ORIGINAL PAPER

Accuracy and sensitivity analysis for 54 models of computinghourly diffuse solar irradiation on clear sky

Viorel Badescu & Christian A. Gueymard &

Sorin Cheval & Cristian Oprea & Madalina Baciu &

Alexandru Dumitrescu & Flavius Iacobescu &

Ioan Milos & Costel Rada

Received: 4 December 2011 /Accepted: 27 April 2012 /Published online: 17 May 2012# Springer-Verlag 2012

Abstract Fifty-four broadband models for computation ofsolar diffuse irradiation on horizontal surface were tested inRomania (South-Eastern Europe). The input data consist ofsurface meteorological data, column integrated data, and dataderived from satellite measurements. The testing procedure isperformed in 21 stages intended to provide information aboutthe sensitivity of the models to various sets of input data.

There is no model to be ranked “the best” for all sets of inputdata. However, some of the models performed better thanothers, in the sense that they were ranked among the best formost of the testing stages. The best models for solar diffuseradiation computation are, on equal footing, ASHRAE 2005model (ASHRAE 2005) and King model (King and Buckius,Solar Energy 22:297–301, 1979). The second best model isMAC model (Davies, Bound Layer Meteor 9:33–52, 1975).Details about the performance of each model in the 21 testingstages are found in the Electronic Supplementary Material.

1 Introduction

Detailed knowledge of solar radiation is of fundamentalimportance in meteorology. However, in most countries,the spatial density of the radiometric stations is inadequateto obey the exigencies of various categories of users. Thishas prompted the development of calculation procedures toprovide radiation estimates for areas where measurementsare not carried out and for situations when gaps in themeasurement records occurred. Only a minority of theseprocedures has been validated by their authors and usuallyunder specific geographical or climatic circumstances only.The correct validation and comparison of radiative modelsraise specific issues. For instance, different statistics may beused to evaluate the bias and random differences betweenthe computed and measured data series. Moreover, variousranking procedures can be used to compare the performanceof models, yielding different results [for more detailed dis-cussions, see Gueymard and Myers (2008)]. To increase theconfidence in modeled data accuracy, there is a need forvalidation by independent groups and at a variety of testsites in many different climatic areas. Most studies on mod-els intercomparison focused on global solar radiation(Davies et al. 1988; Davies and McKay 1989; Gueymard

Electronic supplementary material The online version of this article(doi:10.1007/s00704-012-0667-1) contains supplementary material,which is available to authorized users.

V. Badescu (*) : F. IacobescuCandida Oancea Institute, Polytechnic University of Bucharest,Spl. Independentei 313,Bucharest 060042, Romaniae-mail: badescu@theta.termo.pub.ro

C. A. GueymardSolar Consulting Services,P.O. Box 392, Colebrook, NH 03576, USAe-mail: Chris@SolarConsultingServices.com

S. ChevalNational Research and DevelopmentInstitute for Environmental Protection,Splaiul Independentei nr. 294, Sect. 6,060031 Bucharest, Romaniae-mail: sorincheval@yahoo.com

C. Oprea :M. Baciu :A. Dumitrescu : I. Milos : C. RadaNational Meteorological Administration,97 Sos. Bucuresti-Ploiesti,Bucharest 013686, Romania

A. DumitrescuFaculty of Geography, University of Bucharest,Bucharest, Romania

V. BadescuRomanian Academy,Calea Victoriei 125,Bucharest 010071, Romania

Theor Appl Climatol (2013) 111:379–399DOI 10.1007/s00704-012-0667-1

1993, 2012; Badescu 1997; Battles et al. 2000; Iziomon andMayer 2002; Ineichen 2006; Younes and Muneer 2007;Gueymard and Myers 2008).

There is considerable literature dealing with the computa-tion of diffuse solar radiation. The existing models may beclassified according to their basic principle as well as accord-ing to their main output. Most models belong to the followingcategories: (1) statistical models, which are based on statisticalcorrelations between diffuse solar radiation and other meteo-rological or radiometric quantities, or on autocorrelations ofdiffuse radiation time series, and (2) physical models, whichare based on physics, empirical coefficients, and parametriza-tion relationships for more complicated physical processes.The models output is usually the diffuse irradiance or thediffuse irradiation on various time intervals (hour, day, ormonth).

Most of the existing models have been previously tested,under a few geographical conditions, during different timeintervals and using various testing procedures. How theycompare in practice under specific and identical conditionsis still not known. To answer this question, intercomparisonstudies are necessary. A short survey of such studies fol-lows. Several statistical diffuse solar radiation models havebeen tested in Greece (Katsoulis 1991). The models belongto two categories, i.e., models depending on the fraction ofmonthly average daily diffuse radiation to total solar radia-tion and models that are a function of the sunshine fraction.The performance of ten simple physical models used toestimate hourly and daily diffuse solar irradiation on in-clined surfaces has been evaluated in Spain (Diez-Mediavilla et al. 2005). The best model is that ofMuneer, followed by the Reindl model, for hourly aswell as for daily values. Eight widely used statisticalmodels predicting hourly, daily, and monthly diffuseirradiation values were tested using 5 years of data fromfive selected meteorological stations in Egypt (Elminir2007). The results indicate that the correlations relatingthe diffuse fraction with the clearness index and the sunshinefraction are more reliable for diffuse irradiation predictionsthan those using each variable separately. Several statisticalmodels for estimating the diffuse radiation from global radia-tion have been tested under the Australian climate by Bolandet al. (2008). The authors pointed out the advantage of usingthe logistic function instead of piecewise linear or simplenonlinear functions. Forty statistical models used topredict monthly average daily values of diffuse solarradiation were tested in the three largest towns of Turkey(Ulgen and Hepbasli 2009). The models were grouped intofour categories. The models using the clearness index and thesunshine fraction as input performed better than the othermodels. Several statistical models based on regression rela-tionships for estimating monthly mean daily diffuse irradia-tion were tested using data from nine stations with different

climatic conditions spread all over China (Jiang 2008). Inaddition, an artificial neural network (ANN) model has beentested. The results show that the ANN estimations are in goodagreement with the actual values and are superior to those ofother available models. Ten statistical global radiation modelsand 11 statistical diffuse radiation models were tested underthe Algerian climate in Koussa et al. (2009). These modelsestimate monthly mean daily and hourly global and diffuseirradiations on a horizontal surface from the monthly averageper day of the main meteorological data. The models of Jainand Liu et al. are, respectively, recommended for the recon-stitution of the monthly averages per hour of the global anddiffuse irradiations. A comparison among 17 different statis-tical models for estimating the hourly diffuse fraction of theglobal irradiance has been proposed in Torres et al. (2010).Twelve of them are polynomial correlations of differentorders, two are based on a logistic function, and the last threeones consider the diffuse irradiance values in the previous andposterior hour to that of the calculation. The models Dirint andBRL are the ones recommended since they exhibit the highestprecision and generate a series of hourly diffuse irradiancevalues of which the distribution functions are very similar tothose of the experimental data. Several linear and nonlinearstatistical models for computation of daily diffuse irradiationwere tested for five sites in Malaysia (Khatib et al. 2012).They have been compared with ANN models. The resultsshowed that the ANNmodels are superior to the other simplerstatistical models.

The present investigation considers 54 physical modelsfor clear-sky diffuse solar radiation computation. This is amuch larger sample of what the literature offers than in anyprevious study on models intercomparison under specificclimatic environment. Compared to the existing validationstudies, the present contribution describes some furthersteps that are deemed necessary to better validate and rankthe performance of radiation models under the “typical”conditions encountered by users in practice. In particular,this means that the precise state of the atmosphere cannot beobtained from collocated instruments (such as sun photo-meters), whose availability is normally critical in assessingthe inherent performance of models.

