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Advances in Space Research 53 (2014) 1239–1245

A simple model to predict solar radiation under clear sky conditions

Qiumin Dai, Xiande Fang ⇑

Department of Man, Machine and Environment Engineering, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics,

29 Yudao Street, Nanjing 210016, China

Received 10 September 2013; received in revised form 25 January 2014; accepted 26 January 2014Available online 3 February 2014

Abstract

Solar radiation is one of the major factors that dominate the thermal behaviors of aerostats in the daytime and the primary energysource of high altitude long endurance aerostats. Therefore, it is necessary to propose an accurate model to predict the solar irradiances.A comprehensive review of the well-known solar radiation models is conducted to help develop the new model. Based on the analysis ofthe existing models and the available radiation data, the extensive computer tests of the regression and optimization are conducted, fromwhich the new solar radiation model for direct and diffuse irradiances under clear sky conditions is proposed. The new model has excel-lent prediction accuracy. The coefficient of determination for direct radiation is 0.992, with the root mean square error (RMSE) of16.9 W/m2 and the mean absolute error (MAE) of 2.2%. The coefficient of determination for diffuse radiation is 0.86, withRMSE = 8.7 W/m2 and MAE = 9.9%. Comparisons with the well-known existing models show that the new model is much more accu-rate than the best existing ones.� 2014 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Solar radiation; Direct radiation; Diffuse radiation; Aerostat; Altitude

1. Introduction

An aerostat is a lighter than air vehicle whose lift derivesfrom the buoyancy resulted from the density differencebetween the inner gas and the surrounding air. Its distin-guishing features of long endurance, station keeping, andlow cost-effectiveness make it suitable for transportation,surveillance, telecommunication relay, broadcasting, andmilitary roles, which attract interests all over the world.

In order to predict the thermal performance of an aero-stat, it is important to consider its surrounding thermalenvironment carefully. The solar radiation is one of themajor factors that dominate the thermal behaviors of anaerostat in the daytime. Meanwhile, the solar radiation isthe primary energy source of a high altitude long

http://dx.doi.org/10.1016/j.asr.2014.01.025

0273-1177/$36.00 � 2014 COSPAR. Published by Elsevier Ltd. All rights rese

⇑ Corresponding author. Tel./fax: +86 25 8489 6381.E-mail addresses: daiqiumin913@126.com (Q. Dai), xd_fang@yahoo.

com (X. Fang).

endurance aerostat. The good estimation of solar radiationis crucial for modeling the thermal performance of anaerostat.

Several studies were carried out for modeling solar radi-ation on aerostats. Carlson and Horn (1983) and Wanget al. (2007) assumed the total solar irradiance reachingan aerostat to be constant. Kreith and Kreider (1974)and Farley (2005) neglected the diffuse and reflected irradi-ance. This assumption is unacceptable for low altitude con-ditions, where the magnitude of sum of diffuse and reflectedirradiance can be as high as 400 W/m2. Wang and Yang(2011) employed semi-empirical corrections related to theground measurements which may lead to under-predictionof the solar irradiance for a high altitude aerostat. Xia et al.(2010) used a semi-empirical direct solar model which takesthe effect of altitude into account, but they neglected theinfluence of altitude on diffuse and reflected irradiance.Dai et al. (2012) employed an empirical solar model, wherethe atmospheric transmittance was highly simplified.

rved.

1240 Q. Dai, X. Fang / Advances in Space Research 53 (2014) 1239–1245

The operational altitude of aerostats covers a widerange, from 0 km for tethered aerostats to over 30 kmfor high altitude aerostats. The differences of direct anddiffuse solar irradiances predicted by spectral analysis atdifferent altitudes may as high as 500 and 300 W/m2

(Knaupp and Mundschau, 2004), respectively. In the fore-going solar radiation models, except for Xia et al. (2010)and Dai et al. (2012), all of them do not consider altitude,i.e., they are only suitable for ground applications. On theother hand, Xia et al. (2010) and Dai et al. (2012) consideronly the effect of altitude on direct solar irradiance, whilethe effect of altitude on diffuse solar irradiance isneglected. Therefore, the above-mentioned solar modelsmay lead to remark errors at high altitude. Meanwhile,all of the above mentioned solar radiation models donot consider the effect of atmospheric conditions, suchas aerosol and water vapor. The values of aerosol andwater vapor vary dramatically. At a given altitude, thedifferences of direct and diffuse solar irradiances predictedby spectral analysis at different atmospheric conditionsmay as high as 300 and 200 W/m2, respectively. Therefore,the solar models using constant aerosol and water vaporare not reasonable.

