Estimation of global solar radiation under clear sky radiation in Turkey

Data bank

Estimation of global solar radiation underclear sky radiation in Turkey

Inci Turk TogÆ rula,*, Hasan TogÆ rula, Duygu Evinb

aFõrat University, Faculty of Engineering, Chemical Engineering Department, 23279, ElazõgÆ, TurkeybFõrat University, Faculty of Engineering, Mechanical Engineering Department, 23279, ElazõgÆ, Turkey

Received 9 November 1999; accepted 9 December 1999

Abstract

In this study, the usability of clear sky radiation for predicting the average global solarradiation was investigated. For this aim, the various regression analyses were applied byusing �n= �N and �n= �Nnh parameters. Also, equations which represent the two periods of the

year, winter and summer, were developed by using these parameters.The equations developed by using �n= �N and �n= �Nnh have approximately the same results.Having the better values of the equations developed by using the change of summer and

winter is another result.

In addition, the use of the RMSE and MBE in isolation is not an adequate indicator ofmodel performance. Using the t-statistic method and the harmony of results obtained witheach method will prove that the results are reliable 7 2000 Elsevier Science Ltd. All rights

reserved.

Keywords: Clear sky radiation; Estimation of solar radiation; t-statistic

1. Introduction

In many applications of solar energy, the solar irradiance incident on the

0960-1481/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

PII: S0960 -1481 (99)00128 -7

Renewable Energy 21 (2000) 271±287

www.elsevier.com/locate/renene

* Corresponding author. Tel.: +90-424-237-0015; fax: +90-424-218-1907.

E-mail address: [email protected] (I.T. TogÆ rul).

surface of the earth at the location of interest is an important input

parameter. The temporal and spatial ¯uctuations of such irradiance necessitate

a method to predict them. The systematic variation of solar irradiance outside

the Earth's atmosphere makes it possible to introduce many models for such

prediction [1].

However, at locations on the Earth's surface, the solar radiation is also a

function of such variables as the nature and extent of cloud cover, the aerosol and

water vapour content of atmosphere, etc. Good prediction of the actual value of

Nomenclature

a, b empirical constantsa ', b ' empirical constantsGon the extraterrestrial radiation, measured on the plane normal to the

radiation on the nth day of the year (W mÿ2)Gcb the clear sky beam radiation (W mÿ2)Gcd the clear sky di�use radiation (W mÿ2)�H monthly mean daily global radiation on a horizontal surface (MJ mÿ2)�Hc average clear sky radiation for the location and month in question

(MJ mÿ2)�H0 monthly mean daily extraterrestrial radiation (MJ mÿ2)Ic hourly global clear sky radiation (MJ mÿ2)Icb hourly beam clear sky radiation (MJ mÿ2)Icd hourly clear sky di�use radiation (MJ mÿ2)�n monthly average daily hours of bright sunshine�N monthly average of maximum possible daily hours of bright sunshine

(i.e., day length of the average day of the month)�Nnh the monthly mean sunshine duration taking into account the natural

horizon of the siter correlation coe�cient

Greek symbolsos the sunset hour angle, the angular displacement of the sun east or west

of the local meridian due to rotation of the Earth on its axis at 158 perhour (morning negative, afternoon positive), in degrees

d declination anglef latitude, the angular location north or south of the equator, north

positiveyz zenith angle, the between the vertical and the line to the sun, i.e., the

angle of incidence of beam radiation on a horizontal surfacetb the atmospheric transmittance for beam radiationtd the ratio of di�use radiation to the extraterrestrial (beam) radiation on

the horizontal plane

I.T. TogÆrul et al. / Renewable Energy 21 (2000) 271±287272

solar irradiance for a given location requires, in principal, long-term, averagemeteorological data, which are still scarce for developing countries [2±5]. It is,therefore, not always possible to predict the actual value of solar irradiance for agiven location.

There are several formulae that relate global radiation to other climatologicalparameters such as sunshine hours, relative humidity, max. temperature, andaverage temperature.