The objectives of this investigation are twofold. First, theperformance of these 54 models is analyzed and ranked bycomparing their predictions under identical climate condi-tions and using a unique testing procedure. Second, thesensitivity of each model’s performance on the accuracy inthe atmospheric databases used as inputs is reported. Refer-ence radiometric measurements (used as “ground truth”) areprovided here by a meteorological and radiometric station inSouth-Eastern Europe (Romania). This station has providedgood-quality routine data over many years. It is not high-endresearch-class stations, however, which means that the pres-ent study can be considered representative of what can be

380 V. Badescu et al.

obtained at hundreds of similar stations over the world,rather than at a few specialized sites, like in some previousstudies (e.g., Younes and Muneer 2007; Gueymard 2012).These studies considered the ideal case when all the inves-tigated models’ inputs were measured locally at high fre-quency with high-quality instruments, so as to obtain theintrinsic performance of these models by avoiding propaga-tion of errors. In contrast, the present investigation is muchmore pragmatic, since it uses “normal” radiation measure-ments and time interpolated input data to evaluate the “realworld” performance of models under non-ideal conditions.

2 Radiation models

Fifty-four broadband models for the prediction of diffuseirradiation on horizontal surfaces have been selected for thisstudy. Some of these models were already tested in Gueymardand Myers (2008), Gueymard (2012), and Badescu et al.(2010). All models are briefly described in Appendix 1, usinga call number (G001 to G054) for further reference. Fortranroutines for all models can be found in the Electronic Supple-mentary Material (ESM.pdf). The only input variable that iscommon to all models is the zenith angle, Z, which character-izes the sun position. For those models that refer to the solarconstant, a common value of 1,366.1 W/m2 is used (Gueymardand Kambezidis 2004).

3 Input data

A radiation model is driven by input data that are directly orindirectly related to the optical characteristics of the atmo-sphere for the location and period considered. The mostsophisticated models require a wealth of information aboutvarious atmospheric constituents, such as gases and aerosols,and their vertical distribution. For other models, the inputrequirements may be different, and usually simpler. The inputdata needed during the present work are shown in Table 1.Usually, a particular model does not require all the input datain Table 1. Table 2 shows the entries needed by each model.

The input data have been organized in several subgroups.The most important of these consists of measured radiometricand meteorological data at the ground surface. Data measuredin Cluj-Napoca (46.78° N latitude, 23.57° East longitude,417 ma.s.l. altitude) during year 2009 are used in this study.

3.1 Solar radiation measurements

The Cluj-Napoca radiometric station is provided withCM6B Kipp & Zonen radiometers. The measurement un-certainty is ±5 %. The temperature dependence of sensitivityis ±2 % on the interval −20 to +40°C. On a monthly basis

the bias ranges between −2 and +0.9 % (Myers and Wilcox2009). More information about definition of instrument

Table 1 Input data for models used in this work

Symbol Meaning

Astronomical

Year Number of year

Month Number of the month in the year (1–12)

Day Number of day in the month (1–31)

h Hour (UTC, EET)

δ Sun declination angle (°)

z Zenith angle (°)

Esc Solar constant (W/m2)

En0 Extraterrestrial irradiance (W/m2), corrected forthe actual sun–earth distance

mK Kasten’s air mass

mKY Kasten and Young’s air mass

Geographical

λ Latitude (°)

φ Longitude (°)

hg site's elevation (meter)

ρg ground albedo

Meteorological (surface)

p Surface air pressure (hPa)

T Air temperature, dry-bulb (K)

ΔT Dry-bulb temperature T at time t, T(t), minus T(t−3 h)

U Surface air relative humidity (%)

W Wind speed (m/s)

Meteorological (column integrated)

uo Reduced ozone vertical pathlength (atm-cm)

uN Total NO2 vertical pathlength (atm-cm)

w Precipitable water (cm)

Quantities related to atmospheric turbidity

α Angstrom wavelength exponent

β Angstrom turbidity

α1 Angstrom wavelength exponent for <700 nm

α2 Angstrom wavelength exponent for >700 nm

ϖ1 Aerosol single-scattering albedo, <700 nm

ϖ2 Aerosol single-scattering albedo, >700 nm

TL1 Linke turbidity estimated by using Page formulaand Remund et al (2003) method

TL2 Linke turbidity estimated by using Ineichen (2006)method

TL3 Linke turbidity estimated from the empirical formulaof Dogniaux (1986) as a function of precipitable waterand Angstrom’s β coefficient

TL4 Linke turbidity estimated from the average of four linearrelationships between Angstrom’s β coefficient and Linketurbidity by Hinzpeter (1950), Katz et al. (1982),Abdelrahman et al. (1988) and Grenier et al. (1994).

τa Unsworth–Monteith broadband turbidity coefficient

τ700 Aerosol optical depth at 700 nm, dimensionless

vis Visibility (km)

Analysis of 54 models for clear-sky diffuse solar radiation 381

uncertainty may be found in (Gueymard and Myers 2009;Myers 2009). The radiometers are checked twice per weekand cleaned when necessary.

The measurement methodology is provided by stan-dard procedures prepared at the National MeteorologicalAdministration. Measurements are performed as follows.Solar irradiance (unit, W/m2) is measured at 1-minintervals. The series of irradiance values are averagedover 10, 60, and 1,440 min, respectively. The integralof solar irradiance over a time period is solar irradiation(unit, J/m2). Irradiation values for 10 min, 1 h, and 24 hare obtained by multiplying the appropriate averageirradiance values by the appropriate time duration. Theintegration interval starts 1/2 h before the time stamp inthe files and ends 1/2 h after that time stamp. Civil timeis used in these files.

The radiometers are calibrated once per year throughshaded–unshaded measurements in clear-sky days andthrough direct irradiance measurements on a horizontalsurface with reference to the Linke–Feussner etalonactinometer. The Linke–Feussner etalon is calibratedwith reference to the national etalon, i.e., an Angstrom702 pirheliometer with electric compensation. The na-tional etalon is calibrated once at 5 years with referenceto the World Radiometric Reference at Davos (Swiss).

Partial sky obstructions or shading of the sensors bynatural or artificial structures is low and is not consid-ered. There are <0.1 % missing or suspicious data.

3.2 Surface meteorological data

Routine measurements are performed for air tempera-ture, air pressure, and air relative humidity. The sun-shine duration is calculated from global radiation usingthe World Meteorological Organization sunshine criteri-on (WMO 2008). Then, the “sun is shining” if normalsolar global irradiance exceeds 120 W/m2. Values arerecorded hourly as well as at daily level. Measurementsare reported in tens of hour. Relative sunshine valuesare also reported at hourly and daily level as ratiosbetween the sunshine duration and the length of thereference time interval.

3.3 Column integrated data

3.3.1 Ozone

Long-term measurements of column integrated ozoneare performed in Romania once per day in a singlestation (Bucharest-Băneasa, which is located about400 km south of Cluj-Napoca). Measurements are per-formed usually in the interval 09.00–14.00 (EasternEuropean Time). There are missing days in the

recordings (Saturdays and Sundays as well as otherdays). Here, we are using for Cluj-Napoca data mea-sured in Bucharest during 2009. These data are shownin Fig. S1 (here prefix S comes from the ElectronicSupplementary Material).

3.3.2 Precipitable water

Measurements of column integrated precipitable water areperformed once per day at Cluj-Napoca (0.00 UTC). Dailymeasurements for precipitable water performed during 2009are shown in Fig. S2.

3.4 Satellite-derived data

3.4.1 Albedo

The ground albedo is obtained from satellite images ona monthly basis and 15-km spatial resolution. We haveused the Surface Albedo product of the Satellite ApplicationsFacility on Climate Monitoring (http://www.cmsaf.eu),which is part of the EUMETSAT distributed groundsegment. The product line covers broadband albedoproducts from the SEVIRI instrument aboard Meteosat-9 (Schulz et al. 2009). No information about topographyor terrain roughness around the weather stations wasincluded in the input data files at this stage. Figure S3shows monthly albedo information for Cluj-Napoca asobserved during 2009.

