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Renewable Energy 35 (2010) 1325–1332

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Renewable Energy

journal homepage: www.elsevier .com/locate/renene

Technical Note

Comparative study of various models in estimating hourly diffuse solar irradiance

J.L. Torres a,*, M. De Blas a, A. Garcıa a, A. de Francisco b

a Department of Projects and Rural Engineering, Public University of Navarre, Edificio Los Olivos, 31006 Pamplona, Spainb Department of Agricultural and Forestry Engineering, Polytechnic University of Madrid, 28080 Madrid, Spain

a r t i c l e i n f o

Article history:Received 8 April 2009Accepted 10 November 2009Available online 26 November 2009

Keywords:Hourly diffuse fractionHourly diffuse irradianceModel assessmentsStatistical indicators

* Corresponding author. Tel.: þ34 948 169 175; faxE-mail address: jlte@unavarra.es (J.L. Torres).

0960-1481/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.renene.2009.11.025

a b s t r a c t

This work presents a comparison among seventeen different proposals for estimating the hourly diffusefraction of irradiance. Twelve of them are polynomial correlations of different orders, two are based ona logistic function and the three last ones consider the diffuse irradiance values in the previous andposterior hour to that of the calculation. In general, the proposals showing the more favourable statisticsindexes are those that consider the process dynamics, as they behave better than the rest of the modelseven when the polynomial correlations and the logistic function are calibrated for the experimental dataused in this work.

The models Dirint and BRL are the ones recommended for the data of this study, as they exhibit thehighest precision and generate a series of hourly diffuse irradiance values of which the distributionfunctions are very similar to those of the experimental data.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Most solar radiation data registered in weather stations areglobal irradiance data on the horizontal plane. Notwithstanding,one of both components of the said global irradiance, i.e. direct ordiffuse irradiance, are required for many purposes. For instance,both components are needed for estimating the global irradianceon a tilted plane, and the diffuse irradiance is an input datum formany radiance angular distribution models over the skyhemisphere.

Although some of the applications that need global irradiancedata do not demand a high time resolution, where only daily or theaverage of monthly daily data are required, it is becoming morecommon to need hourly data or even higher frequencies. This is due,on the one hand, to the increasing use of simulation computer toolsthat consider not only the steady state of solar energy installationsbut also transients. On the other hand, new applications are arisingthat intrinsically demand high time resolution data.

Regarding the estimation of global irradiance components,considerable work has been done since the pioneer proposal by Liuand Jordan [1] was published. Many publications are centred ondetermining the hourly diffuse irradiance by means of correlationsthat relate the diffuse fraction kd with the clearness index kt

considering polynomials of different orders. A detailed revision isincluded in Jacovides et al. [2]. Boland et al. [3] proposed a logistic

: þ34 948 169 148.

All rights reserved.

function to obtain the hourly diffuse fraction using the clearnessindex as an input. In the DISC model [4], the hourly normal directirradiance is determined based on physical considerations thatrequire the clearness index and the relative air mass as input data.

The Dirint model [5] is a modification of the DISC model. In thiscase, the direct irradiance calculated by the DISC model is multi-plied by a coefficient from a 6� 6� 5�7 matrix table that dependson the following parameters: the clearness index, the sun elevation,the dew point temperature and a variability index for consideringthe dynamics of the process. This model presents two operationalmodes: on the one hand, the 4-D, which requires the four inputdata, listed above, and on the other hand, the 3-D, which does notneed the dew point temperature data. The described model is usedin both the string of algorithms SoDa (Integration and exploitationof networked Solar radiation Databases for environment moni-toring) [6] and in the database Meteornorm [7]. DirIndex model [8]is an evolution of the Dirint model, where the atmosphericturbidity is considered in a different manner.

Skartveit et al. [9] developed a procedure for determining thehourly diffuse fraction that, as in [5], includes a variability index toconsider the influence on the diffuse fraction of, on the one hand,the changes in the type and location of the clouds and, on the otherhand, the surface albedo. The model is composed of a limitednumber of analytical expressions and needs no coefficient table.This is the model used in the Satel-light Project [10].

Recently, Ridley et al. [11] have improved their logistic model byincluding a persistence index in order to take into account dynamicsof the process. New model is called Boland-Ridley-Lauret (BRL).

Table 1Relevant data of the polynomial correlations applied in this work for determining the diffuse fraction.