The ®rst correlation proposed for estimating the monthly average daily globalirradiation is due to Angstrom [6]. The original Angstrom-type regressionequation related monthly average daily radiation to clear day radiation at thelocation in question and average fraction of possible sunshine hours:

�H= �Hc � a� b� �n= �N� �1�

A basic di�culty with Eq. (1) lies in the ambiguity of the terms �n= �N and �Hc: Page[7] and the others have modi®ed the method to base it on extraterrestrial radiationon a horizontal surface rather than on clear day radiation

�H= �H0 � a 0 � b 0� �n= �N� �2�

where �H0 is the extraterrestrial radiation (MJ/m2) and a ' and b ' are constantdepending on location.

In spite of having complication of �Hc calculations, the better results wereobtained by using �Hc instead of �H0 [8].

The major objective of this article is to investigate usability of clear skyradiation to predict and express the average measured values of solar irradianceon a horizontal surface by using various regression analyses in Turkey.

1.1. Estimation of clear sky radiation

Hottel [9] has presented a method for estimating the beam radiation transmittedthrough clear atmospheres which takes into account zenith angle and altitude fora standard atmosphere and four climate types. The atmospheric transmittance forbeam radiation tb is given in the form:

tb � a0 � a1 exp�ÿk= cos yz� �3�

The constant a0, a1 and k for the standard atmosphere with 23 km visibility arefound from a�0, a

�1 and k�, which are given for altitudes less than 2.5 km by

a�0 � 0:4237ÿ 0:00821�6ÿ A�2 �4a�

a�1 � 0:5055� 0:00595�6:5ÿ A�2 �4b�

I.T. TogÆrul et al. / Renewable Energy 21 (2000) 271±287 273

k� � 0:2711� 0:01858�2:5ÿ A�2 �4c�where A is the altitude of the observer in kilometers.

Correction factors are applied to a�0, a�1 and k� to allow for changes in climate

types.The correction factors r0 � a0=a

�0, r1 � a1=a

�1 and rk � k=k� are given in Table 1.

Thus, the transmittance of this standard atmosphere for beam radiation can bedetermined for any zenith angle and any altitude up to 2.5 km. The clear skybeam radiation (Gcb, W mÿ2) is then

Gcb � Gontb �5�where Gon is the extraterrestrial radiation, measured on the plane normal to theradiation on the nth day of the year and given in the following form (W mÿ2):

Gon � 1367

�1� 0:033 cos

360n 0

365

��6�

The clear sky horizontal ``beam'' radiation is

Gcb � Gontb cos yz �7�It is also necessary to estimate the clear sky di�use radiation on a horizontalsurface to get the total radiation Liu and Jordan [10] developed in an empiricalrelationship between the transmission coe�cient for beam and di�use radiationfor clear days:

td � 0:271ÿ 0:294tb �8�where td is the ratio of di�use radiation to the extraterrestrial (beam) radiation onthe horizontal plane.

The clear sky di�use radiation Gcd (W mÿ2)

Gcd � Gontd cos yz �9�For periods of an hour, the clear sky horizontal beam radiation and clear sky

Table 1

Correction factors for climate types

Climate type r0 r1 rk

Tropical 0.95 0.98 1.02

Midlatitude summer 0.97 0.99 1.02

Subarctic summer 0.99 0.99 1.01

Midlatitude winter 1.03 1.01 1.00


di�use radiation is

Icb � Iontb cos yz �10�and

Icd � Iontd cos yz �11�

2. Data

The objective of this study was to develop some statistical relations to estimatemonthly mean daily global solar radiation by using clear sky radiation in Turkey.

