Evaluation and comparison of hourly solar radiation models

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  • INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2009; 33:538552Published 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 (modied 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 1.98.5% 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 earths surface can be classied 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 earthssurface is less than or equal to 25% of globalradiation, the sky is termed as clear sky.

    Solar radiation available on the Earths 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: jamil.amu@gmail.com

    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 protably 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 rst 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 ve 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 transmissioncoefcient 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 theEarths surface over a given period of time atany location [9].

    Table I. Average hourly global and diffuse radiation (Wm2) 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; 33:538552

    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 modied thevalues of empirical coefcients of ASHRAEmodel valid only for climatic conditions ofBotswana, Namibia and Zimbabwe. This modelgives large error for composite climate of NewDelhi. The modied 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 IN cos yz Id 1

    IN A expB= cos yz 2

    Id CIN 3

    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 sinf: sin d cosf: cos d: coso 4

    Further, solar declination (d) is obtained from

    d 23:45 sin360284 n=365 5

    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 15:012:0 LAT 6

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

    LAT ST ET 4:STL l 7

    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 229:20:000075 0:001868 cosB 0:032077 sinB 0:014615 cos 2B 0:04089 sin 2B 8

    Table II. Average number of days under different weather types in different months during 19912001 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

    Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538552

    DOI: 10.1002/er

  • TableIII.

    (a)EvaluatedvaluesofA,BandCforvariousmodelsand(b)evaluatedvaluesofA,B,CandDfor(a)weathertypea,(b)weathertype

    b,(c)weather

    typecand(d)weather

    typedatNew

    Delhi.

    Months

    Parameter

    Jan

    Feb

    Mar

    Apr

    May

    Jun

    Jul

    Aug

    Sep

    Oct

    Nov

    Dec

    (a)ASH-RAEmodel

    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

    typea

    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

    D39.99

    31.37

    35.77

    30.03

    2.80

    43.83

    297.92

    7.54

    19.55

    107.6

    173.82

    103.58

    Weather

    typeb

    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

    D71.490

    74.033

    59.647

    56.09

    36.99

    2.2115

    9.860

    47.40

    67.01

    1.5204

    37.989

    42.196

    Weather

    typec

    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

    typed

    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

    EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 541

    Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538552

    DOI: 10.1002/er

  • where B n 1360=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 modied 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 ION: expTR=0:9 9:4 cos yz 9

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

    ION ISC1:0 0:033 cos360n=365 10

    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 ION: expm:E:TR 11

    The parameters m and E are expressed as

    m cos yz 0:15 93:885 yz1:2531 12

    and

    E 4:529 104:m2 9:66865 103:m 0:108014 13

    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 a0 a1: expk= cos yz 14

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

    a0 0:4237 0:008216 A2

    a1 0:5055 0:005956:5 A2

    and

    k 0:2711 0:018582:5 A2

    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 modications. 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 modied to give more accurate results and itis given by

    Id CIN D 15

    M. J. AHMAD AND G. N. TIWARI542

    Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538552

    DOI: 10.1002/er

  • 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 (ad) of eachmonth by using data of Table I.

    The equation of time correction (ET) is toconsider small perturbations in the Earths 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 (typesad) 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 ad)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 band 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 band 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 [1827]. 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 dened as

    %RMSE 100

    Gm

    XIi;pre Ii;obs 2h i.

    Nn o1=2

    16

    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 asignicant increase in RMSE.

    EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 543

    Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538552

    DOI: 10.1002/er

  • 4.2. Mean bias error

    The MBE is dened as

    %MBE 100

    Gm

    XIi;pre Ii;obs

    h i.N 17

    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 (Wm2) on a horizontal

    surface for a period of 11 years (19912001) 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.

    Percentage RMSE Percentage MBE

    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

    M. J. AHMAD AND G. N. TIWARI544

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  • 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.

    Percentage RMSE Percentage MBE

    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 IVVIII.

    Nijigorodov model [11] has found the RMSEof 955100% and 6617% for predicting thehourly beam and diffuse radiation respectively(Tables IVV). It yields MBE of 45298% and62 to 7% for predicting the hourly beam anddiffuse radiation, respectively (Tables IVV). 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 55096% and 5224% for predicting the hourlybeam and diffuse radiation, respectively (TablesIVV). It yields MBE of 30493% and 47 to17% for predicting the hourly beam and diffuseradiation, respectively (Tables IVV). 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 3110% and 10112% 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 IVV). It yields MBE of 23 to 2% and96 to 8.7% for predicting the hourly beam anddiffuse radiation, respectively (Tables IVV). 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 of4210% and MBE of 399% while predicting

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

    Kasten and Young model [16] produces RMSEof 4110% and MBE of 388% while predictinghourly beam radiation (Table VI). This modelperforms as good as Perez et al., model. Likewise,

    0

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    1000

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    6 8 10 12 14 16 18Time (hours)

    Sola

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    MeasuredASHRAE (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)

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    Sola

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    2)

    MeasuredASHRAE (r =0.994)Nijegorodov (r =0.994)Machler (r =0.994)Parishwad (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.

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  • its modication is further required to have moreaccurate prediction.

    Hottel model [3] produces RMSE of 272.4% 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 of347.5% while predicting hourly beam radiationand 6538% while predicting hourly diffuseradiation (Tables VII and VIII). It yields MBEof 323% 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 modied for Indian climaticconditions.

    Present model (which is modication ofASHRAE model) produces RMSE of 8.52%

    while predicting hourly beam radiation and 283%while predicting hourly diffuse radiation (Tables VIIand VIII). It yields MBE of 283% 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 7.534.2% RMSE betweenobserved and predicted values of beam radiationusing ASHRAE model for clear days (type a), asshown in Figure 1 and Table VII.

    0

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    Sola

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    diat

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    /m2)

    MeasuredASHRAE (r =0.945)Nijegorodov (r =0.784)Machler (r =0.784)Parishwad (r =0.784)

    0100200300400500600700800900

    6 8 10 12 14 16 18Time (hours)

    Sola

    r rad

    iatio

    n (W

    /m2)

    MeasuredASHRAE (r =0.789)Nijegorodov (r =0.784)Machler (r =0.784)Parishwad (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 aand b, respectively. It is inferred that there is1.98.5% 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

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    Beam (obs) Beam (pre); r =0.969Diffuse (obs)Diffuse (pre); r=0.949

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    Beam (obs)Beam (pre); r=0.859Diffuse (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 (typesad) 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 (Wm2)Id 5 hourly diffuse radiation on the

    horizontal surface (Wm2)IN 5 normal terrestrial beam solar

    radiation at the ground level(Wm2)

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    Sola

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    2)

    Beam (obs)Beam (pre); r=0.989Diffuse (obs)Diffuse (pre); r =0.984

    0

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    6 8 10 12 14 16 18

    Time (hours)

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    2)

    Beam (obs)Beam (pre); r =0.954Diffuse (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 (Wm2)

    ISC 5 solar constant (Wm2)

    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 coefcient 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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