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INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2009; 33:538–552Published online 10 November 2008 in Wiley InterScience(www.interscience.wiley.com). DOI: 10.1002/er.1474

SHORT COMMUNICATION

Evaluation and comparison of hourly solar radiation models

M. Jamil Ahmad�,y and G. N. Tiwari

Centre for Energy Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

SUMMARY

In this paper, an attempt has been made to develop a new model to evaluate the hourly solar radiation for compositeclimate of New Delhi. The comparison of new model for hourly solar radiation has been carried out by using variousmodel proposed by others. The root mean square error (RMSE) and mean bias error (MBE) have been used to comparethe accuracy of new and others model. The results show that the ASHRAE and new proposed model estimate hourlysolar radiation better for composite climate of New Delhi in comparison to other models. Hourly solar radiationestimated by constants obtained by new model (modified ASHRAE model) for composite climate of India is fairlycomparable with measured data. The percentage mean bias error with new constants for New Delhi was found as low as0.15 and 0% for hourly beam and diffuse radiation, respectively. There is a 1.9–8.5% RMSE between observed andpredicted values of beam radiation using new constants for clear days. The statistical analysis has been used for thepresent study. Copyright r 2008 John Wiley & Sons, Ltd.

KEY WORDS: solar radiation; beam radiation; diffuse radiation

1. INTRODUCTION

The solar radiation, through atmosphere, reaching

the earth’s surface can be classified into two

components: beam radiation and diffuse radiation.

Beam radiation is the solar radiation propagating

along the line joining the receiving surface and the

sun. It is also referred to as direct radiation.

Diffuse radiation is the solar radiation scattered by

aerosols, dust and molecules, it does not have a

unique direction. The total radiation is the sum of

the beam and diffuse radiation and is sometimes

referred to as the global radiation. When the

amount of diffuse radiation reaching the earth’ssurface is less than or equal to 25% of globalradiation, the sky is termed as clear sky.

Solar radiation available on the Earth’s surfacedepends on local climatic conditions. Knowledge ofmonthly mean daily global and diffuse radiation onhorizontal surface is essential to design solar energydevices. Further, there is a need to have knowledgeof hourly solar radiation on horizontal surfaces forbetter performance of solar energy devices. Hourlyvalues of solar radiation enable us to derive veryprecise information about the performance of solarenergy systems [1]. Such hourly data is useful for

*Correspondence to: M. Jamil Ahmad, Centre for Energy Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi110016, India.yE-mail: [email protected]

Received 28 May 2008

Revised 8 September 2008

Accepted 13 September 2008Copyright r 2008 John Wiley & Sons, Ltd.

engineers, architects and designers of solar systemsto make effective use of solar energy.

Most locations in India receive abundant solarradiation and hence solar energy technology canbe profitably applied to these regions. The solarradiation data are either obtained fromexperimental measurements of the global anddiffuse radiation or obtained from developedempirical relation for a given latitude. In India,the Indian Meteorology Department (IMD),Government of India, measures sunshineduration, global radiation and diffuse radiationat selected locations. The measured data availablefrom IMD of 11 years have been compiled forpresent study and is given in Table I. Table I givesthe monthly average values of hourly global anddiffuse radiation.

The first attempt to analyse the hourly globalradiation data was made by Whiller [2] and Hotteland Whiller [3]. They have used the data of various

locations in U.S.A., to obtain the variation ofhourly to daily radiation ratio against sunset hourangle. Liu and Jordan [4] have extended the daylength of these variations. By using the correcteddata of five U.S. locations, Collares-Pereira andRabl [5] have developed an analytical expressionfor hourly to daily global radiation ratio interms of sunset hour angle. The hourlycorrelation between daily diffuse transmissioncoefficient and daily clearness index obtainedby Orgill and Hollands [6], Bruno [7] and Bugler[8] can be used to estimate the ratio of hourlydiffuse to hourly global radiation. Liu and Jordan[4] have determined the hourly distribution ofdiffuse radiation from daily radiation. Gopinathan[1] has also obtained the same from sunshinehour. No general formula is available yet forprediction of the solar radiation reaching theEarth’s surface over a given period of time atany location [9].

Table I. Average hourly global and diffuse radiation (Wm�2) in (a) January (b) June for allweather types for New Delhi.

