A simple model to predict solar radiation under clear sky conditions

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    hermsource 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 of

    from the buoyancy resulted from the density dierencebetween the inner gas and the surrounding air. Its distin-

    the primary energy source of a high altitude long

    Several studies were carried out for modeling solar radi-ation on aerostats. Carlson and Horn (1983) and Wang

    (2010) used a semi-empirical direct solar model which takesthe eect of altitude into account, but they neglected theinuence of altitude on diuse and reected irradiance.Dai et al. (2012) employed an empirical solar model, wherethe atmospheric transmittance was highly simplied.

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

    com (X. Fang).

    Available online at www.sciencedirect.com

    ScienceDirect

    Advances in Space Research 53guishing features of long endurance, station keeping, andlow cost-eectiveness 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 is

    et al. (2007) assumed the total solar irradiance reachingan aerostat to be constant. Kreith and Kreider (1974)and Farley (2005) neglected the diuse and reected irradi-ance. This assumption is unacceptable for low altitude con-ditions, where the magnitude of sum of diuse and reectedirradiance 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.the 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 diuse irradiances under clear sky conditions is proposed. The new model has excel-lent prediction accuracy. The coecient 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 coecient of determination for diuse 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; Diuse radiation; Aerostat; Altitude

    1. Introduction

    An aerostat is a lighter than air vehicle whose lift derives

    endurance aerostat. The good estimation of solar radiationis crucial for modeling the thermal performance of anaerostat.A simple model to predict solar ra

    Qiumin Dai,

    Department of Man, Machine and Environment Engineering, College of

    29 Yudao Street, N

    Received 10 September 2013; received in revisedAvailable onlin

    Abstract

    Solar radiation is one of the major factors that dominate the thttp://dx.doi.org/10.1016/j.asr.2014.01.025

    0273-1177/$36.00 2014 COSPAR. Published by Elsevier Ltd. All rights reseiation under clear sky conditions

    iande Fang

    space Engineering, Nanjing University of Aeronautics and Astronautics,

    ing 210016, China

    rm 25 January 2014; accepted 26 January 2014February 2014

    al behaviors of aerostats in the daytime and the primary energy

    www.elsevier.com/locate/asr

    (2014) 12391245rved.

  • tance in the last three decades. The algorithms for eachattenuation process were the basis of the Iqbal (1983)

    h i

    ace Research 53 (2014) 12391245The operational altitude of aerostats covers a widerange, from 0 km for tethered aerostats to over 30 kmfor high altitude aerostats. The dierences of direct anddiuse solar irradiances predicted by spectral analysis atdierent 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 eect of altitude on direct solar irradiance, whilethe eect of altitude on diuse 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 eect of atmospheric conditions, suchas aerosol and water vapor. The values of aerosol andwater vapor vary dramatically. At a given altitude, thedierences of direct and diuse solar irradiances predictedby spectral analysis at dierent 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 dierence of 50 K and the forced convec-tive heat transfer coecient 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 diuse 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 its

    1240 Q. Dai, X. Fang / Advances in Spaccuracy.mR sin h 0:153:885 h1:2531

    3

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

    mA mRp=1013 4where 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 diuse radia-tion (Dumortier, 1995). They can be written as:

    IDN ISUN expmArT L 5Id ISUN 0:0065 0:0646T L2 0:045 sin h

    0:0327T L2 0:014 sin2 h 6where 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 sin h 0:506h 6:081:6361 7

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

    The MAC model provided a dierent treatment of theextinction process involved by water vapor, as expressedin the following:model and the METSTAT model (Maxwell, 1998). Themodel is of the form

    IDN 0:9662ISUNT RT OTMGTW T A 1Id 0:79ISUNT OT W TMGT AA sin h0:51 T R

    0:841 T AS=1 mR m1:02R 2where IDN is the direct irradiance, Id is the diuse irradi-ance, ISUN is the solar constant, TR, TO, TMG, TW andTA are the individual transmission coecient for Rayleighscatting, ozone, mixed gases, water vapor and aerosol, TAAis 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:2. Review of solar radiation models

    2.1. Bird and Hulstrom (1980, 1981) model

    The Bird and Hulstrom model has gained wide accep-IDN ISUN T RT O aW T A 8

  • literature (Gueymard, 2003).

    ace2.6. MRM-5 model (Kambezidis and Psiloglou, 2008;

    Psiloglou et al., 2000)

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

    I I T T T T T 16Id ISUN 0:75TOT R aW 1 T Ag 0:5T O1 T R 9

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

    mR 35=1 1224 sin2 h0:5 10

    2.4. METSTAT model (Maxwell, 1998)

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

    IDN 0:9751ISUNT RT OTMGT W T A 11where the relative air mass mR is determined by Eq. (7).