The heat load caused by solar radiation contributes amajor fraction for high altitude aerostats. For a nearspace aerostat with a diameter of 40 m and a solarabsorptivity of 0.33, its solar absorption can be as highas 400 W/m2 at the operational altitude of 20 km, wherethe direct solar irradiance is about 1300 W/m2, while itsforced convective heat load is only around 100 W/m2 atthe temperature difference of 50 K and the forced convec-tive heat transfer coefficient of 2 W/m2 K. If the meanabsolute errors (MAEs) caused by the convective load cal-culation and the direct solar load calculation are requiredto be equivalent and the MAE of the convective load cal-culation is 20%, the MAE of the direct solar radiation cal-culation should be lower than 5%. All of the abovementioned solar radiation models have an MAE greaterthan 5%.

From the above brief introduction, it can be seen thatan accurate solar radiation model that considers the fac-tors of altitude and atmospheric conditions is needed. Itis the purpose of this paper to propose an accurate modelto predict the direct and diffuse solar irradiances whichtakes into account of altitude and meteorological param-eters. A comprehensive survey of the well-known solarradiation models is conducted. Based on the analysis ofthe existing models and the radiation data from NationalRenewable Energy Laboratory (NREL) (http://www.nrel.gov/midc/srrl_bms), the extensive computer tests of theregression and optimization using the commercialsoftware 1stOpt (7D-Soft High Technology Inc., 2010)are conducted to develop the new solar radiation modelfor clear sky conditions. The new model is compared tothe existing models and reference code to assess itsaccuracy.

2. Review of solar radiation models

2.1. Bird and Hulstrom (1980, 1981) model

The Bird and Hulstrom model has gained wide accep-tance in the last three decades. The algorithms for eachattenuation process were the basis of the Iqbal (1983)model and the METSTAT model (Maxwell, 1998). Themodel is of the form

IDN ¼ 0:9662ISUN T RT OT MGT W T A ð1Þ

Id ¼ 0:79ISUN T OT W T MGT AA sin h½0:5ð1� T RÞþ 0:84ð1� T ASÞ�=ð1� mR þ m1:02

R Þ ð2Þ

where IDN is the direct irradiance, Id is the diffuse irradi-ance, ISUN is the solar constant, TR, TO, TMG, TW andTA are the individual transmission coefficient for Rayleighscatting, ozone, mixed gases, water vapor and aerosol, TAA

is the transmittance of aerosol absorption, TAS is the trans-mittance of aerosol scattering, and XO and XW are theamount of ozone and water vapor in a slant path. The rel-ative air mass mR is determined by the following Kasten(1965) equation:

mR ¼ sin hþ 0:15ð3:885þ hÞ�1:253h i�1

ð3Þ

where h is the solar elevation angle. The absolute air masscan be determined by

mA ¼ mRðp=1013Þ ð4Þ

where p is the atmospheric pressure.

2.2. Heliosat-1 model (Dumortier, 1995; Page, 1996)

The clear sky Heliosat-1 model consists two separatemodels for direct radiation (Page, 1996) and diffuse radia-tion (Dumortier, 1995). They can be written as:

IDN ¼ ISUN expð�mArT LÞ ð5Þ

Id ¼ ISUN ½0:0065þ ð0:0646T L2 � 0:045Þ sin h

� ð0:0327T L2 � 0:014Þ sin2 h� ð6Þ

where TL and TL2 are the turbidity factors, and r is theoptical depth of clean atmosphere. The relative air massis calculated using an expression introduced by Kastenand Young (1989):

mR ¼ ½sin hþ 0:506ðhþ 6:08Þ�1:636��1 ð7Þ

2.3. MAC model (Davies, 1987; Davies et al., 1988)

The MAC model provided a different treatment of theextinction process involved by water vapor, as expressedin the following:

