Energy Conversion and Management 50 (2009) 149–156

Contents lists available at ScienceDirect

Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Diffuse solar radiation estimation models for Turkey’s big cities

Koray Ulgen a,*, Arif Hepbasli b

a Solar Energy Institute, Ege University, 35100 Bornova, Izmir, Turkeyb Mechanical Engineering Department, Engineering Faculty, Ege University, 35100 Bornova, Izmir, Turkey

a r t i c l e i n f o

Article history:Received 8 January 2008Accepted 17 August 2008Available online 8 October 2008

Keywords:Correlation modelsDiffuse solar radiationDiffuse fractionDiffuse coefficientClearness indexCloudness indexSunshine fractionSolar energySustainability

0196-8904/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.enconman.2008.08.013

* Corresponding author. Tel.: +90 232 3886028; faxE-mail address: koray.ulgen@ege.edu.tr (K. Ulgen)

a b s t r a c t

A reasonably accurate knowledge of the availability of the solar resource at any place is required by solarengineers, architects, agriculturists, and hydrologists in many applications of solar energy such as solarfurnaces, concentrating collectors, and interior illumination of buildings. For this purpose, in the past,various empirical models (or correlations) have been developed in order to estimate the solar radiationaround the world. This study deals with diffuse solar radiation estimation models along with statisticaltest methods used to statistically evaluate their performance. Models used to predict monthly averagedaily values of diffuse solar radiation are classified in four groups as follows: (i) From the diffuse fractionor cloudness index, function of the clearness index, (ii) From the diffuse fraction or cloudness index, func-tion of the relative sunshine duration or sunshine fraction, (iii) From the diffuse coefficient, function ofthe clearness index, and (iv) From the diffuse coefficient, function of the relative sunshine duration orsunshine fraction.

Empirical correlations are also developed to establish a relationship between the monthly average dailydiffuse fraction or cloudness index (Kd) and monthly average daily diffuse coefficient (Kdd) with themonthly average daily clearness index (KT) and monthly average daily sunshine fraction (S/So) for thethree big cities by population in Turkey (Istanbul, Ankara and Izmir).

Although the global solar radiation on a horizontal surface and sunshine duration has been measuredby the Turkish State Meteorological Service (STMS) over all country since 1964, the diffuse solar radiationhas not been measured. The eight new models for estimating the monthly average daily diffuse solar radi-ation on a horizontal surface in three big cites are validated, and thus, the most accurate model is selectedfor guiding future projects. The new models are then compared with the 32 models available in the lit-erature in terms of the widely used statistical indicators, namely; the relative percentage error (E), coef-ficient of determination (R2), the mean percentage error (MPE), the mean absolute percentage error(MAPE), the sum of the squares of relative errors (SSRE), the relative standard error (RSE), the mean biaserror (MBE), the root mean square error (RMSE), and the t-statistic (t-stat) method combining the last twoerrors.

It may be concluded that the new models predict the values of cloudness index (Kd) and diffuse coef-ficient (Kdd) as a function of clearness index (KT) and sunshine fraction (S/So) for three big cities in Turkeybetter than other available models, while all the models tested appear to be location independent modelsfor diffuse radiation predictions, at least for three big cities in Turkey. It is also expected that the modelsreviewed and developed will be beneficial to everyone involved or interested in the design and study ofsolar energy.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Renewable energy resources and their utilization are intimatelyrelated to sustainable development. In other words, the develop-ment of renewable energy technology is now widely seen asimportant if the world is to move towards a sustainable approachto energy generation [1,2]. Renewable energy is considered as akey source for the future, not only for Turkey but also for the world.

ll rights reserved.

: +90 232 3886027..

This is primarily due to fact that renewable energy resources havesome advantages when compared to fossil fuels. At the presenttime, with the existing technology, it is impossible to supply thetotal demand of any country from renewable resources [3].

The need for meteorological parameters is essential in the de-sign and study of solar energy conservation devices. In otherwords, a reasonably accurate knowledge of the availability of thesolar resource at any place is required by solar engineers,architects, agriculturists and hydrologists in many applications ofsolar energy such as solar furnaces, concentrating collectors, andinterior illumination of buildings [4,5]. On the other hand, the

Nomenclature

ca average of the calculated value (dimensionless)ci ith calculated value (dimensionless)E relative percentage error (dimensionless)Gon extraterrestrial radiation (W/m2)Gsc solar constant (W/m2)H monthly average daily global solar radiation (MJ/m2)Hd monthly average daily diffuse solar radiation (MJ/m2)Ho monthly average daily extraterrestrial radiation (MJ/m2)Kd (=Hd/H) monthly average daily diffuse fraction or monthly

average daily cloudness (clearness) index (dimension-less)

Kdd (=H/Ho) monthly average daily clearness index (dimension-less)

KT (=H/Ho) monthly average daily clearness index (dimension-less)

ma average of the measured value (dimensionless)MAPE mean absolute percentage error (dimensionless)MBE mean bias error (dimensionless)mi ith measured value (dimensionless)MPE mean percentage error (dimensionless)

n number of calculated and measured values (dimension-less)

nday number of day of the year, starting from the first of Jan-uary (dimensionless)

R2 coefficient of determination (dimensionless)RMSE root mean square error (dimensionless)RSE relative standard error (dimensionless)S monthly average daily measured sunshine duration (h)So monthly average daily maximum possible sunshine

duration (h)SSRE sum of the squares of relative errors (dimensionless)t-stat t-statistic (dimensionless)t-crit critical value for t-statistic method (dimensionless)

Greek symbols/ latitude of site (�)1 � a confidence level (=99%)a statistical significant level (=0.01)d solar declination angle (�)

150 K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156

determination of solar energy capacity effectively through theempirical models plays a key role in developing solar energy tech-nologies and the sustainability of natural resources.

