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Estimation of hourly direct normal from

measured global solar irradiance in Spain

Gabriel Lo pez*, Miguel Angel Rubio, Francisco J. Batlles

Departimento de Fsica Aplicada, Universidad de Almera, 04120, Almera, Spain

Received 18 November 1999; accepted 25 November 1999

Abstract

The availability of a good data set, registered in six Spanish locations, including several

radiometric variables, has been used to test dierent approaches for estimating hourly

direct normal irradiance by decomposition models. Models proposed by dierent authorshave been tested. Following this preliminary study, to improve the kbkt correlations,

another geometric variable has been used as a predictor of hourly beam transmittance, kb,

by means of piecewise correlations. The new beam transmittance correlations, which include

additional geometric information, reduce the root mean square deviation. In addition, they

show a better performance in terms of the determination coecient of the regression

analysis of measured vs calculated values, providing an improved capture of the real world

eects than models that are function of the clearness index only. A new model that uses

only two ranges of clearness index is proposed. The proposed model shows seasonal

dependence and thus we have developed a seasonal version of it. However, the performance

of the seasonal version has proved to be similar to the corresponding annual model. 7 2000

Elsevier Science Ltd. All rights reserved.

Keywords: Direct normal irradiance; Global solar irradiance; Estimation; Regression models

1. Introduction

The ux of solar radiation reaching the surface of the earth is the primary

source among all types of known renewable energies. To calculate the eciency of

0960-1481/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

P I I : S 0 9 6 0 -1 4 8 1 (9 9 )0 0 1 2 1 -4

Renewable Energy 21 (2000) 175186

www.elsevier.com/locate/renene

* Corresponding author. Tel.: +34-950-215295; fax: +34-950-255277.

E-mail address: [email protected] (G. Lo pez).

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solar energy converters, particularly focussing systems, and for simulations of

long-term process operations, the direct normal component of the incident solar

radiation has to be known. Furthermore, hourly values of direct normal

irradiance allow one to derive precise information about the performance of solar

energy systems. Unfortunately, data sets of direct normal irradiance are rarely

available at the location of interest because of maintenance problems and the high

cost of the pyrheliometers, and therefore, the direct normal irradiance must be

estimated from other additional variables that are recorded there. Thus, dierentmodels that relate direct normal irradiance with commonly measurable

atmospheric and meteorological variables have been developed.

In the literature, there are basically two types of models for calculating direct

normal irradiance: (i) radiative transfer models and (ii) decomposition models.

The rst ones take into account interactions on beam solar radiation with the

terrestrial atmosphere, such as atmospheric scattering by air molecules, water and

dust, and atmospheric absorption by O3, H2O and CO2, principally [13]. The

current problem in the use of these models is the large amount of atmospheric

information needed, which is not often measured at the meteorological stations.

On the other hand, the decomposition models relate the direct normal irradiance

with other solar radiation measurements, especially the global solar irradiance on

a horizontal surface [4,5]. Thus, decomposition models require simpler input datathan the parametric models. In fact, decomposition models provide an estimate of

the direct irradiance by means of simple empirical expressions both for cloudless

and cloudy conditions. In this way, these models avoid the diculties associated

with the cloud radiative eect parameterisation.

Historically, one approach to the determination of the hourly direct normal

irradiance from measured global data has been to use a hourly diuse sky model.

Diuse irradiance has been estimated by correlations between the dimensionless

numbers k, hourly diuse fraction (hourly diuse irradiance/hourly global

irradiance), and kt, hourly clearness index (hourly global irradiance/hourly

extraterrestrial horizontal irradiance) [6,7]. Hourly direct irradiance (Id) would

now be calculated from the predicted values of hourly diuse ( D ), recorded values

of hourly global irradiance (G ), and the solar zenith angle (yz) by means of their

relationship:

Id G Da cos yz

However, dierent authors [810] have estimated the direct irradiance from only

measurements of global irradiance on a horizontal surface. In this way, regression

models on dimensionless variables such as the hourly clearness index kt, and

hourly beam transmittance kb (hourly direct irradiance/hourly extraterrestrial

irradiance) have been developed [810]. Moreover, dependencies of the hourly

direct irradiance on other variables besides kt, such as solar elevation, water

vapour content, atmospheric turbidity and climatic variables have been suggested

[1113].