IDN ¼ ISUN ðT RT O � aW ÞT A ð8Þ

Q. Dai, X. Fang / Advances in Space Research 53 (2014) 1239–1245 1241

Id ¼ ISUN ½0:75ðT OT R � aW Þð1� T AÞgþ 0:5T Oð1� T RÞ� ð9Þ

where aw is the extinction coefficient involved by watervapor, and g is the ratio of forward to total scatter by aer-osol. In the MAC model, a different type of relative airmass is used (Rogers, 1967):

mR ¼ 35=ð1þ 1224 sin2 hÞ0:5 ð10Þ

2.4. METSTAT model (Maxwell, 1998)

The METSTAT model was proposed on the basis of theBird and Hulstrom (1980, 1981) model. The modification ismainly focused on the constant for calculating the directirradiance, the transmission coefficients for water vaporand aerosol, and the air mass involved in the model. Theexpressions are given below:

IDN ¼ 0:9751ISUN T RT OT MGT W T A ð11Þ

where the relative air mass mR is determined by Eq. (7).

2.5. MLWT2 model (Gueymard, 2003)

MLWT2 model is a modified version of the MLWT1model (Gueymard, 1998). The MLWT1 model was pro-posed on the concept of multilayer spectral weighting(Gueymard, 1998). It has advantage on avoiding the limita-tion of the Lambert–Beer Law when applied to the broad-band spectrum. The original optical depths and air massesfor each extinction process were introduced in the MLWT1model. The expressions for direct irradiance and optical airmasses are given below:

IDN ¼ ISUN T RT AT OT W T NST NT ð12Þ

mR ¼ ½sin hþ 0:45665ð90� hÞ0:07ð6:4836þ hÞ�1:697��1 ð13Þ

mW ¼ ½sin hþ 0:03314ð90� hÞ0:1ð2:471þ hÞ�1:3814��1 ð14Þ

mNS ¼ ½sin hþ 1:1212ð90� hÞ1:6132ð21:55

þ hÞ�3:2629��1 ð15Þ

where TNS and TNT are transmission coefficients for strato-spheric and tropospheric nitrogen dioxide, respectively. Allnecessary expressions can be found in the above-mentionedliterature (Gueymard, 2003).

2.6. MRM-5 model (Kambezidis and Psiloglou, 2008;

Psiloglou et al., 2000)

The MEM-5 model is the fifth version of the MRMmodel. The main modification is in determining the trans-mission coefficients for uniformly mixed gases, water vaporand ozone. The main equations are provided below:

IDN ¼ ISUN T AT RT OT W T MG ð16ÞId ¼ 0:5ISUN T OT W T MGT AAð1� T AST RÞ sin h ð17Þ

where the relative air mass is determined by Eq. (7). Thetransmittance function for uniformly mixed gases is calcu-lated using five atmospheric gases (CO2, CO, N2O, CH4

and O2). All the transmittance functions for atmosphericgases can be found in Kambezidis and Psiloglou (2008).The transmittance function for aerosol used the Yanget al. (2001) expression which is described below.

2.7. Yang et al. (2001) model

Based on the analysis of spectral model and Angstromcorrelation, Yang et al. (2001) proposed

IDN ¼ ISUN ðT RT OT MGT W T A � 0:013Þ ð18ÞId ¼ ISUN ½T OT MGT W ð1� T RT AÞ þ 0:013� sin h ð19Þwhere the relative air mass is determined by Eq. (3).

3. Data description

The solar datasets used in this paper are from NRELwebsite (http://www.nrel.gov/midc/srrl_bms). The mea-surement station is located in Golden, Colorado, USA(39.74�N, 105.18�W, elevation 1829 m). The solar dataused in this paper include date, time, direct irradiance, dif-fuse irradiance, screen level air temperature, screen levelrelative humidity, AOD at 500 nm, and opaque cloudcover. The data cover the period from January 2012 toDecember 2012 with a time interval of 6 min.