3.4.2 Atmospheric turbidity data

The Ångström turbidity coefficients α and β, as well as theaerosol single-scattering albedo ϖ, are obtained from worlddatasets that have been published in (Gueymard and Theve-nard 2009). These are climatological (long-term monthlyaverage) values, whereas daily values would ideally benecessary. These datasets actually provide the aerosol opti-cal depth at 550 nm, τa550, rather than ß. The latter musttherefore be calculated from Ångström’s law:

b ¼ ta550 � 0:55a ð1ÞData files containing values of α, τa550, and ϖ are

the primary databases that are used here to evaluateaerosol extinction in those models that take it intoaccount. Data are gridded at 1×1° resolution, and nocorrection for elevation is considered. Ideally, a muchfiner spatial resolution (such as 10×10 km) would benecessary for validation purposes, but this is currentlynot available. Figure S3 shows the monthly values of α,τ550, and ϖ at Cluj-Napoca. Since no information aboutthe inputs α1, α2, ϖ1, and ϖ2 in models G040 and

382 V. Badescu et al.

Table 2 Input data for tested models

Model number Model name z m ρg p T U W vis w uo uN α β τa TL ϖ

G001 ASHRAE72 X

G002 ASHRAE05 X

G003 Badescu X X X X X

G004 Basha X X X X X

G005 BCLSM X X X

G006 Biga X

G007 Bird X X X X X X X X

G008 CEM X X X X X

G009 Chandr X X

G010 CLS X X X X X

G011 CPCR2 X X X X X X X X

G012 Dognio X X X X

G013 DPPLT X

G014 ESRA1 X X X X

G015 ESRA2 X X X X

G016 ESRA3 X X X X

G017 ESRA4 X X X X

G018 HLJ X

G019 Ideria X X X X X X

G020 Ineich X X X X

G021 IqbalA X X X X X X X X X

G022 IqbalB X X X X X X X X

G023 IqbalC X X X X X X X X X

G024 Josefs X X X X X

G025 KASM X X X X

G026 Kasten X X X X

G027 King X X X X X X

G028 KZHW X X X X

G029 MAC X X X X X

G030 Machlr X X X X

G031 METSTAT X X X X X X X

G032 MRM4 X X X X X

G033 MRM5 X X X X X X X

G034 Nijego X X X X X X X X

G035 NRCC X X X X X

G036 Paltri X

G037 Perrin X X X X X X

G038 PR X X X X X X X

G039 PSIM X X X X X

G040 REST250 X X X X X X X X X

G041 Rodger X X X X

G042 RSC X X X X X X

G043 Santam X X X X X X

G044 Schulz X

G045 Sharma X

G046 Watt X X X X X X X X

G047 WKB X X X X

G048 Yang X X X X X X

G049 Zhang X X X

Analysis of 54 models for clear-sky diffuse solar radiation 383

G054 is available, the following approximations areused:

a1 ¼ a2 ¼ a ð2Þ

ϖ1 ¼ ϖ2 ¼ ϖ ð3ÞThe monthly data were smoothed to derive daily data.

3.5 Default values

The column-integrated nitrogen dioxide content is not mea-sured. Constant values of 0.0002 atm-cm over rural areasand 0.0005 atm-cm over cities are usually assumed. Thelatter value is adopted here for Cluj-Napoca.

4 Testing procedure

4.1 Compatibility between various input datasets

Since the input datasets come from various sources, it isimportant to check their compatibility.

4.1.1 Time compatibility between measurements datasets

Most meteorological datasets are prepared with reference toUTC. Solar radiation datasets and relative sunshine are reportedby using Central European Time. Time compatibility has beenensured between different datasets. We used different proce-dures to obtain time integrations of various quantities (such asrelative sunshine and solar irradiation). A preprocessing of thedata was therefore needed before it could be used for this study.The quantities were grouped according to their measurementtime stamp, and two or more files with input data were pre-pared; each file contains those measured quantities associatedto the same measurement time.

4.1.2 Compatibility between datasets

Inherent differences exist between two quantities related tothe state of the sky: cloud fraction and relative sunshine.

This is a well-known issue. The differences can beexplained as follows. The cloud cover amount is essentiallyan instantaneous quantity, while the relative sunshine is anintegrated quantity. The sky may be clear at the observationmoment but covered by clouds for some time during theintegration interval, and this yields under unitary values ofrelative sunshine.

4.2 Computation of various quantities

4.2.1 Astronomical quantities

Irradiance is a strong function of the solar zenith angle,Z. Most models take the solar geometry into accountthrough the relative air mass, m, rather than Z. Thespecific relationship between m and Z recommendedby each model is used here. Computations were per-formed only for Z<85° to avoid inaccuracies resultingfrom possible horizon shading or experimental cosineerrors, for instance.

4.2.2 Atmospheric turbidity parameters

A more elaborate value of the Angstrom turbidity coef-ficient ß is used in computation. A first estimate of the

Angstrom turbidity (say bb) is computed using Eq. (1).Next, the modal value β is calculated from

b ¼bb 0:83212þ 3:2104bb� �

1þ 5:852bb0:75 ð4Þ

Some models use a special value of ß that corresponds toa α value fixed at 1.3. It is calculated here using Eq. (1) withα01.3. These various inputs used by different models toevaluate the effects of aerosols are shown in Table 2. Noelevation correction is needed since Cluj-Napoca is lowaltitude.

The parameter β2 for models G040 and G054 corre-sponds to α determined from spectral measurements ofaerosol optical depth (AOD) for wavelengths between

Table 2 (continued)

Model number Model name z m ρg p T U W vis w uo uN α β τa TL ϖ

G050 HS X

G051 ABCGS X

G052 Paulescu X X X X X X

G053 Janjai X X X X X X X

G054 REST281 X X X X X X X X X

384 V. Badescu et al.

700 and 1,000 nm. Since no information about β2 isavailable, the following approximation is used:

b2 ¼ b ð5ÞA simplified way of characterizing the optical depth of

aerosols is through a broadband turbidity index. Well-knownexamples of such are the Linke turbidity factor, TL, and thebroadband aerosol optical depth, τa, also known as the Uns-worth–Monteith coefficient. These indicators are input forsome models as shown in Table 2.

4.3 Accuracy indicators

One considers n couples of measured and computed values,denoted mi and ci (i01, n), respectively. In addition, the meanvalues of measured and computed values are defined as:

m � 1n

Pni¼1

mi; c � 1n

Pni¼1

ci ð6; 7Þ

The way of defining the error ei of the ith computed valueis:

ei � ci � mi ð8ÞNote that the error ei has the same physical dimension

like mi and ci.The most common bulk performance statistics are the

mean bias error (MBE) and the root mean square error(RMSE), which are defined as

MBE ¼ 1nm

Pni¼1

ei; RMSE ¼ 1m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Pni¼1

e2i

s; ð9; 10Þ

Equations (9,10) provide dimensionless quantities sincetheir right hand side has been divided by the mean measuredirradiance or irradiation

MBE and RMSE do not characterize the same aspect of theoverall errors’ behavior. Therefore, when comparing variousmodels against the same reference dataset, the ranking that isobtained from MBE in ascending order (of absolute value) isfrequently different from the ranking obtained from RMSE.

4.4 Broad criteria for model performance

A model designed to compute hourly solar irradiation pro-vides good performance if the MBE, RMSE, and the stan-

dard deviation σ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRMSE2 �MBE2

phave as low values

as possible. The following criteria are used in this work tostratify the diffuse hourly irradiation models:

& “Good model” means |MBE|<10 % and RMSE<30 %& “Good enough model” means 10 %<|MBE|<20 % and

30 %<RMSE <40 %& “Bad model” means |MBE|>10 % and RMSE>40 %.

5 Results and discussions

The testing procedure is performed in several stagesintended to provide information about the accuracy andsensitivity of the models to various sets of input data. Anoverall conclusion about models’ performance is alsodrawn. In the following, we denote by σ the hourly relativesunshine (between 0 and 1) and by C the total cloud coveramount (instantaneous quantity, between 0 and 1).

5.1 Accuracy and sensitivity analysis

5.1.1 Stage 1

A first test (so called stage 1) consisted of checking the proce-dure. A large number of subsequent tests have been performedfor Cluj-Napoca using data from 2009. The most relevant stepsare reported here.

5.1.2 Stage 2

This testing stage is associated to the best available inputdata. Only data associated to C00 and σ01 were used. Atotal of 133 hourly recordings were used in calculations.