In this work, we estimate the hourly direct irradiance by means of kbkt

G. Lopez et al. / Renewable Energy 21 (2000) 175186176

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correlations. Firstly, following previous work developed by Vignola and

McDaniels [9], the direct transmittance is computed from only the hourly

clearness index. Next, the solar zenith angle is used as an additional predictor

variable in order to obtain a better performance of the beam transmittance and

capture the real world eects in a better way. In this sense, we have selected two

annual models and a seasonal model to compare their performances. The selected

models were formulated by Louche et al. [10], Maxwell [13] and Rerhrhaye et al.

[14], respectively. The rst one is a kbkt correlation which involves a polynomialof order ve, and the second one is termed as a `quasi-physical' model as it

combines a physical clear sky model with experimental ts for other conditions,

including the clearness index kt and air mass as inputs to the model. The third

model estimates the hourly direct irradiance from three seasonal kbktcorrelations.

Several authors have suggested the convenience of a seasonal modelling [9,14].

We have tested the seasonal performance for the proposed models using the data

set registered at Granada. Based on these tests, we have developed two models for

estimating the hourly direct irradiance at Granada: one involving seasonal

dependence and the other without. A comparison between the performances of

both models has been carried out.

2. Data set

In this work, we have used data registered in six Spanish radiometric stations,

located in areas characterised by dierent climatic conditions. Table 1 summarises

the geographical locations, the number of data and the date of the measurements

used for each station. In the case of Oviedo only the rst semester of 1991 has

been used to exclude data aected by the enhancement of stratospheric aerosols

due to the volcanic eruption of Mt. Pinatubo. There are coastal locations,

Almera, and inland locations with dierent degrees of continentality. For the

dierent locations, we found rather dierent cloud regimes. The measurements

include global and diuse solar irradiance on a horizontal surface, which were

registered by means of pairs of Kipp & Zonen pyranometers, one of them ttedwith a polar axis shadowband. In Granada and Almera stations CM-11

Table 1

Geographical location, period of measure and number of data for each radiometric station

Latitude (8N) Longitude (8W) Elevation (m) Years N

Almera 368 50 ' 28 25 ' 0 19941996 8461

Granada 378 10 ' 38 34 ' 660 19941995 7442

Murcia 388 0 ' 18 10 ' 69 1987 3779

Madrid 408 29 ' 38 45 ' 664 1981 2625

Logron o 428 28 ' 28 41 ' 372 1990 856

Oviedo 438 21 ' 58 21 ' 348 1991 1984

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pyranometers have been used, whereas the other radiometric stations used CM-5

pyranometers. For all the variables, hourly values have been obtained. Calibration

constants of the radiometric devices used at Almera and Granada have been

checked periodically by our research group. For the rest of the stations the

Instituto Nacional de Meteorologa (INM) is responsible for the corresponding

calibration checks. Analytical checks, for measurement consistency, were carried

out to remove problems associated with shadowband misalignments, and other

questionable data. Due to cosine response problems we have used only cases atsolar zenith angles less than 858. The diuse irradiance, measured by shadowband,

has been corrected using the model developed by Batlles et al. [15]. Lastly, the

hourly direct irradiance is obtained from their relationship with the hourly global

and diuse irradiance.

3. Development of new beam transmittance correlations

The whole data set has been divided into two subsets by means of a random

selection. Around two thirds of the whole data set have been used to develop the

new correlations, whereas the remaining third is reserved for validation purposes.

Fig. 1 shows kb- vs kt-values. Firstly, in a previous analysis, we have grouped thedata into kt bins 0.020 in width. For each bin, the mean of kb has been obtained.

A graphical analysis of these mean values of kb vs kt has shown a linear

dependence of kb against kt, for kt-values less than 0.220, while a quadratic

relationship dependence has been found for the following kt-values. Thus, the

following have been derived using standard regression procedures upon the two

kt-regions, which we call MODEL 1:

kb=0.014 kt kt 0.220

kb=0.0120.252 kt+1.455 k2t

ktr0.220 R2=0.88

where R 2 is the determination coecient, which accounts for the fraction of the

variance in the two variables that is shared.

