prediction of global irradiance on inclined surfaces from ...hera.ugr.es/doi/1501518x.pdfprediction...

Energy 24 (1999) 689–704www.elsevier.com/locate/energy

Prediction of global irradiance on inclined surfaces fromhorizontal global irradiance

F.J. Olmo, J. Vida, I. Foyo, Y. Castro-Diez, L. Alados-Arboledas*

Dpto. Fısica Aplicada, Universidad de Granada, Campus Fuentenueva, s/n 18071, Granada, Spain

Received 23 June 1998

Abstract

Knowledge of the radiation components incoming at a surface is required in energy balance studies,technological applications such as renewable energy and in local and large-scale climate studies. Experi-mental data of global irradiance on inclined planes recorded at Granada (Spain, 37.08°N, 3.57°W) havebeen used in order to study the pattern of the angular distribution of global irradiance. We have modelledthe global irradiance angular distribution, employing horizontal global irradiance as the only radiometricinput, and geometric information. We have obtained good results (root mean square deviation about 5%),except for surfaces affected by artificial horizon effects, which are not allowed for in this new model. TheSkyscan’834 data set has also been used in order to test the model under completely different conditionsfrom those in Granada, with respect to the amount of cloud, local peculiarities, experimental design andinstrumentation. The results prove the validity of our model, even when compared with the Perez et al.model. The model offers a reliable tool for use when solar radiance data are scarce or limited to globalhorizontal irradiance. 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

Knowledge of short-wave global irradiance incoming on inclined surfaces is often necessaryin order to study the surface energy balance, the local and large-scale climate, or to design techno-logical applications such as renewable energy conversion systems, either in urban or in ruralzones. Historically, at many national meteorological stations, global irradiance has been measuredonly on horizontal surfaces and rarely on inclined ones. Thus different estimation methods have

* Corresponding author. Tel.:1 34-958-244024; fax:1 34-958-243214; e-mail: [email protected]

0360-5442/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0360-5442(99)00025-0

690 F.J. Olmo et al. /Energy 24 (1999) 689–704

been developed, such as Refs. [1–6]. Nevertheless, calculations using these kinds of models arenot simple, since they require information of the direct or diffuse irradiance. This informationcan be obtained from different techniques that involve horizontal diffuse irradiance measurements,as in Refs. [7–10]. A detailed analysis of the contributions received on an arbitrarily orientedsurface suggests that the parameters involved in these models must depend on the atmosphericstate, properties of the adjacent surfaces, topography and geometrical factors.

Different authors have tested the results of inclined surfaces models against experimental data[1,6,11–15]. Feuermann and Zemel [6] have shown that a combined application of the Perez etal. model [3] with empirical correlations to estimate the direct component from horizontal globalirradiance, can yield accurate predictions for locations where only global horizontal data are avail-able.

There has been growing interest in recent decades about the use of satellite data to obtain solarirradiance at a surface. This method provides the possibility of continuous and global monitoringof this radiation flux [16–19]. Although this technique seems powerful, the use of remote sensingdata in the case of complex topography terrains requires additional efforts [20,21]. The algorithmsthat deal with the influence of complex topography in the solar irradiance field require the con-sideration of solar irradiance impinging on non-horizontal surfaces. This information must beacquired with the unique knowledge of horizontal solar global irradiance.

In this paper, we present a simple model to estimate global irradiance on inclined surfaces,under all weather conditions, which only requires the horizontal global irradiance, and the sun’selevation and azimuth as input parameters. The added value of this model lies in its applicabilityto sites where only horizontal global irradiance is measured, as is the case at most conventionalmeteorological stations, or satellite-derived global irradiance data. In addition, this method allowsfor the estimation of the global irradiance distribution by means of the horizontal global irradiancein a way that may on occasions be more practical than the methods of Feuermann and Zemel [6]or Perez et al. [3]. These latter methods require a previous estimation of direct and diffuseirradiance from the horizontal global irradiance.