Weather type

a b c d

Time Total Diffuse Total Diffuse Total Diffuse Total Diffuse

(a) January8 132.99 52.60 119.58 52.75 71.11 64.16 51.20 48.169 355.56 86.28 332.50 102.57 235.55 146.66 140.11 107.6710 554.69 107.29 516.25 123.09 360.00 195.56 237.11 175.6611 680.73 121.53 650.41 149.46 457.78 220.00 301.78 221.0012 726.74 126.39 708.75 155.32 515.55 226.12 379.92 246.5013 733.85 136.63 723.33 161.18 515.55 226.12 379.92 255.0014 656.08 128.30 650.41 155.32 462.22 210.84 328.72 240.8315 500.00 110.94 498.75 128.94 353.34 180.28 261.36 187.0016 311.46 90.28 315.00 96.71 217.78 122.22 161.67 138.8317 106.42 41.84 110.84 46.88 71.11 51.94 45.80 42.50

(b) June8 436.67 123.89 433.34 198.33 358.33 277.77 235.12 169.569 637.22 149.44 641.34 250.83 555.56 350.70 350.12 251.3110 802.22 157.22 794.45 277.08 727.78 378.47 454.88 360.3111 915.00 158.89 912.89 297.50 816.67 416.66 595.44 405.7212 951.67 167.78 999.55 300.42 833.33 434.03 672.12 454.1713 946.11 185.00 996.66 335.41 861.11 423.61 682.34 481.4214 882.78 180.56 912.89 315.00 763.89 402.78 631.22 448.1115 765.56 176.11 808.89 291.67 688.89 385.41 536.66 393.6116 611.67 142.78 635.55 274.17 538.89 347.22 426.78 330.0317 420.00 116.11 416.00 207.08 333.33 246.53 281.12 260.39

EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 539

Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552

DOI: 10.1002/er

The hourly solar radiation calculated fordifferent locations in India by ASHRAE modelpredicts higher beam radiation and lower diffuseradiation [10]. This may be due to the fact that theASHRAE model has been developed for clear skycondition. Nijigorodov [11] has modified thevalues of empirical coefficients of ASHRAEmodel valid only for climatic conditions ofBotswana, Namibia and Zimbabwe. This modelgives large error for composite climate of NewDelhi. The modified ASHRAE models by Machlerand Iqbal [12] and Parishwad et al. [13] do notvalidate the measured data of climatic conditionsof New Delhi (latitude: 28.581N; longitude:77.021E; elevation: 216m above msl).

The objective of the present study is to developa new model based on ASHRAE for different skyconditions to estimate hourly global (I) and diffuse(Id) radiation on a horizontal surface. The analysishas been done for the following four types ofweather conditions.

(a) Clear day (blue sky): If diffuse radiation is lessthan or equal to 25% of global radiation andsunshine hour is more than or equal to 9 h.

(b) Hazy day(fully): If diffuse radiation is less than50% or more than 25% of global radiation andsunshine hour is between 7 and 9h.

(c) Hazy and cloudy (partially): If diffuse radiationis less than 75% or more than 50% of globalradiation and sunshine hour is between 5 and 7h.

(d) Cloudy day (fully): If diffuse radiation is morethan 75% of global radiation and sunshinehour is less than 5 h.

The above four conditions constitute thecomposite climate of New Delhi [14].

Table II gives the average number of daysunder different types of weather conditions in eachmonth.

2. EXISTING MODELS

2.1. ASHRAE model

By using ASHRAE model [10], the hourly globalradiation (I), hourly beam radiation in direction ofrays (IN) and hourly diffuse radiation (Id) on thehorizontal surface on a clear day are calculated byusing the following equations:

I ¼ IN cos yz þ Id ð1Þ

IN ¼ A exp½�B= cos yz� ð2Þ

Id ¼ CIN ð3Þ

where the values of the constants A, B and C aregiven in Table III(a).

yz is the zenith angle, which depends upon thelatitude of the location (f), hour angle (o) andsolar declination (d), and is evaluated from thefollowing equation:

cos yz ¼ sinf: sin dþ cosf: cos d: coso ð4Þ

Further, solar declination (d) is obtained from

d ¼ 23:45 sin½360ð284þ nÞ=365� ð5Þ

The hour angle (o) is an angular measure of timeand is equivalent to 151 per hour. It is measuredfrom noon-based local apparent time (LAT) fromthe following equation

o ¼ 15:0ð12:0� LATÞ ð6Þ

LAT value is obtained from the standard time(ST) by using the following relation

LAT ¼ STþ ET� 4:ðSTL� lÞ ð7Þ

where STL is standard meridian for the local timezone (For India, its value is 811540), l is thelongitude of the location and E is the equation oftime correction (in minutes) given as

E ¼ 229:2ð0:000075þ 0:001868 cosB� 0:032077 sinB� 0:014615 cos 2B� 0:04089 sin 2BÞ ð8Þ

Table II. Average number of days under different weather types in different months during 1991–2001 for New Delhi.

Weather Jan Feb March April May June July Aug Sep Oct Nov Dec

a 3 3 5 4 4 3 2 2 7 5 6 3b 8 4 6 7 9 4 3 3 3 10 10 7c 11 12 12 14 12 14 10 7 10 13 12 13d 9 9 8 5 6 9 17 19 10 3 2 8

M. J. AHMAD AND G. N. TIWARI540


DOI: 10.1002/er

TableIII.