From Fig. 1, it is noted that values of kb go quickly to zero as kt is less thanaround 0.200, corresponding to hazy conditions, whereas there is a wide range of

values attained by kb for a given kt at intermediate clearness indices. However,

only one hourly beam transmittance value is related with one kt-value by kbktcorrelations. It is obvious that the necessity of additional independent variables

account for this spread of kb- vs kt-values. Maxwell [13] introduced the air mass

in his model as the main parameter aecting the relationship between beam

transmittance and clearness index. Jeter and Balaras [5] used the air mass as an

additional variable to predict kb by means of an own developed surface-tting

method and showed that the air mass dependence becomes dominant at high

clearness indices. Based on these previous works, we have analysed the

dependence of solar zenith angle on hourly kbkt correlations. In order to test this

dependence, we have divided the data set into six cos yz intervals. The boundaries,


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displayed in Table 2, have been chosen in order to provide a quasi-uniform

distribution of the data into the intervals. We have used a similar technique to

that employed to develop MODEL 1 for each cos yz interval, obtaining the

following kbkt correlations, which we call MODEL 2:

cos yz 0.343 kb=0.031 kt kt 0.240

kb= 0.145+0.476 kt+0.663 k2t

ktr0.240 R2=0.82,

SE=30%

0.343 < cos yz 0.500 kb=0.015 kt kt 0.275

kb= 0.081 0.074 kt+1.394 k2t

ktr0.275 R2=0.87,

SE=21%

0.500 < cos yz 0.643 kb=0.033 kt kt 0.320

kb= 0.075 0.240 kt+1.586 k2t

ktr0.320 R2=0.90,

SE=17%

0.643 < cos yz 0.766 kb=0.051 kt kt 0.280

kb=0.090 0.856 kt+2.093 k2t

ktr0.280 R2=0.90,

SE=15%

0.766 < cos yz 0.866 kb=0.046 kt kt 0.230

kb=0.139 1.085 kt+2.292 k2t

ktr0.230 R2=0.90,

SE=13%

0.866 < cos yz kb=0.081 kt kt 0.300

kb=0.254 1.584 kt+2.726 k2t

kt 0.300 R2=0.90,

SE=12%

where SE is the standard error of the estimates for each considered interval. The

standard error has been calculated as the square root of the mean error and has

been expressed as a percentage of the mean measured kb-value for each interval of

cos yz for a better comparison. The eect of the solar zenith angle on correlationsis noted in Fig. 2. As it may be seen, the tting curves are clearly dierent for

each interval of cos yz and the spread of the data is partially explained by means

of the parameterisation on cos yzX The determination coecient varies from 0.82,

for the rst interval, to 0.90 for the intervals with values of cos yz higher than

0.500, pointing out to a signicant dependence of the kbkt correlations on solar

zenith angle. Furthermore, there is a dramatic reduction of the spread of the

estimated vs measured kb-values, from the rst to the last interval, given the basis

of the reduction of the standard error from 30 to 12%. The large standard error

for cos yz < 0.343 is reasonable because we have used both kb and ktmeasurements corresponding to dierent locations with dierent atmospheric

conditions, e.g., dierent amounts of water vapour and aerosol contents in the

atmosphere. At high solar zenith angles, the beam solar radiation is aected, on a


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wider scale, by these dierent conditions, which cannot be reproduced using only

global solar radiation. In addition, this interval contains around 60% of the kt-

values that are less than 0.500. Partly cloudy skies are the prevailing weather

conditions in this region of kt-values and the clearness index is not suitable to

parameterise the eect of the clouds on direct irradiance. As cos yz values increase,

the percentage of kt-values over 0.500 also increase leading to cloudless skyconditions, whereas the relationship among hourly beam transmittance and hourly

clearness index is more close, and thus, the standard error decreases.

Nevertheless, the complexity of the model encourages us in the search of a more

compact form that could take into account the solar zenith angle dependence in

an analytical way. We have carried out a stepwise regression using dierent degree

order polynomials in kt and a linear dependence in cos yzX We have examined the

residual dierences of the MODEL 1 against cos yzX For this purpose, the residual

Fig. 1. Measured hourly beam transmittance vs hourly clearness index for one third of the whole used

data set.

Table 2

Boundaries used to dene the six categories for cosyz

0.343 0.500 0.643 0.766 0.866


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dierences have been averaged on intervals of 0.050 step in cos yz and plotted

against it (Fig. 3). From Fig. 3 it is noted the dependence of kbkt correlations on

cos yz and it is shown that this dependence follows a linear tendency. Thus, the

inclusion of cos yz in the stepwise regression by means of a linear dependence is

justied. The best results are obtained for the following set of expressions, which

we call MODEL 3:

kb= k

2

t (0.9280.909cos y

z) kt

0.325kb=0.0690.475 kt+1.733 k2t

0.096 cos yz ktr0.325 R2=0.90

The determination coecient obtained in the stepwise regression indicates a

better tting than that carried out by a kbkt correlation (MODEL 1). In

determining whether the model can be simplied, notice that all terms are

statistically signicant at the 99% condence level.