The total radiation on a horizontal surface is recorded at a largenumber of locations, while diffuse radiation, needed in many solarenergy applications, is measured in comparatively few locations[4,6,7]. The diffuse fraction under clear-sky conditions may be cal-culated theoretically [8]. However, it is common practice for thelarge number data to be condensed and presented in simple use-able form obtained from the measurements for various types ofusers [6]. Correlations used for predicting monthly average dailyvalues of diffuse radiation may be classified in four groups as: (i)from the clearness index (KT = H/Ho), (ii) from the relative sunshineduration or sunshine fraction (S/So), (iii) from the diffuse coefficient(Kdd = Hd/Ho), and (iv) from the diffuse fraction or cloudiness index(Kd = Hd/H) [4,6,7].

The main objectives of this study are as follows: (i) to compre-hensively review diffuse solar radiation estimation models by clas-sifying them in four groups, (ii) to develop some empiricalrelations for three big cities by population (Istanbul, Ankara andIzmir) in Turkey in order to estimate the daily diffuse fraction asa function of the daily clearness index and (iii) to compare theresults of this study with those obtained using other correlationsavailable in the literature.

2. Experimental data and statistical analysis methods

The data of the monthly average daily global solar radiation on ahorizontal surface and sunshine duration measured by the TurkishState Meteorological Service (STMS) over a sixteen-year periods1990-2006.

In the literature, there are several statistical test methods usedto statistically evaluate the performance of the models of solarradiation estimations. Among these, the relative percentage error(E), coefficient of determination (R2), the mean percentage error(MPE), the mean absolute percentage error (MAPE), the sum ofthe squares of relative errors (SSRE), the relative standard error(RSE), the mean bias error (MBE), the root mean square error(RMSE), and the t-statistic (t-stat) are the most commonly usedones to compare the results statistically [9–11].

2.1. Relative percentage error (E)

The relative percentage is given by

E ¼ ci �mi

mi

� �100 ð1Þ

where ci the ith calculated value and mi the ith measured value. TheE provides the percentage deviation between the calculated andmeasured data. The ideal value of E equals zero.

2.2. Coefficient of determination (R2)

The coefficient of determination can be used to test determiningthe linear relation between calculated and measured values, whichcan be obtained from the following equation:

R2 ¼Pn

i¼1ðci � caÞ � ðmi �maÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ðci � caÞ2

h i Pni¼1ðmi �maÞ2

h ir ð2Þ

where ca and ma are the average of the calculated and measured val-ues, respectively.

2.3. Mean percentage error (MPE)

The mean percentage error can be defined as the percentagedeviation of the monthly average daily radiation values esti-mated by the proposed equations from the measured values, as gi-ven by

MPE ¼Pn

i¼1En

ð3Þ

where n is the number of calculated and measured values.

2.4. Mean absolute percentage error (MAPE)

The mean percentage error is expressed as the absolute averagevalue of percentage deviation between estimated and measuredsolar radiations and is given by

K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156 151

MAPE ¼ ABSPn

i¼1En

� �ð4Þ

2.5. The sum of the square of relative error (SSRE)

The sum of the square of relative error is calculated by

SSRE ¼Xn

i¼1

ci �mi

mi

� �2

ð5Þ

which gives the positive value of sum of squares of relative devia-tion throughout the year. The ideal value of SSRE equals zero.

2.6. The relative standard error (RSE)

The relative standard error is calculated by

RSE ¼ffiffiffiffiffiffiffiffiffiffiffiSSRE

n

rð6Þ

which provides the degree of accuracy of estimation of correlations.

2.7. Mean bias error (MBE)

The mean bias error is given by

MBE ¼ 1n

Xn

_I¼1

ðci �miÞ ð7Þ

The MBE provides information on the long-term performance of thecorrelations by allowing a comparison of the actual deviationbetween calculated and measured values term by term. The idealvalue of MBE is ‘zero’.

2.8. Root mean square error (RMSE)

The root mean square error may be computed from the follow-ing equation

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Xn

i¼1

ðci �miÞ2vuut ð8Þ

which provides information on the short-term performance. Thevalue of RMSE is always positive, representing ‘zero’ in the ideal case.

2.9. t-statistic (t-stat) method

To determine whether or not the equation estimates are statis-tically significant, i.e., not significantly different from their actualcounterparts, at a particular confidence level, Stone [12], proposedthe t-statistic as [13]:

t-stat ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn� 1ÞMBE2

RMSE2 �MBE2

" #vuut ð9Þ

Using published data in the literature, Stone [12], demonstratedthat MBE and RMSE separately do not represent a reliable assess-ment of the model’s performance and can lead to the false selectionof the best model from a set of candidate ones.

In order for that equation estimates to be significant, the t-statvalue produced by Eq. (9) must be smaller than the value for thatconfidence level given in standard statistical tables. In all the abovestatistical tests of accuracy, except R2, the smaller the value, thebetter is the model performance [13].

3. Models used for estimating the diffuse solar radiation

When the solar radiation enters the earth’s atmosphere, a partof the incident energy is removed by scattering and a part byabsorption, both of which influence the extraterrestrial spectrumby considerably modifying the spectral energy as it passes throughthe atmosphere.

The scattered radiation is called diffuse radiation. In the past,several empirical models have been developed to predict themonthly average diffuse solar radiation. These models expressedthe monthly average daily diffuse solar radiation as a function ofmeasured sunshine duration, measured global solar radiation andextraterrestrial solar radiation.