The air temperature and relative humidity can be usedto determine the vertical water vapor column based on alocally adjusted model. In Golden area, the best fit to thetwo years of available data is (Gueymard, 2012)

w ¼ 0:1849ðRhpw;sÞ1:0049 ð20Þ

where Rh is the relative humidity, and pw,s is the saturatewater vapor pressure in mbar and can be calculated with(Buck, 1981)

pw;s ¼ 6:112 expð17:5t=ð241þ tÞÞ ð21Þ

where t is the air temperature in �C.Cloud is an important factor affecting the solar irradi-

ance. Even small amount of clouds may become a seriousfluctuation. All occurrences of clouds must be removedto obtain valid performance results (Gueymard, 2012),and quality control tests should be conducted to eliminatethe effect of clouds. A datum will be rejected during thequality tests if

(a) there is an indication of instrumentation malfunctionor power failure,

(b) the direct irradiance is less than 10 W/m2,(c) the opaque cloud cover is higher than 5%,(d) the relative humidity is higher than 90%,(e) the direct irradiance sharply fluctuates, or(f) the AOD is higher than 0.5.

After the quality test, a total of 9660 qualified datapoints are selected for the nonlinear regression.

1242 Q. Dai, X. Fang / Advances in Space Research 53 (2014) 1239–1245

4. Development of the new solar radiation model

4.1. Consideration of solar radiation at the top of the

atmosphere

The yearly average value of extraterrestrial solar con-stant is taken as 1367 W/m2 according to the current WorldMeteorological Organization (1981) recommendation.Because the earth’s orbit is slightly oval and the sun-earthdistance varies throughout the year, a correction factor(Wertz, 1985) is employed for calculating solar radiationat the top of the atmosphere, ISUN.

ISUN ¼ 1367ð1:017þ 0:0174 cos fÞ2 ð22Þ

where f is the true anomaly, which can be calculated with

f ¼ MAþ 0:0334 sinðMAÞ þ 0:000349 sinð2MAÞ ð23Þ

where MA = 2np/365, and n is the day number in a year.

4.2. Consideration of direct solar radiation

The direct solar irradiance can reach approximately 80%of the total irradiation at sea level under cloudless condi-tions, and thus it is the key parameter in solar irradiancepredictions. The direct solar irradiance depends on thesolar elevation angle, atmospheric pressure, and severalmeteorological parameters describing the environmentalconditions.

Under clear sky conditions, the attenuation is mostlycaused by scattering of air molecules, water vapor, andaerosols and absorption related to aerosols and watervapor (Duffie and Beckman, 2006). Therefore, the physicalmodel of the direct solar radiation can be expressed interms of

IDN ¼ ISUN T RT W T A ð24Þ

where TR, TW and TA are transmission coefficients forRayleigh scattering, water vapor and aerosol, respectively.All the individual transmission coefficients are proposed onthe basis of Lambert–Beer law. They can be written as

T R ¼ expða1mn1A Þ ð25Þ

T W ¼ expða2wn2 mn3R Þ ð26Þ

T A ¼ expða3sn4 mn5

R Þ ð27Þ

where a1, a2, a3, n1, n2, n3, n4 and n5 are the constants need-ing to be determined from the solar data. The relative airmass is determined by Eq. (3).

The AOD can be calculated from the spectral opticaldepths at 500 nm on the basis of Angstrom’s law as the fol-lowing (Bird and Hulstrom, 1981):

s ¼ 0:744s500 ð28Þ

Because the AOD and water vapor column are mea-sured on the ground, the vertical distribution of AODand water vapor column should be considered when themodel is used for predicting the direct irradiance at high

altitude. The vertical distributions of AOD and watervapor column can be calculated from aerosol attenuationcoefficient and vertical distribution of water density. There-fore, the vertical distributions of AOD and water vaporcolumn for American standard atmosphere are related tothe values of the ground measurements and the heightabove the measurement site (Elterman, 1970; McClatcheyet al., 1971).

s ¼ sm expð�0:691DHÞ ð29Þw ¼ wm expð�0:44DHÞ ð30Þwhere s is the AOD at different altitude, sm is the AOD atmeasurement site, w is the water vapor column at differentaltitude, wm is the water vapor column at measurement site,and DH is the height above the measurement site in km.