Models and locations a1 a2 a3 a4 a5 a6 a7 a8 Second range of kt

First order modelsOrgill and Hollands [16] (Toronto) 1 �0.249 1.557 �1.84 0 0 0 0.177 0.35� kt� 0.75Reindl et al. [17] (Several sites) 1.02 �0.248 1.45 �1.67 0 0 0 0.147 0.3< kt< 0.78Model 1 Pamplona 1.0058 �0.2195 1.3264 �1.512 0 0 0 �0.49 0.24< kt< 0.75

Second order modelHawlader [18] (Singapore) 0.915 0 1.135 �0.9422 �0.3878 0 0 0.18 0.225< kt< 0.775Model 2 Pamplona 0.992 �0.098 1.2158 �1.0467 �0.448 0 0 0.1787 0.22< kt< 0.75

Third order modelsMiguel et al. [19] (North Mediterranean) 0.995 �0.081 0.724 2.738 �8.32 4.96 0 0.19 0.21< kt� 0.76Karatasou et al. [20] (Athens) 0 0 0.9995 �0.05 �2.4156 1.4926 0 0.78 0< kt� 0.78Jacovides et al. [2] (Cyprus) 0.987 0 0.94 0.937 �5.01 3.32 0 0.177 0.1< kt� 0.8Model 3 Pamplona 0.9923 �0.098 1.1459 �0.5612 �1.4952 0.7103 0 0.1755 0.22<kt� 0.755

Fourth order modelsErbs et al. [21] (USA) 1 �0.09 0.951 �0.1640 4.388 �16.638 12.336 0.165 0.22< kt� 0.8Oliveira et al. [22] (Sao Paolo) 1 0 0.97 0.8 �3.0 �3.1 5.2 0.17 0.17< kt< 0.75Model 4 Pamplona 0.9943 �0.1165 1.4101 �2.9918 6.4599 �10.329 5.514 0.18 0.225< kt� 0.755

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–13321326

Other assessments of both global to beam and diffuse irradiancemodels which consider databases from different sites, includingsome Spanish stations, can be found in [12–14].

The objective of this study is to perform a comparison amongdifferent existing models for determining the hourly diffuse irra-diance in Pamplona (Navarre); a city located in the Spanish regionwhere important developments in photovoltaic solar energy

0.2 0.4 0.6 0.8 1.0kt

0.2

0.4

0.6

0.8

1.0

Diffuse fractionexperimental

a b

c

0.2 0

0.2

0.4

0.6

0.8

1.0

Diffuse fractionwindow size 100

Fig. 1. Evolution of the diffuse fraction with the clearness index. Experimental values

installations are situated. Additional parameters based on theKolmogorov–Smirnov test have been considered [15] in order tocompare and validate models using experimental data, in additionto usual parameters such as RMSDr or MBDr. In this way, thebenchmarking results were applied to an area which has particularinterests in PV. The results available on related literature arecomplemented with the application of second order statistics.

0.2 0.4 0.6 0.8kt

0.2

0.4

0.6

0.8

1.0

Diffuse fractionwindow size 25

.4 0.6 0.8kt

(a); application of a moving average with a window size of 25 (b) and of 100 (c).

0

5

10

15

20

25

30

35

40

45

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Azimut (º)

Ele

va

tio

n(º)

Fig. 2. Sky line of the measuring location.

100 200 300 400

Gdcal Dirint

100

200

300

400

500

Gdmes

Fig. 3. Measured hourly average diffuse irradiance values (Gdmes, Wm�2) with respectto the ones calculated by applying the Dirint model.

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–1332 1327

2. Analysed models

2.1. Polynomial correlations

General expressions for the correlations between the diffusefraction (kd) and the clearness index (kt) are collected in Eq. (1). It iscommon practice to divide the total range of clearness indexes intothree intervals, taking into account the increasing kt values,wherein the different kd–kt regression for each interval are estab-lished. The polynomial order in the second zone characterizes thecorrelation.

kd ¼ a1 þ a2$kt first zone of ktkd ¼ a3 þ a4$kt þ a5$k2

t þ a6$k3t þ a7$k4

t second zone of ktkd ¼ a8 third zone of kt

(1)

Table 1 shows the values of coefficients ai corresponding to thedifferent correlations considered in this research. It can clearly beseen that some of these coefficients are zero, as a function of theconsidered correlation order, as well as of the weather conditions of

Table 2Statistical performance of the models for calculating the diffuse irradiance.