(a)Evaluatedvalues

ofA,BandC

forvariousmodelsand(b)evaluatedvalues

ofA,B,C

andD

for(a)weather

type‘a’,(b)weather

type

‘b’,(c)weather

type‘c’and(d)weather

type‘d’atNew

Delhi.

Months

Parameter

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

(a)ASH-R

AE

model

A1230

1215

1186

1136

1104

1088

1085

1107

1152

1193

1221

1234

B0.142

0.144

0.156

0.180

0.196

0.205

0.207

0.201

0.177

0.160

0.149

0.142

C0.058

0.060

0.071

0.097

0.121

0.134

0.136

0.122

0.092

0.073

0.063

0.057

Nijigorodovmodel

A1163

1151

1142

1146

1152

1157

1158

1152

1150

1156

1167

1169

B0.177

0.174

0.170

0.165

0.162

0.160

0.159

0.164

0.167

0.172

0.174

0.177

C0.114

0.112

0.110

0.105

0.101

0.098

0.100

0.103

0.107

0.111

0.113

0.115

MachlerandIqbalmodel

A1202

1187

1164

1130

1106

1092

1093

1107

1136

1166

1190

1204

B0.141

0.142

0.149

0.164

0.177

0.185

0.186

0.182

0.165

0.152

0.144

0.141

C0.103

0.104

0.109

0.120

0.130

0.137

0.138

0.134

0.121

0.111

0.106

0.103

Parishwadet

al.model

A610.00

652.20

667.86

613.35

558.39

340.71

232.87

240.80

426.21

584.73

616.60

622.52

B0.000

0.010

0.036

0.121

0.200

0.428

0.171

0.148

0.074

0.020

0.008

0.000

C0.242

0.249

0.299

0.395

0.495

1.058

1.611

1.624

0.688

0.366

0.253

0.243

(b)Weather

type‘a’

A1100.6

1095.8

1065.1

1017.4

1058.3

953.7

873.7

836.8

949.2

1148.6

861.9

914.9

B0.1137

0.1715

0.205

0.212

0.286

0.202

0.225

0.205

0.178

0.299

0.075

0.082

C0.176

0.195

0.224

0.251

0.214

0.274

0.721

0.243

0.223

0.315

0.379

0.264

D�39.99

�31.37

�35.77

�30.03

2.80

�43.83

�297.92

7.54

�19.55

�107.6

�173.82

�103.58

Weather

type‘b’

A1014.4

1059.1

1057.5

1065.7

1021.7

990.9

942.7

996.0

901.3

846.5

943.0

101.2

B0.115

0.171

0.2078

0.2443

0.4375

0.3854

0.4540

0.4298

0.2362

0.2628

0.3492

0.1855

C0.2585

0.3068

0.3033

0.3235

0.4006

0.4667

0.5529

0.3444

0.4166

0.3701

0.3116

0.2722

D�71.490

�74.033

�59.647

�56.09

�36.99

�2.2115

�9.860

�47.40

�67.01

1.5204

�37.989

�42.196

Weather

type‘c’

A685.4

698.1

783.2

832.7

1049.4

1028.9

770.2

681.6

700.9

829.5

534.3

658.9

B0.3001

0.3912

0.4384

0.6050

0.7414

0.8589

0.5810

0.6334

0.4030

0.4384

0.3780

0.2056

C0.4624

0.4723

0.4607

0.5653

0.5743

0.5788

0.7477

0.7021

0.6569

0.3821

0.6260

0.5686

D17.044

41.541

41.899

73.302

117.74

170.93

31.482

116.3

44.39

46.75

41.82

59.32

Weather

type‘d’

A300.6

320.7

770.9

976.2

959.7

580.2

321.9

375.1

447.1

1135.0

4700.4

362.9

B0.3768

0.5669

0.7787

0.8686

1.1016

1.0200

0.6642

0.6850

0.6928

1.2596

1.6837

0.4351

C1.1618

1.1219

0.9108

0.7022

0.9495

1.4352

2.1369

1.8470

1.3885

0.5660

0.3151

0.6024

D28.6590

75.8745

79.1801

125.685

150.0405

130.221

54.70

46.664

84.379

194.02

152.108

129.588



DOI: 10.1002/er

where B ¼ ðn� 1Þ360=365 and n5 nth day ofthe year.

We have also calculated constants A, B ofEquation (2) for composite climate of New Delhi.The results are given in Table III(b).

2.2. Nijigorodov model

Nijiorodov [11] has revised the constants A, B andC (of ASHRAE model) for clear days in Botswanafrom analysis of different solar radiation compo-nents recorded at the university of Botswana,Botswana Technology Centre and some synopticstations. The results are given in Table III(a).