Besides the parameters mentioned above, the ozone andoxynitride absorption has minor influence on direct irradi-ance. This influence varies slightly throughout the year(Kambezidis and Psiloglou, 2008). Meanwhile, their den-sity is difficult to be measured at the same high frequencyas the solar and meteorological parameters. By assumingthat the component of air molecules along with the altitudeis constant, the effect of ozone and oxynitride on directirradiance is simplified and merged into the transmissioncoefficient of the Rayleigh scattering in this paper.

4.3. Consideration of diffuse solar radiation

When passing through the atmosphere, solar radiationis attenuated by the atmosphere, and part of the radiationlost in the direct beam is re-distributed as diffuse radiation.It goes in all directions and is treated as ideally isotropicunder clear sky conditions. This energy can be higher than200 W/m2 and amount to approximately 20% of the totalhorizontal radiation at sea level. The diffuse radiation iscaused by the scatter effect of molecular, water vapor andaerosol, and thus it decreases with altitude increasing.

Liu and Jordan (1960) found that the atmospheric trans-mittance for diffuse radiation was corresponding to a fixedvalue of atmospheric transmittance for direct radiationdepending upon the solar elevation angle. The experimen-tal data showed that this simplification had gained ade-quate accuracy, and it has been used in various ways toevaluate diffuse irradiance from many areas in the world(Kumar et al., 1997; Togrul et al., 2000). Therefore, the dif-fuse part of the new model is developed on the basis of theLiu and Jordan (1960) model as the following:

Id ¼ ðaþ b sin hþ cwþ dsÞðISUN � IDN Þ sin h ð31Þwhere a, b, c and d are the best fitting parameters.

4.4. Results and analysis

Regression analysis is the most widely used statisticaltechnique for investigating and modeling the relationshipbetween variables. Normal regression models are usuallyapplied in science and engineering to model data for which

Fig. 1. Predicted vs. measured direct solar irradiance.

Fig. 2. Predicted vs. measured diffuse solar irradiance.

Q. Dai, X. Fang / Advances in Space Research 53 (2014) 1239–1245 1243

linear or nonlinear functions of unknown parameters areused.

The measured data were used in multiple regressionanalysis to obtain the best fitting constants in Eqs. (25)–(27) and (31). Regression analysis was carried out withthe software 1stOpt (7D-Soft High Technology Inc.,2010). Least squares method was utilized to judge whichthe best fitting parameters are. It indicates that the sumof the squares of the residuals should be least.

The statistical criteria used are the coefficient of determi-nation (R2), the root mean square error (RMSE), and themean absolute error (MAE), as defined in the following:

R2 ¼PN

i¼1ðF Pred;i � �F PredÞðF Meas;i � �F MeasÞ� �2PNi¼1ðF Pred;i � �F PredÞ2

PNi¼1ðF Meas;i � �F MeasÞ2

ð32Þ

RMSE ¼PN

i¼1ðF Pred;i � F Meas;iÞ2

N

" #0:5

ð33Þ

MAE ¼ 1

N

XN

i¼1

jF Pred;i � F Meas;ijF Meas;i

ð34Þ

where FPred,I and FMeas,i are the ith predicted and measureddata, �F Pred and �F Meas are the mean values of the predictedand measured data, and N is the number of the data.

The regression yields the following correlations:

IDN ¼ ISUN exp½�0:103m0:571A � 0:081ðwmRÞ0:213

� s0:91m0:87R � ð35Þ

Id ¼ ð0:143þ 0:113 sin h� 0:0485wþ sÞðISUN � IDN Þ� sin h ð36Þ

where w is the vertical water vapor column, s is the AOD,and h is the solar elevation angle. The air mass mR is deter-mined by Eq. (3).