For this aim, the data obtained from the observation stations in Antalya, I.zmir,

Ankara, Yenihisar (Aydõn) and Yumurtalõk (Adana) and the measured values inElazõgÆ were used. The global solar radiation was measured by CM11 model Kipppyranometer in observation stations, and by CM6 model Kipp pyranometer inElazõgÆ . The geographical location of the six cities are shown in Table 2. �Hc valuescalculated for six cities are given in Table 3. The correlations have been studied byusing both the ratios �n= �N and �n= �Nnh, where �Nnh is the monthly mean sunshineduration taking into account the natural horizon of the site, and is given in thefollowing equation [11]

1

�Nnh

� 0:8706�N� 0:0003 �12�

In this work, we developed equations to the estimate monthly mean global solarradiation �H, applying various regression types to parameters such as; �n= �N, �n= �Nnh:

The e�ects which the solar radiation is exposed to until it reaches to the earthfrom the atmosphere change a lot for winter and summer. Therefore, it will be atrue approach to compute the reaching monthly mean global solar radiation forboth summer and winter. Thus, the relation between �H= �Hc � f� �n= �N� and �H= �Hc �

Table 2

Geographical location of six cities used in this study

Station Longitude (8) Latitude (8) Altitude (m)

Antalya 30.42 36.53 54

I.zmir 27.10 38.26 29

Ankara 32.52 39.57 891

Yenihisar (Aydõn) 27.16 37.23 56

Yumurtalõk (Adana) 35.47 36.46 27

ElazõgÆ 39.14 38.40 1067


f � �n= �Nnh� were investigated by di�erent regression analyses both of two periods,too.

The values of �H were estimated by using these equations. These values werethen compared with original measured values for each city.

3. Equations

The following equations were obtained when we investigated the relation

Table 3�Hc values for each city

Months Antalya I.zmir Ankara Yenihisar (Aydõn) Yumurtalõk (Adana) ElazõgÆ

January 10.062 9.261 9.811 9.576 10.048 10.564

February 13.729 12.958 13.835 13.442 13.704 14.604

March 18.663 17.996 19.300 18.425 18.621 20.041

April 23.021 22.577 24.293 22.870 22.960 24.884

May 26.367 26.159 28.265 26.323 26.292 28.734

June 27.606 27.506 29.782 27.607 27.523 30.187

July 26.953 26.806 28.997 26.934 26.874 29.429

August 24.335 24.006 25.876 24.239 24.269 26.413

September 19.979 19.408 20.810 19.780 19.934 21.483

October 15.130 14.382 15.380 14.850 15.101 16.134

November 10.921 10.128 10.760 10.620 10.905 11.514

December 9.061 8.260 8.720 8.754 9.050 9.461

Fig. 1. The scatter of �H= �Hc vs �n= �N (* Antalya, w I.zmir, Q Ankara, q Yenihisar (Aydõn), R

Yumurtalõk (Adana), r ElazõgÆ ).


between �n= �N and �H= �Hc by trying di�erent regression types. The scatter of themonthly mean values of �n= �N and �H= �Hc are given in Fig. 1

�H= �Hc � 0:5917� 0:4279� �n= �N� �13�

�H= �Hc � 0:9896� �n= �N�0:2828 �14�

�H= �Hc � 0:596 e0:5821� �n= �N� �15�

�H= �Hc � 0:2036 ln� �n= �N� � 0:9615 �16�

�H= �Hc � 0:3646� 1:4291� �n= �N� ÿ 0:977� �n= �N�2 �17�

�H= �Hc � 0:1987� 2:6683� �n= �N� ÿ 3:6445� �n= �N�2 � 1:732� �n= �N�3 �18�The results of regression analyses applied for winter (January±March andOctober±December intervals) and summer (April±September) are given below