Daily measurements for precipitable water performed onceper day at 0.00 UTC at Cluj-Napoca were used. Daily measure-ments for ozone performed once per day at 12.00 UTC atBucharest were used for Cluj-Napoca. For intermediate hours,precipitable water and ozone values were linearly interpolatedbetween two consecutive measured values. Satellite data wereused to derive monthly averaged values for albedo, α, τ550, andϖ for Cluj-Napoca. Each value derived from satellite data wasassociated to the 15th day in a month. For intermediate days,values were linearly interpolated. A constant value is assumedduring the day.

Figure 1 shows the results for the models fulfillingthe performance criteria of Section 4.4. Models CG029,CG006, and CG052 provide the best performance. Figure 2shows the comparison between the measured data and theresults predicted by models CG006 and G052. Both modelsunderestimate solar diffuse irradiation. However, modelCG006 provides results with smaller bias (see Table 3 andFig. 2).

5.1.3 Stage 3

This testing stage estimates models’ performance sensi-tivity to input data related to visual (direct) estimationof the state of the sky. Therefore, only data associatedto C00 were used, while the values of σ are free tovary (Fig. S4). A few data points associated to C00and σ00 exist. They may occur in clear days near thesunrise and sunset, when the global radiation is small

Analysis of 54 models for clear-sky diffuse solar radiation 385

and the WMO “sun shining” criterion (WMO 2008)may not work. Other input data are similar to thosefor stage 2. A total of 166 hourly recordings were usedin calculations.

Table S2 shows that the accuracy of some models(i.e., CG002, CG010, CG020, CG027, CG038, andCG045) decreases as compared to stage 2 (see Table 3).For other models, the accuracy increases. Table 4 pro-vides a comparison between results obtained for stages3 and 2 for models with good performance. Generally,both MBE and RMSE values slightly decrease in stage3 as reported to stage 2. The slight decrease in MBEvalues in stage 3 may be the result of the larger numberof data points as compared to stage 2. This is theknown effect of recurrence to mediocrity (Stigler 1997). Fur-thermore, the standard deviation decreases for all models instage 3.

5.1.4 Stage 4

This testing stage estimates models’ performance sensitivityto input data related to (indirect) estimation of the state ofthe sky (i.e., by means of sunshine recordings). Indeed, theapproximation C01−σ is sometimes used in literature whendata on cloud cover amount is missing. Only data associatedto σ01 were used while the values of C are free to vary(Fig. S5). Many of the cloud cover values equal 0, asexpected. However, positive values of C are also pres-ent. Such sort of situations may be imagined. Indeed,the sun may shine continuously for 1 h despite someparts of the sky being covered by clouds. Other inputdata are similar to those used for stage 2. A total of775 hourly recordings were used in calculations.

Table S3 shows that models’ accuracy significantlydecreases as compared to stage 2 (see Table 3). Most ofthe models do not obey the accuracy criteria. This provesthat using clear sky models under cloudy sky conditions isnot recommended.

5.1.5 Stage 5

This testing stage estimates models’ performance sensi-tivity to input data related to ozone content in theatmosphere. A constant value uo00.3 atm-cm was as-sumed all over the year. Other input data are similar tothose of stage 2. A total of 133 hourly recordings wereused in calculations.

The models may be stratified as: (a) not using ozonedata and (b) using ozone data. The accuracy of category(a) of models is of course the same in both stages 2 and5. Table S4 shows that in case (b), the models’ accura-cy slightly decreases using a fix amount of ozone asinput data (compare with Table 3). There is, however,one model (CG038) whose accuracy slightly increases(Table 5).

5.1.6 Stage 6

This testing stage estimates models’ performance sensitivityto input data related to precipitable water. The precipitablewater content has been computed by using Leckner formula(1978). Other input data are similar to those used for stage 2.A total of 133 hourly recordings were used in calculations.

The models may be stratified as: (a) not using precipita-ble water data and (b) using precipitable water data. Theaccuracy of category (b) of models is of course the same inboth stages 2 and 6. Table S5 shows that, in case (b), theaccuracy of some models (i.e., CG010, CG020, and CG038)decreases when using computed instead of measured valuesof precipitable water (compare with Table 3). The accuracyof a few other models increases (Table 6).

Fig. 1 MBE and standard deviation for the models fulfilling theperformance criteria for computation of diffuse hourly irradiation atCluj-Napoca in 2009. The model number is attached to the column

Fig. 2 Comparison between results predicted by model CG006 andCG052 for diffuse irradiation and measured data at Cluj-Napoca in2009

386 V. Badescu et al.

Table 3 Accuracy indicators at Cluj-Napoca for computing diffuse solar hourly irradiation in 2009

Best model

Good model |MBE|<10%; RMSE<30%

Good enough model 10%<|MBE|<20%;30%<RMSE<40%

Bad model |MBE|>10%; RMSE>40%

Worst model

Model number Model name MBE(%) RMSE(%)

G001 ASHRAE72 -9.830 29.740

G002 ASHRAE05 6.880 26.180

G003 Badesc 46.060 52.550

G004 Basha -16.600 30.110

G005 BCLSM -5.690 45.630

G006 Biga 2.120 24.960

G007 Bird 23.940 35.780

G008 CEM -33.460 41.500

G009 Chandr 81.840 89.110

G010 CLS 4.860 25.080

G011 CPCR2 27.080 37.900

G012 Dognio -8.950 26.530

G013 DPPLT 53.900 65.510

G014 ESRA1 -4.560 25.020

G015 ESRA2 3.780 25.120

G016 ESRA3 22.080 35.310

G017 ESRA4 16.760 30.240

G018 HLJ -13.530 28.250

G019 Ideria 215.430 230.920

G020 Ineich -8.740 26.730

G021 IqbalA 26.240 37.180

G022 IqbalB -41.570 49.300

G023 IqbalC 21.670 34.210

G024 Josefs -10.720 26.860

G025 KASM 12.720 27.840

G026 Kasten -31.690 44.850

G027 King 6.750 25.630

G028 KZHW 61.100 84.880

G029 MAC 0.650 24.660

G030 Machlr 104.160 112.930

G031 METSTAT -17.630 31.090

G032 MRM4 159.100 166.140

G033 MRM5 -14.000 28.930

G034 Nijego 38.140 47.620

G035 NRCC 25.810 49.500

G036 Paltri -12.800 27.920

G037 Perrin -14.830 32.170

G038 PR -9.190 27.010

G039 PSIM 10.100 27.790

G040 REST250 18.050 31.660

G041 Rodger -18.080 30.480

G042 RSC -41.990 49.180

G043 Santam 51.530 57.450

G044 Schulz -23.740 34.260

G045 Sharma -9.640 26.310

G046 Watt 2.130 32.400

G047 WKB -33.700 42.860

G048 Yang -12.560 28.550

G049 Zhang 79.220 98.020

G050 HS -23.740 34.260

G051 ABCGS -23.740 34.260

G052 Paulescu 3.590 25.540

G053 Janjai -42.210 52.970

G054 REST281 4.080 25.980

Analysis of 54 models for clear-sky diffuse solar radiation 387

5.1.7 Stage 7

This testing stage estimates models’ performance sensitivityto input data related to ground albedo. A constant value 0.2has been used for the ground albedo all over the year. Otherinput data are similar to those used for stage 2. A total of133 hourly recordings were used in calculations.

The models may be stratified as: (a) not using albedo dataand (b) using albedo data. The accuracy of category (a) ofmodels is of course the same in both stages 2 and 7 (seeTables S6 and S3). For most models of category (b), bothMBEand RMSE values slightly increase in stage 7 as compared tostage 2 (Table 7). Thus, the accuracy is a bit better in stage 2.

5.1.8 Stage 8

This testing stage estimates models’ performance sensitivityto input data related to visual estimation of the state of thesky and ozone. Only data for C00 were used (i.e., thevalues of the sunshine fraction are free to vary). Other inputdata are similar to those used for stage 5 (where a constantozone value has been used as input). A total of 166 hourlyrecordings were used in calculations.

There is no general tendency for models’ accuracy whenchanging the input data from stage 5 to 8 (Table S7). TheMBE and RMSE values increase for some models butdecrease for others.