For Eq. (35), R2 = 0.992, RMSE = 16.9 W/m2, andMAE = 2.2%. For Eq. (36), R2 = 0.86, RMSE = 8.7W/m2, and MAE = 9.9%. Comparisons of the predictedresults vs. the measured data are shown in Figs. 1 and 2.Fig. 1 compares the predicted direct irradiance to the mea-sured direct irradiance. Fig. 2 shows the predicted diffuseirradiance plotted against the measured diffuse irradiance.The results show that the new model is able to accuratelypredict the direct and diffuse irradiance under clear skyconditions.

5. Evaluation of solar radiation models

The new model is compared to the existing high qualitymodels. The effects of the ozone and oxynitride absorptionare considered in several models. These effects are relativelysmall and vary slightly throughout the year. Fixed valuesof 0.34 atm-cm for ozone and 0.204 matm-cm for oxynit-ride are used in evaluating the related models (Kambezidisand Psiloglou, 2008). The comparisons of the direct solarradiation models and the diffuse solar radiation modelsare shown in Tables 1 and 2, respectively.

For the direct radiation, the new model has the highestprediction accuracy, with R2 = 0.992, RMSE = 16.9 W/m2,and MAE = 2.2%. The models of MLWT2, MRM-5, andYang et al. are the best existing ones, with the R2 of0.976, 0.98, and 0.976, respectively and the MAE of3.9%, 5.4%, and 5.4%, respectively. From Table 2 it is clearthat the prediction performances of the diffuse solar radia-tion models differ from one another dramatically. The newmodel is much better than any existing models, withR2 = 0.86, RMSE = 8.7 W/m2, and MAE = 9.9%. TheMETSTAT model and MRM-5 model are the two bestexisting models. The excellent performance of the newmodel may result from the high quality data measured byNREL, rigorous data filter and regression method involvedin this paper.

The applicability of the new model to other places is val-idated by comparing its predictions with the calculationswith the reference code SMARTS (Gueymard, 1995,2001). This code has gained wide acceptance in atmosphereradiation computation. The comparisons of the direct and

Table 1Comparison of direct solar radiation models.

Model R2 RMSE(W/m2)

MAE(%)

New model 0.992 16.9 2.2MLWT2 (Gueymard, 2003) 0.976 33 3.9MRM-5 (Kambezidis and Psiloglou, 2008;

Psiloglou et al., 2000)0.98 45.4 5.4

Yang et al. (2001) 0.976 45.9 5.4METSTAT (Maxwell, 1998) 0.928 72.5 7.7Bird and Hulstrom (1980, 1981) 0.982 84.6 10.1MAC (Davies, 1987; Davies et al., 1988) 0.975 92.3 11.6Helosat-1 (Dumortier, 1995; Page, 1996) 0.932 99.1 12.4

Table 2Comparison of diffuse solar radiation models.

Model R2 RMSE(W/m2)

MAE(%)

New model 0.86 8.7 9.9METSTAT (Maxwell, 1998) 0.803 10.5 15.9MRM-5 (Kambezidis and Psiloglou, 2008;

Psiloglou et al., 2000)0.8 10.8 15.9

Helosat-1 (Dumortier, 1995; Page, 1996) 0.739 15.6 21.6Bird and Hulstrom (1980, 1981) 0.809 26.3 40.9MAC (Davies, 1987; Davies et al., 1988) 0.81 26.7 41.6Yang et al. (2001) 0.807 90.8 154

Table 3Comparisons of the solar irradiances predicted by the new model andSMARTS for different elevation angles at sea level.

h (�) Direct irradiance (W/m2) Diffuse irradiance (W/m2)

New model SMARTS New model SMARTS

5 82.0 93.7 39.2 36.410 241.1 233.3 70.3 70.315 374.8 355.1 94.8 99.820 476.7 452.8 115.3 124.130 615.2 592.4 149.0 161.345 733.3 716.4 188.1 197.260 796.5 784.1 216.3 21875 828.4 818.6 233.6 229.190 838.3 828.7 239.5 232.9

Table 4Comparisons of the solar irradiances predicted by the new model andSMARTS for different altitudes (h = 60�).