This work represents a previous stage in the study of topographic effects on the solar irradiancefield, with the goal of mapping solar radiation on local and large scales. At present we only intendto test the validity of the inclined global irradiance model that we propose in contrast to othertype of models—such as the one used in [20] or [21]. These authors use the Erbs et al. model[22], which can present errors of about 37% and 22% when splitting global into diffuse and directirradiance, respectively [23].

2. Data base

Measurements of incoming global irradiance on surfaces with different slopes and orientationshave been carried out at Granada (Spain). The measurement system consists of a pyranometerKipp-Zonen CM-5, mounted on a device with the ability to vary both the elevation (0, the horizon-tal, to 90°, at 15° intervals) and azimuth (0, the south meridian, to 360°, at 45° intervals) of theinclined surface. This system has been installed on the terrace of the Science Faculty of theUniversity of Granada (650 m a.s.l.) [24,25]. The stability of the radiometer calibration factorhas been periodically tested against a reference pyranometer used as substandard and not exposedto the sun except during the intercomparison trials.

691F.J. Olmo et al. /Energy 24 (1999) 689–704

Each sky scan includes 49 instantaneous measurements, acquired during an interval of 45 min,that we standardise for the solar height changes during the experiment (refer to 2D polar plot inFig. 1). The data base consists of a total of 114 clear sky experiments distributed over the year(all months) and carried out with the same sequence of measurements at different times of theday distributed uniformly around solar noon. Thus, a complete range of solar azimuth (from247.5° to 104°) and solar height combinations (from 10° to 76°) are included in our data base.

In order to carry out a test of the model, trying to avoid the limitation of our data set (clearskies only) and possible local dependencies from Granada in our results, we have used the Skys-can’834 data set [26,27]. The measurement system consists of an Eppley Precision SpectralPyranometer (PSP) installed on the Mechanical Engineering Building rooftop (University ofToronto, 43.7°N, 79.4°W, 111 m a.s.l.). The pyranometer was calibrated by the National Atmos-pheric Radiation Centre in July 1982. This is a well-known data set, widely used and referencedin this type of works. Skyscan’834 covers a full year of measurements in a completely differentenvironment to that of our trials in Granada. The main limitation is that the Skyscan’834 dataset only contains slope irradiance measurements for the case of surfaces oriented to the south withan elevation angle of 44°. On the other hand, the system has the advantage that the pyranometer isshielded from the radiation reflected from the ground [26]. We have characterised the differentatmospheric conditions by means of the clearness index,kt, defined as the global horizontalirradiance to extraterrestrial horizontal irradiance ratio.kt ranges from 0.4 to 0.8 in Granada andfrom 0 to 1 in the Skyscan’834 data set. In the latter, we observe certain values for which anincrease of the diffuse fraction comes together with an increase ofkt. This fact is associated withpartly cloudy conditions with some clouds located near the sun. Under these circumstances, cloudborders reflect the direct irradiance during periods when the sun is unshaded by the surroundingclouds, thus enhancing both global and diffuse irradiance, and these situations are associated withchanging sky conditions [28,29].

3. Model formulation

In order to represent graphically the measurements of global irradiance obtained, we have usedtwo types of diagram: polar and three dimensional. Fig. 1 presents a three-dimensional and polarglobal irradiance diagram for one of our experiments. In the figure caption, specific features ofthis experimental series are explained. In short, this diagram facilitates the visual information ofeach experience. Both diagrams have been obtained from experimental values, using the kriginginterpolation method [30].

The complete Granada data set has been analysed by means of this type of diagram. In thissense, Fig. 1 can be considered as an example of the general performance of the irradiance distri-bution. These diagrams show the existence of a dependence between global irradiance values andthe angular distance subtended by the normal to the inclined plane and the sun (c). To evidencethis correlation we have restricted our analysis to the solar zenith plane. This is the plane thatcontains the zenith and the sun’s position (Fig. 2). Fig. 3 shows the global irradiance values inthe solar zenith plane (Gcsz) versus the angular distance to the sun’s position (csz), which, forthis plane, is the difference between the surface zenith angle and the sun’s zenith angle. In thisway, we eliminate the azimuth dependence. In this figure, we have plotted only the values that


Fig. 1. (a) Three-dimensional; (b) projection on a horizontal plane; (c) detailed polar representation of the globalirradiance for one scan. Sun elevation, 74°; sun azimuth, 341°.


