A simple model to predict solar radiation under clear sky conditions

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<ul><li><p>dX</p><p>Aero</p><p>anj</p><p>foe 3</p><p>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</p><p>from the buoyancy resulted from the density dierencebetween the inner gas and the surrounding air. Its distin-</p><p>the primary energy source of a high altitude long</p><p>Several studies were carried out for modeling solar radi-ation on aerostats. Carlson and Horn (1983) and Wang</p><p>(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.</p><p> Corresponding author. Tel./fax: +86 25 8489 6381.E-mail addresses: daiqiumin913@126.com (Q. Dai), xd_fang@yahoo.</p><p>com (X. Fang).</p><p>Available online at www.sciencedirect.com</p><p>ScienceDirect</p><p>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.</p><p>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</p><p>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.</p><p>Keywords: Solar radiation; Direct radiation; Diuse radiation; Aerostat; Altitude</p><p>1. Introduction</p><p>An aerostat is a lighter than air vehicle whose lift derives</p><p>endurance aerostat. The good estimation of solar radiationis crucial for modeling the thermal performance of anaerostat.A simple model to predict solar ra</p><p>Qiumin Dai,</p><p>Department of Man, Machine and Environment Engineering, College of</p><p>29 Yudao Street, N</p><p>Received 10 September 2013; received in revisedAvailable onlin</p><p>Abstract</p><p>Solar radiation is one of the major factors that dominate the thttp://dx.doi.org/10.1016/j.asr.2014.01.025</p><p>0273-1177/$36.00 2014 COSPAR. Published by Elsevier Ltd. All rights reseiation under clear sky conditions</p><p>iande Fang </p><p>space Engineering, Nanjing University of Aeronautics and Astronautics,</p><p>ing 210016, China</p><p>rm 25 January 2014; accepted 26 January 2014February 2014</p><p>al behaviors of aerostats in the daytime and the primary energy</p><p>www.elsevier.com/locate/asr</p><p>(2014) 12391245rved.</p></li><li><p>tance in the last three decades. The algorithms for eachattenuation process were the basis of the Iqbal (1983)</p><p>h i</p><p>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</p><p>(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.</p><p>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%.</p><p>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</p><p>1240 Q. Dai, X. Fang / Advances in Spaccuracy.mR sin h 0:153:885 h1:2531</p><p>3</p><p>where h is the solar elevation angle. The absolute air masscan be determined by</p><p>mA mRp=1013 4where p is the atmospheric pressure.</p><p>2.2. Heliosat-1 model (Dumortier, 1995; Page, 1996)</p><p>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:</p><p>IDN ISUN expmArT L 5Id ISUN 0:0065 0:0646T L2 0:045 sin h</p><p> 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):</p><p>mR sin h 0:506h 6:081:6361 7</p><p>2.3. MAC model (Davies, 1987; Davies et al., 1988)</p><p>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</p><p>IDN 0:9662ISUNT RT OTMGTW T A 1Id 0:79ISUNT OT W TMGT AA sin h0:51 T R</p><p> 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</p><p>2.1. Bird and Hulstrom (1980, 1981) model</p><p>The Bird and Hulstrom model has gained wide accep-IDN ISUN T RT O aW T A 8</p></li><li><p>literature (Gueymard, 2003).</p><p>ace2.6. MRM-5 model (Kambezidis and Psiloglou, 2008;</p><p>Psiloglou et al., 2000)</p><p>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:</p><p>I I T T T T T 16Id ISUN 0:75TOT R aW 1 T Ag 0:5T O1 T R 9</p><p>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):</p><p>mR 35=1 1224 sin2 h0:5 10</p><p>2.4. METSTAT model (Maxwell, 1998)</p><p>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:</p><p>IDN 0:9751ISUNT RT OTMGT W T A 11where the relative air mass mR is determined by Eq. (7).</p><p>2.5. MLWT2 model (Gueymard, 2003)</p><p>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:</p><p>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</p><p> 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</p><p>Q. Dai, X. Fang / Advances in SpDN SUN A R O W MG</p><p>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.</p><p>2.7. Yang et al. (2001) model</p><p>Based on the analysis of spectral model and Angstromcorrelation, Yang et al. (2001) proposed</p><p>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).</p><p>3. Data description</p><p>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.</p><p>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)</p><p>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)</p><p>pw;s 6:112 exp17:5t=241 t 21where t is the air temperature in C.</p><p>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</p><p>(a) there is an indication of instrumentation malfunctionor power failure,</p><p>(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.</p><p>Research 53 (2014) 12391245 1241After the quality test, a total of 9660 qualied datapoints are selected for the nonlinear regression.</p></li><li><p>ace4. Development of the new solar radiation model</p><p>4.1. Consideration of solar radiation at the top of the</p><p>atmosphere</p><p>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.</p><p>ISUN 13671:017 0:0174 cos f2 22where f is the true anomaly, which can be calculated with</p><p>f MA 0:0334 sinMA 0:000349 sin2MA 23where MA = 2np/365, and n is the day number in a year.</p><p>4.2. Consideration of direct solar radiation</p><p>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.</p><p>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</p><p>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</p><p>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).</p><p>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):</p><p>s 0:744s500 28Because the AOD and water vapor column are mea-</p><p>sured on the ground, the vertical distribution of AOD</p><p>1242 Q. Dai, X. Fang / Advances in Spand water vapor column should be considered when themodel is used for predicting the direct irradianc...</p></li></ul>