�H= �Hc � 0:5361� 0:5615� �n= �N� for winter �19a�

�H= �Hc � 0:7225� 0:2227� �n= �N� for summer �19b�

�H= �Hc � 0:222 ln� �n= �N� � 0:9809 for winter �20a�

�H= �Hc � 0:1424 ln� �n= �N� � 0:9311 for summer �20b�

�H= �Hc � 0:2384� 2:1498� �n= �N� ÿ 1:8664� �n= �N�2 for winter �21a�

�H= �Hc � 0:5771� 0:6843� �n= �N� ÿ 0:3544� �n= �N�2 for summer �21b�

�H= �Hc � 0:3081� 1:4933� �n= �N� ÿ 0:0839� �n= �N�2 ÿ 1:4513� �n= �N�3

for winter�22a�

�H= �Hc � 0:4356� 1:3939� �n= �N� ÿ 1:4988� �n= �N�2 � 0:5954� �n= �N�3

for summer�22b�

�H= �Hc � 0:5451 e0:7981� �n= �N� for winter �23a�


�H= �Hc � 0:7295 e0:2633� �n= �N� for summer �23b�

�H= �Hc � 1:0302� �n= �N�0:3204 for winter �24a�

�H= �Hc � 0:9337� �n= �N�0:1688 for summer �24b�The linear regression analyses to investigate the usability of �n= �Nnh ratio forcomputing the monthly mean global solar radiation were made. The results ofthese analyses can be seen as the following equations. The e�ect of �n= �Nnh on the�H= �Hc is given in Fig. 2.

�H= �Hc � 0:592� 0:4889� �n= �Nnh� �25�

�H= �Hc � 0:2034 ln� �n= �Nnh� � 0:9887 �26�

�H= �Hc � 1:0277� �n= �Nnh�0:2825 �27�

�H= �Hc � 0:5962 e0:665� �n= �Nnh� �28�

�H= �Hc � 0:3653� 1:6322� �n= �Nnh� ÿ 1:2759� �n= �Nnh� 2 �29�

�H= �Hc � 0:1987� 3:0544� �n= �Nnh� ÿ 4:7764� �n= �Nnh� 2 � 2:5987� �n= �Nnh�3 �30�

Fig. 2. The variation of �H= �Hc vs �n= �Nnh (* Antalya, w I.zmir, Q Ankara, q Yenihisar (Aydõn), R

Yumurtalõk (Adana), r ElazõgÆ ).


The e�ect of �n= �Nnh ratio on �H= �Hc ratio for both winter (January±March andOctober±December intervals) and summer (April±September intervals) isinvestigated by various regression analyses and the results are given below.

�H= �Hc � 0:5362� 0:6423� �n= �Nnh� for winter �31a�

�H= �Hc � 0:7225� 0:2567� �n= �Nnh� for summer �31b�

�H= �Hc � 0:2219 ln� �n= �Nnh� � 1:0107 for winter �32�

�H= �Hc � 0:1425 ln� �n= �Nnh� � 0:9502 for summer �32a�

�H= �Hc � 0:2385� 2:4604� �n= �Nnh� ÿ 2:445� �n= �Nnh� 2 for winter �33a�

�H= �Hc � 0:577� 0:7825� �n= �Nnh� ÿ 0:4631� �n= �Nnh� 2 for summer �33b�

�H= �Hc � 0:3077� 1:7137� �n= �Nnh� ÿ 0:1251� �n= �Nnh� 2 ÿ 2:1612� �n= �Nnh�3

for winter�34a�

�H= �Hc � 0:4356� 1:5935� �n= �Nnh� ÿ 1:9584� �n= �Nnh� 2 � 0:8893� �n= �Nnh�3

for summer�34b�

�H= �Hc � 0:5453 e0:9128� �n= �Nnh� for winter �35a�

�H= �Hc � 0:729 e0:3012� �n= �Nnh� for summer �35b�

�H= �Hc � 1:0755� �n= �Nnh�0:3202 for winter �36a�

�H= �Hc � 0:9552� �n= �Nnh�0:1689 for summer �36b�

4. The comparison methods

In this study, two statistical tests, root mean square error (RMSE) and meanbias error (MBE), and t-statistic were used to evaluate the accuracy of thecorrelations described above.


4.1. Root mean square error

The root mean square error is de®ned as

RMSE � 1

n

Xni�1

d 2i

!1=2

�37�

where n is the number of data pairs and di is the di�erence between ith estimatedand ith measured values.