Fig. 2. Sky dome showing the solar zenith plane geometry:csz, angular distance, in the zenith plane, of the normalto the considered surface (n9) with respect to the sun’s position;c, angular distance, for an arbitrary plane, of thenormal to the considered surface (n) with respect to the sun’s position;u andus, zenith angles for the normal (n9) andthe sun, respectively;a and as, azimuth angles for the normal (n9) and the sun, respectively.

correspond to the intervals 60–80 and 20–30° of solar elevation in the Granada data set. It is wellknown that direct irradiance on an inclined plane can be written as direct irradiance on a horizontalsurface multiplied by a geometrical factor [31]. This geometrical factor is a ratio of cosines. Inorder to investigate if a similar ratio could be applied when dealing with global irradiance let usrefer to Fig. 3. In this figure, in whichGn is the global irradiance in the surface normal to thesun beam, we investigate the ratioGczs/Gn, which suggests a dependence in terms of angulardistance to the sun’s position (csz). An additional analysis shown in Fig. 4 indicates that this ratiodepends also on the clearness index,kt (global to extraterrestrial horizontal irradiance ratio). Thisis especially true for inclined planes far from the sun’s position (greatercsz). In our case, wehave taken the global irradiance at normal incidence to the sun,Gn, as maximum (Fig. 3). Thenwe have modelled the global irradiance distribution in the solar zenith plane in terms of thismaximum value and an exponential function that includes the clearness index,kt, and the angulardistance to the sun’s position. The exponential function is adopted by convenience, as it will beexplained later, even though we could also model the global irradiance in the solar zenith planeas a function of cosines with similar behaviours. In this procedure, we have only used 8% of thedata set, i.e. data lying exactly in the solar zenith plane.

Thus, we propose the following expression:

Gczs 5 Gn exp(2 ktc2zs) (1)

whereczs is expressed in radians. Thekt value takes into account the influence of sky conditions,turbidity and clouds, as a modulating function in the solar zenith plane.

In previous works [32,33], the effects of turbidity on the global solar irradiance measurements


Fig. 3. Global irradiance on solar zenith plane vs. angular distance to the sun’s position for two extreme solar elev-ation intervals.

for inclined and oriented surfaces have been studied. It has been shown that the influence ofturbidity on global solar irradiance has great importance for surfaces normal to the sun.

Moreover, the symmetries observed in Fig. 2 suggest the possibility of extending geometricallythis procedure to the entire hemisphere (by means of the experimental function adopted). For thispurpose the angular distancec can be evaluated as follows:

cosc 5 sin u sin us 1 cosu cosus cos(as 2 a) (2)

whereu represents the zenith angle anda the azimuth. The subscript s refers to the sun’s position.We should point out that the angular distancec is the so-called scattering angle.

The scheme developed provides a good representation of the global irradiance distribution onthe complete hemisphere. Nevertheless, the global irradiance on a surface normal to the sun is notmeasured in most radiometric networks. Therefore, considering that horizontal global irradiance isthe usually available term, we have modified the proposed scheme in order to use as input thehorizontal global irradiance.

For horizontal global irradiance our model reads:

GH 5 Gn exp(2 ktc2H) (3)

wherecH denotes the angular distance between the normal direction to the horizontal plane andthe sun’s position, that is,cH reduces to the solar zenith angleus.


Fig. 4. Normalised global irradiance on solar zenith plane vs. quadratic angular distance to the sun’s position, fordifferent clearness index intervals.

From Eq. (3) and extending Eq. (1) to the whole hemisphere, we can obtain:

Gc 5 GH exp(2 kt(c2 2 c2H)) (4)

wherec andcH are expressed in radians.This simple equation enables us to calculate the global irradiance distribution using as inputs

the horizontal global irradiance and the solar position.

4. Performance assessment

At first, we tested the model against our Granada data base. Considering that the model develop-ment has been carried out using only data in the solar zenith plane, which represents about 8%of the data base, this part of the data base has not been used in the testing of the model.