Correlation used for predicting monthly average daily values ofdiffuse solar radiation may be classified in four groups as follows:

Group I: From the diffuse fraction or cloudness index, functionof the clearness index

Kd ¼Hd

H

� �� f KT ¼

HHo

� �:

Group II: From the diffuse fraction or cloudness index, functionof the relative sunshine duration or sunshine fraction

Kd ¼Hd

H

� �� f

SSo

� �:

Group III: From the diffuse coefficient, function of the clearnessindex

Kdd ¼Hd

Ho

� �� f KT ¼

HHo

� �:

Group IV: From the diffuse coefficient, function of the relativesunshine duration or sunshine fraction

Kdd ¼Hd

Ho

� �� f

SSo

� �

where Ho is the monthly average daily extraterrestrial radiation(MJ/m2), H is the monthly average daily global solar radiation(MJ/m2), Hd is the monthly average daily diffuse solar radiation(MJ/m2), S is the monthly average daily measured sunshine dura-tion (h) and So is the monthly average daily maximum possible sun-shine duration (h).

The models proposed by the many national and internationalinvestigators are presented in the following:

Group I: From the diffuse fraction or cloudness index, function ofthe clearness index Model 1 [14]:

Kd ¼ 1:00� 1:13ðKTÞ ð10Þ

Model 2 [15]:

Kd ¼ 1:0492� 1:3246ðKTÞ ð11Þ

Model 3 [16]:

Kd ¼ 0:789� 0:869ðKTÞ ð12Þ

Model 4 [17]:

Kd ¼ 1:0212� 1:1672ðKTÞ ð13Þ

Model 5 [15]:

Kd ¼ 1:0896� 1:4797ðKTÞ þ 0:1471ðKTÞ2 ð14Þ

152 K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156

Model 6 [18]:

Kd ¼ 1:039� 1:741ðKTÞ2 ð15Þ

Model 7 [19]:

Kd ¼ �0:193þ 0:3431

KT

� �ð16Þ

Model 8 [20]:

Kd ¼ 0:9885� 1:4276ðKTÞ þ 0:5679ðKTÞ2 ð17Þ

Model 9 [17]:

Kd ¼ 1:1244� 1:5582ðKTÞ þ 0:3635ðKTÞ2 ð18Þ

Model 10 [21]:

Kd ¼ 1:39� 4:027ðKTÞ þ 5:531ðKTÞ2 � 3:108ðKTÞ3 ð19Þ

Model 11 [22]:

Kd ¼ 1:317� 3:023ðKTÞ þ 3:372ðKTÞ2 � 1:769ðKTÞ3 ð20Þ

Model 12 [15]:

Kd ¼ 13:9375� 76:276ðKTÞ þ 144:3846ðKTÞ2 � 92:148ðKTÞ3 ð21Þ

Model 13 [18]:

Kd ¼ �5:759þ 35:093ðKTÞ � 61:052ðKTÞ2 þ 33:115ðKTÞ3 ð22Þ

Model 14 [5]:

Kd ¼ 0:583þ 0:9985ðKTÞ � 5:24ðKTÞ2 þ 5:322ðKTÞ3 ð23Þ

Model 15 [20]:

Kd ¼ 1:027� 1:6582ðKTÞ þ 1:1018ðKTÞ2 � 0:4019ðKTÞ3 ð24Þ

Model 16 [17]:

Kd ¼ 1:711� 4:9062ðKTÞ þ 6:6711ðKTÞ2 � 3:9235ðKTÞ3 ð25Þ

Model 17 [4]:

Kd ¼ 1:6932� 8:2262ðKTÞ þ 25:5532ðKTÞ2 � 37:807ðKTÞ3

þ 19:8178ðKTÞ4 ð26Þ

Group II: From the diffuse fraction or cloudness index, function ofthe relative sunshine duration or sunshine fractionModel 18 [15]:

Kd ¼ 0:6603� 0:5272SSo

� �ð27Þ

Model 19 [17]:

Kd ¼ 0:663� 0:4883SSo

� �ð28Þ

Model 20 [19]:

Kd ¼1

0:631þ 4:22 SSo

� � ð29Þ

Model 21 [15]:

Kd ¼ 0:7434� 0:8203SSo

� �þ 0:2454

SSo

� �2

ð30Þ

Model 22 [17]:

Kd ¼ 0:6492� 0:4323SSo

� �� 0:0512

SSo

� �2

ð31Þ

Model 23 [17]:

Kd ¼ 0:5562þ 0:1536SSo

� �� 1:2027

SSo

� �2

þ 0:7122SSo

� �3

ð32Þ

Group III: From the diffuse coefficient, function of the clearnessindex Model 24 [17]:

Kdd ¼ 0:331� 0:233ðKTÞ ð33Þ

Model 25 [17]:

Kdd ¼ 0:0511þ 0:8267ðKTÞ � 0:9854ðKTÞ2 ð34Þ

Model 26 [17]:

Kdd ¼ 0:3276� 0:7515ðKTÞ þ 1:9883ðKTÞ2 � 1:8497ðKTÞ3 ð35Þ

Group IV: From the diffuse coefficient, function of the relative sun-shine duration or sunshine fractionModel 27 [15]:

Kdd ¼ 0:2626� 0:1391SSo

� �ð36Þ

Model 28 [19]:

Kdd ¼ 0:293� 0:135SSo

� �ð37Þ

Model 29 [17]:

Kdd ¼ 0:2593� 0:0978SSo

� �ð38Þ

Model 30 [15]:

Kdd ¼ 0:2205� 0:0126SSo

� �� 0:1292

SSo

� �2

ð39Þ

Model 31 [17]:

Kdd ¼ 0:2142þ 0:0863SSo

� �� 0:1684

SSo

� �2

ð40Þ

Model 32 [17]:

Kdd ¼ 0:2427� 0:0933SSo

� �þ 0:1846

SSo

� �2

� 0:2184SSo

� �3

ð41Þ

4. Application of models

The data of the monthly average daily global solar radiation andsunshine duration measured by the DMI in the years of 1990–2006were used in this illustrative example. The monthly average dailyextraterrestrial radiation (Ho) and the maximum possible sunshineduration (So) for each month of the three big cities (Ankara, Istan-bul, and Izmir) in Turkey were calculated from the fundamentalmathematical expressions.

