Atmospheric Research 150 (2014) 69–78

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Atmospheric Research

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Evaluation of errors made in solar irradiance estimation due toaveraging the Angstrom turbidity coefficient

Delia-Gabriela Calinoiu a,b, Nicoleta Stefu c,⁎, Marius Paulescu c, Gavrilă Trif-Tordai a, Oana Mares c,Eugenia Paulescu c, Remus Boata c,d, Nicolina Pop b, Angel Pacurar c

a Mechanical Engineering Faculty, “Politehnica” University of Timisoara, Mihai Viteazu Ave. 1, 300222 Timisoara, Romaniab Department of Physical Foundations of Engineering, “Politehnica” University of Timisoara, V. Parvan Ave. 2, 300223 Timisoara, Romaniac Physics Department, West University of Timisoara, V. Parvan Ave. 4, 300223 Timisoara, Romaniad Astronomical Institute of the Romanian Academy, Timisoara Astronomical Observatory, A. Sever Sq. 1, 300210 Timisoara, Romania

a r t i c l e i n f o

⁎ Corresponding author.E-mail address: nstefu@gmail.com (N. Stefu).

http://dx.doi.org/10.1016/j.atmosres.2014.07.0150169-8095/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

Article history:Received 31 March 2014Received in revised form 10 July 2014Accepted 11 July 2014Available online 23 July 2014

Even though themonitoring of solar radiation experienced a vast progress in the recent years bothin terms of expanding themeasurement networks and increasing the data quality, the number ofstations is still too small to achieve accurate global coverage. Alternatively, various models forestimating solar radiation are exploited inmany applications. Choosing amodel is often limited bythe availability of the meteorological parameters required for its running. In many cases thecurrent values of the parameters are replaced with daily, monthly or even yearly average values.This paper deals with the evaluation of the error made in estimating global solar irradiance byusing an average value of the Angstrom turbidity coefficient instead of its current value. A simpleequation relating the relative variation of the global solar irradiance and the relative variation ofthe Angstrom turbidity coefficient is established. The theoretical result is complemented by aquantitative assessment of the errors made when hourly, daily, monthly or yearly average valuesof the Angstrom turbidity coefficient are used at the entry of a parametric solar irradiance model.The study was conducted with data recorded in 2012 at two AERONET stations in Romania. It isshown that the relative errors in estimating global solar irradiance (GHI) due to inadequateconsideration of Angstrom turbidity coefficient may be very high, even exceeding 20%. However,when an hourly or a daily average value is used instead of the current value of the Angstromturbidity coefficient, the relative errors are acceptably small, in general less than 5%. All resultsprove that in order to correctly reproduce GHI for various particular aerosol loadings of theatmosphere, the parametric models should rely on hourly or daily Angstrom turbidity coefficientvalues rather than on themore usualmonthly or yearly average data, if currentlymeasured data isnot available.

© 2014 Elsevier B.V. All rights reserved.

Keywords:AerosolAngstrom turbidity coefficientSolar irradianceError

1. Introduction

The knowledge of solar energy at ground is a key issue inmany fields, such as agriculture (estimation of the crop yield,e.g. Trnka et al., 2007), public health care (balancing exposure

to UV, e.g. Malinovic-Milicevic and Mihailovic, 2011) andengineering (operating the photovoltaic power plants, e.g.Paulescu et al., 2013). The spatial density of meteorologicalstations equipped for solar radiation monitoring is far less thanrequired. Currently, approaches based on satellite observationturn into a suitable alternative,with global coverage.Most of theequations that govern the satellite-basedmodels for computingsolar irradiance employ information from satellite images andestimates of the atmospheric parameters. Before 2000 it was

70 D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

demonstrated that compared to groundmeasurements, satellite-derived hourly solar irradiation is a more accurate option forsites locatedmore than 25 km away from a ground station (e.g.Zelenka et al., 1999). However, in many applications the solarradiation is estimated using equations originating either fromthe simple class of empirical site-specific models or from themore general class of parametric models. The name parametricsignifies the usage of atmospheric parameters (e.g. ozonecolumn content, precipitable water, atmospheric pressure,Angstrom turbidity coefficient) at the model entry, aiming toincrease its accuracy in specific weather conditions.

The accuracy of amodel in computing solar irradiance reliesboth on the complexity of the model and on the quality of themeasured data used at input. A comprehensive study on thisline was reported by Badescu et al. (2013). After testing fifty-four broadband models for global and diffuse solar irradianceon various sets of data, the authors concluded that there is noparticular model that can be ranked as the best for all sets ofinput data. Very simple empirical models or more complexparametricmodelsmight aswell belong to the category of goodmodels, but the latter generally perform better. In practice thechoosing of a parametric model is limited by the availability ofdata needed at input. Sometimes a parametric model is appliedto estimate solar irradiance, even if one ormore parameters arenot regularly measured by the local meteorological service. Themissing data is replaced by monthly or yearly mean values, ifavailable, or even climatological values. It is obvious that suchreplacements may generate errors in the estimation of solarirradiance if the substitutedmeteorological parameter is highlyvariable.

