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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: 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 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
Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552
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
EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 541
Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552
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
M. J. AHMAD AND G. N. TIWARI542
Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552
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 (‘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.
EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 543
Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552
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.
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
Copyright r 2008 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:538–552
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.
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 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)
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.
EVALUATION AND COMPARISON OF HOURLY SOLAR RADIATION MODELS 547
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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)
Nijegorodov (r =0.784)
Machler (r =0.784)
Parishwad (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)
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.
M. J. AHMAD AND G. N. TIWARI548
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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
adia
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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