The monthly average daily extraterrestrial radiation on a hori-zontal surface (Ho) was obtained from the following equations [9]:

Ho ¼24p

Gon cos / cos d sin xs þp

180xs sin / sin d

� �ð42Þ

where Gon is the extraterrestrial radiation, computed from Helwaet al. [23], and

Gon ¼ Gsc 1þ 0:033 cos360 nday

365

� �� �ð43Þ

where Gsc is the solar constant (=1367 W/m2), / is the latitude ofthe site, d is the solar declination angle, xs is the sunshine hour an-gle for the month and nday is the number of the day of the year,starting from the first of January.

The solar declination (d), the main sunshine hour angle for themonth (xs) and the maximum possible sunshine duration daylength (So) was calculated from Cooper [24]:

Table 1Geographical locations of three big cities in Turkey

Station Longitude Latitude Altitude (m)

Ankara 32�530 39�570 894Istanbul 29�050 40�580 39Izmir 27�100 38�240 15

K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156 153

d ¼ 23:45 sin360365ð284þ ndayÞ

� �ð44Þ

xs ¼ cos�1ð� tan / tan dÞ ð45Þ

So ¼2

15xs ð46Þ

Since the monthly average daily diffuse solar radiation measure-ment data (Hd) for the three big cities in Turkey are not available,the monthly average daily diffuse solar radiations were predictedusing Eqs. (10)–(41). The average values of the estimations by allthese thirty-two models for the H

Hoand S

Sovalues were used to cali-

brate the new correlation models presented in this work. The aimof this approach was to eliminate the differences among the predic-tions of the thirty-two models because the selection of the mostappropriate model was impossible without using experimentaldata.

5. Results and discussion

The stations of the three big cities in Turkey considered areshown in Fig. 1, while the geographical information on these citiesis presented in Table 1. The measured data of the monthly averagedaily global solar radiation (H) and the sunshine duration (S) ofthree big cities are also given in Table 2, respectively.

The eight new hybrid models are developed by taking linear andthird order polynomial forms of each group. These new modelsdeveloped are based on the average values predicted by thethirty-two current models, as given below and shown in Figs. 2–5.Group I:

Kd ¼ 0:6772� 0:4841ðKTÞ; R2 ¼ 0:9707 ð47Þ

Kd¼0:981�1:9028ðKTÞþ1:9319ðKTÞ2�0:6809ðKTÞ3; R2¼0:9979

ð48Þ

Fig. 1. Turkey’s map indicating t

Group II:

Kd ¼ 0:5456� 0:2242SSo

� �; R2 ¼ 0:9037 ð49Þ

Kd ¼ 0:6595� 0:7841SSo

� �þ 0:7461

SSo

� �2

� 0:2579SSo

� �3

;

R2 ¼ 0:9722 ð50Þ

Group III:

Kdd ¼ 0:1155þ 0:1958ðKTÞ; R2 ¼ 0:9965 ð51Þ

Kdd ¼ 0:0273þ 0:727ðKTÞ � 1:0411ðKTÞ2 þ 0:6659ðKTÞ3;R2 ¼ 0:9974 ð52Þ

Group IV:

Kdd ¼ 0:1677þ 0:0926SSo

� �; R2 ¼ 0:9662 ð53Þ

Kdd ¼ 0:1437þ 0:2151SSo

� �� 0:1748

SSo

� �2

þ 0:0697SSo

� �3

;

R2 ¼ 0:9820 ð54Þ

In the scope of this study, the meteorological data for the provinceof Ankara, Istanbul and Izmir were used. The accuracy between theinvestigated and present models was determined. The resultsobtained from the statistical test methods are given in Table 3.

he three big cities stations.

Table 2Monthly average daily solar radiations on a horizontal surface and monthly averagedaily sunshine duriation of the three big cities in Turkey

Months Ankara Istanbul Izmir

H (MJ/m2day)

S (h) H (MJ/m2day)

S (h) H (MJ/m2day)

S (h)

January 6.10 2.60 5.50 2.60 8.30 4.40February 9.10 4.00 7.40 2.90 10.80 5.10March 13.40 5.40 11.90 4.60 15.70 6.90April 17.50 6.40 16.30 5.90 19.20 7.40May 22.10 8.50 20.40 7.60 23.00 9.00June 25.30 10.40 23.90 9.70 26.30 11.80July 25.60 11.00 23.80 10.10 26.10 12.20August 23.60 10.70 21.70 9.60 24.00 11.50September 18.60 9.30 16.90 8.10 19.20 9.80October 12.70 6.80 10.50 5.00 13.80 7.40November 7.50 4.00 6.40 3.10 9.50 5.30December 5.10 2.00 4.30 1.80 7.20 3.60

Annual 15.55 6.76 14.08 5.92 16.93 7.87

Kd = -0.4841(KT) + 0.6772R2 = 0.9707

Kd = -0.6809(KT)3 + 1.9319(KT)2 - 1.9028(KT) + 0.981R2 = 0.9979

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.30 0.40 0.50 0.60 0.70KT

Kd

Fig. 2. The linear and third order polynomial relationships between the monthlyaverage values of (Kd) vs. (KT).