2.3. Machler and Iqbal model

Machler and Iqbal [12] have modified the con-stants A, B and C (of ASHRAE model), whichtake into account the advancement in the solarradiation research over past decades. The resultsobtained for A, B and C of Equations (1)–(3) forCanada are given in Table III(a).

2.4. Parishwad et al. model

Parishwad et al. [13] have evaluated the constantsA, B and C (of ASHRAE model) using regressionanalysis of measured solar radiation data of sixcities of India. The results are given in Table III(a).

2.5. Perez et al. model

Perez et al. [15] proposed the correlation to predictdirect normal terrestrial solar radiation. Theexpression for direct normal terrestrial radiationis given by

IN ¼ ION: exp½�TR=ð0:9þ 9:4 cos yzÞ� ð9Þ

where TR is Linke turbidity factor and ION isnormal extraterrestrial solar radiation which isexpressed as

ION ¼ ISC½1:0þ 0:033 cosð360n=365Þ� ð10Þ

where ISC is solar constant.

2.6. Kasten and Young model

Kasten and Young [16] have also developed anempirical relation for direct terrestrial solar radia-tion in terms of air mass m, integrated Rayleigh

scattering optical thickness of atmosphere E andLinke turbidity factor TR. An expression for IN isgiven as

IN ¼ ION: expð�m:E:TRÞ ð11Þ

The parameters m and E are expressed as

m ¼ ½cos yz þ 0:15� ð93:885� yzÞ�1:253��1 ð12Þ

and

E ¼ 4:529� 10�4:m2 � 9:66865� 10�3:mþ 0:108014 ð13Þ

2.7. Hottel model

Hottel [3] has presented a model to estimate thebeam radiation transmitted through clear atmo-sphere in terms of zenith angle and altitude for astandard atmosphere and for four climate types.The atmospheric transmittance tb is IN=ION and itis given by

tb ¼ a0 þ a1: expð�k= cos yzÞ ð14Þ

The constants a0, a1 and k are functions of thealtitude of the location, which are given by

a0 ¼ 0:4237� 0:00821ð6� AÞ2

a1 ¼ 0:5055þ 0:00595ð6:5� AÞ2

and

k ¼ 0:2711þ 0:01858ð2:5� AÞ2

where A is altitude in km.

2.8. Present model

It is based on the ASHRAE model describedabove in Section 2.1. Since the evaluated constantsA, B and C in Equations (2) and (3) are notvalidating the data for composite climate, hence itrequires modifications. The expressions for hourlyglobal and radiation are same as Equation (1) and(2). The values of constants A and B have beenrevised by using regression analysis of the solarradiation data

The expression for hourly diffuse radiation hasbeen modified to give more accurate results and itis given by

Id ¼ CIN þD ð15Þ



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where C and D are constants whose values havebeen determined from regression analysis of solarradiation data. In this case the constants A, B, Cand D have been evaluated for composite climateof New Delhi. The results are given in Table III(b).If D becomes zero, then Equation (15) reduces toEquation (3) of ASHRAE model.

3. CALCULATION PROCEDURE FORPRESENT MODEL

As recommended by ASHRAE [10], the hourlyglobal radiation (I), hourly beam radiation in thedirection of rays (IN) and hourly diffuse radiation(Id) on the horizontal surface on a clear day arecalculated using Equations (1), (2) and (10) whereA, B, C and D are constants. The values have beenobtained for four weather types (‘a’–‘d’) of eachmonth by using data of Table I.

The equation of time correction (ET) is toconsider small perturbations in the Earth’s orbitand rate of rotation. It was taken from the tablegiven by Tiwari [17]. The second correction arisesbecause of the difference between the longitude oflocation (l) and standard time longitude (STL). Asthe longitude of New Delhi is 77.21E, ST at Delhiis based on 77.21E (STL). The negative sign in thiscorrection is applicable for the eastern hemisphere,while the positive is for the western hemisphere.For India, the negative sign is applicable as it liesin the eastern hemisphere.

In order to evaluate constants A and B ofEquation (2), for the month of January and June,concept of regression analysis has been applied.For regression analysis, the data of Table I havebeen used. Similarly the constants A and B forother months have also been obtained. The resultsfor each month and all weather conditions (types‘a’–‘d’) are given in Table III(b), which can be usedto generate hourly beam radiation data for NewDelhi.

The constants C and D for diffuse radiation inEquation (15) have again been obtained byregression analysis from the data of Table I andother months. The results for C and D for eachmonth and all weather conditions (types ‘a’–‘d’)are given in Table III(b), which can be used to

generate the hourly diffuse radiation data for NewDelhi.

It can be further seen that the constant A isminimum for cloudy days (type ‘d’) due toattenuation of radiation in the atmosphere,unlike for clear days (type ‘a’). The value ofconstant A for other weather conditions (types ‘b’and ‘c’) lies between these two extreme values, asexpected.