Models MBD(Wm�2)

rMBD(%)

RMSD(Wm�2)

rRMSD(%)

R2

First order modelsOrgill and Hollands [16] 10.90 7.05 54.34 35.14 0.747Reindl et al. [17] 2.34 1.52 53.12 34.36 0.747Model 1 Pamplona �2.42 �1 55.87 36.10 0.742

Second order modelHawlader [18] �0.77 �0.50 55.42 35.85 0.724Model 2 Pamplona �4.54 �2.94 54.98 35.57 0.751

Third order modelMiguel et al. [19] 4.06 2.62 52.98 34.26 0.748Karatasou et al. [20] �0.02 �0.01 58.70 37.98 0.712Model 3 Pamplona �4.16 �2.69 55.06 35.62 0.7499

Fourth order modelErbs et al. [21] 2.05 0.13 54.086 35.00 0.737Oliveira et al. [22] �8.46 �5.47 57.87 37.43 0.728Jacovides et al. [2] 0.04 0.02 56.97 36.85 0.728Model 4 Pamplona �5.37 3.47 55.29 35.77 0.747

Model considering a logistic functionBoland et al. [3] 8.50 5.49 54.89 35.51 0.737Model 5 Pamplona 8.49 5.49 54.89 35.51 0.738

Models considering the process dynamicsDirint [5] 3.98 2.57 45.53 29.38 0.816Skartveit et al. [10] 9.24 5.98 46.57 30.13 0.813BRL [11] 2.29 1.49 48.12 31.4 0.787

the site where the regression was undergone. For instance, for thesecond order Hawlader regression [16], a6 and a7 coefficients arenull. Coefficients of models 1–4 Pamplona have been obtained byapplying a least squared fitting to experimental data of Pamplona.

2.2. Logistic functions

In order to analyse the functional relationship between theexperimental diffuse fraction and the clearness index, observeddata are commonly filtered by a moving average method. Fig. 1ashows the evolution of the experimental diffuse fraction vs. theclearness index. Fig. 1b is the result of applying a moving averagewith a window size of 25 (as in [2]), whereas Fig. 1c considersa window size of 100 (as in [23]).

It is clear that the higher the moving average window sizebecomes, the greater the relationship between the diffuse fractionand the clearness index results in softening, which is more proxi-mate to an analytical function. Although this behaviour can facili-tate the identification of the underlying functional kd–kt

relationship, there is a significant loss of information with respectto the experimental truth. As a consequence, the model fitted to thecurves shown in Fig. 1b,c, cannot explain an essential part of thevariability observed in Fig. 1a.

When the moving average is applied, the resultant curve provesan evolution of the concavity that led Boland et al. [3] to propose

Table 3Statistical performance of the models for calculating the diffuse fraction.

Models rMBD (%) RMSD rRMSD % k BIC (�104)

First order modelsReindl et al. [17] �4.85 0.140 27.89 5 �1.497Model 1 Pamplona �0.43 0.135 26.95 5 �1.525

Second order modelsHawlader [18] 1.19 0.138 27.62 6 �1.507Model 2 Pamplona �0.02 0.135 26.89 6 �1.524

Third order modelsMiguel et al. [19] �5.7 0.141 28.09 7 �1.490Model 3 Pamplona 0.03 0.135 26.89 7 �1.523

Fourth order modelsErbs et al. [21] �6.07 0.146 29.13 8 �1.462Model 4 Pamplona �0.09 0.135 26.89 8 �1.522

Models considering a logistic functionBoland et al. [3] �7.44 0.145 28.90 2 �1.466Model 5 Pamplona 0.78 0.137 27.29 2 �1.509

Models considering the process dynamicsDirint [5] 4.44 0.124 24.80 5 �1.532Skartveit et al. [9] 9.66 0.143 28.48 20 �1.461BRL [11] 3.17 0.1254 25.41 9 �1.523