Kd = -0.2242(S/So) + 0.5456R2 = 0.9037

Kd = -0.2579(S/So)3 + 0.7461(S/So)2 - 0.7841(S/So) + 0.6595R2 = 0.9722

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90S/So

Kd

Fig. 3. The linear and third order polynomial relationships between the monthlyaverage values of (Kd) vs. (S/So).

Kdd = 0.1958(KT) + 0.1155R2 = 0.9964

Kdd = 0.6659(KT)3 - 1.0411(KT)2 + 0.727(KT) + 0.0273R2 = 0.9974

0.150.160.170.180.190.200.210.220.230.240.25

0.30 0.40 0.50 0.60 0.70

KT

Kdd

Fig. 4. The linear and third order polynomial relationships between the monthlyaverage values of (Kdd) vs. (KT).

Kdd = 0.0926(S/So) + 0.1677R2 = 0.9662

Kdd = 0.0697(S/So)3 - 0.1748(S/So)2 + 0.2151(S/So) + 0.1437R2 = 0.982

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90S/So

Kdd

Fig. 5. The linear and third order polynomial relationships between the monthlyaverage values of (Kdd) vs. (S/So).

154 K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156

For the three big cities studied, the following main results wereobtained from the evaluation of the values presented in Table 3:

Evaluating group I:

(i) For the province of Ankara: The best results forMBE = �0.005 MJ/m2, RMSE = 0.024 MJ/m2, MPE = �0.071%,MAPE = 0.381%, SSRE = 0.0, RSE = 0.005, and R2 = 1.000 wereobtained from the third order polynomial model (Eq. (48)),while the best result for t-critic = 0.481 was calculated fromthe linear model (Eq. (47)) belonging to the province ofAnkara.

(ii) For the province of Istanbul: The best results forMBE = �0.004 MJ/m2, RMSE = 0.023 MJ/m2, MAPE = 0.423%,SSRE = 0.0, RSE = 0.005, R2 = 1.000 and t-critic = 0.537 wereobtained from the third order polynomial model (Eq. (48)),while the best result for MPE = �0.040% was calculated fromthe linear model (Eq. (47)) belonging to the province ofIstanbul.

(iii) For the province of Izmir: The best results forMBE = 0.006 MJ/m2, RMSE = 0.023 MJ/m2, MPE = 0.105%,MAPE = 0.279%, SSRE = 0.0, RSE = 0.003, and R2 = 1.000 wereobtained from the third order polynomial model (Eq. (48)),while the best result for t-critic = 0.557 was calculated fromthe linear model (Eq. (47)) belonging to the province ofIzmir.

Evaluating group II:

(iv) For the province of Ankara: The best results forMBE = 0.017 MJ/m2, RMSE = 0.055 MJ/m2, SSRE = 0.002,RSE = 0.014, and R2 = 1.000 were obtained from the thirdorder polynomial model (Eq. (50)), while the best result forMPE = 0.335%, MAPE = 0.335%, and t-critic = 0.673 were cal-culated from the linear model (Eq. (49)) belonging to theprovince of Ankara.

(v) For the province of Istanbul: The best results forMBE = �0.023 MJ/m2, MPE = �1.187%, MAPE = 1.187%,SSRE = 0.003, RSE = 0.015, and R2 = 1.000 were obtainedfrom the third order polynomial model (Eq. (50)), whilethe best result for RMSE = 0.079 MJ/m2, and t-critic = 0.878were calculated from the linear model (Eq. (49)) belongingto the province of Istanbul.

Table 3The MBE, RMSE, MPE, MAPE, SSRE, RSE, R2 and t-stat values of the equations developed for each city