5.1.9 Stage 9

This testing stage estimates models’ performance sensitivityto input data related to visual estimation of the state of thesky and precipitable water. Only data for C00 were used(i.e., the values of the sunshine fraction are free to vary).

Other input data are similar to those used for stage 6 (wherethe precipitable water has been computed using Leckner’sformula). A total of 166 hourly recordings were used incalculations.

There is no general tendency for models’ accuracy whenchanging the input data from stage 6 to 9 (Table S8). TheMBE and RMSE values increase for some models butdecrease for others.

5.1.10 Stage 10

This testing stage estimates models’ performance sensitivityto input data related to visual estimation of the state of thesky and ground albedo. Only data for C00 were used (i.e.,the values of the sunshine fraction are free to vary). Otherinput data are similar to those used for stage 7 (when a fixedvalue of ground albedo has been used). A total of 166 hourlyrecordings were used in calculations.

There is no general tendency for models’ accuracy whenchanging the input data from stage 7 to 10 (Table S9). TheMBE and RMSE values increase for some models butdecrease for others.

5.1.11 Stage 11

This testing stage estimates models’ performance sensitivitywhen measurements about ozone, precipitable water, andground albedo are missing, but surface measurement dataare available. A constant ozone amount 0.3 atm-cm has beenassumed. Precipitable water has been computed by usingLeckner formula. A constant ground albedo value (0.2) hasbeen assumed. Other input data are similar to those used forstage 2. A total of 133 hourly recordings were used incalculations.

Table 4 Comparison between results obtained at Cluj-Napoca for stages 2 and 3, respectively; Δ(MBE/RMSE/σ)≡(MBE/RMSE/σ)3−(MBE/RMSE/σ)2. Color code similar to Table 3

Model number

Model name

MBE 3

(%) MBE RMSE 3

(%) RMSE 3 (%)

G001 ASHRAE72 -8.84 0.99 28.77 -0.97 27.3782 -0.6903

G006 Biga 0.87 -1.25 24.01 -0.95 23.9942 -0.8756

G020 HLJ -9.38 -0.64 25.96 -0.77 24.2061 -1.0546

G052 Paulescu 2.53 -1.06 24.45 -1.09 24.3187 -0.9677

G054 REST281 2.19 -1.89 24.95 -1.03 24.8537 -0.8039

Table 5 Comparison between results obtained at Cluj-Napoca for stages 2 and 5, respectively. Only models using ozone as input data are shown.Δ(MBE/RMSE)≡(MBE/RMSE)5−(MBE/RMSE)2. Color code similar to Table 3

Model number

MBE 5

(%) MBE RMSE 5

(%) RMSEG038 -8.99 0.2 26.97 -0.04 G052 3.77 0.18 25.59 0.05 G054 4.37 0.29 26.06 0.08

388 V. Badescu et al.

There is no general tendency for models’ accuracy whenchanging the input data from stage 2 to 11 (Table 8). TheMBE and RMSE values increase for some models butdecrease for others.

5.1.12 Stage 12

This testing stage estimates models’ performance sensitivitywhen measurements about ozone, precipitable water, andground albedo are missing and only visual informationabout the state of the sky exists. Only data for C00 wereused (i.e., the values of the sunshine fraction are free tovary). Other input data are similar to those used for stage 11.A total of 166 hourly recordings were used in calculations.

Table S10 shows that for most models both the MBE andRMSE values decrease in stage 12 as compared to stage 11.However, exceptions exist.

5.1.13 Stage 13

This testing stage is associated to the best available inputdata during the extended warm season (April–October).Only data associated to C00 and σ01 were used. Otherinput data are similar to those used for stage 2. A total of129 hourly recordings were used in calculations.

Table S11 shows that the MBE values increase while theRMSE values slightly decrease in stage 13 as compared tostage 2. Thus, the accuracy is generally worse in stage 13.

5.1.14 Stage 14

This testing stage is associated to the best available inputdata during the extended cold season (November–March).Only data associated to C00 and σ01 were used. Otherinput data are similar to those used for stage 2. A total offour hourly recordings were used.

Table S12 shows a great variety of results. The MBE andRMSE values increase for some models but decrease forothers. These results should be taken with caution due to thesmall number of recordings.

5.1.15 Stage 15

This testing stage estimates models’ performance sensitivityduring the extended warm season (April–October) to inputdata related to visual estimation of the state of the sky. There-fore, only data associated to C00 were used while the valuesof σ are free to vary. Other input data are similar to those usedfor stage 3. A total of 161 hourly recordings were used.

Table S13 shows a great variety of results. The MBE andRMSE values increase for somemodels but decrease for others.

5.1.16 Stage 16

This testing stage estimates models’ performance sensi-tivity during the extended cold season (November–March) to input data related to visual estimation of the

Table 6 Comparison between results obtained at Cluj-Napoca for stages 2 and 6, respectively. Only models using precipitable water as input dataare shown. Δ(MBE/RMSE)≡(MBE/RMSE)6−(MBE/RMSE)2. Color code similar to Table 3

Model number

Model name

MBE6

(%) MBERMSE6

(%) RMSEG010 CLS 7.06 2.2 25.61 0.53 G020 Ineich -9.04 -0.3 26.77 0.04 G027 King 6.75 0 25.63 0 G029 MAC 0.31 -0.34 24.59 -0.07 G038 PR -9.78 -0.59 27.09 0.08 G052 Paulescu 3.06 -0.53 25.35 -0.19 G054 REST281 3.75 -0.33 25.85 -0.13

Table 7 Comparison between results obtained at Cluj-Napoca for stages 2 and 7, respectively. Only models using ground albedo as input data areshown. Δ(MBE/RMSE)≡(MBE/RMSE)7−(MBE/RMSE)2. Color code similar to Table 3

Model number

Model name

MBE7

(%) MBERMSE7

(%) RMSEG010 CLS 5.66 0.8 25.28 0.2 G027 King 8.09 1.34 26.07 0.44 G029 MAC 1.5 0.85 24.76 0.1 G038 PR -8.35 0.84 26.76 -0.25 G054 REST281 4.97 0.89 26.25 0.27

Analysis of 54 models for clear-sky diffuse solar radiation 389

state of the sky. Therefore, only data associated to C00were used while the values of σ are free to vary. Other

input data are similar to those used for stage 3. A totalof five hourly recordings were used.

Table 8 Comparison between results obtained at Cluj-Napoca for stages 2 and 11, respectively. Δ(MBE/RMSE)≡(MBE/RMSE)11−(MBE/RMSE)2. Color code similar to Table 3

Model number Model name MBE 11 (%) MBE RMSE 11 (%) RMSEG001 ASHRAE72 -9.83 0 29.74 0