0.2 0.4 0.6 0.8 1.0

kt

0.2

0.4

0.6

0.8

1.0

Diffuse fraction

a b

dc

e

0.0 0.2 0.4 0.6 0.8 1.0

kt0.0

0.2

0.4

0.6

0.8

1.0

Diffuse fraction

0.0 0.2 0.4 0.6 0.8 1.0

kt0.0

0.2

0.4

0.6

0.8

1.0

Diffuse fraction

0.0 0.2 0.4 0.6 0.8 1.0

kt0.0

0.2

0.4

0.6

0.8

1.0

Diffuse fraction

0.0 0.2 0.4 0.6 0.8 1.0

kt0.0

0.2

0.4

0.6

0.8

1.0

Diffuse fraction

Fig. 4. (a) kd calculated for Dirint model as a function of kt; (b) kd calculated for Skartveits model as a function of kt; (c) kd calculated for BRL model as a function of kt; (d) curve of thelogistic model Model 5 Pamplona overlapped on the resulting point cloud of measured kd values vs. kt; (e) curves of the polynomial correlations Model 1 and Model 4 Pamplonaoverlapped on the resulting point cloud of measured kd values vs. kt.

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–13321328

a mathematical model, based on a logistic function in which theinitial exponential growth softens as a consequence of certaindegrees of saturation. The proposed model, Eq. (2), is parsimoniousas only two parameters are needed.

kd ¼1

1þ exp b1ðkt þ b2Þ(2)

For Victoria (Australia) data, b1 and b2 coefficients become 7.997and �0.586, respectively. Notwithstanding, Boland and Ridley [23]

proposed that the values 8.6 and�0.581 can have a general validity.The model called Model 5 Pamplona has been obtained fromexperimental irradiance data of Pamplona. In this case, resultingcalibrated coefficients are b1 ¼ 7.027 and b2 ¼ �0.552.

2.3. Dirint model [5]

Eq. (3) proves the basic expression used to obtain the normaldirect irradiance (Gb,n). The diffuse irradiance can be determinedfrom this value.

Table 4Performance of the models for calculating the diffuse irradiance based on secondorder statistics.

Models DK-S OVER% KSI%

First order modelsReindl et al. [17] 0.116 101.45 189.57Model 1 Pamplona 0.142 174.26 263.8

Second order modelsHawlader [18] 0.150 186.31 274.09Model 2 Pamplona 0.1287 154.57 244.828

Third order modelsMiguel et al. [19] 0.152 116.88 208.18Model 3 Pamplona 0.132 158.12 247.97

Fourth order modelsErbs et al. [21] 0.137 91.51 179.11Model 4 Pamplona 0.125 148.185 237.87

Models considering a logistic functionBoland et al. [3] 0.154 105.46 190.28Model 5 Pamplona 0.153 105.45 190.22

Models considering the process dynamicsDirint [5] 0.054 10.28 55.28Skartveit et al. [9] 0.118 49.49 99.10BRL [11] 0.087 28.99 88.54

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–1332 1329

Gb;n ¼ Gb;n;disc$X�

k0

t ; qz;W;Dk0

t

�(3)

Gb,n,disc is the normal direct irradiance calculated with the DISCmodel [4], dependant on the clearness index and the solar zenithangle through the relative air mass. The value of Xðk0t ; qz;W;Dk

0

tÞfunction is extracted hourly from a four dimensional look-up tableconsisting of a 6� 6� 5�7 matrix. For doing so, the followingparameters must be previously determined: k

0

t , correction of theclearness index to make it independent on the sun’s position (Eq.(4)); qz, solar zenith angle; W, precipitable water in the atmosphere(obtained from the dew point temperature); and Dk

0

t , stabilityindex that depends on the k

0

t values corresponding to the previous,current and following hour (Eq. (5)). As mentioned earlier,version$3-D of the model does not need dew point data.

k0

t ¼ kt=½1:031$exp� 1:4=ð0:9þ 9:4=mÞ þ 0:1� (4)

Dk0

t ¼0:5����k0ti

� ktiþ10���þ ���k0ti

� kti�10����

or Dk0

t ¼���k0ti� kti�1

0��� if either the preceding or

following hourly record is missing ð5Þ

2.4. Skartveit et al. model [9]

This model is composed of a high number of analytical func-tions. Input data for its application are similar to the ones for Dirintmodel described above. In this case, in order to consider the processdynamics, an hourly variability index (s3) is included. The modeldistinguishes between two kinds of hours, those called invariablehours (when s3 z 0) and the variable hours (with s3> 0). Inaddition, the effect of the surface albedo on the diffuse fraction istaken into account, by considering or not a snow free terrestrialenvironment.