Eqs. No. ANKARA ISTANBUL IZMIR

MBE RMSE MPE MAPE SSRE RSE R2 ta MBE RMSE MPE MAPE SSRE RSE R2 ta MBE RMSE MPE MAPE SSRE RSE R2 ta

Group I10 �0.653 1.253 �5.295 5.295 0.293 0.156 1.000 2.024 �0.151 0.739 1.873 1.873 0.170 0.119 1.000 0.693 �1.044 1.456 �12.501 12.501 0.334 0.167 1.000 3.41211 �1.592 2.315 �18.317 18.317 0.880 0.271 0.999 3.141 �0.880 1.523 �8.428 8.428 0.418 0.187 1.000 2.350 �2.165 2.680 �28.418 28.418 1.218 0.319 1.000 4.54712 �1.649 2.047 �22.407 22.407 0.744 0.249 1.000 4.505 �1.215 1.538 �17.136 17.136 0.441 0.192 1.000 4.270 �1.995 2.309 �27.680 27.680 1.001 0.289 1.000 5.68713 �1.583 2.302 �18.157 21.386 0.873 0.270 0.999 3.140 �0.875 1.524 �8.223 15.422 0.424 0.188 1.000 2.324 �2.156 2.663 �28.340 28.340 1.206 0.317 1.000 4.57614 1.106 1.748 26.352 31.499 1.662 0.372 0.999 2.710 1.909 2.080 38.765 38.765 2.232 0.431 0.999 7.666 0.520 1.398 14.184 22.972 0.802 0.259 1.000 1.33015 0.368 0.835 12.008 16.605 0.540 0.212 1.000 1.627 0.857 1.055 21.713 22.135 1.001 0.289 1.000 4.621 �0.032 0.573 2.284 8.411 0.109 0.095 1.000 0.18516 �0.586 0.903 �6.297 8.487 0.137 0.107 1.000 2.827 �0.287 0.579 �2.063 6.347 0.068 0.075 1.000 1.892 �0.821 1.041 �10.570 10.570 0.180 0.122 1.000 4.25617 �1.270 1.605 �16.777 16.777 0.449 0.193 1.000 4.292 �0.898 1.198 �11.667 11.951 0.255 0.146 1.000 3.753 �1.566 1.815 �21.703 21.703 0.617 0.227 1.000 5.66218 �0.540 0.979 �4.613 10.187 0.180 0.122 1.000 2.192 �0.154 0.604 1.184 8.769 0.118 0.099 1.000 0.876 �0.841 1.131 �10.343 10.501 0.204 0.130 1.000 3.68219 �1.916 3.256 �17.766 33.181 2.095 0.418 0.930 2.414 �0.366 1.863 9.788 33.702 2.520 0.458 0.857 0.665 �2.914 3.990 �36.779 36.779 2.483 0.455 0.995 3.54620 1.689 2.170 29.675 30.429 1.581 0.363 0.999 4.110 2.066 2.563 29.397 35.442 1.762 0.383 0.997 4.515 1.504 1.952 29.217 30.736 1.642 0.370 1.000 4.00721 0.751 1.146 8.915 8.988 0.206 0.131 1.000 2.878 0.359 0.590 4.154 4.612 0.057 0.069 1.000 2.544 1.034 1.396 12.937 12.937 0.306 0.160 1.000 3.65622 �0.501 0.831 �4.950 7.997 0.119 0.100 1.000 2.504 �0.202 0.509 �0.691 6.150 0.061 0.072 1.000 1.430 �0.732 0.961 �9.179 9.179 0.149 0.112 1.000 3.89523 �0.068 0.912 3.627 13.355 0.258 0.147 1.000 0.247 0.396 0.663 10.043 12.259 0.239 0.141 1.000 2.468 �0.406 0.977 �2.606 10.627 0.181 0.123 1.000 1.51524 �0.649 1.286 �4.972 4.972 0.316 0.162 1.000 1.939 �0.124 0.760 2.574 2.574 0.193 0.127 1.000 0.550 �1.059 1.491 �12.573 12.573 0.350 0.171 1.000 3.34325 �0.642 1.275 �4.756 14.073 0.321 0.163 1.000 1.934 �0.120 0.783 2.991 12.160 0.219 0.135 1.000 0.514 �1.058 1.470 �12.658 13.358 0.342 0.169 1.000 3.43326 �0.637 1.282 �4.490 14.315 0.335 0.167 1.000 1.898 �0.090 0.789 3.928 12.939 0.260 0.147 1.000 0.383 �1.063 1.483 �12.725 13.329 0.346 0.170 1.000 3.40847 �0.017 0.118 �0.106 1.318 0.003 0.015 1.000 0.481 0.024 0.076 �0.040 1.177 0.004 0.017 1.000 1.085 �0.023 0.139 0.151 1.746 0.004 0.019 1.000 0.55748 �0.005 0.024 �0.071 0.381 0.000 0.005 1.000 0.770 �0.004 0.023 �0.066 0.423 0.000 0.005 1.000 0.537 0.006 0.023 0.105 0.279 0.000 0.003 1.000 0.898

Group II27 �1.103 1.648 �12.533 12.533 0.451 0.194 1.000 2.986 �0.749 1.176 �8.589 8.589 0.230 0.139 1.000 2.739 �1.613 2.243 �19.231 19.231 0.779 0.255 1.000 3.43228 �1.072 1.638 �11.497 11.497 0.467 0.197 1.000 2.874 �0.724 1.216 �7.290 7.290 0.260 0.147 1.000 2.461 �1.565 2.136 �18.733 18.733 0.720 0.245 1.000 3.56929 �1.222 1.759 �13.028 13.028 0.620 0.227 1.000 3.203 �1.237 1.702 �14.445 14.445 0.500 0.204 1.000 3.511 �1.692 2.095 �21.747 21.747 0.745 0.249 0.999 4.54330 �0.687 1.183 �6.726 6.726 0.244 0.143 1.000 2.367 �0.410 0.793 �3.694 3.694 0.112 0.096 1.000 2.004 �1.112 1.702 �12.239 12.239 0.446 0.193 1.000 2.86431 �0.688 1.191 �6.783 6.783 0.244 0.143 1.000 2.349 �0.405 0.785 �3.731 3.731 0.108 0.095 1.000 2.001 �1.123 1.736 �12.287 12.287 0.462 0.196 1.000 2.81432 �0.697 1.194 �7.020 7.020 0.241 0.142 1.000 2.382 �0.429 0.821 �4.234 4.234 0.115 0.098 1.000 2.035 �1.072 1.637 �11.706 11.706 0.419 0.187 1.000 2.87549 0.023 0.114 0.335 0.335 0.003 0.015 1.000 0.673 �0.023 0.091 �1.187 1.187 0.010 0.028 1.000 0.878 �0.012 0.281 1.046 1.046 0.021 0.042 1.000 0.13850 0.017 0.055 0.385 0.385 0.002 0.014 1.000 1.046 �0.071 0.079 �1.336 1.336 0.003 0.015 1.000 6.945 0.036 0.072 0.998 0.998 0.004 0.018 1.000 1.939