    2.5. MLWT2 model (Gueymard, 2003)

    MLWT2 model is a modied 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 LambertBeer 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 ISUNT RT AT OT W T NST NT 12mR sin h 0:4566590 h0:076:4836 h1:6971 13mW sin h 0:0331490 h0:12:471 h1:38141 14mNS sin h 1:121290 h1:613221:55

    h3:26291 15where TNS and TNT are transmission coecients for strato-spheric and tropospheric nitrogen dioxide, respectively. Allnecessary expressions can be found in the above-mentioned

    Q. Dai, X. Fang / Advances in SpDN SUN A R O W MG

    Id 0:5ISUNT OT W TMGT AA1 T AST R sin h 17where the relative air mass is determined by Eq. (7). Thetransmittance function for uniformly mixed gases is calcu-lated using ve atmospheric gases (CO2, CO, N2O, CH4and 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 ISUN T RT OTMGTW T A 0:013 18Id ISUN T OTMGT W 1 T RT A 0:013 sin h 19where 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(39.74N, 105.18W, 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 t to thetwo years of available data is (Gueymard, 2012)

    w 0:1849Rhpw;s1:0049 20where Rh is the relative humidity, and pw,s is the saturatewater vapor pressure in mbar and can be calculated with(Buck, 1981)

    pw;s 6:112 exp17:5t=241 t 21where t is the air temperature in C.

    Cloud is an important factor aecting the solar irradi-ance. Even small amount of clouds may become a seriousuctuation. All occurrences of clouds must be removedto obtain valid performance results (Gueymard, 2012),and quality control tests should be conducted to eliminatethe eect 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 uctuates, or(f) the AOD is higher than 0.5.

    Research 53 (2014) 12391245 1241After the quality test, a total of 9660 qualied datapoints are selected for the nonlinear regression.

  • ace4. 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 earths 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 13671:017 0:0174 cos f2 22where f is the true anomaly, which can be calculated with

    f MA 0:0334 sinMA 0:000349 sin2MA 23where 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 (Due and Beckman, 2006). Therefore, the physicalmodel of the direct solar radiation can be expressed interms of

    IDN ISUNT RT W T A 24where TR, TW and TA are transmission coecients forRayleigh scattering, water vapor and aerosol, respectively.All the individual transmission coecients are proposed onthe basis of LambertBeer law. They can be written as

    T R expa1mn1A 25TW expa2wn2mn3R 26T A expa3sn4mn5R 27where 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 Angstroms law as the fol-lowing (Bird and Hulstrom, 1981):

    s 0:744s500 28Because the AOD and water vapor column are mea-

    sured on the ground, the vertical distribution of AOD

    1242 Q. Dai, X. Fang / Advances in Spand water vapor column should be considered when themodel is used for predicting the direct irradiance at highaltitude. The vertical distributions of AOD and watervapor column can be calculated from aerosol attenuationcoecient 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 sm exp0:691DH 29w wm exp0:44DH 30where s is the AOD at dierent altitude, sm is the AOD atmeasurement site, w is the water vapor column at dierentaltitude, 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 inuence on direct irradi-ance. This inuence varies slightly throughout the year(Kambezidis and Psiloglou, 2008). Meanwhile, their den-sity is dicult 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 eect of ozone and oxynitride on directirradiance is simplied and merged into the transmissioncoecient of the Rayleigh scattering in this paper.

    4.3. Consideration of diuse 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 diuse 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 diuse radiation iscaused by the scatter eect of molecular, water vapor andaerosol, and thus it decreases with altitude increasing.

    Liu and Jordan (1960) found that the atmospheric trans-mittance for diuse radiation was corresponding to a xedvalue of atmospheric transmittance for direct radiationdepending upon the solar elevation angle. The experimen-tal data showed that this simplication had gained ade-quate accuracy, and it has been used in various ways toevaluate diuse 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 a b sin h cw dsISUN IDN sin h 31where a, b, c and d are the best tting parameters.

    4.4. Results and analysis

    Regression analysis is the most widely used statisticaltechnique for investigating and modeling the relationship

    Research 53 (2014) 12391245between variables. Normal regression models are usuallyapplied in science and engineering to model data for which

  • 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 diuse solar radia-tion models dier 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 lter 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,

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

    acelinear or nonlinear functions of unknown parameters areused.