This paper is focused on evaluating the errors in solarirradiance estimation caused by using mean values (hourly,daily,monthly, yearly) of theAngstrom turbidity coefficient. TheAngstrom turbidity coefficient β (Angstrom, 1961) is a measureof the turbidity of the atmosphere, defined as the aerosol opticaldepth (AOD) at a wavelength of 1 μm. This parameter is highlyvariable, depending on the aerosol load of the atmosphere, andit ranges between 0 (clear atmosphere) and 0.5 (high aerosolload). Sometimes, mainly around desert regions, β may exceed0.5 (Rahoma and Hassan, 2012; Djafer and Irbah, 2013). Thisstudy is motivated by several facts, addressed in the following.

We have studied the variability in β, considering asample of 320 daily values of beta, available for Timisoara,for the time period 2011–2012, and a sample of 442 dailyvalues of beta, for Cluj-Napoca, from 2011 to 2012. Thevalues for Timisoara are within the range [0.01496–0.44730]with amean valueβ ¼ 0:09029, with a 95% confidence interval[0.08388, 0.09670], and a standard deviation of 0.05826. Thevalues for Cluj-Napoca extend over the range [0.01174–0.35130]with a mean β ¼ 0:09029 , a 95% confidence interval of[0.073820-0.08326], and a standard deviation of 0.05053. Takinginto account these results, the probability for a researcher to befar from the reality is quite large, when using the mean valueinstead of the real value.

Currently, the subject of correct representation of aerosolsin models for estimating solar irradiance is investigated onboth facets: measurements and modeling. Suri et al. (2009)compared the values of direct-normal irradiance (DNI) pro-vided by five databases Meteonorm, Satel-Light, NASA-SSE,SOLEMI and PVGIS. The results revealed a large differencebetween DNI values provided by these databases: in some

regions of Europe, the uncertainty of DNI expressed by relativestandard deviation may reach 17% (DNI estimates may deviateup to 33% from average). In part, this uncertainty can beconnected to an inaccurate description of aerosol properties(Cebecauer et al., 2011).

Ruiz-Arias et al. (2013) assessed the uncertainty inducedin the predicted DNI when the Level-3 MODIS AOD product(MODIS, 2013) is used as input into radiative transfer models.The authors found that the induced uncertainty inDNI is smallerthan≈15% for AOD values below 0.5 (90% of the global dataset)and a solar zenith angle of 30°. Under the same condition, therelative uncertainty induced in the global horizontal irradiance(GHI) is always below 5%. This study and other (e.g. Gueymard,2012) show that DNI is more sensitive than GHI to AOD.

However, there are many applications where GHI isrequired, rather than DNI. For example, the amazing increase ofthe photovoltaic sector is well known (see e.g. EPIA, 2013). Forsizing andoperating thephotovoltaic plants, accurate knowledgeof the solar energy collectable on the PV modules surface isrequired (generally the PV modules are set facing South andtilted to an angle equal to the local latitude). Since very fewmeteorological stations are equipped for monitoring solarradiation on tilted surface, generally the collectable solarenergy is estimated starting from the horizontal componentsof solar irradiation (e.g. Ch. 2 in Duffie and Beckman, 2013).

Many simple models for estimating GHI are developedempirically requiring only geographical coordinates and tem-poral reference as input (Pandey and Soupir, 2012). Othermodels use a singlemeteorological parameter for describing theatmospheric transmittance (e.g. air temperature in El-Metwally,2004). Generally the parametricmodels for estimatingGHI use aset of atmospheric parameters for modeling the extinction ofsolar radiation in the atmosphere. Several parameters are usedto describe the effects of aerosols on the solar radiation flux:AOD (Maxwell et al., 1998), AOD at 700 nm (Ineichen, 2006),the Linke turbidity coefficient (Rigollier et al., 2000; Ineichenand Perez, 2002) or Angstrom turbidity coefficient (Paulescuand Schlett, 2003; Yang et al., 2006).

Errors made in estimating GHI by the inadequate modelingof aerosols have been signaled before in several papers (e.g.Batlles et al., 2000; Ineichen, 2006; Andrada et al, 2008;Gueymard and Thevenard, 2013). A brief discussion on the lasttwo references follows. Andrada et al. (2008) reported thatwhen the real values for the aerosols' optical parametersprovided by AERONET were incorporated in the TUV model(Madronich and Flocke, 1997), the UVB estimates showed asignificant improvement that in some cases reached 27%.Gueymard and Thevenard (2013) found that AOD does notfollow a normal distribution over monthly periods, but rather alog-normal one. Consequently, using themedian of AOD insteadof its mean provides better results in the estimation of solarirradiance under clear sky conditions.

Sometimes the reverse modeling is required: the solarradiation models are used to indirectly determine the atmo-spheric turbidity coefficients (El-Metwally, 2013; Bilbao et al.,2014).

