrial Phys

2005;67(14):1331–42.[19] Togrul IT, Onat E. A comparison of estimated and measured values of solar radiation in Elazig, Turkey. Renew Energy

2000;20:243–52.[20] Togrul IT, Togrul H, Evin D. Estimation of global solar radiation under clear sky radiation in Turkey. Renew Energy

2000;21:271–87.[21] Halawa EEH, Sugiyatno. Estimation of global solar radiation in the Indonesian climatic region. Renew Energy 2001;24:197–206.[22] Souza JL, Nicacio RM, Moura MAL. Global solar radiation measurements in Maceio, Brazil. Renew Energy 2005;30:1203–20.[23] Almorox J, Benito M, Hontoria C. Estimation of monthly Angstrom–Prescott equation coefficients from measured daily data in

Toledo, Spain. Renew Energy 2005;30:931–6.[24] Almorox J, Hontoria C, Benito M. Statistical validation of day length definitions for estimation of global solar radiation in Toledo,

Spain. Energy Convers Manage 2005;46:1465–71.[25] Elagib N, Mansell MG. New approaches for estimating global solar radiation across Sudan. Energy Convers Manage

2000;41:419–34.[26] Rehman S, Halawani T. Global solar radiation estimation. Renew Energy 1997;12:369–85.[27] Togrul IT, Onat E. A study for estimating solar radiation in Elazig using geographical and meteorological data. Energy Convers

Manage 1999;40:1577–84.[28] Togrul IT, Togrul H. Global solar radiation over Turkey: comparison of predicted and measured data. Renew Energy

2002;25:55–67.[29] Tiba C, Aguiar R, Fraidenraich N. Analysis of a new relationship between monthly global irradiation and sunshine hours from a

database of Brazil. Renew Energy 2005;30:957–66.[30] Chegaar M, Chibani A. Global solar radiation estimation in Algeria. Energy Convers Manage 2001;42:967–73.[31] Ibrahim SMA. Predicted and measured global solar radiation in Egypt. Solar Energy 1985;35:185–8.[32] Soler A. Dependence on cloudiness of the relation between the ratio of diffuse to global radiation and the ratio of global to

extraterrestrial radiation for average daily values. Solar Energy 1990;44:179–81.[33] Ogelman H, Ecevit A, Tasdemiroglu E. A new method for estimating solar radiation from bright sunshine data. Solar Energy

1984;33:619–25.[34] Bahel V, Bakhsh H, Srinivasan R. A correlation for estimation of global solar radiation. Energy 1987;12:131–5.[35] Bashahu M, Nkundabakura P. Analysis of daily global irradiation data for five sites in Rwanda and one in Senegal. Renew Energy

1994;4:425–35.[36] Kholagi A. Solar radiation over Sudan—comparison of measured and predicted data. Solar Energy 1983;31:45–53.[37] Raja IA. Insolation sunshine relation with site elevation and latitude. Solar Energy 1994;53:53–6.[38] Alnaser WE. New model to estimate the solar global irradiation using astronomical and meteorological parameters. Renew Energy

1993;3:175–7.[39] Raja IA, Twidell JW. Distribution of global insolation over Pakistan. Solar Energy 1990;44:63–71.[40] Raja IA, Twidell JW. Diurnal variation of global insolation over five locations in Pakistan. Solar Energy 1990;44:73–6.[41] Ododo JC, Sulaiman AT, Aidan J, Yuguda MM, Ogbu FA. The importance of maximum air temperature in the parameterisation of

solar radiation in Nigeria. Renew Energy 1995;6:751–63.


[42] Akinoglu BG, Ecevit A. A further comparison and discussion of sunshine based models to estimate global solar radiation. Energy1990;15:865–72.

[43] Hussain M, Rahman L, Rahman MM. Techniques to obtain improved predictions of global radiation from sunshine duration.Renew Energy 1999;18:263–75.

[44] Lin W, Lu E. Validation of eight sunshine based global radiation models with measured data at seven places in Yunan Province,China. Energy Convers Manage 1999;40:519–25.