4. Model performance

The performance of the proposed models is compared with the original version

of previous decomposition models by Louche et al. [10], Maxwell [13] and

Rerhrhaye et al. [14]. Among the performance indicators we have chosen the

Fig. 2. kbkt diagram for MODEL 2. Solar zenith angle dependence on kbkt correlations is clearly

observed.


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regression analysis of measured values vs calculated ones, in terms of the

intercept, a, slope, b, and determination coecient, R 2. We have also included the

root mean square deviation (RMSD) and the mean bias deviation (MBD) as

indicators of the spread of the deviation between measured and calculated values

and as a test for existence of systematic tendencies, respectively. Table 3 shows the

statistical results obtained when a performance test of the models is applied to the

remaining third of the database, reserved for validation purposes.

The correlations relying only on kt, both Louche's model and the proposed

model, MODEL 1, provides similar global results. Although the intercept and the

Fig. 3. Averaged residuals from estimate of the hourly beam transmittance by MODEL 1 as a function

of the cos yzX The error bars denote one standard deviation from the mean value of the averaged

residuals.

Table 3

Statistical results of selected and proposed models for estimating hourly direct normal irradiancea

a (W m2) b R 2 MBE (W m2) RMSE (W m2)

Louche's model 20.6 0.95 0.899 4.2 99.2

Maxwell's model 32.7 0.90 0.900 20.5 95.7

Rerhrhaye's model 32.5 0.82 0.882 127.3 165.7

Model 1 52.9 0.90 0.901 1.1 96.5

Model 2 45.2 0.90 0.915 9.6 89.9

Model 3 52.6 0.89 0.914 11.1 90.8

a

Mean value of measured direct irradiance: 532.4 W m2

; N=7007.


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slope for Louche's model is more close to the 1:1 line of perfect t, MODEL 1

shows a slight improvement in terms of both RMSD and MBD, corresponding to

0.5% reduction in both deviations over the mean value of the measured hourly

direct irradiance. On the other hand, the fth order polynomial dependence on ktgiven by Louche's model is removed and replaced for only a quadratic

dependence on kt for ktr 0.220 and a linear dependence on kt for kt 0.220.

Rerhrhaye's model, which is a kbkt correlation on summer, winter and spring/

autumn separately, shows the worst statistical results with the highest value of theRMSD and a large underestimation. Dierent atmospheric conditions from

Spanish locations can possibly lead to these results.

The inclusion of cos yz as an additional estimator leads to the improved results

of models, in terms of determination coecient and RMSD, but with a slight

degradation in terms of tuning of the model, as indicated by slope, intercept and

MBD. Obviously, the calibration of the cos yz dependence performed for two

thirds of our data set, provides the lowest RMSD together with the higher

determination coecient. In this way, the kbkt-cos yz model parameterised in six

intervals of cos yz, MODEL 2, shows the best results. The variance in kb explained

by the variation in both kt and cos yz is increased by 1.5% compared with the

models that do not incorporate cos yzX Similarly, the spread of deviation between

calculated vs measured hourly direct irradiance is reduced and the RMSD reachesa minimum value that represents a 16.9% over the mean value of the measured

Fig. 4. Estimated hourly direct normal irradiance by MODEL 3 vs measured ones using one third of

the whole data set. The solid line denotes the line 1:1 of the perfect t.


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hourly direct irradiance. However, the inclusion of cos yz as a variable in the

stepwise regression, MODEL 3, provides a model with similar values in all terms,

except MBD, which is only 0.3% higher than that obtained by MODEL 2. Fig. 4

shows the estimated hourly direct irradiance by MODEL 3 vs measured ones.

Maxwell's model, which uses an exponential relationship between air mass and

transmittance parameterised in kt, presents worse results than MODELS 2 and 3

with both determination coecient and RMSD close to those obtained from

Louche's model and MODEL 1. Furthermore, Maxwell's model exhibits apronounced underestimation with an MBD value that is around 3.4% higher than

those obtained by Louche's model and MODEL 1 and around 2% higher than

those obtained by MODELS 2 and 3. Thus, based on these statistical results,

MODEL 3 seems to capture in a better and more compact way the behaviour of

the real processes.

5. Seasonal performance

Dierent authors have suggested the convenience of seasonal modelling [9,14].

Vignola and McDaniels [9] studied the time dependence of the average residuals

between the measured and calculated values from both daily and monthlyaveraged correlations, and found sinusoidal behaviours typical for each considered

site. Rerhrhaye et al. [14] divided the year into three periods and developed three

dierent kbkt correlations, one for each period.