This paper reports an evaluation of the errors made in theestimation of GHIwhen using hourly, daily,monthly and yearlymean values of the Angstrom turbidity coefficient. The analysiswas performed using a parametric model, previously devel-oped by our group by integrating the spectral transmittances

71D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

from the SMARTS2 model (Gueymard, 1995) and presented indetail in Calinoiu et al. (2013). The model takes into accountthe most important attenuators in the atmosphere via theirtransmittances, averaged over the entire solar spectrum, and ituses at input the following parameters: the total O3, NO2 andwater vapor column contents and the Angstrom turbiditycoefficient. In the following this model will be referred to asSIMv.1 (solar irradiation model, version 1).

2. SIMv.1 model

In themodel SIMv.1 GHI is computed as the sum of beam Gb

and diffuse Gd components:

G ¼ Gb þ Gd: ð1Þ

Gb and Gd are expressed as functions of the sun elevation angleh with the equations:

Gb ¼ GSCετb sinh ð2aÞ

Gd ¼ GSCετd sinh ð2bÞ

where GSC = 1366.1 W/m2 is the solar constant and ε is acorrection factor according to the Sun–Earth distance. τb and τdstand for the beam and diffuse atmospheric transmittances,averaged with respect to wavelength, respectively.

The beam transmittance τb takes into account the mostimportant attenuators in the atmosphere via their specifictransmittance: ozone ( τO3 ), nitrogen dioxide ( τNO2 ), watervapor (τw), trace gas absorption (τg), Rayleigh scattering (τR)and aerosol extinction (τa ). Thus, the beam transmittance isgiven by:

τb ¼ τO3τNO2

τwτgτRτa: ð3Þ

Diffuse radiation at ground level is treated as a sum of twoindividual components corresponding to the Rayleigh (τd;R )and aerosols ( τd;a ) scattering processes (see e.g. Gueymard(1995) for more details on this topic):

Gd ¼ G0 ε τd;R sinhþ G0 ε τd;a sinh ð4Þ

where

τd ¼ τd;R þ τd;a ¼ γRτO3τNO2

τwτgτaa 1−τRð Þþ γaτO3

τNO2τwτgτRτaa 1−τasð Þ ð5Þ

γR and γa are the fractions of the scattered radiation that istransmitted downward, τaa is the transmittance for aerosolsabsorption and τas ¼ τa=τaa is the transmittance for aerosolscattering.

The accuracy of SIMv.1 is briefly assessed in Appendix A.

3. Effect of the Angstrom turbidity coefficient on GHI. Asimplified model

A relation between the relative variations of GHI, dG/G as afunction of the relative variation of the Angstrom turbiditycoefficient dβ/β is derived. In order to do that, two simplifyinghypothesis are made: (i) we assume a perfectly non-absorbing

aerosol, i.e.τaa ¼ 1 and τas ¼ τa; (ii) we assume the same valuefor the downward fractions, i.e. γR = γa = γ. Taking intoaccount these simplifications, Eq. (5) writes:

τd ¼ γτO3τNO2

τwτg 1−τRτað Þ: ð6Þ

By using Eqs. (1), (3) and (6) and denoting C ¼GSCετO3τNO2τwτg sinh, we obtain:

G ¼ C γ þ τRτa 1−γð Þ½ �: ð7Þ

In Eq. (7) only τa depends on β. By differentiating withrespect toβ, after a simple calculation, the following equation isobtained:

dGG

¼ K βð Þdββ

ð8Þ

where

K βð Þ ¼ β τR 1−γð Þγ þ τR τa 1−γð Þ

∂τa∂β ð9Þ

is a measure of the correlation strength between dG/G anddβ/β. The equations for τa and τR inferred in the model SIMv.1are (Calinoiu et al., 2013):

τa xað Þ ¼ 1−0:046mβ1þ 1:73849mβ þ 0:79081m2β2 ð10Þ

τR xrð Þ ¼ 1þ 0:1564mþ 0:001m2

1þ 0:26038mþ 0:00697m2 : ð11Þ

In Eqs. (10) and (11)m stands for the atmospheric air mass.Finally, by differentiating Eq. (10) with respect to β and using itin Eq. (9), one obtains:

K βð Þ ¼ − β τR τa 1−γð Þγ þ τR τa 1−γð Þ

0:046m1−0:046mβ

þ 1:73849mþ 1:5816m2β1−0:046mβ

τa

!:

ð12Þ

Fig. 1 shows the dependence of K(β) on β for variousconditions of radiative transfer in the atmosphere, conditionsgiven by the parameters γ and m. It can be seen that |K(β)|increases almost linearly with β for a large set of elevationangles (an atmospheric mass of 2.5 corresponds to a sunelevation angle of ≈23°). There also can be noticed that,regardless of the value of the downward factor, |K(β)| increaseswith the increase of atmospheric mass, and, respectively, withthe decrease of the downward fraction.

Fig. 1 highlights the strong dependence of GHI on theAngstrom turbidity coefficient. This proves that using imprecisevalues for β at the input of the parametric models may generatesignificant errors in estimating solar irradiance. For instance, at45°N latitude, for a sun elevation angle of 43° (m = 1.5) andconsidering a scattering Rayleigh atmosphere (γ = 0.5), arelative error of 10% in estimatingβ aroundβ=0.1will producea relative error of around −1% in estimating G. But the realitycan bemuchdifferent. There are timeswhen the relative error inevaluating β may be 200% or 300%, and in these situations therelative error in estimating GHI becomes important: e.g. in

Fig. 1.Dependence ofK(β) on the Angstrom turbidity coefficient for different values of the downward fraction: (a) γ=0.4; (b)γ=0.5; and (c) γ=0.6. The parameterof the curves is the atmospheric mass.