G002 ASHRAE05 6.88 0 26.18 0

G003 Badesc 45.65 -0.41 52.13 -0.42

G004 Basha -16.6 0 30.11 0

G005 BCLSM -6.95 -1.26 46.45 0.82

G006 Biga 2.12 0 24.96 0

G007 Bird 24.51 0.57 36.16 0.38

G008 CEM -32.84 0.62 41.02 -0.48

G009 Chandr 81.84 0 89.11 0

G010 CLS 7.88 3.02 25.88 0.8

G011 CPCR2 28.05 0.97 38.68 0.78

G012 Dognio -5.06 3.89 25.71 -0.82

G013 DPPLT 53.9 0 65.51 0

G014 ESRA1 -3.12 1.44 24.91 -0.11

G015 ESRA2 5.62 1.84 25.61 0.49

G016 ESRA3 25.38 3.3 37.52 2.21

G017 ESRA4 16.76 0 30.24 0

G018 HLJ -13.53 0 28.25 0

G019 Ideria 215.43 0 230.92 0

G020 Ineich -9.04 -0.3 26.77 0.04

G021 IqbalA 26.68 0.44 37.48 0.3

G022 IqbalB -41.69 -0.12 49.35 0.05

G023 IqbalC 22.25 0.58 34.58 0.37

G024 Josefs -10.25 0.47 26.66 -0.2

G025 KASM 12.3 -0.42 27.57 -0.27

G026 Kasten -31.69 0 44.85 0

G027 King 8.09 1.34 26.07 0.44

G028 KZHW 61.1 0 84.88 0

G029 MAC 1.15 0.5 24.67 0.01

G030 Machlr 104.09 -0.07 112.85 -0.08

G031 METSTAT -16.66 0.97 30.63 -0.46

G032 MRM4 159.1 0 166.15 0.01

G033 MRM5 -13.52 0.48 28.63 -0.3

G034 Nijego 37.8 -0.34 47.31 -0.31

G035 NRCC 25.23 -0.58 49.15 -0.35

G036 Paltri -12.8 0 27.92 0

G037 Perrin -13.62 1.21 31.57 -0.6

G038 PR -8.76 0.43 26.79 -0.22

G039 PSIM 10.16 0.06 27.7 -0.09

G040 REST250 18.95 0.9 32.24 0.58

G041 Rodger -16.9 1.18 29.9 -0.58

G042 RSC -41.44 0.55 48.72 -0.46

G043 Santam 51.62 0.09 57.49 0.04

G044 Schulz -23.74 0 34.26 0

G045 Sharma -9.64 0 26.31 0

G046 Watt 8.04 5.91 32.87 0.47

G047 WKB -33.7 0 42.86 0

G048 Yang -12.97 -0.41 28.64 0.09

G049 Zhang 79.22 0 98.02 0

G050 HS -23.74 0 34.26 0

G051 ABCGS -23.74 0 34.26 0

G052 Paulescu 3.24 -0.35 25.4 -0.14

G053 Janjai -21.31 20.9 32.72 -20.25

G054 REST281 4.94 0.86 26.2 0.22

390 V. Badescu et al.

For most models the MBE values increase in stage 16 ascompared to stage 3 (Table S14). The RMSE values increasesor decreases as compared to stage 3. These results should betaken with caution due to the small number of recordings.

5.1.17 Stage 17

This testing stage estimates models’ performance sensitivityduring the extended warm season (April–October) to input datarelated to indirect estimation of the state of the sky. Only dataassociated to σ01 were used in this stage while the values of Care free to vary. Other input data are similar to those used forstage 4. A total of 695 hourly recordings were used incalculations.

Table S15 shows a rather small difference in models’ accu-racy in stages 4 and 17, respectively. All models have poorperformance in Stage 17.

5.1.18 Stage 18

This testing stage estimates models’ performance sensitivityduring the extended cold season (November–March) to inputdata related to indirect estimation of the state of the sky. Onlydata associated to σ01 were used in this stage while the valuesof C are free to vary. Other input data are similar to those usedfor stage 4. A total of 65 hourly recordings were used incalculations.

Table S16 shows that for most models the accuracydecreases considerably in stage 18 as compared to stage 4.All models have poor performance in stage 18.

5.1.19 Stage 19

This testing stage estimates models’ performance sensitivitywhen measurements about ozone, precipitable water, and

ground albedo are missing, but surface measurement dataare available for the extended warm season (April–October).Other input data are similar to those used for stage 11. Atotal of 129 hourly recordings were used in calculations.

Table S17 shows a rather small difference in models’accuracy in stages 19 and 11, respectively. For most models,the accuracy is slightly smaller in stage 19.

5.1.20 Stage 20

This testing stage estimates models’ performance sensitivitywhen measurements about ozone, precipitable water, andground albedo are missing, but surface measurement dataare available during the extended cold season (November–March). Other input data are similar to those used for stage11. A total of four hourly recordings were used incalculations.

Table S18 shows that the accuracy of some modelssignificantly decreases in stage 20 as compared to stage11. However, the reversed situation happens for othermodels.

5.1.21 Stage 21

This testing stage estimates models’ performance sensitivityduring the extended warm season (April–October) whenmeasurements about ozone, precipitable water, and groundalbedo are missing, but visual information about the state ofthe sky is available. Only data for C00 were used (i.e., thevalues of the sunshine fraction are free to vary). Other inputdata are similar to those used for stage 12. A total of 161hourly recordings were used in calculations.

Table S19 shows a comparable models’ accuracy in stage21 as compared to stage 12. The RMSE values generallydecrease in stage 21.

5.1.22 Stage 22

This testing stage estimates models’ performance sensi-tivity during the extended cold season (November–March) when measurements about ozone, precipitablewater, and ground albedo are missing, but visual infor-mation about the state of the sky is available. Only datafor C00 were used (i.e., the values of the sunshinefraction are free to vary). Other input data are similarto those used for stage 12. A total of five hourlyrecordings were used in calculations.

For some models, the difference in accuracy is largebetween stage 12 and 22 (Table S20). In some cases, theMBE and/or RMSE values increase in stage 22. The re-versed situation also happens. These results should be takenwith caution due to the small number of recordings.

Fig. 3 Number of testing stages per model with accuracy criteria fulfilledfor both MBE and RMSE indicators. Diffuse solar radiation in 2009 atCluj-Napoca is considered. The maximum number of stages is 21

Analysis of 54 models for clear-sky diffuse solar radiation 391

5.2 Overview

An overview of the results presented in Section 5.1 is useful.We have counted the number of testing stages per modelwith accuracy criteria fulfilled for both MBE and RMSEindicators. This way of classification stratified the models inseveral accuracy groups, and the first nine best groups areshown in Fig. 3. Twelve models belong to the first three bestgroups. Three of these models (i.e., G001, G002, and G006)are very simple models (i.e., they do not require meteoro-logical data as input).

Two models belonging to the first three best groups (i.e.,G020 and G054) were found among the five best models as aresult of the more accurate analysis described in (Gueymard2012). However, another model (i.e., G007) found among thefive best models in (Gueymard 2012) performed rather poorhere.

The fact that very simple models may be found in thethree best groups of models is remarkable and raises thequestion of why the community developed more advancedmodels, including complex radiative transfer codes such asLibradtran, Modtran, Lowtran, or Sciatran. A few commentson this line follow. Complex models need as input a precisedescription of the state of the atmosphere. Furthermore, theyprovide much better results than simple models when theaverage irradiance is computed on short time intervals, suchas minutes or seconds. Moreover, accurate measurementsare necessary to appropriately validate the performance ofmore complex models. Here, we accumulate some condi-tions that are not favorable for obtaining the best of what thecomplex models can do. Indeed, we compute hourly aver-aged irradiances, and we face some uncertain inputs (partic-ularly about aerosols). In addition, we obtain some missinginput data by interpolation, extrapolation, or estimation(thus introducing larger uncertainties in the predicted irradi-ances). Finally, the response of CM11 device for the mea-sured irradiance may be up to 5 % too low due to its thermalimbalance. Proper ranking of complex models may beobtained by running them against “optimal/ideal” databases,such as that described in Gueymard (2012). The high qualityof the database is given by the fact that each recordingconsists of several input and output data measured almostsimultaneously by high quality instruments. However, sucha database is inherently very small, containing less thantwenty recordings. The present study shows how existingmodels perform with “real world” (i.e., imperfect) inputdata. Depending on the degree of imperfection of theseinputs, the performance of a complex “good” model maydeteriorate, and vice versa, the performance of simpler mod-els may improve.

Using complex models and codes is recommended sincemultiple-scattering effects by aerosols have strong influenceon solar irradiance, particularly on cloudy sky (which,

however, is of smaller interest for this study referring toclear sky models). This has been observed from studies ofirradiance interannual variability, showing a marked season-ality (Pozo Vázquez et al. 2011). Scattering effects by aero-sols may be considered in the simple radiation schemesthrough parameterization procedures, for instance.

6 Conclusions

Fifty-four broadband models for computation of diffuseirradiation on horizontal surface were selected. These mod-els were tested in Romania (South-Eastern Europe). Theinput data consist of surface meteorological data, columnintegrated data and data derived from satellite measure-ments. The testing procedure is performed in 21 stagesintended to provide information about the accuracy andsensitivity of the models to various sets of input data.

The state of the sky may be evaluated either directly(using observations of cloud cover C) or indirectly (usingrecordings of relative sunshine σ). Testing stage 2 refers tothe best available input data. Data associated to C00 andσ0were used there. The results of stage 2 (see Table 3) maybe used as a reference for the other testing stages, whenadditional constraints have been imposed to the input data.No obvious general dependence of models accuracy onseason has been observed (stages 13 and 14).