We have tested the seasonal performance of MODEL 3 using the data set

registered at Granada. We have taken the data set corresponding to only one site

to avoid possible dierent seasonal behaviours depending on each location, as

Vignola and McDaniels noted. Table 4 shows the statistical results of MODEL 3

for the four seasons, evaluated with validation subset (one-third of the whole

Granada's data set). Summer, autumn and winter seasons present underestimation

of the hourly direct irradiance, whereas there is an overestimation tendency for

spring. The RMSD for winter months is 81.0 W m2, which have the best global

results, increasing to 94.7 W m2

for the autumn months. Dierent cloud regimesand dierent atmospheric compositions for each season can lead to these dierent

degrees of spread. Moreover, we note that the winter season has a percentage of

kt-values around 20% for kt < 0.500, whereas this percentage for autumn season

Table 4

Statistical results of MODEL 3 for each season using Granada's database


Spring 73.4 0.88 0.935 6.42 83.3

Summer 85.0 0.82 0.918 21.3 84.2

Autumn 53.1 0.83 0.929 26.5 94.7

Winter 52.0 0.89 0.948 11.5 81.0


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rises to 40%. As was noted in section two, the increase of kt-values falling below

0.500 leads to worse estimations of hourly direct irradiance. An equivalent study

performed upon MODEL 1 conrms these results.

Thus, this analysis suggests the convenience of trying seasonal modelling. First,

we have developed an annual model for Granada, which we call MODEL 4,

similar to MODEL 3 in order to carry out a comparison with the seasonal

version. MODEL 4, which has been evaluated for each season, presents similar

performance to that of MODEL 3, though with better statistical results as is

expected since overall data corresponds to the same location. Thus, for example,

MODEL 4 shows the highest RMSD value for autumn and the lowest RMSD

value for winter. Next, we have developed the seasonal version of the precedingmodel. For this purpose, we have divided two thirds of the Granada's data set

into four subsets corresponding to each season. Expressions equivalent to

MODEL 4, which we call the SEASONAL MODEL, have been obtained for each

seasonal subset. The set of correlation equations for both models is not shown

because we are only interested in their performance. Lastly, both models have

been tested with the remaining third of Granada's data set. Table 5 includes the

results of the statistical analysis for MODEL 4 and the SEASONAL MODEL.

Both models present similar values for all statistical parameters with a slight

improvement for the SEASONAL MODEL as indicated by slope, intercept and

RMSE. However, the RMSE reduction represents only a 0.5% over the mean

value of the measured hourly direct irradiance. Anyway, we conclude that a

seasonal modelling does not improve substantially the estimation of hourly directirradiance against annual modelling.

6. Conclusions

This work has shown the eciency of the solar zenith angle as estimator of the

hourly direct normal irradiance. We have developed two decomposition models

based on kbkt correlations that incorporate a solar zenith angle dependence.

Their performances have been carried out using data sets registered in six Spanish

locations, covering a variety of climatic and latitudinal conditions. The

performances have been compared against three kbkt correlation based models,

one of them also developed from our data set, that do not have any dependence

Table 5

Statistical results of annual model, MODEL 4, and its corresponding seasonal version, SEASONAL

MODEL, for Granada's databasea


Model 4 57.3 0.90 0.923 3.6 83.5

Seasonal model 51.2 0.91 0.929 2.5 80.8

a Mean value of measured direct irradiance: 550.2 W m2; N= 2356.


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on solar zenith angle and a fourth model with an air mass dependence. Inclusion

of cos yz as an additional independent variable in kbkt correlations accounts for

some of the variance of the data and improves the performances of kbktcorrelation based models. In this way, an analytical equation between kb and ktand cos yz using two kt-intervals has been provided as the better option to

estimate the hourly direct normal irradiance. This has been observed in terms of

the root mean square deviation and determination coecient of the regression

analysis of measured vs calculated values. The statistical results of the proposedmodels suggest that the existence of a wide range of kb-values for a given kt could

be partially due to variations in the solar zenith angle.

We have also analysed the seasonal performance of the proposed models using

Granada's database. This analysis has shown a signicant seasonal dependence.

However, the seasonal version of an annual model for Granada performs similarly

to that annual model. Thus, we consider that the seasonal versions are not able to

improve the estimation of the hourly direct normal irradiance.

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2000 Lopez (Ren Energy_21)