72 D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

Timisoara (see Section 5.1) in 2012 the average value wasβyear = 0.078, while during an episode of heavy aerosol loadwith Saharan dust the Angstrom turbidity coefficient hasreached a value β = 0.37 (November 19, 2012).

Finally, it is worth mentioning that an equation relating therelative variation of DNI to the variation of β has been reportedbefore in Gueymard (2012). There the proportionality factorwas referred to as Aerosol Sensitivity Index (ASI). In thesimplified formalism presented here, a quantity equivalent toASI but for GHI can be simply defined as γGHI = |K(β)/β |.

4. Virtual experiment

Let's assume that an engineer needs to estimate GHI in alocation placed at 45°N latitude, using the parametric modelSIMv.1. In addition to geographical coordinates and temporalinformation, the computation of GHI requires four otherparameters: ozone column content lO3 , nitrogen dioxide columncontent lNO2 , water vapor column content w and the Angstromturbidity coefficientβ. Let's assume that themeasurements for allthe above mentioned parameters are available to the engineer,except for the Angstrom turbidity coefficient, for which only themulti-annual mean is known βc = 0.079. Let us evaluate theerror made by the engineer in the estimation of GHI in variouscases of aerosol loadings in the atmosphere. In order to completethis task, we have run the SIMv.1 parametric model in twodifferent situations: first, with the data known by the engineer,and second using measured meteorological parameters, andtaking possible real values for β, ranging between 0.05 and 0.25.

The results for June 21st, (j=172) for three different positionsof the Sun in the sky, (zenithal angles θz = 30°, 45° and 60°) arepresented in Fig. 2. In order to compare results,wehave used fixed

values for lO3 ¼ 0:332 cm � atm, lNO2 ¼ 2:085� 10−4 cm � atm andw=2.3 g/cm2. Basically, Fig. 2 illustrates three facets of the samereality. Fig. 2a depicts the dependence of GHI on β. It can benoticed that the decrease of GHIwith the increase ofβ occurswiththe same rate, regardless of the zenithal angle. The influence of βon GHI is significant. For instance, let us consider an atmosphericaerosol load, leading to the doubling of the Angstrom turbiditycoefficient, β = 2βc, that occurs for θz = 45°. The hypotheticalengineer, using βc as input into the model, estimates a value of664.1W/m2 for GHI, which is 39.1W/m2more than the real valueof 625.0 W/m2. The relative error RE = 100 × (GHI(βc) −GHI(β))/GHI(β) versus β is presented in Fig. 2b. It can beseen that RE increases linearly with β, the larger thezenithal angle, the larger the increase rate. The crossingpoint of the three lines corresponds to the abscissaβc, where RE=0. It can be seen that the engineer will make errors up to 5% inestimating GHI by using a mean value, in clear atmosphereconditions, with small aerosol loadings (β b 0.05). In case ofheavier aerosol loadings (β N 0.25) the error in estimating GHI canreach 20%. Fig. 2c presents RE versus the relative error made inusingβc insteadof the current valueβ. Anonlinear behavior canbenoticed: the heavier the atmospheric aerosol load (the larger therelative error in assuming β) and the lower the sun in the sky(larger zenithal angle), the larger the relative error in estimatingGHI.

5. Case study

In this section we present a case study. Errors made inestimating GHI are being evaluated when hourly, daily,monthly and yearly mean values of the Angstrom turbiditycoefficient are used as input to the SIMv.1model, instead of the

Fig. 2. (a) GHI versus the Angstrom turbidity coefficient; (b) relative error in estimatingGHI versus the Angstrom turbidity coefficient; and (c) relative error in estimatingGHI versus the relative error in assuming the Angstrom turbidity coefficient. The zenithal angle is the curve parameter. GHI was estimated with the model SIMv.1.

Table 1Daily average of relative sunshine (σ), sunshine stability number (ζ), aerosoloptical depth at 440 nm (AOD) and Angstrom turbidity coefficient in thetesting days. N stands for the number of measurements performed in a day.

Date Site N σ ζ AOD β

June 09 Timisoara 13 0.88 0.0049 0.32 0.14Cluj-Napoca 14 0.75 NAa 0.31 0.09

June 15 Timisoara 19 0.92 0.0034 0.10 0.03Cluj-Napoca 17 0.75 NA 0.14 0.03

June 16 Timisoara 30 0.95 0.0002 0.08 0.03Cluj-Napoca 29 0.81 NA 0.11 0.03

September 09 Timisoara 36 0.96 0.0056 0.13 0.04Cluj-Napoca 31 0.81 NA 0.07 0.02

a NA— not available.

73D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

current value. First the data base is described, then thecomputing method is presented, and finally the results arebeing discussed.