Group III33 �0.598 1.249 �3.129 3.129 0.398 0.182 1.000 1.807 �0.045 0.942 6.804 6.804 0.512 0.207 1.000 0.157 �1.059 1.389 �13.146 13.146 0.315 0.162 1.000 3.90634 �0.640 1.276 �4.681 4.681 0.323 0.164 1.000 1.923 �0.112 0.783 3.216 3.216 0.227 0.138 1.000 0.479 �1.057 1.472 �12.637 12.637 0.342 0.169 1.000 3.42135 �0.630 1.280 �4.311 4.311 0.340 0.168 1.000 1.875 �0.074 0.797 4.463 4.463 0.288 0.155 1.000 0.310 �1.058 1.479 �12.657 12.657 0.344 0.169 1.000 3.39951 �0.005 0.026 �0.101 0.101 0.000 0.006 1.000 0.684 �0.002 0.029 0.017 0.017 0.000 0.006 1.000 0.195 0.009 0.023 0.131 0.131 0.000 0.003 1.000 1.46752 �0.006 0.024 �0.076 0.076 0.000 0.004 1.000 0.802 �0.005 0.022 �0.085 0.085 0.000 0.005 1.000 0.731 0.006 0.024 0.102 0.102 0.000 0.003 1.000 0.850

Group IV36 �1.225 1.864 �12.562 12.562 0.648 0.232 1.000 2.892 �0.689 1.408 �3.787 3.787 0.496 0.203 1.000 1.861 �1.788 2.241 �23.019 23.019 0.838 0.264 1.000 4.39137 �0.299 1.156 2.433 2.433 0.508 0.206 1.000 0.888 0.217 0.963 11.769 11.769 0.694 0.241 1.000 0.769 �0.835 1.351 �8.509 8.509 0.301 0.158 1.000 2.60538 �1.683 2.336 �19.894 19.894 0.928 0.278 1.000 3.444 �1.094 1.750 �11.089 11.089 0.567 0.217 1.000 2.656 �2.320 2.870 �30.392 30.392 1.387 0.340 1.000 4.55939 �0.622 1.212 �4.030 4.030 0.351 0.171 1.000 1.983 �0.174 0.908 3.749 3.749 0.371 0.176 1.000 0.646 �1.072 1.450 �12.938 12.938 0.337 0.167 1.000 3.64240 �0.627 1.236 �4.540 4.540 0.316 0.162 1.000 1.952 �0.155 0.841 3.051 3.051 0.271 0.150 1.000 0.624 �1.116 1.608 �13.011 13.011 0.399 0.182 1.000 3.19941 �0.622 1.236 �4.359 4.359 0.325 0.165 1.000 1.930 �0.141 0.829 3.433 3.433 0.286 0.154 1.000 0.572 �1.141 1.655 �13.304 13.304 0.419 0.187 1.000 3.15653 �0.029 0.040 �0.405 0.405 0.001 0.007 1.000 3.477 0.042 0.060 1.117 1.117 0.004 0.018 1.000 3.323 0.001 0.106 �0.563 0.563 0.004 0.019 1.000 0.03354 �0.027 0.058 �0.480 0.480 0.001 0.009 1.000 1.786 0.055 0.064 1.008 1.008 0.002 0.012 1.000 5.363 �0.009 0.047 �0.463 0.463 0.002 0.012 1.000 0.629

a tcrit = 3.106 (a = 0.01).

K.U

lgen,A.H

epbasli/EnergyConversion

andM

anagement

50(2009)

149–156

155

156 K. Ulgen, A. Hepbasli / Energy Conversion and Management 50 (2009) 149–156

(vi) For the province of Izmir: The best results for RMSE =0.072 MJ/m2, MPE = 0.998%, MAPE = 0.998%, SSRE = 0.004,RSE = 0.018, and R2 = 1.000 were obtained from the thirdorder polynomial model (Eq. (50)), while the best result forMBE = �0.012 MJ/m2, and t-critic = 0.138 were calculatedfrom the linear model (Eq. (49)) belonging to the provinceof Izmir.

Evaluating group III:

(vii) For the province of Ankara: The best results for RMSE =0.024 MJ/m2, MPE = �0.076%, MAPE = 0.076%, SSRE = 0.0,RSE = 0.004, and R2 = 1.000 were obtained from the thirdorder polynomial model (Eq. (52)), while the best result forMBE = �0.005 MJ/m2, and t-critic = 0.684 were calculatedfrom the linear model (Eq. (51)) belonging to the provinceof Ankara.

(viii) For the province of Istanbul: The best results for RMSE =0.079 MJ/m2, SSRE = 0.0, RSE = 0.005, and R2 = 1.000 wereobtained from the third order polynomial model (Eq. (52)),while the best result for MBE = �0.002 MJ/m2, MPE =�0.085%, MAPE = 0.085%, and t-critic=0.195 were calculatedfrom the linear model (Eq. (51)) belonging to the province ofIstanbul.

(ix) For the province of Izmir: The best results for MBE =0.006 MJ/m2, MPE = 0.102%, MAPE = 0.102%, SSRE = 0.0,RSE = 0.003, R2 = 1.000, and t-critic = 0.850 were obtainedfrom the third order polynomial model (Eq. (52)), whilethe best result for RMSE = 0.023 MJ/m2, was calculated fromthe linear model (Eq. (51)) belonging to the province ofIzmir.

Evaluating group IV:

(x) For the province of Ankara: The best results for MBE =�0.027 MJ/m2, SSRE = 0.001, R2 = 1.000, and t-critic = 1.786were obtained from the third order polynomial model (Eq.(54)), while the best result for RMSE = 0.040 MJ/m2, MPE =�0.480%, MAPE = 0.480%, and RSE = 0.007 were calculatedfrom the linear model (Eq. (53)) belonging to the provinceof Ankara.