Fig. 5 shows the scatter plot of calculated (Eq. (4)) versus measured global irradiance for caseswith the sun’s elevations in the range 60–80° at Granada. We must take into account that thissubset includes data for different ranges of clearness index and the fact that we represent experi-mental instantaneous values.

Nevertheless, it must be pointed out that in the present formulation the model does not allow


Fig. 5. Global irradiance: measured vs. calculated values from Eq. (4); 60–80° sun elevation range.

for the effect of ground reflected radiation. In this sense, it could be worthy to include a factorthat considers the effect of anisotropic reflections. The multiplying factor that we propose is amodified version of that proposed by Temps and Coulson [34]:

Fc 5 1 1 r sin2(c/2) (5)

wherer is the albedo of the underlying surface. In our case, for uncoloured concrete,r 5 0.35[31]. Finally, our model reads:

Gc 5 GH exp(2 kt(c2 2 c2H))Fc (6)

Taking into account the expression forFc, the correction is stronger for surfaces at 180° azimuthangle from the sun, that present the greater contribution of ground reflected radiation. The use ofthe anisotropic reflection factor (Eq. (6)) provides an improvement over the results shown in Fig.5, as we can see in Fig. 6.

Fig. 7 shows the scatter plot of measured versus calculated global irradiance for the whole database (excluding data in the solar zenith plane). In Table 1 we show the model’s statistical resultsfor six different solar elevation ranges, wherea is the slope,b the intercept andr the correlationcoefficient of the experimental versus calculated values [35]. The correlation coefficient gives anevaluation of the experimental data variance explained by the model, while the other two provideinformation about the tendency to over- or under-estimate in a particular range. Moreover, themodel performance was evaluated using the root mean square deviation (RMSD) and the mean


Fig. 6. Global irradiance: measured vs. calculated values from Eq. (4) taking into account the effect of ground aniso-tropic reflection (Fc); 60–80° sun elevation range.

bias deviation (MBD) [35]. These statistics allow for the detection of differences between theexperimental data and the model estimates and the existence of systematic over- or under-esti-mation tendencies, respectively. The slopes and the intercept variability can be explained becauseof the influence of almost vertical surfaces. These surfaces present anisotropic ground reflectanceeffects. This fact explains the great accumulation of values in the lowest part of the plot in Figs.6 and 7. We must point out that the terrace of the Science Faculty presents a horizon obstructedby buildings, which reaches in the worse case elevations of 20° (over the terrace level), especiallyin the north and west directions. Furthermore, most of the surrounding buildings are painted white.Because of this situation, we find great changes for some surfaces when the solar elevation islower than 40°. In these cases, the reflection from the buildings around is very important, andwe have not introduced these effects on the model. The departure of the slope from the idealvalue in the 10–20° category can be explained by this fact.

We must remember that the global irradiance values correspond to instantaneous measurements.On the other hand, although the solar azimuth angle changes during the experiment, we use inEq. (4) a mean value for all the experimental points in each scan. Moreover, as already mentioned,our experimental measurements recorded in each scan correspond to an interval of 45 min. Duringthis time, the global irradiance variation can reach 20% for mean solar elevations. These circum-stances can explain the RMSD between model and experimental data, which increases when thesolar elevation decreases. Nevertheless, as suggested by Table 1 and Figs. 6 and 7, the model


Fig. 7. Global irradiance: measured vs. calculated values for our data set, taking into account the effect of groundanisotropic reflection (Fc).

Table 1Statistical model results for different solar elevations in Granada data set; wherea is the slope,b the intercept,r thecorrelation coefficient of the experimental vs. calculated values, MBD the mean bias deviation and RMSD the rootmean square deviation

Solar elevation a b r MBD RMSD(Wm−2) (%) (%)

10–20° 0.888 19.5 0.887 2 1.2 38.020–30° 1.009 2 1.0 0.936 0.6 26.930–40° 1.104 2 20.9 0.969 5.3 17.840–50° 1.041 2 22.5 0.963 0.7 16.050–60° 1.081 2 53.0 0.972 2 2.1 13.060–80° 0.998 9.3 0.982 1.3 8.1

Complete data set 1.027 10.1 0.966 0.2 17.8


provides a good estimation for solar elevations greater than 40°. The relatively marked overestim-ation in 30–40° may be explained as a result of the shorter number of cases in this category,which corresponds to afternoon series.