[45] Supit I, Kappel RRV. A simple method to estimate global radiation. Solar Energy 1998;63:147–60.[46] Meza F, Varas E. Estimation of mean monthly solar global radiation as a function of temperature. Agric Forest Meteorol

2000;100:231–41.[47] Shaltout MAM, Hassan AH, Fathy AM. Study of the solar radiation over Menia. Renew Energy 2001;23:621–39.[48] Trabea AA, Shaltout MAM. Correlation of global solar radiation with meteorological parameters over Egypt. Renew Energy

2000;21:297–308.[49] Hepbasli A, Ulgen K, Eke R. Solar energy applications in Turkey. Energy Source 2004;26:551–61.[50] Evrendilek F, Ertekin C. Assessing the potential of renewable energy sources in Turkey. Renew Energy 2003;28:2303–15.[51] WEC-TNC. Turkish energy report. Ankara: World Energy Council Turkish National Committee, 2000 [in Turkish].[52] WEC-TNC. Turkish energy report. Ankara: World Energy Council Turkish National Committee, 2002 [in Turkish].[53] Annoymous. Energy statistics. Ankara, 2005. Available from: http://www.enerji.gov.tr/.[54] Angstrom A. Solar and terrestrial radiation. Quart J Roy Met Soc 1924;50:121–5.[55] Prescott JA. Evaporation from water surface in relation to solar radiation. Trans Roy Soc Austr 1940;46:114–8.[56] Page JK. The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine

records for latitudes 40�N–40�S. In: Proceedings of UN conference on new sources of energy, 1961. p. 378–90.[57] Duffie JA, Beckman WA. Solar engineering of thermal process. New York: Wiley; 1991.[58] Tiris M, Tiris C, Erdalli Y. Water heating systems by solar energy. Marmara Research Centre, Institute of Energy Systems and

Environmental Research, NATO TU-COATING, Gebze, Kocaeli, Turkey, 1997 [in Turkish].[59] Bahel V, Srinivasan R, Bakhsh H. Solar radiation for Dhahran, Saudi Arabia. Energy 1986;11:985–9.[60] Louche A, Notton G, Poggi P, Simonnot G. Correlations for direct normal and global horizontal irradiation on a French

Mediterranean site. Solar Energy 1991;46:261–6.[61] Benson RB, Paris MV, Sherry JE, Justus CG. Estimation of daily and monthly direct, diffuse and global solar radiation from

sunshine duration measurements. Solar Energy 1984;32:523–35.[62] Dogniaux R, Lemoine M. Classification of radiation sites in terms of different indices of atmospheric transparency. Solar energy

research and development in the European Community, Series F, vol. 2. Dordrecht, Holland: Reidel; 1983.[63] Soler A. Monthly specific Rietveld’s correlations. Solar Wind Technol 1990;7:305–8.[64] Gopinathan KK. A simple method for predicting global solar radiation on a horizontal surface. Solar Wind Technol 1988;5:581–3.[65] Zabara K. Estimation of the global solar radiation in Greece. Solar Wind Technol 1986;7:267–72.[66] Kilic A, Ozturk A. Solar energy. Istanbul: Kipas Yayincilik; 1983 [in Turkish].[67] Gariepy J. Estimation of global solar radiation. International Report, Service of Meteorology, Government of Quebec, Canada,

1980.[68] Rietveld M. A new method for estimating the regression coefficients in the formula relating solar radiation to sunshine. Agric

Meteorol 1978;19:243–52.[69] Gopinathan KK. A general formula for computing the coefficients of the correlations connecting global solar radiation to sunshine

duration. Solar Energy 1988;41:499–502.[70] Ampratwum DB, Dorvlo ASS. Estimation of solar radiation from the number of sunshine hours. Appl Energy 1999;63:161–7.[71] Samuel TDMA. Estimation of global radiation for Sri Lanka. Solar Energy 1991;47:333–7.[72] Glower J, McGulloch JSG. The empirical relation between solar radiation and hours of sunshine. Quart J R Met Soc 1958;84:172.[73] Newland FJ. A study of solar radiation models for the coastal region of South China. Solar Energy 1988;31:227–35.[74] Allen R. Self calibrating method for estimating solar radiation from air temperature. J Hydrol Eng 1997;2:56–67.[75] Hargreaves GL, Hargreaves GH, Riley P. Irrigation water requirement for the Senegal River Basin. J Irrigat Drain Eng ASCE

1985;111:265–75.[76] Bristow KL, Champbell GS. On the relationship between incoming solar radiation and daily maximum and minimum temperature.