From Table III(b), it can be seen that theconstant B is maximum for cloudy days (type ‘d’)due to attenuation of radiation in the atmosphere,unlike for clear days (type ‘a’). The value ofconstant B for other weather conditions (types ‘b’and ‘c’) lies between these two extreme values, asexpected.

The values of constants C and D for eachmonth vary according to the weather conditionsand instability in them [Table III(b)].

4. STATISTICAL METHODS USED

There are numerous statistical methods availablein solar energy literature, which deal with theassessment and comparison of solar radiationestimation models [18–27]. In the present studystatistical indicators, namely root mean squareerror (RMSE) and mean bias error (MBE) havebeen used.

4.1. Root mean square error

The RMSE is defined as

%RMSE ¼100

Gm

XIi;pre � Ii;obs� �2h i.

Nn o1=2

ð16Þ

where Ii;pre is ith predicted value, Ii;obs is ithobserved value, N is total number of observationsand Gm is mean of N measured values. The RMSEis always positive, a zero value is ideal. This testprovides information on the short-term perfor-mance of the models by allowing a term-by-termcomparison of actual deviation between thecalculated value and the measured value. How-ever, a few large errors in the sum can produce asignificant increase in RMSE.



DOI: 10.1002/er

4.2. Mean bias error

The MBE is defined as

%MBE ¼100

Gm

XðIi;pre � Ii;obsÞ

h i.N ð17Þ

This test provides information on the long-termperformance. A low MBE is desired. Ideally a zerovalue of MBE should be obtained. A positive valuegives the average amount of over-estimation in thecalculated value and vice versa. One drawback ofthis test is that over-estimation of an individualobservation will cancel under-estimation in aseparate observation.

5. EXPERIMENTAL DATA

For the present study, the data of the hourly global

and diffuse solar radiation (Wm�2) on a horizontal

surface for a period of 11 years (1991–2001) have

been used (Table I). The data have been obtained

from the India Meteorological Department, Pune,

India. The data for composite climate of New

Delhi have been obtained using a thermoelectric

pyranometer with (diffuse) and without (global) a

shade ring. The shade ring factor has been used to

make corrections for shaded sky assuming that sky

radiation is isotropic. The pyranometers used are

Table IV. Percentage root mean square error (RMSE) between predicted results and measured monthly mean hourlybeam radiation for New Delhi.

Percentage RMSE Percentage MBE

Month Machler Parishwad Nijigorodov Machler Parishwad Nijigorodov

Jan 176.8 31.2 220.7 144.1 �23.4 171.4Feb 135.8 26.4 152.0 122.2 �18.9 133.8Mar 112.7 22.8 118.2 106.2 �17.9 110.6Apr 100.6 21.2 103.7 96.5 �17.8 99.6May 102.6 19.1 106.1 �98.9 �14.7 102.8Jun 95.9 18.4 100.0 93.4 �16.2 98.0Jul 121.9 10.5 125.8 119.4 �5.4 123.8Aug 135.5 10.3 138.1 132.2 �2.2 135.3Sep 123.1 20.0 126.6 116.7 �12.6 120.1Oct 167.6 22.0 184.5 148.1 �10.4 159.5Nov 406.3 23.7 599.2 248.1 �10.6 322.5Dec 550.2 25.6 955.3 304.5 �11.9 452.4

Table V. Percentage root mean square error (RMSE) between predicted results and measured monthly mean hourlydiffuse radiation for New Delhi.


Month Machler Parishwad Nijigorodov Machler Parishwad Nijigorodov

Jan 23.8 93.3 16.9 �18.2 92.2 �9.5Feb 34.3 57.8 29.1 �32.3 57.4 �27.1Mar 39.5 37.6 38.9 �38.1 37.4 �37.6Apr 43.9 15.9 51.0 �43.0 15.0 �50.1May 40.6 11.8 54.0 �40.2 11.3 �53.6Jun 37.4 15.8 55.4 �36.0 13.1 �54.2Jul 51.8 19.4 65.6 �48.0 -8.7 �62.3Aug 50.7 22.4 62.6 �44.3 0.6 �57.2Sep 39.3 26.2 46.4 �37.9 24.3 �45.0Oct 38.4 53.5 38.4 �30.6 51.3 �30.6Nov 34.9 77.4 31.2 �23.6 74.5 �18.5Dec 37.4 101.4 33.4 �16.7 95.7 �7.0



DOI: 10.1002/er

calibrated once a year with reference to the WorldRadiometric Reference. The estimated uncertaintyin the measured data is about 75%. For thecomputation of constants A, B, C and D the beam

radiation data have been derived from measuredhourly global and diffuse radiation data. For everymonth over period of 11 years, the average numberof days falling under different weather conditions

Table VI. Percentage root mean square error (RMSE) between predicted results and measured monthly mean hourlybeam radiation for New Delhi.