Testing stage 3 shows that using as input only the visual(direct) estimation of the state of the sky increases theaccuracy of models with good performance. Testing stage4 shows that using as input only the indirect estimation ofthe state of the sky decreases significantly the models’accuracy. This applies for both the warm season (stage 17)and the cold season (stage 18). A consequence is that usingclear sky models under moderate cloudy sky conditions isnot recommended.

Using estimations instead of measured values for severalinput quantities generally decreases models’ accuracy. Thishas been proved in case of ozone (stage 5) and precipitablewater (stage 6). Using a fixed ground albedo value as inputslightly increases the accuracy (stage 7). Note, however, thatthe value used in the reference stage 2 for ground albedo isderived from monthly averaged value by interpolation. Noobvious general effect on models’ accuracy is observedwhen estimated values instead of measured values are usedsimultaneously for ozone, precipitable water, and groundalbedo during the whole year (Stage 11) or during the warmand cold seasons (stages 19 and 20, respectively).

No obvious general effect on models’ accuracy is ob-served when visual estimates for the state of the sky areused, together with estimated values instead of measuredvalues for ozone (stage 8), precipitable water (stage 9), andground albedo (stage 10). However, the accuracy slightly

392 V. Badescu et al.

increases in this case when visual estimates for the state ofthe sky are used during the warm season (stage 21).

There is no model to be ranked “the best” for all sets ofinput data. However, some of the models performed betterthan others, in the sense that they were ranked among thebest for most of the testing stages. Figure 3 shows that thesemodels can be grouped in three categories. The best modelsare G002 and G027. The second best model is G029. Thethird best models are G001, G006, G010, G014, G015,G020, G038, G052, and G054.

Details about the accuracy of each model can be found inthe paper and in the Electronic Supplementary Material.

Acknowledgments This work was supported in part by a grant of theRomanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0089 and by theEuropean Cooperation in Science and Technology project COSTES1002. The authors thank the reviewers for useful comments andsuggestions.

Appendix 1

Clear sky models to computed hourly diffuse solarirradiation

G001: ASHRAE 1972

This is a historical model still widely used by engineers tocalculate solar heat gains and cooling loads in buildings, orinsolation of simple solar systems. It is based on originalempirical work conducted in the 1950s and 1960s (ASHRAE1972). For more details, see Gueymard (1993, 2012).

G002: ASHRAE 2005

ASHRAE 2005 is similar to ASHRAE 1972, but with somerevised coefficients that appeared in ASHRAE (2005).

G003: Badescu

This model is based on MAC (entry G029 in this list) andwas proposed by Badescu (2008).

G004: Bashahu

The composite model of Bashahu and Laplaze (1994) isbased on the direct irradiance model of King and Buckius(1979); see entry G027 below.

G005: BCLSM

This is the model originally proposed by Barbaro et al.(1979) and later modified in Davies et al. (1988) and

Badescu (1987, 1997). The original Barbaro equations areused here, with three modifications: (1) The units in Barbaro'sEq. (10) are converted from cal/(cm2 min) to W/m2, using1 cal04.184 J; (2) a multiplier of 0.1 that was apparentlymissing in the second part of Eq. (10) has been added; and(3) Barbaro's Eq. (11) has been corrected for the missing cosZterm involving the zenith angle Z.

G006: Biga

This is the model proposed by Biga and Rosa (1979).

G007: Bird model

This is the broadband transmittances and turbidity modelaccording to Bird and Hulstrom (1980, 1981). For moredetails, see Gueymard (2003a, 2012).

G008: CEM

This is the broadband model proposed in Atwater and Ball(1978, 1979). See details in Gueymard (2003a).

G009: Chandra

In this model proposed by Chandra (1978), the Linke tur-bidity coefficient, TL, was originally a necessary input. Thiscoefficient is obtained here from an average of four linearrelationships between Angstrom’s β coefficient and TL, asproposed in Hinzpeter (1950), Katz et al. (1982), Abdelrahmanet al. (1988), and Grenier et al. (1994):

TL ¼ 2:1331þ 19:0204b ð11ÞTL must be constrained to the range 2–5 to avoid diver-

gence in Chandra’s model. Chandra reported his results asabsolute irradiances in cal/(cm2 min) and used a solar con-stant of 1.94 cal/(cm2 min) or 1,353 W/m2. His results aretherefore divided here by 1.94 to obtain transmittances. Thisempirical model is based on measurements that certainlyused the IPS56 radiometric scale; hence, the multiplicationby a correction factor of 1.022 (Iqbal 1983) to comply withthe current WRR scale (whose announcement by the WorldMeteorological Organization in 1978 is posterior to thehistorical data used for the model’s development).

G010: CLS

This is the Cloud Layer Sunshine model developed bySuckling and Hay (1976, 1977). Note that precipitable watermust be in centimeters, contrarily to what is indicated in theoriginal papers.

Analysis of 54 models for clear-sky diffuse solar radiation 393

G011: CPCR2

This is the original CPCR2 model by Gueymard (1989) withrevised optical masses (Gueymard 1993).

G012: Dogniaux

This combines the models that have been proposed byDogniaux for direct radiation (Dogniaux 1976) and fordiffuse radiation (Dogniaux 1970). The expression for TLas a function of β is taken here from Dogniaux (1975). SeeGueymard (2003a) for details.

G013: DPP

This is the Daneshyar–Paltridge–Proctor model tested inBadescu (1997), Goswami and Klett (1980), and alsoreviewed in Festa and Ratto (1993). The DPP acronym isfrom Badescu (1997). The direct irradiance at normal inci-dence is here from the original paper by Daneshyar (1978)who used the model of Paltridge and Proctor (1976), withcorrected unit for Z. (It is actually degrees rather thanradians as Badescu or Daneshyar suggested). Coefficientsfor diffuse irradiance are given by Daneshyar as 0.218 and0.299 cal/(cm2 h) for the USA, as cited from Kreith andKreider (1975). Daneshyar also suggests 0.123 and 0.181for Iran. However, their Eq. (5a) suggests that Z is inradians, which is not correct. Goswami and Klett (1980)used the USA values, which are used here, too, after con-version from cal/(cm2 h) to W/m2. They used the correctunit for Z. Festa and Ratto (1993) mention the Iran values,but with Z in radians rather than degrees, due to the confu-sion in the original papers. Badescu (1997) also used radi-ans, but with considerably larger values for the coefficients.To avoid further confusion, the modified equations as usedhere are provided below:

Ebn ¼ 950 1� exp �0:075hð Þ½ � ð12Þ

Ed ¼ 2:534þ 3:475h ð13ÞHere Ebn and Ed stand for direct normal irradiance and

diffuse irradiance, respectively, both measured in W/m2.The sun altitude angle is given by h090−Z where Z is indegrees.

G014: ESRA1

This is the model that was used to develop the EuropeanSolar Radiation Atlas, with the broadband transmittancesand turbidity computed according to Rigollier et al. (2000)and ESRA (2010). Here, the Linke turbidity coefficient for

an air mass of 2 is obtained from the new Page's formula,Eq. (19) of Remund et al. (2003). This revision was sug-gested by J. Remund (personal communication, 2006). Formore details, see Gueymard (2012).

G015: ESRA2

This model is the same as ESRA1, except that TL is nowobtained using formulae proposed in Molineaux et al.(1998) and Ineichen (2006).

G016: ESRA3

This model is similar to ESRA1 or ESRA2, but TL isobtained from the empirical formula of Dogniaux (1986)as a function of precipitable water and Angstrom’s βcoefficient.

G017: ESRA4

This model is again similar to ESRA1, but TL is hereobtained from Eq. (11).

G018: HLJ

This is the broadband model developed by Hottel (1976) fordirect irradiance and later modified in De Carli et al. (1986),Jafarpur and Yaghoubi (1989), Aziz (1990), Khalil andAlnajjar (1995), and Togrul et al. (2000), who all addedthe diffuse transmittance formula of Liu and Jordan (1960),based on Hottel's own recommendation. The Hottel equa-tions corresponding to a visibility of 23 km are used here,which is consistent with the literature cited. For moredetails, see Gueymard (2012).