Altitude (km) Direct irradiance (W/m2) Diffuse irradiance (W/m2)

New model SMARTS New model SMARTS

0 797 783.7 216.1 218.62 1046.2 1016.8 77.5 97.74 1150 1128.9 46.2 526 1199.2 1198.1 34.9 36.38 1229.4 1226.3 28.6 29.8

10 1252.2 1255.1 23.9 23.812 1271 1270.7 20 19.815 1293.4 1285.4 15.3 14.920 1319.9 1299.5 9.8 8.925 1337.3 1307.3 6.2 5.430 1347.9 1311.6 4 3.2

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diffuse solar irradiances are shown in Table 3. The identicalatmospheric conditions are US standard atmosphere,w = 1.416 cm, rural aerosol 23 km meteorological rangesor s = 0.2688.

It can be seen form Table 3 that the direct and diffuseparts predicted by the new model agree with those ofSMARTS very well. The MAEs of the new model againstSMARTS for direct and diffuse irradiances are 4.1% and4.2%, respectively. The relative error between the newmodel and SMARTS increases slight with decreasing eleva-tion angle, and reaches 12.5% for direct irradiance and7.7% for diffuse irradiance at h = 5�. However, the absoluteerrors are still acceptable, with 11.7 W/m2 for direct irradi-ance and 2.8 W/m2 for diffuse irradiance. At the altitudefrom sea level up to 30 km, comparisons of the predictions

of the new model with those of the SMARTS are shown inTable 4, from which it can be seen that the direct and dif-fuse irradiances predicted by the new model at differentaltitudes agree with those of SMARTS very well. TheMAEs of the new model against SMARTS for direct anddiffuse irradiances at different altitudes are 1.3% and8.6%, respectively. The MAE for diffuse irradiance is rela-tively higher than that of the direct irradiance. It is mainlyresulting from the low values of diffuse irradiance at highaltitude. The solar irradiance on the seal level (0 km) andat high altitude (30 km) varies dramatically. Under clearsky conditions, the difference of direct solar irradiancecan be as high as 500 W/m2, while the difference of diffusesolar irradiance is over 200 W/m2. Therefore, altitude is anon-ignorable parameter in evaluating the solar irradiance.

6. Conclusion

A simple and accurate model has been proposed to pre-dict direct and diffuse solar radiations under cloudless con-ditions at various altitudes. Due to many atmosphericvariables affecting these values and the functional complex-ity involved between them, the method based on the non-linear regression is used. The dataset used is from NREL.The input parameters include solar elevation angle, pres-sure, AOD at 500 nm, air temperature, and relativehumidity.

Among the well-known solar models reviewed, theMLWT2 model (Gueymard, 2003) and METSTAT model(Maxwell, 1998) perform best in predicting the direct anddiffuse irradiances, respectively. The MLWT2 model fordirect radiation has a R2 of 0.976, a RMSE of 33 W/m2,and an MAE of 3.9%. The METSTAT model for diffuseradiation has a R2 of 0.803, a RMSE of 10.5 W/m2, andan MAE of 15.9%.

The new model for direct solar radiation has a R2 of0.992, a RMSE of 16.9 W/m2, and an MAE of 2.2%, andthe new model for diffuse radiation has a R2 of 0.86, aRMSE of 8.7 W/m2, and an MAE of 9.9%. Therefore,

Q. Dai, X. Fang / Advances in Space Research 53 (2014) 1239–1245 1245

the accuracy of the new model is remarkably higher thanthe best existing models.

The applicability of the new model in different placesand at different altitudes is validated by the reference codeof SMARTS. The results show that the new model can beused to predict the solar irradiances in different localities atdifferent altitudes. The new solar radiation model can beprogrammed as a subroutine of the simulation code. Com-bining this subroutine with other heat load subroutinessuch as infrared and convective heat transfer, the instanta-neous thermal characteristics of aerostats can be predicted.

Acknowledgments

This work was supported by Funding for OutstandingDoctoral Dissertation in NUAA(BCXJ10-02), the Funda-mental Research Funds for the Central Universities andthe Priority Academic Program Development of JiangsuHigher Education Institutions. The authors also wish toacknowledge the NREL for maintaining a qualifieddatabase.

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