(xi) For the province of Istanbul: The best results for MBE =0.035 MJ/m2, MPE = 1.008%, MAPE = 1.008%, SSRE = 0.002,RSE = 0.012, and R2 = 1.000 were obtained from the thirdorder polynomial model (Eq. (54)), while the best result forRMSE = 0.060 MJ/m2, was calculated from the linear model(Eq. (53)) belonging to the province of Istanbul. However,in both models the values larger than t-crit were obtained.

(xii) For the province of Izmir: The best results for RMSE =0.047 MJ/m2, MPE = �0.463%, MAPE = 0.463%, SSRE = 0.002,RSE = 0.012, and R2 = 1.000, were obtained from the thirdorder polynomial model (Eq. (54)), while the best result forMBE = 0.001 MJ/m2, and t-critic = 0.033 were calculatedfrom the linear model (Eq. (53)) belonging to the provinceof Izmir.

6. Conclusions

Solar energy occupies one of the most important places amongthe various possible alternative energy sources. In the design and

study of solar energy, information on solar radiation and its com-ponents at a given location is very essential. Solar energy technol-ogies offer a clean, renewable and domestic energy source, and areessential components of a sustainable energy future.

Turkey lies in a sunny belt between 36� and 42�N latitudes andis geographically well situated with respect to solar energy poten-tial. In the design and evaluation of solar energy, information onsolar radiation and its components at a given location is needed.In this regard, solar radiation models are of big importance. Itmay be concluded that the eight new models developed by theauthors were found to be reasonably good for all the test methodsto estimate or predict the solar radiation in Turkey and possiblyelsewhere with similar climatic conditions.

References

[1] Dincer I. Renewable energy and sustainable development: a crucial review.Renew Sust Energy Rev 2000;4:157–75.

[2] Elliott D. Renewable energy and sustainable futures. Futures 2000;32:261–74.[3] Ediger VS, Kentel E. Renewable energy potential as an alternative to fossil fuels

in Turkey. Energy Convers Manage 1999;40:743–55.[4] Tasdemiroglu E, Sever R. Estimation of monthly average daily, horizontal

diffuse radiation in Turkey. Energy 1991;16(4):787–90.[5] Tiris M, Tiris C, Ture IE. Correlations of monthly-average daily global, diffuse

and beam radiations with hours of bright sunshine in Gebze, Turkey. EnergyConvers Manage 1996;37(9):1417–21.

[6] Jacovides CP, Hajioannou L, Pashiardis S, Stefanou L. On the diffuse fraction ofdaily and monthly global radiation for the island of Cyprus. Solar Energy1996;56(6):565–72.

[7] Gopinathan KK, Soler A. Diffuse Radiation models and monthly-average, daily,diffuse data for a wide latitude range. Energy 1995;20(7):65–7.

[8] Duffie JA, Beckman WA. Solar engineering of thermal processes. 2nd ed. NewYork: Wiley Inc.; 1991.

[9] Ulgen K, Hepbasli A. Comparison of solar radiation correlation for Izmir,Turkey. Int J Energ Res 2002;26:413–30.

[10] Ulgen K, Hepbasli A. Solar radiation models Part 2: Comparison and developingnew models. Energy Sources 2004;26:521–30.

[11] Haydar A, Balli O, Hepbasli A. Global solar radiation potential, Part 2: Statisticalanalysis. Energy Sources 2006;1:317–26.

[12] Stone RJ. Improved statistical procedure for the evaluation of solar radiationestimation models. Solar Energy 1993;51(4):288–91.

[13] Elagib NA, Alvi SH, Mansell MG. Correlations between clearness index andrelative sunshine duration for Sudan. Renew Energy 1999;7:473–98.

[14] Page JK. The estimation of monthly mean values of daily total short-waveradiation on vertical and inclined surfaces from sunshine records for latitudes40_N–40_S. In: Proceedings of UN conference on new sources of energy, vol. 4,August 21–31, Rome, Italy, United Nations, Paper No. 598; 1961. p. 378–90.

[15] Barbaro S, Cannato G, Coppolina S, Leone C, Sinagra E. Diffuse solar radiationstatistics for Italy. Solar Energy 1981;26:429–35.

[16] Kaygusuz K, Ayhan T. Analysis of solar radiation data for Trabzon, Turkey.Energy Convers Manage 1999;40:545–56.

[17] Haydar A, Balli O, Hepbasli A. Estimating the horizontal diffuse solar radiationover the Central Anatolia region of Turkey. Energy Convers Manage2006;47:2240–9.

[18] Elhadidy MA, Abdel-Nabi DY. Diffuse fraction of daily global radiation atDhahran, Saudi Arabia. Solar Energy 1991;46(2):89–95.

[19] Jain PC. A model for diffuse and global irradiation on horizontal surfaces. SolarEnergy 1990;45:301–408.

[20] Tarhan S, Sari A. Model selection for global and diffuse radiation over theCentral Black Sea (CBS) region of Turkey. Energy Convers Manage2005;46:605–13.

[21] Liu BYH, Jordan RC. The relationship and characteristics distribution of directdiffuse and total radiation. Solar Energy 1960;4(3):1–19.

[22] Erbs DG, Klein SA, Duffie JA. Estimation of the diffuse radiation fraction forhourly, daily and monthly average global radiation. Solar Energy1982;28:293–302.

[23] Helwa NH, Baghdad ABG, El Shafee AMR, El Shenawy ET. Computation of thesolar energy captured by different solar tracking systems. Energy Sources2000;22:35–44.

[24] Cooper PI. The absorption of solar radiation in solar stills. Solar Energy1969;12(3):333–46.