Agric Forest Meteorol 1984;31:159–66.[77] Bennett I. Monthly maps of mean daily insolation for United States. Solar Energy 1965;9:145–59.[78] Abdalla YAG. New correlation of global solar radiation with meteorological parameters for Bahrain. Int J Solar Energy

1994;16:111–20.[79] Ojosu JO, Komolafe LK. Models for estimating solar radiation availability in South Western Nigeria. Nig J Solar Energy

1987;6:69–77.[80] Garg HP, Garg ST. Prediction of global solar radiation from bright sunshine hours and other meteorological parameters. Solar-

India, Proceedings on national solar energy convention. New Delhi: Allied Publishers; 1982. p. 1.004–7.[81] Klabzuba J, Bures R, Koznarova V. Model calculation of daily sums of global radiation used in growth models. In: Proceedings of

the bioclimatology labour hours, Zvolen, 1999. p. 121–2.[82] Badescu V. Correlations to estimate monthly mean daily solar global irradiation: application to Romania. Energy 1999;24:883–93.[83] Black JN. The distribution of solar radiation over the earth’s surface. Arch Met Geoph Biokl 1956;7:165–1889.[84] Lewis G. Estimates of irradiance over Zimbabwe. Solar Energy 1983;31:609–12.

http://www.enerji.gov.tr/


[85] El-Metwally M. Simple new methods to estimate global solar radiation based on meteorological data in Egypt. Atmos Res2004;69:217–39.

[86] Swartman RK, Ogunlade O. Solar radiation estimates from common parameters. Solar Energy 1967;11:170–2.[87] Zeroual A, Ankrim M, Wilkinson AJ. The diffuse global correlation: its application to estimating solar radiation on tilted surfaces in

Marrakesh, Morocco. Renew Energy 1996;7:1–13.[88] Hutchinson MF, Booth TH, McMahon JP, Nix HA. Estimating monthly mean values of daily total solar radiation for Australia.

Solar Energy 1984;32:277–90.[89] Iziomon MG, Mayer H. Performance of solar radiation models—a case study. Agric Forest Meteorol 2001;110:1–11.[90] Tymvios FS, Jacovides CP, Michaelides SC, Scouteli C. Comparative study of Angstrom and artificial neural networks

methodologies in estimating global solar radiation. Solar Energy 2005;78:752–62.[91] Notton G, Cristofari C, Muselli M, Poggi P. Calculation on an hourly basis of solar diffuse irradiations from global data for

horizontal surfaces in Ajaccio. Energy Convers Manage 2004;45:2849–66.[92] Mediavilla MD, Miguel A, Bilbao J. Measurement and comparison of diffuse solar irradiance models on inclined surfaces in

Valladolid, Spain. Energy Convers Manage 2005;46:2075–92.[93] Soares J, Oliveira AP, Boznar MZ, Mlakar P, Escobedo JF, Machado AJ. Modeling hourly diffuse solar radiation in the city of Sao

Paulo using a neural network technique. Appl Energy 2004;79(2):201–14.[94] Tiba C. Solar radiation in the Brazilian Northeast. Renew Energy 2001;22:565–78.[95] Halouani N, Nguyen CT, Vo-Ngoc D. Calculation of monthly average global solar radiation on horizontal surfaces using daily

hours of bright sunshine. Solar Energy 1993;50:247–58.[96] Ulgen K, Hepbasli A. Comparison of diffuse fraction of daily and monthly global radiation for Izmir, Turkey. Energy Source

2003;25:637–49.[97] Ulgen K, Hepbasli A. Solar radiation models. Part 2: Comparison and developing new models. Energy Source 2004;26:521–30.[98] Abdul-Aziz J, Nagi AA, Zumailan AAR. Global solar radiation estimation from relative sunshine hours in Yemen. Renew Energy

1993;3:645–53.[99] Kamel MA, Shalaby SA, Mostafa SS. Solar radiation over Egypt: comparison of predicted and measured meteorological data. Solar

Energy 1993;50:463–7.[100] Lingamgunta C, Veziroglu TN. A universal relationship for estimating daily clear sky insolation. Energy Convers Manage

2004;45:2313–34.[101] Ma CCY, Iqbal M. Statistical comparison of solar radiation correlations. Solar Energy 1984;33:143–8.[102] Elminir HK, Areed FF, Elsayed TS. Estimation of solar radiation components incident on Helwan site using neural networks. Solar

Energy 2004;79(3):270–9.

Documents

Evaluation of global solar radiation models for Konya, Turkey