This test provides information on the short-term performance of thecorrelations by allowing a term by term comparison of the actual deviationbetween the calculated value and the measured value. The smaller the value, thebetter the model's performance. However, a few large errors in the sum canproduce a signi®cant increase in RMSE.

4.2. Mean bias error

The mean bias error is de®ned as

MBE � 1

n

Xni�1

di �38�

This test provides information on the long-term performance. A low MBE isdesired. A positive value gives the average amount of over-estimation in thecalculated value and vice versa. A drawback of this test is that over-estimation ofan individual observation will cancel under-estimation in a separate observation.

It is obvious that each test by itself may not be an adequate indicator of amodel's performance. It is possible to have a large RMSE value and at the sametime a small MBE (a large scatter about the line of perfect estimation). On theother hand, it is also possible to have a relatively small RMSE and a relativelylarge MBE (a consistently small over- or under-estimation).

However, although these statistical indicators generally provide a reasonableprocedure to compare models, they do not objectively indicate whether a model'sestimates are statistically signi®cant, i.e., not signi®cantly di�erent from theirmeasured counterparts. In this article, an additional statistical indicator, the t-statistic, was used. This statistical indicator allows models to be compared and atthe same time indicate whether or not a model's estimates are statisticallysigni®cant at a particular con®dence level [12]. It was seen that the t-statistic usedin addition to the RMSE and MBE gave more reliable and explanatory results[13].

4.3. Derivation of the t-statistic from the RMSE and MBE

The t-statistic is de®ned as [14];


t �

1

n

Xni�1

di

S

n1=2

�39�

where S is the standard deviation of the di�erences between estimated andmeasured values and is given by:

S 2 �nXni�1

di ÿ Xn

i�1di

!2

n�nÿ 1� �40�

Rearranging Eqs. (29) and (30) gives;

Xni�1

d 2i � nRMSE2 �41�

Xni�1

di � nMBE �42�

Combining Eqs. (34) and (31) yields;

t � MBE

S

n1=2

�43�

Using Eqs. (33) and (34) in conjunction with Eq. (32) gives:

S ��n�RMSE2 ÿMBE2�

nÿ 1

�1=2

�44�

Substituting for S in Eq. (35) yields;

t �� nÿ 1�MBE2

RMSE2 ÿMBE2

�1=2�45�

The smaller the value of t, the better is the model's performance. To determinewhether a model's estimates are statistically signi®cant, one simply has todetermine a critical t value obtainable from standard statistical tables, i.e., ta/2 atthe a level of signi®cance and (n ÿ 1) degrees-of-freedom. For the model'sestimates to be judged statistically signi®cant at the 1ÿa con®dence level, thecalculated t value must be less than the critical t value.


5. Results and discussion

If we group the developed equations which include di�erent variables and twoperiods of year, the comparison will be more detailed.

GROUP I [Eqs. (13)±(18)]: the developed equations which include the wholeyear by using �n= �N parameter.GROUP II [Eqs. (19)±(24)]: the developed equations for both winter andsummer by using �n= �N parameter.GROUP III [Eqs. (25)±(30)]: the developed equations which include thewhole year by using �n= �Nnh variable.GROUP IV [Eqs. (31)±(36)]: the developed equations for two periods ofyear, summer and winter, by using �n= �Nnh parameter.

It was shown in Table 4 that the good results were not seen in the short term,but, relatively good results were seen in the long term. Eq. (16) has the best resultamong the equations developed in Group I. Similarly, the best MBE and RMSEvalues were obtained by Eqs. (23)±(24) and Eq. (21) in Group II, respectively.