As mentioned above, we have also carried out a second study involving the Skyscan’834 dataset [26,27] to gain general applicability of the model. We have also considered the Perez model[3] in parallel testing with our model, in order to have a comparison with one of the most reliableexisting methodologies.

It is important to point out that our model can be used with either instantaneous measurementsof global horizontal irradiance or any kind of averaged values. In contrast, the Perez model needshourly values. If we don’t know the direct and diffuse horizontal irradiance values, they have tobe calculated by means of the model described in [36]. In order for the Perez method to calculatethe direct and diffuse irradiance on a horizontal surface, knowledge of the hourly global irradiancein the previous, next and actual hour of the considered period is required. As the horizontal globalirradiance has to be the only radiometric input, several previous steps had to be carried out. Usingthe Perez model to obtain the direct irradiance and the diffuse irradiance on a horizontal surface[36], we had to compute both magnitudes employing Skyscan’834, even though both magnitudeswere available in the data set. We also had to carry out the proper averaging process in theSkyscan’834 data set that the Perez model demands. After this it was possible to compute thedirect irradiance projection and the diffuse irradiance on a tilted surface, the latter by means ofthe method described in [3], to get the global irradiance on a tilted surface by adding the othertwo and having the same initial conditions for both models, that is, global horizontal irradianceas the only input.

In Figs. 8 and 9, we can observe the performance of the Perez model and our model forGc

hourly values, respectively. We have not used the factorFc when testing the model against theSkyscan’834 data set, because of the very well shielded pyranometer as mentioned above. As thedata set covers a full year period, a variety of solar elevation angles as well as cloudy and turbidityconditions are included. Just one thing has to be pointed out before we analyse the statistics forboth models. The Skyscan’834 data set is composed of 5845 instantaneous measurements for thesloping global irradiance, but they turn into 1029 after the averaging process.

Having this in mind, in Table 2 we show the statistical results of the regression analyses forboth models. As can be observed in Table 2, together with Figs. 8 and 9, our model offers anover-estimation of 22.6 W/m2 (4.8% of the averaged measured slope irradiance) in contrast to anunder-estimation of the Perez model of only 3.9%. The RMSD follows a similar behaviour, being1% greater for our model than for the Perez model. But considering the statistic as a whole, ourmodel gives good results as well, with a slope close to 1 and a general performance that, togetherwith its simple formulation and input requirements, makes it quite useful and reliable.

If we test our model against the instantaneous measurements, we find similar results (Fig. 10).Considering the whole data set (all types of skies), the results show a general over-estimation of5.2%, a correlation coefficient greater than 0.99 and a slope close to 1. Thus, we can concludethat our model offers a reliable tool for use when solar radiation data are scarce or limited toglobal horizontal irradiance. Additionally, a fast and simple estimation is often necessary andpreferred. Compared with the Perez model, our model gives a similar performance but has theadvantage of its simpler formulation.


Fig. 8. The Perez et al. model for the estimation of global slope irradiance vs. Skyscan’834 hourly values for allweather conditions.

5. Conclusions

As our study shows, it is possible to obtain the global irradiance angular distribution fromknowledge of the global horizontal irradiance and the astronomic parameters, by means of a modelthat has been proposed for our latitude. This model depends on local atmospheric conditions bymeans ofkt, and avoids the classical partitioning of the global solar irradiance into direct anddiffuse components. This fact gives a general applicability character to the model, which couldbe used in sites where only horizontal global irradiance is measured.

This model has been developed taking into account the exponential pattern shown by the globalirradiance angular distribution and its symmetry using polar diagrams. Our model takes intoaccount the ground reflection effects by means of an anisotropic factor, although the effects ofan artificial horizon are not taken into account. This provides a good estimation for instantaneousglobal irradiance values on inclined surfaces.