2.5. BRL model [11]

Eq. (6) shows the expression used to obtain the diffuse fraction

kd¼1

1þexp�5:38þ6:63ktþ0:006AST�0:007aSþ1:75Ktþ1:31j

(6)

AST is the apparent solar time, in decimal hours; kt is the hourlyclearness index; Kt is the daily clearness index; aS is the solarelevation in degrees and j is the persistence index, calculated withEq. (7).

8<:

kt�1 þ ktþ1

2sunrise < t < sunset

ktþ1 t ¼ sunrisekt�1 t ¼ sunset

(7)

3. Radiation data

For this research, experimental global, direct and diffuse irra-diance data with a 10 min frequency were recorded in Pamplona,Spain (42.83� N, 1.6� W, 435 a.m.s.l, Mediterranean/continentalclimate) from October 2006 up to May 2008. Two Kipp&Zonen CM-11 pyranometers were employed for measuring global and diffuseirradiance, whereas a CH-1 pyranometer was used for the directirradiance. The three instruments were installed in a Kipp&Zonnen2Ap 2 axis tracker/positioner, provided with a ball for shadowingthe pyranometer that measures the diffuse irradiance. All sensorsare periodically calibrated following a controlled maintenance

routine to minimize measuring errors. Fig. 2 shows the elevationcontour line observed from the measuring station.

The corresponding hourly values were obtained by averagingthe six registered values corresponding to 10 min time fractions.

For quality control, hourly irradiance data were eliminated,either Gd> 1.1 G, G>1.2 G0, Gd> 0.8 G0, G< 5 W m�2or Gb>G0 [2].Resulting data after this filter also satisfied the conditions imposedin [17]. For this period, the measured average global irradiance was406 W/m2. Hourly average diffuse irradiance was 155 W/m2.

4. Results

Table 2 shows the absolute and percentage values of MBD (MeanBias Difference) and RMSD (Root Mean Square Difference) resultingfrom comparing the experimental hourly average diffuse irradiancewith the one calculated by applying the different analysed models.The R2 value, collected in the last column, refers to the determi-nation coefficient between calculated and measured diffuse irra-diance values, as Fig. 3 shows for the case of the Dirint model.

The applied statistics indicate that an increase in the correlationorder does not significantly improve the diffuse irradiance esti-mation. For instance, Reindl [17] first order correlation presentsrRMSD and R2 values, which are very similar to third order corre-lations, as the one of Miguel et al. [19], or even better than Erbs et al.[21] fourth order correlation. Models that consider the calibratedcoefficients for the diffuse fraction in Pamplona (Model 1–4 Pam-plona) do not offer better results for the diffuse irradiance thantheir respective correlations of equivalent order obtained from theliterature.

The logistic model does not improve the results of thecorrelations.

Regarding the three models that consider the dynamics of theprocess, i.e. Skartveit et al. Dirint and BRL, although they exhibithigher MBDs than some correlations and show a certain tendencyto overestimate the diffuse irradiance, they also offer noticeablybetter rRMSD values than the rest of the models, being even lowerthan 30% in the case of the Dirint model. In addition, the higher R2

values of these three models show that the resulting diffuseradiance values obtained fit better to the experimental measure-ments than when the rest of the models are applied. These resultsare coherent with what Ineichen [13] concludes. Ineichen [13]

0 100 200 300 400

Gd Wm-2

0.0

0.2

0.4

0.6

0.8

1.0

Cumulative frequency

0 100 200 300 400Gd Wm

-20.00

0.05

0.10

0.15

0.20

Dna

0 100 200 300 400

Gd Wm-2

0.0

0.2

0.4

0.6

0.8

1.0

Cumulative frequency

0 100 200 300 400Gd Wm

-20.00

0.05

0.10

0.15

0.20

Dnb

0 100 200 300 400 500

Gd Wm-2

0.0

0.2

0.4

0.6

0.8

1.0

Cumulative frequency

0 100 200 300 400 500Gd Wm

-20.00

0.05

0.10

0.15

0.20

Dnc

0 100 200 300 400

Gd Wm-2

0.0

0.2

0.4

0.6

0.8

1.0

Cumulative frequency

0 100 200 300 400Gd Wm

-20.00

0.05

0.10

0.15

0.20

Dnd

0,00

0,20

0,40

0,60

0,80

1,00

0

Gd

(W·m-2

)

Cumulative

frequency

Exper. data

Reindl Model 1 Pamplona HawladerModel 2 Pamplona Miguel Model 3 Pamplona Erbs Model 4 Pamplona

0,00

0,05

0,10

0,15

0,20

Gd

(W·m-2

)

Dn Vc

Reindl

Model 1 Pamplona

Hawlader

Model 2 Pamplona

Miguel

Model 3 Pamplona

Erbs

Model 4 Pamplona

e

100 200 300 500400

0 100 200 300 500400

Fig. 5. Cumulative distribution functions (left) and distances (right) for the calculated and measured hourly average diffuse irradiance values. (a) Dirint model; (b) Skartveit’smodel; (c) BRL model; (d) Model 5 Pamplona; (e) Correlations.