When we compared the developed equations in Group III, we saw that the best

Table 4

RMSE and MBE for the equation developed in Turkey

Equation number MBE RMSE

13 ÿ0.563 1.569

14 ÿ0.560 1.469

15 ÿ0.582 1.649

16 ÿ0.553 1.435

17 ÿ0.572 1.373

18 ÿ0.579 1.404

19 0.031 1.005

20 0.029 0.947

21 0.017 0.908

22 0.017 0.913

23 0.001 1.044

24 0.004 0.970

25 0.562 1.568

26 ÿ0.552 1.434

27 ÿ0.559 1.469

28 ÿ0.582 1.648

29 ÿ0.571 1.372

30 ÿ0.579 1.404

31 0.032 1.005

32 ÿ0.677 1.808

33 0.017 0.908

34 0.018 0.913

35 ÿ0.533 1.543

36 0.005 0.970


MBE and RMSE values were seen in Eqs. (26) and (29), respectively. Similarly,

the best MBE and RMSE values in Group IV were obtained by Eq. (36) and Eqs.

(33) and (34), respectively.

When we investigated all the equations, we saw that although the best MBE

value was seen in Eq. (23), the results of Eqs. (24) and (36) were good too. The

best RMSE values were obtained with Eqs. (21) and (33) which have the same

results. In addition, both the MBE and RMSE values of Eqs. (22) and (34) which

have close results were good.

Considering the whole country and each city, the performance of developed

equations are di�erent. Therefore, the MBE and RMSE values of the developed

equations for each city were calculated. The results of the statistical comparison

are given in Table 5.

At the ®rst view, it was seen that the MBE and RMSE values of Table 5 are

higher than Table 4. Each equation according to city was compared with the

equations in its groups. The results obtained were set up in order below.

Antalya: Eq. (17) in Group I has the best result. Similarly, the results of Eq.

(20) in Group II are better. A relatively good result was obtained by Eq. (29) in

Group III. The best MBE and RMSE values in Group IV were obtained by

Eqs. (31) and (36), and Eqs. (33) and (34), respectively.

I.zmir: The best results in Groups I and II were obtained by Eqs. (17) and (18),

and Eqs. (19) and (23), respectively. Similarly, the best MBE and RMSE values

in Groups III and IV were obtained by Eqs. (29), (31) and (36), respectively.

Ankara: Eqs. (14) and (22) showed good results among the equations developed

in Groups I and II, respectively. Eqs. (27) and (30) in Group III and Eqs. (34)

and (35) in Group IV showed better results.

Yenihisar (Aydõn): Eq. (15) in Group I, Eq. (21) in Group II, Eqs. (25) and (28)

in Group III, and Eqs. (33) and (34) in Group IV showed relatively good

results.

Yumurtalõk (Adana): The equations which give the better results in Groups I±

IV were Eqs. (17), (22), (29), (30) and (34), respectively.

ElazõgÆ : Eqs. (17), (20, (29) and (35) showed the lowest errors in Groups I±IV,

respectively.

Although the models give good results for the whole country (Table 4), the

highest errors were obtained for cities (Table 5). These tables did not include

adequate information about performance of developed equations. In addition to

the above-mentioned investigation of results of the t-statistic method (Table 5),

which is applied to the equation developed, can be useful. The critical t-values are

shown in Table 6. t-values higher than critical t-values show that the equation has

no statistical signi®cance.

Having higher t-values than critical t-values, a lot of equations are statistically

signi®cant in Table 6. Although the equations in Groups I and III had completely

unlogical results, the equations in Groups II and IV showed completely good and

logical results in Antalya and I.zmir.


Table

5

TheMBEandRMSEvalues

oftheequationsdeveloped

foreach

city

Equationnumber

Antalya

I. zmir

Ankara

Yenihisar(A

ydõn)

Yumurtalõk(A

dana)