5.1. Database

Data recorded at two AERONET stations in Romania,Timisoara and Cluj-Napoca are used in this study. Situatedon the southeastern edge of the Pannonia plain, Timisoara(45°46′ N, 21°26′ E) lies at an altitude of 85 m. Timisoara ischaracterized by a warm temperate climate, fully humid(Köppen climate classification Cfb — based on the Kottek et al.(2006) digital Köppen–Geiger world map on climate classifi-cation)with warm summer, typical for the Pannonia Basin. Thedominating temperate air masses during spring and summerare of oceanic origin and comewith precipitations. Cluj-Napoca(46°76′ N, 23°58′ E) lies at an altitude of 385 m, situated at theconfluence of the Apuseni Mountains (belonging to theWesternCarpathianMountains) and the Transylvanian plain. Cluj-Napocais classified as Dfb by Köppen climate classification. This locationhas a humid continental climatewithwarm summers and severewinters. The specific climate is influenced by the mountainsproximity and urbanization.

The AERONET stations are maintained by the “Politehnica”University of Timisoara and Babes-Bolyai University ofCluj-Napoca. The measurements are performed at both stationswith a Cimel automatic sun tracking photometer CE 318 (Holbenet al., 1998). It makes two basic measurements, e.g. direct-normal spectral irradiance and spectral sky radiance, bothwithinseveral automatic sequences. The direct-normal solar irradianceis measured on nine spectral channels (340, 380, 440, 500, 670,

870, 940, 1020 and 1640 nm) while the diffuse sky radiance ismeasured on four channels (440, 670, 870 and 1020 nm)employing the almucantar procedure. At Cluj-Napoca the1640 nm channel is unavailable. An uncertainty of approximate-ly 0.01–0.02 in AOD (wavelength dependent) due to calibrationuncertainty for the field instruments is specified on theAERONETwebsite [AERONET, 2014].

The following quantities derived from the sun-photometermeasurements are used in this study: Angstrom turbiditycoefficient β, water vapor column content w, ozone columncontent lO3 and nitrogen dioxide column content lNO2 .Measurements performed in four days of 2012 are considered(Table 1). All days were sunny days (daily relative sunshineranges between 0.75 and 0.96) with a stable radiative regime.

The stability of the radiative regime was quantified bymeans of the sunshine stability number ζ, a measure of thefrequency of sun appearance or disappearance on/from the sky,

74 D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

defined in Paulescu and Badescu (2011). ζ ranges between 0(fully stable solar radiative regime) and 1/2 (fully unstablesolar radiative regime). In this case ζ ranges between 0.0002and 0.0056, indicated a stable radiative regime. The AOD valuesfor June 15–16 and September 9 lie between 0.07 and 0.14,indicating days without significant aerosol load. The sizedistribution shows that the particles belong to the fine mode.On June 9, AOD reached a value of 0.32 at Timisoara and 0.31 atCluj-Napoca and SSA was higher than 0.95 for both stations,increasing with the wavelength. Running the HYSPLIT model(Draxler and Rolph, 2014), the back-trajectories indicated aSaharan origin of the dust.

Fig. 3. a. Relative error in estimating solar irradiance as function of the measured Angb. Relative error in estimating solar irradiance as function of the zenithal angle (degestimating solar irradiance as function of the solar irradiance in Timisoara and Cluj-Na

5.2. Methodology

In order to evaluate the errors made in estimating GHI dueto the averaging of the Angstrom turbidity coefficient, theparametric model SIMv.1 was run in five cases. First, the GHIvalues were evaluated in the chosen test days, by inputting themeasured values of the four parameters into themodel: lO3, lNO2,w and β. In the next four cases, the model was run withmeasured values for lO3, lNO2, andw, but the instantaneous valueof the Angstrom turbidity coefficient β was replaced in turnswith the hourly meanβh, the daily meanβd, themonthly meanβm and the yearly mean βy.

strom turbidity coefficient in Timisoara and Cluj-Napoca in four days of 2012.rees) in Timisoara and Cluj-Napoca in four days of 2012. c. Relative error inpoca in four days of 2012.

Fig. 3 (continued.)

75D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

The error made in GHI estimation using an averaged valueof the Angstrom turbidity coefficient was assessed relatively tothe GHI estimation using the measured value of the Angstromturbidity coefficient by the equation:

REk ¼GHI βk

� �−GHI βð Þ�� ��

GHI βð Þ � 100 ; k ¼ h;d;m; y: ð13Þ

5.3. Results and discussions

Results are comprised in Fig. 3. REk is presented as a functionof the measured value of the Angstrom turbidity coefficient(Fig. 3a), as a function of the zenithal angle (Fig. 3b) and as afunction of GHI (Fig. 3c). The non-linearity of the curves in Fig.3a may seem odd at first glance, if compared to the ones inFig. 2b. This behavior can be easily explained, though, by thefact that in evaluating GHI (β) for each point in Fig. 3a,measured values of all parameters were used, while in Fig. 2blO3 , lNO2 andwwere kept constant. One can see in Fig. 3 that theerror amplitude ranges between zero and approximately 17%.REk decreases with increasing β (Fig. 3a), it increases with thezenithal angle (Fig. 3b) and decreases with increasing solarirradiance. These trends are obvious in clear sky days (β b βc )and less obvious in the high aerosol load episode on June 9. Nosignificant differences can be observed between the errorsevaluated in both locations, although their climate is quitedifferent, as shown in Section 5.1.