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–13321330

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–1332 1331

pointed out that the Dirint and the Skartveit et al. models arecomparable in precision.

When instead of analysing the diffuse irradiance behaviour, thediffuse fraction is analysed, which is something rudimentary inprevious research in order to obtain the correlations that have beenapplied in this work, the values that indicate the quality of themodels change. Table 3 shows the resulting values for the diffusefraction. As proven, in this case the number of polynomial corre-lations from the literature considered was reduced to four, one pereach correlation order, after having selected the ones that showedthe best behaviour according to the results collected in Table 2. Thequality analysis has also been applied to the new models whosecoefficients were calculated by applying a least squared fitting toexperimental data of Pamplona (models 1–5 Pamplona).

In general, the rRMSD values obtained in this study are lowerthan those obtained by Jacovides et al. [2] in their comparativeanalysis; both in the correlations and in the logistic model. Newmodels (1–5) with calibrated coefficients for Pamplona exhibitbetter RMSD and rRMSD values when they are compared with theirrespective models of the literature. Notwithstanding, it can beobserved that the models BRL and Dirint which consider thedynamics of the process give better results, wherein the results forthe diffuse fraction were also better, in comparison to the rest of themodels, even in the case of the models recalibrated for the local data.

Fig. 4 shows that dynamic models generate diffuse irradiancevalues that exhibit a degree of scattering, which come closer toa more realistic situation. In contrast, in the case of the correlationsor of the logistic model, the said scattering does not take placewhile resulting diffuse irradiance fraction is represented by a curve.

In order to complete the analysis, besides the parameters basedon first order statistics, those based on second order statisticsshown in Table 4 were also calculated. These statistics are obtainedby comparing the distribution functions of the diffuse irradiancevalues measured and calculated by the different models. Theirmeaning and mathematical expression are collected in Appendix A

According to the amount of data observed and to Eq. A.2, none ofthe analysed models pass the Kolmogorov–Smirnov test (K–S test)to a confidence level (a)of 99.9%, since all DK-S shown in Table 4become higher than the critical distance value (see Vc Ec. A.2) forthe considered a, which results in Vc ¼ 0.026. Notwithstanding, itcan be observed that the DK-S for the Dirint and BRL modelsbecomes noticeably lower than in the rest of the models.

Fig. 5 (left) shows the distribution functions of the diffuse irra-diance values measured and calculated with the different models,whereas Fig. 5 (right) shows the distances (Dn) (see Eq. A.3)between the distribution functions corresponding to the measuredand modelled diffuse irradiance as a function of the diffuse irradi-ance. The horizontal line in these graphs is the critical value Vc.

The model Dirint exhibits the lowest area of the curve Dn abovethe critical level. In fact, the said curve only overpasses the criticallevel in this model for hourly average diffuse irradiances rangingfrom 40 to 80 W m�2, whereas this range becomes much higher forthe rest of the models, except for BRL that also proves to actsatisfactorily. As a result, the parameter OVER% becomes noticeablylower for Dirint than for the rest of the models. The KSI% parameteralso becomes the lowest for the Dirint and BRL models.

In order to study the trade-off between the goodness of fit andthe complexity of the models the Bayesian Information Criterion(BIC) was calculated, with Eq. (8).

BIC ¼ n ln�

RSSn

�þ k lnðnÞ (8)

Where RSS is the residual sum of squares, n is the number of datapoints and k is the number of parameters estimated. Table 3

indicates k values considered in each model and the BIC results. Itcan be observed that the Dirint model exhibits the lowest value ofBIC, which indicates that it is the best model according to thiscriterion, although differences among models under study aresmall.