Elazõg Æ

MBE

RMSE

MBE

RMSE

MBE

RMSE

MBE

RMSE

MBE

RMSE

MBE

RMSE

13

ÿ1.503

1.995

ÿ1.039

1.457

0.632

0.950

ÿ0.557

0.936

ÿ1.049

1.101

1.518

1.593

14

ÿ1.400

1.885

ÿ0.983

1.378

0.469

0.792

ÿ0.566

0.893

ÿ0.955

0.995

1.340

1.495

15

1.560

2.083

ÿ1.118

1.505

0.588

0.866

ÿ0.439

1.028

ÿ1.123

1.170

1.700

1.843

16

ÿ1.371

1.828

ÿ0.919

1.362

0.504

0.894

ÿ0.670

0.883

ÿ0.903

0.955

1.184

1.377

17

ÿ1.228

1.650

ÿ0.818

1.300

0.398

1.008

ÿ0.945

1.100

ÿ0.750

0.791

0.667

1.461

18

ÿ1.312

1.709

ÿ0.817

1.361

0.387

1.076

ÿ0.914

1.055

ÿ0.786

0.873

0.865

1.322

19

ÿ0.228

0.965

0.033

0.738

0.545

1.099

ÿ0.894

1.068

ÿ0.983

1.031

1.081

1.305

20

ÿ0.244

0.873

0.119

0.718

0.472

1.024

ÿ0.840

0.995

ÿ0.880

0.943

0.975

1.310

21

ÿ0.316

0.772

0.220

0.725

0.379

1.015

ÿ0.759

0.895

0.727

0.830

0.899

1.346

22

ÿ0.322

0.776

0.227

0.730

0.356

1.021

ÿ0.766

0.905

ÿ0.702

0.811

0.931

1.357

23

ÿ0.220

1.058

ÿ0.034

0.765

0.484

1.068

ÿ0.932

1.118

ÿ1.040

1.084

1.068

1.311

24

ÿ0.240

0.938

0.059

0.732

0.416

0.992

ÿ0.871

1.034

ÿ0.922

0.981

0.980

1.311

25

ÿ1.501

1.993

ÿ1.037

1.456

0.634

0.951

ÿ0.556

0.936

ÿ1.049

1.100

1.519

1.594

26

ÿ1.371

1.827

ÿ0.919

1.362

0.504

0.895

ÿ0.671

0.884

ÿ0.903

0.955

1.184

1.376

27

ÿ1.400

1.884

ÿ0.982

1.378

0.469

0.793

ÿ0.566

0.893

ÿ0.955

0.994

1.340

1.494

28

ÿ1.560

2.081

ÿ1.117

1.504

0.588

0.867

ÿ0.438

1.029

ÿ1.124

1.170

1.701

1.844

29

ÿ1.226

1.648

ÿ0.817

1.298

0.400

1.009

ÿ0.944

1.099

ÿ0.749

0.790

0.668

1.462

30

ÿ1.312

1.709

ÿ0.817

1.361

0.387

1.077

ÿ0.914

1.055

0.787

0.874

0.866

1.322

31

ÿ0.227

0.965

0.034

0.738

0.546

1.099

ÿ0.893

1.067

ÿ0.984

1.031

1.082

1.306

32

ÿ0.537

1.269

ÿ0.368

0.995

ÿ0.497

2.412

ÿ2.500

3.254

ÿ1.750

2.326

ÿ0.295

1.786

33

ÿ0.316

0.771

0.220

0.725

0.380

1.015

ÿ0.758

0.893

ÿ0.727

0.830

0.900

1.346

34

ÿ0.322

0.776

0.227

0.730

0.356

1.020

ÿ0.765

0.904

ÿ0.702

0.811

0.932

1.357

35

ÿ0.438

1.273

ÿ0.421

0.928

ÿ0.137

1.824

ÿ2.174

2.628

ÿ1.626

1.956

ÿ0.060

1.618

36

ÿ0.239

0.937

0.060

0.733

0.416

0.992

ÿ0.869

1.032

ÿ0.922

0.980

0.982

1.311


Ankara has the least unpractical equations. Only Eqs. (13)±(15), (25), (27) and(28) are not statistically signi®cant.

The number of equations being statistically insigni®cant and being statisticallysigni®cant were 15 and 9 in Yenihisar (Aydõn), respectively. Eqs. (13)±(16), (25)±(28) and (32) were the usable equations.

It was seen that Adana includes the most insigni®cant equations. Only theequations having statistical signi®cance were Eqs. (22), (32), (34) and (35).