Month Kasten Perez Hottel Kasten Perez Hottel

Jan 10.4 10.8 22.4 8.5 8.7 �20.9Feb 23.0 23.5 9.4 21.5 21.9 �9.0Mar 25.7 26.3 2.4 24.2 24.7 �0.5Apr 23.8 24.5 6.4 22.2 22.8 4.4May 23.8 24.5 10.1 22.8 23.5 9.3Jun 20.2 20.9 9.6 18.6 19.4 8.1Jul 34.5 35.4 23.8 31.9 32.7 21.6Aug 41.2 42.0 26.9 38.1 38.9 24.2Sep 27.9 28.7 10.0 25.6 26.3 6.9Oct 33.7 34.3 12.4 28.8 29.3 3.6Nov 32.3 32.8 9.7 27.4 27.9 �4.4Dec 27.4 27.8 13.5 22.6 23.1 �11.1

Table VII. Percentage root mean square error (RMSE) between predicted results and measured monthly mean hourlybeam radiation for location New Delhi.

‘a’ Type weather ‘b’ Type ‘c’ Type ‘d’ Type

ASHRAE model Parishwad model New Cons. New Cons. New Cons. New Cons.

Jan RMSE 7.6 31.2 5.1 3.2 9.4 16.8MBE 3.6 �23.4 �0.2 �0.1 �1.7 �2.2

Feb RMSE 17.7 26.4 3.0 2.1 10.7 28.1MBE 17.4 �18.9 �0.1 �0.0 �0.8 �2.4

Mar RMSE 21.2 22.8 1.9 1.1 2.8 9.7MBE 21.1 �17.9 �0.0 �0.0 �0.1 0.2

Apr RMSE 17.5 21.2 4.9 1.7 3.4 6.8MBE 16.8 �17.8 �0.1 �0.0 0.1 �0.1

May RMSE 18.3 19.1 4.1 1.4 6.0 10.4MBE 17.7 �14.7 �0.1 �0.0 0.0 �0.1

Jun RMSE 14.2 18.4 4.4 4.4 8.6 17.1MBE 13.4 �16.2 �0.1 �0.1 �0.7 �1.5

Jul RMSE 27.4 10.5 4.0 4.4 7.8 13.0MBE 27.1 �5.4 �0.1 �0.1 �0.2 �0.9

Aug RMSE 33.3 10.3 4.6 3.1 9.3 14.1MBE 32.8 �2.2 �0.1 �0.1 �0.6 �0.9

Sep RMSE 25.2 20.0 4.9 6.5 5.9 19.8MBE 23.1 �12.6 �0.1 �0.3 �0.3 �1.7

Oct RMSE 32.1 22.0 8.5 3.4 4.6 23.4MBE 24.9 �10.4 �0.4 �0.2 �0.7 �1.8

Nov RMSE 34.2 23.7 6.7 1.9 11.4 36.3MBE 18.0 �10.6 0.3 �0.1 �0.8 5.1

Dec RMSE 28.7 25.6 5.7 6.6 8.4 21.8MBE 13.8 �11.9 �0.2 �0.3 �0.5 �3.0



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has been given in Table II. The average number ofdays falling under different weather conditions ineach month has been obtained on the basis ofrecorded weather observations, given total sun-shine hours and daily global radiation. Table Igives the average hourly measured data fortotal and diffuse radiation for typical months ofJanuary (winter conditions) and June (summerconditions), respectively. The data of Table I havebeen used in evaluating constants A, B, C and D.Similar data for other months have also beenobtained and used.

6. RESULTS AND DISCUSSION

The constants of various models discussed inSection 2 have been used to estimate the IN andId for composite climate of New Delhi. The RMSEand MBE for each model have been given inTables IV–VIII.

Nijigorodov model [11] has found the RMSEof 955–100% and 66–17% for predicting thehourly beam and diffuse radiation respectively(Tables IV–V). It yields MBE of 452–98% and�62 to �7% for predicting the hourly beam anddiffuse radiation, respectively (Tables IV–V). Thismodel may be limited to Botswana, it is notfeasible for Indian climatic conditions due to veryhigh RMSE and MBE.

Machler and Iqbal model [12] produces RMSEof 550–96% and 52–24% for predicting the hourlybeam and diffuse radiation, respectively (TablesIV–V). It yields MBE of 304–93% and �47 to�17% for predicting the hourly beam and diffuseradiation, respectively (Tables IV–V). This modelmay also be limited to Canada, it is not feasible forIndian climatic conditions due to very high RMSEand MBE.

Parishwad et al., model [13] produces RMSEof 31–10% and 101–12% for predicting thehourly beam and diffuse radiation, respectively

Table VIII. Percentage root mean square error (RMSE) between predicted results and measured monthly mean hourlydiffuse radiation for location New Delhi.