5. Conclusions

Regarding the observed data in Pamplona for the time periodunder analysis, it can be concluded that:

1. The models applied for estimating the hourly diffuse irradiancethat consider the values corresponding to the previous andposterior hours to the one under analysis, exhibit noticeablylower rRMSD values than the ones based on polynomialcorrelations or in the logistic function. In contrast, rMBDsvalues become lower in several polynomial correlations than inthe rest of the models; therefore, the said polynomial correla-tions better estimate the average value of the diffuse irradiancewithin the considered period.

2. A similar conclusion related to rRMSD for the Dirint and BRLmodels is applicable to the case in which the variable underestimation is the diffuse fraction instead of the diffuse irradi-ance. This conclusion maintains its validity even when both thecorrelations and logistic models are specifically calibrated forthe data observed in this research.

3. In addition, the analysis of the parameters based on secondorder statistics reveals that calculated series data with thedynamic models better approximate the observed diffuseirradiance, given that the cumulative frequencies of occurrenceof the different values fit better.

4. The models Dirint and BRL exhibit the best general behaviour,as practically all their statistical indicators are better than therest of the considered models. These are the recommendedmodels for estimating the hourly diffuse irradiance valuesunder the observed sky conditions. Regarding the calculationprocess, the BRL model is more easily applied than the Dirintmodel.

Acknowledgements

The present study was carried out under the research project‘‘Modelado de la distribucion de radiancia en el cielo para la eval-uacion de la irradiancia en proyectos de sistemas de captacion solar,arquitectura bioclimatica y planificacion energetica’’ supported bythe Ministry of Education and Science (CICYT, ref. ENE2007-64413/ALT)

Nomenclature

a.m.s.l. above mean sea levelAST apparent solar time, decimal hoursBIC Bayesian Information CriterionGb,n hourly normal direct irradiance, W m�2

Gb,n,disc hourly normal direct irradiance calculated with DISCmodel, W m�2

G hourly total irradiance on a horizontal surface, W m�2

Gb hourly beam irradiance on a horizontal surface, W m�2

Gd hourly diffuse irradiance on a horizontal surface, W m�2

G0 hourly extraterrestrial irradiance on a horizontal surface,W m�2

Kt daily clearness indexk number of parameters estimated in a modelkt hourly clearness indexk0

t clearness index independent on the sun position

J.L. Torres et al. / Renewable Energy 35 (2010) 1325–13321332

k0

ticlearness index independent on the sun position for thecurrent hour

k0

tiþ1clearness index independent on the sun position for thefollowing hour

k0

ti�1clearness index independent on the sun position for thepreceding hour

kd hourly diffuse fractionm relative air mass [24].MBDr relative mean bias deviationn number of data points considered to apply a modelPV photovoltaic solar energyRMSDr relative root mean square deviationRSS residual sum of squaresR2 coefficient of determinationW precipitable water in the atmosphereqz solar zenith angleaS solar elevation in degreesj persistence index in the BRL models3 Hourly variability index in Skartveits modelDk

0

t stability index, k0

t dependant

Appendix A. Meanings and expressions of parameters basedon second order statistics

A.1. The DK-S is defined as the maximum value of the absolutedifference between two distribution functions with a single vari-able (for instance, the diffuse irradiance). Mathematically:

DK � S ¼ maxjF � Q j (A.1)

where F and Q are the distribution functions of the observed andmeasured values, respectively.

It can be considered that observed and modelled values comefrom the same distribution when the DK-S is inferior to a criticalvalue given by:

Vc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lnða=2Þ=2n

pðn > 100Þ (A.2)

where a is the level of confidence and n is the number of data of theconsidered sample.

A.2. In other to obtain further information, apart from thecalculation of the simple distance DK-S, a number of statistics Dncan be obtained according to [25,26]:

Dn ¼ jFðxnÞ � QðxnÞj (A.3)

where xn is every interval in which the values of the consideredvariable are grouped.

Fig. 5 (right) shows graphs of the Dn as a function of the hourlyaverage diffuse irradiance.

A.3. KSI (Kolmogorov–Smirnov test Integral) parameter.

KSI% ¼ KSIacritical

$100

KSI ¼Z

Dn$dx

acritical ¼ Vc$ðxmax � xminÞ

(A.4)

xmax and xmin are the maximum and minimum values of xn.

A.4. OVER parameter.

OVER% ¼ OVERacritical

$100

OVER ¼Z

aux$dx

aux ¼�

Dn � Vc Dn > Vc0 Dn � Vc

(A.5)

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