The equations showing the best performance for each city within theequations which are statistically signi®cant can be found by selecting thelowest t-values.

The equations having the best performance were Eq. (23) in Antalya, Eqs.(19), (23) and (31) in I

.zmir, Eq. (35) in Ankara, Eqs. (15) and (28) in

Yenihisar (Aydõn), Eq. (32) in Yumurtalõk (Adana) and Eq. (35) in ElazõgÆ .

Table 6

Critical t-values and the results of t-statistic analyses for each city

Equation number Antalya I.zmir Ankara Yenihisar (Aydõn) Yumurtalõk (Adana) ElazõgÆ

13 6.577 4.659 3.566 1.816 7.025 10.410

14 6.370 4.659 2.936 2.005 7.646 6.709

15 6.494 5.084 3.692 1.157 7.692 7.932

16 6.511 4.187 2.732 2.854 6.496 5.593

17 6.393 3.713 1.720 4.104 6.648 1.701

18 6.886 3.436 1.540 4.242 4.618 2.870

19 1.395 0.207 2.285 3.752 7.076 4.892

20 1.674 0.773 2.080 3.864 5.083 3.697

21 2.583 1.458 1.611 3.932 4.057 2.974

22 2.619 1.500 1.489 3.885 3.855 3.131

23 1.219 0.202 2.032 3.699 7.602 4.661

24 1.519 0.372 1.846 3.824 6.163 3.735

25 6.580 4.656 3.571 1.809 7.060 10.419

26 6.517 4.188 2.731 2.854 6.526 5.597

27 6.377 4.660 2.936 2.004 7.691 6.713

28 6.501 5.083 3.693 1.153 7.738 7.927

29 6.390 3.709 1.727 4.104 6.679 1.705

30 6.892 3.437 1.539 4.247 4.625 2.876

31 1.393 0.209 2.287 3.751 7.093 4.901

32 2.681 1.821 0.842 2.938 2.554 0.555

33 2.582 1.460 1.613 3.932 4.059 2.979

34 2.616 1.501 1.491 3.887 3.867 3.136

35 2.104 2.334 0.302 3.608 3.346 0.124

36 1.514 0.377 1.851 3.821 6.182 3.745

Critical t 2.720 2.830 2.920 3.500 4.030 3.110


When Table 5 is compared with Table 6, we can say that the results are

not harmonious with each other, only in Antalya and Yumurtalõk (Adana),

but are harmonious with each other in the other cities.

A summary of the developed equations which gave the better results

according to cities and the whole country can be clearly seen in Table 7.

As can be understood from Table 7, the equations which include two periods of

the year, summer and winter, should be preferred because of having the best

results.

Having approximately the same performance, comparison of �n= �N and �n= �Nnh

ratios is more di�cult.

6. Conclusions

In this study, ®rst of all it was seen that the clear sky radiation can be used to

estimate the global radiation in Turkey.

It was seen that the equations which include the summer and winter periods

gave better results than the others in all of the developed equations. It is a

predictable result that the performance of the equations are di�erent for the whole

of Turkey, and for the cities.

Eqs. (21)±(24), (33), (34) and (36) gave the best results in all of the developed

equations. It can be said that all the analyses were harmonious except for the

linear and logarithmic ones.

Finally, these results clearly indicate that reliance on the RMSE and MBE used

separately can lead to a wrong decision in selecting the best model suited from

candidate models and that the use of the RMSE and MBE in isolation is not an

adequate indicator of model performance. Therefore, the t-statistic should be used

in conjunction with these two indicators to better evaluate a model's performance.

Table 7

Results of the compilations

Location Equation number

The whole country 23, 24, 36, 21(33), 22(34)

Antalya 23

I.zmir 19, 23, 31

Ankara 35

Yenihisar (Aydõn) 15, 28

Yumurtalõk (Adana) 32

ElazõgÆ 35


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Estimation of global solar radiation under clear sky radiation in Turkey