‘a’ Type weather ‘b’ Type ‘c’ Type ‘d’ Type

ASHRAE model Parishwad model New Cons. New Cons. New Cons. New Cons.

Jan RMSE 58.1 93.3 9.6 5.9 11.2 14.4MBE �53.9 92.2 00 00 00 00

Feb RMSE 63.2 57.8 3.3 3.9 9.8 21.6MBE �60.9 57.4 00 00 00 00

Mar RMSE 61.1 37.6 4.3 2.9 4.7 9.5MBE �59.7 37.4 00 00 00 00

Apr RMSE 54.8 15.9 5.2 5.9 5.8 10.4MBE �53.9 15.0 00 00 00 00

May RMSE 44.8 11.8 3.3 3.3 12.8 8.3MBE �44.4 11.3 00 00 00 00

Jun RMSE 38.8 15.8 4.4 4.8 5.2 17.5MBE �37.4 13.1 �0.1 00 00 00

Jul RMSE 52.6 19.4 13.2 8.7 11.5 12.0MBE �48.7 -8.7 00 00 00 00

Aug RMSE 55.2 22.4 19.2 5.9 7.3 14.7MBE �49.3 0.6 00 00 00 00

Sep RMSE 54.1 26.2 5.7 9.2 9.9 16.7MBE �52.7 24.3 00 00 00 00

Oct RMSE 60.7 53.5 12.1 2.7 4.3 13.1MBE �54.3 51.3 00 00 00 00

Nov RMSE 62.3 77.4 12.4 1.8 8.9 24.9MBE �54.6 74.5 00 00 00 00

Dec RMSE 65.6 101.4 28.4 4.6 8.4 24.3MBE �53.9 95.7 00 00 00 00



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(Tables IV–V). It yields MBE of �23 to �2% and96 to �8.7% for predicting the hourly beam anddiffuse radiation, respectively (Tables IV–V). Thismodel may be limited to other climatic conditionsin India, it is not feasible for composite climate ofNew Delhi due to high RMSE and MBE.

Perez et al., model [15] produces RMSE of42–10% and MBE of 39–9% while predicting

hourly beam radiation (Table VI). Although thismodel gives better performance than the earlierthree models, its modification is required to havemore accurate prediction.

Kasten and Young model [16] produces RMSEof 41–10% and MBE of 38–8% while predictinghourly beam radiation (Table VI). This modelperforms as good as Perez et al., model. Likewise,

0

200

400

600

800

1000

1200

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Measured

ASHRAE (r =0.996)

Nijegorodov (r =0.988)

Machler (r =0.996)

Parishwad (r =0.996)

Perez (r =0.997)

Kasten (r =0.997)

Hottel (r =0.996)

0

200

400

600

800

1000

1200

1400

1600

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Measured

ASHRAE (r =0.994)


Machler (r =0.994)


Perez (r =0.995)

Kasten (r =0.995)

Hottel (r =0.994)

(a)

(b)

Figure 1. (a) Hourly variation in beam radiation with time for the month of January (type ‘a’) weather conditionusing various models and (b) hourly variation in beam radiation with time for the month of June (type ‘a’)

weather condition using various models.



DOI: 10.1002/er

its modification is further required to have moreaccurate prediction.

Hottel model [3] produces RMSE of 27–2.4% andMBE of 24 to �21% while predicting hourly beamradiation (Table VI). This model performs betterthan Kasten and Young, and Perez et al. model.

ASHRAE model [10] produces RMSE of34–7.5% while predicting hourly beam radiationand 65–38% while predicting hourly diffuseradiation (Tables VII and VIII). It yields MBEof 32–3% while predicting hourly beam radiationand �61 to �37% while predicting hourly diffuseradiation (Tables VII and VIII). In order to havemore accurate prediction, ASHRAE model isrequired to be modified for Indian climaticconditions.

Present model (which is modification ofASHRAE model) produces RMSE of 8.5–2%

while predicting hourly beam radiation and 28–3%while predicting hourly diffuse radiation (Tables VIIand VIII). It yields MBE of 28–3% while predictinghourly beam radiation and 0% while predictinghourly diffuse radiation (Tables VII and VIII).

Figures 1(a,b) give hourly variation in observedand predicted beam radiation using variousmodels for typical months of January (winter)and June (summer), respectively, and for weathertypes ‘a’ only. Figures 2(a,b) give the hourlyvariation in observed and predicted diffuseradiation using various models for the typicalmonths of January (winter) and June (summer),respectively and for weather types ‘a’ only. It isinferred that there is a 7.5–34.2% RMSE betweenobserved and predicted values of beam radiationusing ASHRAE model for clear days (type ‘a’), asshown in Figure 1 and Table VII.