Generally, REk increases with the time interval over whichβ was averaged. For hourly or daily means, the errors aregenerally smaller than 5%, excepting some cases even smallerthan 2.5%.Whenmonthly or yearlymeans of βwere employed,the errors extend over the range from 5 to 17%. From this, an

important conclusion can be drawn: hourly or daily means ofthe Angstrom turbidity coefficient may be used in evaluatingGHI, and the accuracy of the estimation will still be withinreasonable range. Using monthly or yearly mean values of theAngstrom turbidity coefficient in estimating GHI may lead tosignificant errors, in aerosol loaded as well as in aerosol freeatmosphere.

6. Conclusions

The error in estimating GHI with a parametric model due tothe unavailability of the Angstrom turbidity coefficient wasevaluated. The calculations are based on the model reported inCalinoiu et al. (2013). Since the model performance is in thesame rangewith the one of other parametricmodels, the resultspresented in this paper can be considered as general.

A new equation relating the relative variation of GHI tothe relative variation of the Angstrom turbidity coefficientwas established. It shows a nonlinear behavior of the relativeerror in GHI with respect to the relative error in Angstromturbidity coefficient: the heavier the aerosol load and the lowerthe sun in the sky, the larger the relative error in estimatingGHI.

The studies on virtual and real data demonstrated that theerrors in estimating GHI due to inadequate consideration ofAngstrom turbidity coefficientmay be very high. By inputting amonthly or yearly mean values of the Angstrom turbiditycoefficient into a parametricmodel instead of the current value,a relative error as large as 20% may be produced in estimatingGHI, during an episode of heavy aerosol loading of theatmosphere or in a very clear atmosphere, alike. Differently,when the current value of the Angstrom turbidity coefficient isreplaced with an hourly or a daily value, the relative errors are

76 D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

acceptably small, in general less than 5%. Therefore, if currentlymeasured data is not available, in order to correctly reproduceGHI for various particular aerosol loadings of the atmosphere,the parametric models should rely on hourly or daily Angstromturbidity coefficient values rather than on the more usualmonthly or yearly average data.

Table A1Statistical indicators of the accuracy of the SIMv.1 model applied to estimate global sodaily average of the Angstrom turbidity coefficient.

Station Latitude Longitude Altitude[m]

Moldova 47°00′N 28°48′E 248

Toravere 58°15′N 26°27′E 70

Bozeman 45°39′N 111°82′W 1587

GSFC 38°59′N 76°58′W 87

a Relative mean bias error rMBE ¼ 100�∑N

i¼1Fi−yið Þ=∑

N

i¼1yi %½ �. yi and Fi are i-th m

measurements taken into account.b Relative root mean square error rrmse ¼ 100� N �∑

N

i¼1Fi−yið Þ2

" #1=2=∑

N

i¼1yi %½ �:

Acknowledgments

We thank the principal investigators and their staff forestablishing and maintaining the Bozeman, Cluj_UBB, GSFC,Moldova, Timisoara and Toravere AERONET sites used in thisinvestigation.

Appendix A. On the SIMv.1 accuracy

An evaluation of the SIMv.1 performance in Timisoara is discussed in Calinoiu et al. (2013). Here the discussion on the SIMv.1accuracy in relation to the atmospheric aerosol loading is extended. The tests were carried out on two levels: (i) field tests againstmeasured data and (2) comparison with other parametric models.

The SIMv.1 accuracy at four AERONET stations, two located in Europe and two in North America, is assessed in Table A1. At eachstation, along with the parameters retrieved from the sun photometer, global solar irradiation is measured and displayed on theAERONET website. The stations are located between 38°N and 58° Northern latitudes and between 70 m and 1587 m altitude. SIMv.1was tested in two sunny days (solar irradiance measured during the day has a nice bell shape) at each station. Statistical indicatorspresented in Table 1 (rrmse range between 0.8% and 4.8%) demonstrate a good quality of SIMv.1 model.

lar irradiance in clear sky conditions at four AERONET stations. β stands for the

Date[yyyy/mm/dd]

β rmbea

[%]rrmseb

[%]

2012/06/17 0.025 1.5 1.72012/09/03 0.025 1.8 1.92012/09/12 0.082 −0.4 3.42012/07/28 0.094 1.0 1.12012/08/16 0.043 −0.5 0.82012/08/29 0.102 1.3 1.72012/09/12 0.021 3.9 3.92012/09/13 0.033 4.8 4.8

easured and computed quantities, respectively, while N is the number of

Appendix A2. Comparison with other models

Fig. A1 displays GHI calculated with SIMv.1 and other three parametric models with respect to the Angstrom turbidity coefficient.The other threemodels are: (1) PS (Paulescu and Schlett, 2003). Themodelwas calibratedwith data recorded also in Timisoara but bythe local meteo station. The PS model was identified among the top-performers of 54 clear-sky models by Badescu et al. (2013);(2) hybrid (Yang et al., 2001, 2006) was found to be of high performance in many locations around the world, e.g. Japan (Yang et al,2001), Eastern Europe (Paulescu and Schlett, 2004) and Northern Africa (Madkour et al., 2006); and (3) semi-empiric (Janjai et al.,2011) fitted and validated with data recorded at the tropics.