G019: Ideriah

This model is proposed in Ideriah (1981) based on the directirradiance model by King and Buckius (see entry G027below).

G020: Ineichen

This model comes from Ineichen's parameterization of theSOLIS spectral model (Ineichen 2008). For more details, seeGueymard (2012).

G021. Iqbal A

This is “model A” from Iqbal (1983). It is adapted from theMcMaster (MAC) model (Davies 1975). The original for-mulation for the Rayleigh transmittance is used here, rather

394 V. Badescu et al.

than Iqbal's Eq. (7.4.8), which contains a typo. For moredetails, see Gueymard (1993, 2012).

G022. Iqbal B

This is “model B” from Iqbal (1983). It is adapted from theHoyt (1978) model. For more details, see Gueymard (1993,2012).

G023: Iqbal C

This is “model C” from Iqbal (1983). It is adapted fromBird's model (entry G005 above). For more details, seeGueymard (1993, 2003, 2012).

G024: Josefsson

This is the model by Josefsson (W. Josefsson, unpublishedmanuscript, 1985), as described, used and/or modified inDavies et al. (1988) and Davies and McKay (1989). SeeGueymard (1993) for details.

G025: KASM

This is the modified Kasten (1983) model according toBadescu (1997).

G026: Kasten

This is another modification of the original Kasten (1983)model, this time following Davies and McKay (1989)andKrarti et al. (2006). TL is obtained from Eq. (A1).

G027: King

This model uses the broadband transmittances and turbidityfunctions according to King and Buckius (1979) and Buckiusand King (1978), respectively. The former reference providesand expression that can be used to calculate direct irradiance(Gueymard 2003a). In the latter reference, the diffuse irradi-ance is to be calculated from its Eq. (36), but two free param-eters remain, namely, κL and a1. For this investigation, thefunction κL has been fitted to the numerical values provided inBuckius and King (1978) for five discrete values of ß, such as

kL ¼ 0:8336þ 0:17257b � 0:64595

� exp �7:4296b1:5� �

: ð14ÞSimilarly, the value of coefficient a1 was only described

as varying between 0 and 1. The average value a100.5 isassumed here in the absence of more specific indications.

G028: KZHW

This model was proposed by Krarti et al. (2006) as acombination of those of Zhang et al. (2002) model for globalradiation and Watanabe et al. (1983) for the direct/diffusecomponent separation. The empirical coefficients as modi-fied in Krarti et al. (2006) are used here.

G029: MAC

The McMaster model of Davies (1975) was later used and/ormodified in Davies et al. (1988) and Davies and Uboegbulam(1979). The formulation (particularly for the Rayleigh andaerosol transmittances) used here is as described in Davies etal. (1988). A corrected Rayleigh transmittance expression isused here since it was misprinted in the latter report. For moredetails, see Gueymard (1993, 2012).

G030: Machler

This is the model proposed by Machler and Iqbal (1985).

G031: METSTAT

This is a modified version of Bird’s model (entry G007above) according to Maxwell (1998). For more details, andhow to evaluate the Unsworth–Monteith turbidity coeffi-cient it uses, see Gueymard (1993, 2003a, 2012).

G032: MRM4

This model is described by Muneer et al. (1998) andKambezidis (personal communication, 2002). The nu-merical coefficients considered here are for the USAand southern Europe, as given in Muneer (2004, p.73). For more details, and discussion of this model’sshortcomings, see Gueymard (2003b, 2012).

G033: MRM5

More than a mere revision of MRM4 (entry G032), this isactually a completely different algorithm. It is adapted herefrom Fortran code, version 5, by Kambezidis and Psiloglou(2008). Additional information came from Kambezidis (per-sonal communication, 2007). Ozone, precipitable water,Angstrom’s coefficient ß, and albedo are the input variableshere, as discussed in Gueymard (2012).

G034: Nijegorodov

This model (Nijegorodov et al 1997) uses the air massformula from Nijegorodov and Luhanga (1996) and trans-mittance expressions from Bird's model (entry G007 above).

Analysis of 54 models for clear-sky diffuse solar radiation 395

It is assumed here that the Earth radius is 6,367 km and thatthe effective atmosphere thickness is 29.7 km.

G035: NRCC

The original model by Belcher and DeGaetano (2005, 2007)was slightly modified according to B. Belcher (personalcommunication, 2005).

G036: Paltridge

This is the empirical model of Paltridge and Platt (1976).

G037: Perrin

This is the broadband model of Perrin de Brichambaut andVauge (1982), as discussed in Gueymard (2003a).

G038. PR

A part of this model is described in Psiloglou et al. (2000).However, the aerosol transmittance expression is here fromthe REST model (Gueymard 2003a) per Psiloglou's request(Psiloglou, personal communication, 2006), with the pur-pose to improve the model's performance. The new acronymstands for “Psiloglou-REST.”

G039: PSIM

This model is described by Gueymard (2003b), and itsperformance for direct irradiance predictions was discussedin Gueymard (2003a,b).

G040: REST250

This is version 5.0 of the REST2 model, as described byGueymard (2008).

G041: Rodgers

This model uses the Unsworth–Monteith turbidity coeffi-cient. It is described by Rodgers et al. (1978). See alsoGueymard (2003a).

G042: RSC

Carroll (1985) combined earlier algorithms by Robinson(1966) and Sellers (1965) to derive this model, hence itsacronym.

G043: Santamouris

This model is based on atmospheric transmittances fromPsiloglou et al. (2000) (entry G029 above), following theadvice of M. Santamouris (personal communication, 2002).A pressure correction was added where needed for consis-tency, as also discussed in Gueymard (2003a).

G044: Schulze

This model was proposed by Schulze (1976).

G045: Sharma

This empirical model was proposed by Sharma and Pal(1965). They used a solar constant of 2 cal/(cm2 min), whichis replaced here by the more recent value of 1,366.1 W/m2

(Gueymard and Kambezidis 2004).

G046: Watt

This model was proposed by Watt (1978), and its perfor-mance was studied by Bird and Hulstrom (1981). Theseauthors, however, seem to have misinterpreted some equa-tions, due to Watt’s non-explicit use of the decimal loga-rithm. This can explain the poor performance results of theBird and Hulstrom study. The required extinction layerheights are derived here from the original author’s Fig.4.0.2. A fixed stratospheric turbidity of 0.02 is assumed.

G047. Wesely

This model by Wesely and Lipschutz (1976a, b) uses visi-bility, which is derived here from Angstrom’s β coefficientusing the formula of King and Buckius (1979) in reversemode.

G048: Yang

The original model of Yang et al. (2001) is used here withthe corrections described in Gueymard (2003a), which wereeventually included in a later description of the model (Yanget al. 2006).

G049. Zhang

The model proposed by Zhang et al. (2002) and Zhang (2006)described site-specific coefficients empirically derived fromradiation measurements in China. For this study, the averageof the coefficients tabulated for 24 sites (Zhang 2006) wasrather used for improved universality. The model’s Gompertzfunction that separates the direct and diffuse componentsappeared to generate unphysical values, which was caused

396 V. Badescu et al.

by its coefficient a4 being misprinted. The correct value (2.99)is used here (Zhang, personal communication, 2008).

G050: HS

This is a combination between the model by Hourwitz(1945, 1946) to predict direct irradiance and the model bySchulze (1976) to predict diffuse irradiance.

G051: ABCGS

This is a combination between the model by Adnot et al.(1979) to predict direct irradiance and the model by Schulze(1976) to predict diffuse irradiance.

G052: Paulescu and Schlett

In this model by Paulescu and Schlett (2003), the value γ00.5 is adopted here to approximate a Rayleigh atmosphere,based on the suggestion by Paulescu (personal communica-tion, 2009).

G053: Janjai

This model is based on publications by Janjai (2010) andJanjai et al. (2011).

G054: REST281

This is version 8.1 of REST2 model, which contains a fewcorrections compared to the entry G040 above and theoriginal description by Gueymard (2008). For more details,see Gueymard (2012).

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