    The measured data were used in multiple regressionanalysis to obtain the best tting 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 tting parameters are. It indicates that the sumof the squares of the residuals should be least.

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

    R2 PN

    i1F Pred;i F PredF Meas;i F Meas 2PNi1F Pred;i F Pred2

    PNi1F Meas;i F Meas2

    32

    RMSE PN

    i1F Pred;i F Meas;i2N

    " #0:533

    MAE 1N

    XNi1

    jF Pred;i F Meas;ijF Meas;i

    34

    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 ISUN exp0:103m0:571A 0:081wmR0:213

    s0:91m0:87R 35Id 0:143 0:113 sin h 0:0485w sISUN IDN

    sin h 36where 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 diuseirradiance plotted against the measured diuse irradiance.The results show that the new model is able to accuratelypredict the direct and diuse irradiance under clear skyconditions.

    5. Evaluation of solar radiation models

    The new model is compared to the existing high qualitymodels. The eects of the ozone and oxynitride absorptionare considered in several models. These eects 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 solar

    Q. Dai, X. Fang / Advances in Spradiation models and the diuse solar radiation modelsare shown in Tables 1 and 2, respectively.Fig. 1. Predicted vs. measured direct solar irradiance.

    Research 53 (2014) 12391245 12432001). This code has gained wide acceptance in atmosphereradiation computation. The comparisons of the direct and

  • aceTable 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

    1244 Q. Dai, X. Fang / Advances in Spdiuse 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 diuseparts predicted by the new model agree with those ofSMARTS very well. The MAEs of the new model againstSMARTS for direct and diuse 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 diuse 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 diuse irradiance. At the altitudefrom sea level up to 30 km, comparisons of the predictions

    MAEs of the new model against SMARTS for direct and

    Table 2Comparison of diuse 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 dierent elevation angles at sea level.

    h () Direct irradiance (W/m2) Diuse 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 828.4 818.6 233.6 229.190 838.3 828.7 239.5 232.9diuse irradiances at dierent altitudes are 1.3% and8.6%, respectively. The MAE for diuse irradiance is rela-tively higher than that of the direct irradiance. It is mainlyresulting from the low values of diuse irradiance at highaltitude. The solar irradiance on the seal level (0 km) andat high altitude (30 km) varies dramatically. Under clearsky conditions, the dierence of direct solar irradiancecan be as high as 500 W/m2, while the dierence of diusesolar 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 diuse solar radiations under cloudless con-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 dierentaltitudes agree with those of SMARTS very well. The

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

    Altitude (km) Direct irradiance (W/m2) Diuse 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 29.810 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

    Research 53 (2014) 12391245ditions at various altitudes. Due to many atmosphericvariables aecting 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 anddiuse 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 diuseradiation 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 diuse radiation has a R2 of 0.86, aRMSE of 8.7 W/m2, and an MAE of 9.9%. Therefore,

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

    The applicability of the new model in dierent placesand at dierent altitudes is validated by the reference code

    Gueymard, C.A., 2001. Parameterized transmittance model for directbeam and circumsolar spectral irradiance. Sol. Energy 71, 325346.

    Gueymard, C.A., 2003. Direct solar transmittance and irradiance predic-

    Q. Dai, X. Fang / Advances in Space Research 53 (2014) 12391245 1245of SMARTS. The results show that the new model can beused to predict the solar irradiances in dierent localities atdierent 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 qualieddatabase.

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    A simple model to predict solar radiation under clear sky conditions1 Introduction2 Review of solar radiation models2.1 Bird and Hulstrom (1980, 1981) model2.2 Heliosat-1 model (Dumortier, 1995; Page, 1996)2.3 MAC model (Davies, 1987; Davies et al., 1988)2.4 METSTAT model (Maxwell, 1998)2.5 MLWT2 model (Gueymard, 2003)2.6 MRM-5 model (Kambezidis and Psiloglou, 2008; Psiloglou et al., 2000)2.7 Yang et al. (2001) model

    3 Data description4 Development of the new solar radiation model4.1 Consideration of solar radiation at the top of the atmosphere4.2 Consideration of direct solar radiation4.3 Consideration of diffuse solar radiation4.4 Results and analysis

    5 Evaluation of solar radiation models6 ConclusionAcknowledgmentsReferences

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