GHI estimates are compared at 45°N for the same set of the input meteorological parameters on two different days: 1 January (Fig.A1a, c) and 1 July (Fig A1b, d) and for two values of the water vapor column contentw=2.3 g/cm2 (Fig. A1a, b) andw=4.0 g/cm2 (FigA1c, d). The comparison ismade at different times of the day (indicated on the graphs in terms of hour angle,ω). Visual inspection showsthat different models roughly estimate the same value for GHI and that there are nomajor differences in GHI trajectories. Therefore, it isexpected that the results of this study are preserved when GHI is estimated with other parametric models.

Fig. A1. Global solar irradiance estimated by different clear sky parametric models (SIMv.1 Calinoiu et al. (2013); PS (Paulescu and Schlett, 2003).; hybrid (Yang et al.,2001, 2006); and semi-empiric (Janjai et al., 2011) in (a, c) 1 January and (b, d) 1 July and for twodifferent values of thewater vapor column content (a, b) 2.3 g/cm2 and(c, d) w = 4.0 g/cm2.

77D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

References

AERONET, 2014. System descriptions and calibration. http://aeronet.gsfc.nasa.gov/new_web/system_descriptions_calibration.html (accessed June, 2014).

Andrada, G.C.,Palancar, G.G.,Toselli, B.M., 2008. Using the optical properties ofaerosols from the AERONET database to calculate surface solar UV-Birradiance in Cordoba, Argentina: comparison with measurements. Atmos.Environ. 42, 6011–6019.

Angstrom, A., 1961. Techniques of determining the turbidity of the atmo-sphere. Tellus 13, 214–223.

Badescu, V., Gueymard, C.A., Cheval, S., Oprea, C., Baciu, M., Dumitrescu, A.,Iacobescu, F., Milos, I., Rada, C., 2013. Computing global and diffuse solarhourly irradiation on clear sky. Review and testing of 54 models. Renew.Sust. Energ. Rev. 16, 1636–1656.

Batlles, F.J., Olmo, F.J., Tovar, J., Alados-Arboledas, L., 2000. Comparison ofcloudless sky parameterizations of solar irradiance at various Spanishmidlatitude locations. Theor. Appl. Climatol. 66, 81–93.

Bilbao, J.,Román, R.,Miguel, A., 2014. Turbidity coefficients from normal directsolar irradiance in Central Spain. Atmos. Res. 143, 73–84.

Calinoiu, D.,Paulescu, M., Ionel, I.,Stefu, N.,Pop, N.,Boata, R.,Pacurar, A.,Gravila,P., Paulescu, E., Trif-Tordai, G., 2013. Influence of aerosols pollution on theamount of collectable solar energy. Energy Convers. Manag. 70, 76–82.

Cebecauer, T., Suri, M., Gueymard, C., 2011. Uncertainty sources in satellite-derived direct normal irradiance: how can prediction accuracy beimproved globally. Proceedings of the SolarPACES Conference, Granada,Spain, 20–23 Sept 2011.

Djafer, D., Irbah, A., 2013. Estimation of atmospheric turbidity over Ghardaiacity. Atmos. Res. 128, 76–84.

Draxler, R.R., Rolph, D.G., 2014. Hysplit: Hybrid Single-Particle LagrangianIntegrated Trajectory. NOAA Air Resources Laboratory, Silver Spring, MD(http://ready.arl.noaa.gov/HYSPLIT.php (Accessed January 2014)).

Duffie, J.A., Beckman, W.A., 2013. Solar Engineering of Thermal Processes, 4thed. Wiley, Hoboken.

El-Metwally, M., 2004. Simple new methods to estimate global solar radiationbased on meteorological data in Egypt. Atmos. Res. 69, 217–239.

El-Metwally, M., 2013. Indirect determination of broadband turbidity coeffi-cients over Egypt. Meteorol. Atmos. Phys. 119, 71–90.

EPIA, 2013. European Photovoltaic Industry Association (EPIA) GlobalMarket Outlook for Photovoltaics 2013–2017. http://www.epia.org/news/publications/global-market-outlook-for-photovoltaics-2013-2017 (accessed December 2013).

Gueymard, C.A., 1995. SMARTS2 a simple model of the atmospheric radiativetransfer of sunshine: algorithms and performance assessment. FloridaSolar Energy Center Rep. FSEC-PF-270-95.

Gueymard, C.A., 2012. Temporal variability in direct and global irradiance atvarious time scales as affected by aerosols. Sol. Energy 86, 3544–3553.

Gueymard, C.A.,Thevenard, D., 2013. Revising ASHRAE climatic data for designand standards — Part II, clear-sky solar radiation model (1613-RP).ASHRAE Trans. 119, 194–209.