In order to establish its applicability, the model had to be validated against other experimentaldata sets. To this end, the Skyscan’834 data set has been used. This is a well-known data base,which includes information of slope irradiance under different types of cloud cover and turbidityconditions. The results corroborate our predictions and show that when used with either instan-taneous or averaged measurements, the new model performs well.

Owing to the appropriate performance of the model for different solar elevations and inclined


Fig. 9. The new model for the estimation of the global slope irradiance vs. Skyscan’834 hourly values for allweather conditions.

Table 2Statistical results obtained from testing the Perez model and the new model against Skyscan’834 hourly and instan-taneous values

Model a b r MBD RMSD(W/m2) (%) (%)

Hourly valuesPerez model 0.975 2 6.0 0.994 2 3.9 8.3New model 1.01 19.5 0.993 4.8 9.3

Instantaneous valuesNew model 1.00 22.7 0.993 5.2 10.1

surfaces we deem that it could be a good tool for the study of hourly and daily values of solarirradiance on inclined surfaces, using the horizontal global irradiance as unique input. The esti-mations provided by the model can be used for the estimation of the energy balance, in technologi-cal applications or in local and large-scale climate studies. On the other hand, this work representsa previous stage in the study of topographic effects on the solar irradiance field, with the goal ofmapping solar radiation on local and large scales.


Fig. 10. The new model for the estimation of the global slope irradiance vs. Skyscan’834 instantaneous values forall weather conditions.

Acknowledgements

This work was supported by La Direccio´n General de Ciencia y Tecnologı´a from the Educationand Research Spanish Ministry through the project No. CLI98-0912-C02-01. We would like tothank sincerely Dr. Alfred Brunger for lending us the Skyscan’834 data set and for the permissionto use it. We are also very grateful to Dr. Richard Perez for sending us the Perez model code.

References

[1] Hay JE, McKay DC. Estimating solar radiance on inclined surfaces: a review and assessment of methodologies.Int. J. Solar Energy 1985;3:203–40.

[2] Skartveit A, Olseth JA. Modelling slope irradiance at height latitudes. Solar Energy 1986;36:333–44.[3] Perez R, Ineichen P, Seals R, Michalsky J, Stewart R. Modelling daylight availability and irradiance components

from direct and global irradiance. Solar Energy 1990;44:271–89.[4] Burlon R, Bivona S, Leone C. Instantaneous hourly and daily radiation on tilted surfaces. Solar Energy

1991;47(2):83–9.[5] Gopinathan KK. Solar radiation on variously oriented sloping surfaces. Solar Energy 1991;47(3):173–9.[6] Feuermann D, Zemel A. Validation of models for global irradiance on inclined planes. Solar Energy

1992;48(1):59–66.[7] Klein SA. Calculation of monthly average insolation of tilted surfaces. Solar Energy 1977;19:325–9.


[8] Butera F, Fiesta R, Ratto CF. Calculation of the monthly average of hourly and daily beam insolation on tiltedsurfaces. Solar Energy 1982;28:547–50.

[9] Zelenka A. Asymmetrical analytically weightedRb factors. Solar Energy 1988;41:405–15.[10] Reindl DT, Beckman WA, Duffie JA. Evaluation of hourly tilted surface radiation models. Solar Energy

1990;45:1–7.[11] Davies JA, McKay DC. Evaluation of selected models for estimating solar radiation on horizontal surfaces. Solar

Energy 1989;43:153–68.[12] Reindl DT, Beckman WA, Duffie JA. Evaluation of hourly tilted surface radiation models. Solar Energy

1990;45:9–17.[13] Utrillas MP, Martınez-Lozano JA, Casanovas AJ. Evaluation of models for estimating solar irradiation on vertical

surfaces at Valencia, Spain. Solar Energy 1991;47:223–9.[14] Kambezidis HD, Psiloglou BE, Gueymard C. Measurements and models for total solar irradiance on inclined

surfaces in Athens. Solar Energy 1994;53:177–85.[15] Olivier HR. Studies of surface energy balance of sloping terrain. Int. J. Climatol. 1992;12:55–68.[16] Pinker RT, Laszlo I. Modelling surface solar irradiance for satellite applications on a global scale. J. Appl. Meteo-

rol. 1992;31:194–211.[17] Pinker RT, Frouin R. A review of satellite methods to derive surface shortwave irradiance. Remote Sens. Environ.