0

50

100

150

200

250

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Measured

ASHRAE (r =0.945)


Machler (r =0.784)


0

100

200

300

400

500

600

700

800

900

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Measured

ASHRAE (r =0.789)


Machler (r =0.784)


(a)

(b)

Figure 2. (a) Hourly variation in diffuse radiation with time for the month of January (type ‘a’ weather condition)using various models and (b) hourly variation in diffuse radiation with time for the month of June (type ‘a’ weather

condition) using various models.



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Figures 3 and 4 give hourly variation in observedand predicted beam and diffuse radiation using newconstants for typical months of January (winter) andJune (summer), respectively and for weather types ‘a’and ‘b’, respectively. It is inferred that there is1.9–8.5% RMSE between observed and predictedvalues of beam radiation using new constants for cleardays (type ‘a’), as shown in Figure 3 and Table VII.

For new constants the evaluated values ofpercentage RMSE and percentage MBE forbeam radiation have been given in Table VII foreach month and each type of weather.

For new constants the evaluated values ofpercentage RMSE and percentage MBE for

diffuse radiation have been given in Table VIIIfor each month and each type of weather.

The new constants generally give better resultsfor clear sky conditions of Indian regions. The lowMBEs are particularly remarkable. Therefore,their use is recommended for composite climateof New Delhi.

7. CONCLUSIONS ANDRECOMMENDATION

ASHRAE model can be applied to estimate thehourly beam radiation for composite climate of

0

100

200

300

400

500

600

700

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Beam (obs)

Beam (pre); r =0.969

Diffuse (obs)

Diffuse (pre); r=0.949

0

100

200

300

400

500

600

700

800

900

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

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tion

(W/m

2)

Beam (obs)

Beam (pre); r=0.859

Diffuse (obs)

Diffuse (pre); r=0.788

(a)

(b)

Figure 3. (a) Hourly variation in beam and diffuse radiation with time for the month of January (type ‘a’ weathercondition) using new constants and (b) hourly variation in beam and diffuse radiation with time for the month of June

(type ‘a’ weather condition) using new constants.



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New Delhi by assigning new values to constants Aand B. Moreover, to estimate hourly diffuseradiation for composite climate of New Delhi,one more constant D has been introduced. Byassigning new values to constants C and D, moreaccurate prediction of diffuse radiation can bemade. The new values of constants A, B, C and Dfor each month and all weather conditions (types‘a’–‘d’) are given in Table III(b), which can be usedto generate the hourly beam radiation data forNew Delhi. The present studies should be extendedto the other climatic conditions of India.

As indicated in Table VIII that almostall MBEs are of zero for the four types ofweather conditions. It may be due to thefact that the model development and modelvalidation were conducted using the same

database (11-year-measured data). It is suggestedthat independent sets of measured data should beused for the model evaluation for future work.

NOMENCLATURE

A 5 altitude of the location in kilo-meters

ET 5Equation of time correction (min)I 5 hourly global radiation on the

horizontal surface (Wm�2)Id 5 hourly diffuse radiation on the

horizontal surface (Wm�2)IN 5 normal terrestrial beam solar

radiation at the ground level(Wm�2)

0

100

200

300

400

500

600

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Beam (obs)

Beam (pre); r=0.989

Diffuse (obs)

Diffuse (pre); r =0.984

0

100

200

300

400

500

600

700

800

6 8 10 12 14 16 18

Time (hours)

Sol

ar r

adia

tion

(W/m

2)

Beam (obs)

Beam (pre); r =0.954

Diffuse (obs)

Diffuse (pre); r =0.950

(a)

(b)

Figure 4. (a) Hourly variation in beam and diffuse radiation with time for the month of January (type ‘b’ weathercondition) using new constants and (b) hourly variation in beam and diffuse radiation with time for the month of June

(type ‘b’ weather condition) using new constants.



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ION 5 normal extraterrestrial solarradiation (Wm�2)

ISC 5 solar constant (Wm�2)Ii;pre 5 ith predicted value of solar radiationIi;obs 5 ith observed value of solar radiationl 5 longitude of the location (degrees

west)LAT 5 local apparent time (degree)m 5 air mass (dimensionless)n 5 day of the year, starting from 1st

JanuaryN 5 total number of observationsr 5 coefficient of correlationST 5 standard timeSTL 5 standard time latitude

Greek symbols

d 5 solar declinatione 5 integrated Rayleigh scattering

optical thicknessyz 5 zenith angle (degree)tb 5 atmospheric transmittancef 5 latitude of the locationo 5 hour angle

ACKNOWLEDGEMENTS

The authors are grateful to the Indian MeteorologicalDepartment, Pune, India for providing the hourly globaland diffuse radiation data for the period of 11 yearsfrom 1991 to 2001. The authors are grateful to Prof.Ibrahim Dincer for his valuable suggestions.

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