Holben, B.N., Eck, T.F., Slutsker, I., Tanre, D., Buis, J.P., Setzer, A., Vermote, E.,Reagan, J.A.,Kaufman, Y.,Nakajima, T.,Lavenu, F., Jankowiak, I.,Smirnov, A.,1998. AERONET — a federated instrument network and data archive foraerosol characterization. Remote Sens. Environ. 66, 1–16.

Ineichen, P., 2006. Comparison of eight clear sky broadband models against 16independent data banks. Sol. Energy 80, 468–478.

Ineichen, P., Perez, R., 2002. A new airmass independent formulation for theLinke turbidity coefficient. Sol. Energy 73 (3), 151–157.

Janjai, S.,Sricharoen, K.,Pattarapanichai, S., 2011. Semi-empirical models for theestimation of clear sky solar global and direct normal irradiances in thetropics. Appl. Energy 88, 4749–4755.

Kottek, M., Grieser, J., Beck, C., Rudolf, B., Rubel, F., 2006. World map of theKöppen–Geiger climate classification updated. Meteorol. Z. 15, 259–263.

Madkour, M.A., El-Metwally, M., Hamed, A.B., 2006. Comparative study ondifferent models for estimation of direct normal irradiance (DNI) overEgypt atmosphere. Renew. Energy 31, 361–382.

Madronich, S., Flocke, S., 1997. Theoretical estimation of biologically effectiveUV radiation at the Earth's surface. In: Zerefos, C. (Ed.), Solar UltravioletRadiation - Modeling, Measurements and Effects, NATO ASI Series, vol. I52.Springer, Berlin.

Malinovic-Milicevic, S.,Mihailovic, D.T., 2011. The use of NEOPLANTAmodel forevaluating the UV index in the Vojvodina region (Serbia). Atmos. Res. 101,621–630.

Maxwell, M.L.,George, R.L.,Wilcox, S.M., 1998. A climatological solar radiationmodel. In: Campbell-Howe, R., Cortez, T., Wilkins-Crowder, B. (Eds.),Proceedings of the 1998 American Solar Energy Society annual conference,Albuquerque, NM, June 14–17, 1998. American Solar Energy SocietyBoulder, Colorado, pp. 505–510.

MODIS, 2013. Moderate-Resolution Imaging Spectroradiometer (MODIS).http://modis-atmos.gsfc.nasa.gov/products.html.

Pandey, P.K.,Soupir, M.L., 2012. A new method to estimate average hourlyglobal solar radiation on the horizontal surface. Atmos. Res. 114–115,83–90.

78 D.-G. Calinoiu et al. / Atmospheric Research 150 (2014) 69–78

Paulescu, M., Badescu, V., 2011. New approach to measure the stability of thesolar radiative regime. Theor. Appl. Climatol. 103, 459–470.

Paulescu, M.,Schlett, Z., 2003. A simplified but accurate spectral solar irradiancemodel. Theor. Appl. Climatol. 75, 203–212.

Paulescu, M., Schlett, Z., 2004. Performance assessment of global solarirradiation models under Romanian climate. Renew. Energy 29, 767–777.

Paulescu, M.,Paulescu, E.,Gravila, P.,Badescu, V., 2013. Weather Modeling andForecasting of PV Systems Operation. Springer, Berlin.

Rahoma, U.A.,Hassan, A.H., 2012. Determination of atmospheric turbidityand its correlation with climatologically parameters. Am. J. Environ.Sci. 8, 597–604.

Rigollier, C., Bauer, O., Wald, L., 2000. On the clear sky model of the 4thEuropean Solar Radiation Atlas with respect to the Heliosat method. Sol.Energy 68 (1), 33–48.

Ruiz-Arias, J.A.,Dudhia, J.,Gueymard, C.A., Pozo-Vazquez, D., 2013. Assessmentof the Level-3 MODIS daily aerosol optical depth in the context of surface

solar radiation and numerical weather modeling. Atmos. Chem. Phys.Theor. Appl. Climatol. 13, 675–692.

Suri, M., Remund, J., Cebecauer, T., Hoyer-Klick, C., Dumortier, D., Huld, T.,Stackhouse Jr., P.W., Ineichen, P., 2009. Comparison of direct normalirradiation maps for Europe. SolarPACES Conference “Concentrated SolarPower and Chemical Energy Systems”, 15–18 Sep. 2009, Berlin, Germany.

Trnka, M., Eitzinger, J., Kapler, P., Dubrovský, M., Semerádová, D., Žalud, Z.,Formayer, H., 2007. Effect of estimated daily global solar radiation data onthe results of crop growth models. Sensors 7, 2330–2362.

Yang, K., Huang, G.W., Tamai, N., 2001. A hybrid model for estimating globalsolar irradiance. Sol. Energy 70, 13–22.

Yang, K., Koike, T., Ye, B., 2006. Improving estimation of hourly, daily, andmonthly solar radiation by importing global data sets. Agric. For. Meteorol.137, 43–55.

Zelenka, A.,Perez, R.,Seals, R.,Renne, D., 1999. Effective accuracy of the satellite-derived hourly irradiance. Theor. Appl. Climatol. 62, 199–207.