1995;51:108–24.[18] Olmo FJ, Pozo D, Pareja R, Alados-Arboledas L. Estimating surface photosynthetically active radiation from

Meteosat data. In: Proceedings of the 1996 Meteorological Satellite Data Users’ Conference. EUMETSAT, Vienna,Austria, 16–20 September, 1996:459–63.

[19] Olmo FJ, Foyo I, Vida J, Pareja R, Alados-Arboledas L. Obtencioń de la irradiancia global y de la irradianciafotosinteticamente activa a partir de ima´genes Meteosat. In: Hernańdez C, Arias JE, editors. TeledetecciońAplicada a la Gestioń de Recursos Naturales y Medio Litoral Marino. Santiago de Compostela, Spain, 1997:273–6.

[20] Fu H, Tajchman SJ, Kochenderfer JN. Topography and radiation exchange of a mountainous watershed. J. Appl.Meteorol. 1995;34:890–901.

[21] Dubayah R, Loechel S. Modelling topographic solar radiation using GOES data. J. Appl. Meteorol.1977;36:141–54.

[22] Erbs DG, Klein SA, Duffie JA. Estimation of the diffuse radiation fraction for hourly, daily and monthly-averageglobal radiation. Solar Energy 1982;28:293–302.

[23] Olmo FJ, Batlles J, Alados-Arboledas L. Performance of global to direct/diffuse decomposition models beforeand after the eruption of Mt. Pinatubo, June 1991. Solar Energy 1996;57:433–43.

[24] Castro-Dıéz Y, Jimenez JI. Solar radiation upon slopes: an experimental study. In: Coleman MJ, editor. Proceed-ings of the 1988 Annual Meeting, ASES, 1988. Cambridge: EEUU, 1988:250–6.

[25] Jimenez JI, Castro-Dıéz Y, Vida J, Foyo-Moreno I. On the performance of the Helios model for estimating theangular distribution fluxes. In: Sayigh A, editor. Proceedings of the 1st WREC, 1990, Reading, UK, 1990:3127–31.

[26] Brunger AP. The magnitude, variability and angular characteristics of the shortwave sky radiance. PhD Thesis,Departament of Mechanical Engineering, University of Toronto, 1987.

[27] Brunger AP, Hooper FC. Measurements shortwave sky radiance in an urban atmosphere. Solar Energy1991;47(2):137–42.

[28] Orgill JF, Hollands KGT. Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy1977;19:357–9.

[29] Vazquez M, Ruiz V, Perez R. The roles of scattering, absorption, and air mass on the diffuse-to-global correlations.Solar Energy 1991;47:181–8.

[30] Ripley BP. Statistics spatial. New York: John Wiley, 1981.[31] Iqbal M. An introduction to solar radiation. New York: Academic Press, 1983.[32] Olmo FJ, Vida J, Castro Y, Alados L, Jimeńez JI, Valko P. Contribution of the graphical analysis of the sky

radiance and angular distribution of the global irradiance. In: Burley S, Coleman MJ, editors. Proceedings of the1990 Annual Conference, ASES, Solar’90, Austin, Texas, 1990:461–73.

[33] Olmo FJ, Vida J, Foyo I, Jimeńez JI. A general new relationship between solar irradiance and sun height. In:Sayigh A, editor. Proceedings of the Renewable Energy Technology and the Environment, WREC-92, Reading,UK, 1992:2760–4.


[34] Temps RC, Coulson KL. Solar radiation incident upon slopes of different orientations. Solar Energy1977;19:179–84.

[35] IEA. Solar R&D, Handbook on methods of estimating solar radiation. Task V, Subtask B, Stockholm, Sweden,1984:33–41.

[36] Perez R, Ineichen P, Maxwell E, Seals R, Zelenka A. Dynamic global-to-direct irradiance conversion models.ASHRAE Trans. 1991;3578(RP–644):354–69.

prediction of global irradiance on inclined surfaces from ...hera.ugr.es/doi/1501518x.pdfprediction...

Documents