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

Prediction of global irradiance on inclined surfaces from horizontal global irradiance F.J. Olmo, J. Vida, I. Foyo, Y. Castro-Diez, L. Alados-Arboledas* Dpto. Fı´sica 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. Experimental data of global irradiance on inclined planes recorded at Granada (Spain, 37.08°N, 3.57°W) have been used in order to study the pattern of the angular distribution of global irradiance. We have modelled the global irradiance angular distribution, employing horizontal global irradiance as the only radiometric input, 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. The Skyscan’834 data set has also been used in order to test the model under completely different conditions from those in Granada, with respect to the amount of cloud, local peculiarities, experimental design and instrumentation. 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 global horizontal irradiance. 1999 Elsevier Science Ltd. All rights reserved.

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

* Corresponding author. Tel.: ⫹ 34-958-244024; fax: ⫹ 34-958-243214; e-mail: [email protected] 0360-5442/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 9 ) 0 0 0 2 5 - 0

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been developed, such as Refs. [1–6]. Nevertheless, calculations using these kinds of models are not simple, since they require information of the direct or diffuse irradiance. This information can 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 oriented surface suggests that the parameters involved in these models must depend on the atmospheric state, 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 et al. model [3] with empirical correlations to estimate the direct component from horizontal global irradiance, can yield accurate predictions for locations where only global horizontal data are available. There has been growing interest in recent decades about the use of satellite data to obtain solar irradiance at a surface. This method provides the possibility of continuous and global monitoring of this radiation flux [16–19]. Although this technique seems powerful, the use of remote sensing data in the case of complex topography terrains requires additional efforts [20,21]. The algorithms that deal with the influence of complex topography in the solar irradiance field require the consideration of solar irradiance impinging on non-horizontal surfaces. This information must be acquired 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’s elevation and azimuth as input parameters. The added value of this model lies in its applicability to sites where only horizontal global irradiance is measured, as is the case at most conventional meteorological stations, or satellite-derived global irradiance data. In addition, this method allows for the estimation of the global irradiance distribution by means of the horizontal global irradiance in 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 diffuse irradiance from the horizontal global irradiance. This work represents a previous stage in the study of topographic effects on the solar irradiance field, with the goal of mapping solar radiation on local and large scales. At present we only intend to test the validity of the inclined global irradiance model that we propose in contrast to other type 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 direct irradiance, respectively [23]. 2. Data base Measurements of incoming global irradiance on surfaces with different slopes and orientations have been carried out at Granada (Spain). The measurement system consists of a pyranometer Kipp-Zonen CM-5, mounted on a device with the ability to vary both the elevation (0, the horizontal, to 90°, at 15° intervals) and azimuth (0, the south meridian, to 360°, at 45° intervals) of the inclined surface. This system has been installed on the terrace of the Science Faculty of the University of Granada (650 m a.s.l.) [24,25]. The stability of the radiometer calibration factor has been periodically tested against a reference pyranometer used as substandard and not exposed to the sun except during the intercomparison trials.

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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 in Fig. 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 the day distributed uniformly around solar noon. Thus, a complete range of solar azimuth (from 247.5° 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 (clear skies only) and possible local dependencies from Granada in our results, we have used the Skyscan’834 data set [26,27]. The measurement system consists of an Eppley Precision Spectral Pyranometer (PSP) installed on the Mechanical Engineering Building rooftop (University of Toronto, 43.7°N, 79.4°W, 111 m a.s.l.). The pyranometer was calibrated by the National Atmospheric Radiation Centre in July 1982. This is a well-known data set, widely used and referenced in this type of works. Skyscan’834 covers a full year of measurements in a completely different environment to that of our trials in Granada. The main limitation is that the Skyscan’834 data set only contains slope irradiance measurements for the case of surfaces oriented to the south with an elevation angle of 44°. On the other hand, the system has the advantage that the pyranometer is shielded from the radiation reflected from the ground [26]. We have characterised the different atmospheric conditions by means of the clearness index, kt, defined as the global horizontal irradiance to extraterrestrial horizontal irradiance ratio. kt ranges from 0.4 to 0.8 in Granada and from 0 to 1 in the Skyscan’834 data set. In the latter, we observe certain values for which an increase of the diffuse fraction comes together with an increase of kt. This fact is associated with partly cloudy conditions with some clouds located near the sun. Under these circumstances, cloud borders reflect the direct irradiance during periods when the sun is unshaded by the surrounding clouds, thus enhancing both global and diffuse irradiance, and these situations are associated with changing sky conditions [28,29]. 3. Model formulation In order to represent graphically the measurements of global irradiance obtained, we have used two types of diagram: polar and three dimensional. Fig. 1 presents a three-dimensional and polar global irradiance diagram for one of our experiments. In the figure caption, specific features of this experimental series are explained. In short, this diagram facilitates the visual information of each experience. Both diagrams have been obtained from experimental values, using the kriging interpolation method [30]. The complete Granada data set has been analysed by means of this type of diagram. In this sense, Fig. 1 can be considered as an example of the general performance of the irradiance distribution. These diagrams show the existence of a dependence between global irradiance values and the angular distance subtended by the normal to the inclined plane and the sun (). To evidence this correlation we have restricted our analysis to the solar zenith plane. This is the plane that contains the zenith and the sun’s position (Fig. 2). Fig. 3 shows the global irradiance values in the solar zenith plane (Gsz) versus the angular distance to the sun’s position (sz), which, for this plane, is the difference between the surface zenith angle and the sun’s zenith angle. In this way, we eliminate the azimuth dependence. In this figure, we have plotted only the values that

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

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Fig. 2. Sky dome showing the solar zenith plane geometry: sz, angular distance, in the zenith plane, of the normal to the considered surface (n⬘) with respect to the sun’s position; , angular distance, for an arbitrary plane, of the normal to the considered surface (n) with respect to the sun’s position; and s, zenith angles for the normal (n⬘) and the sun, respectively; ␣ and ␣s, azimuth angles for the normal (n⬘) and the sun, respectively.

correspond to the intervals 60–80 and 20–30° of solar elevation in the Granada data set. It is well known that direct irradiance on an inclined plane can be written as direct irradiance on a horizontal surface multiplied by a geometrical factor [31]. This geometrical factor is a ratio of cosines. In order to investigate if a similar ratio could be applied when dealing with global irradiance let us refer to Fig. 3. In this figure, in which Gn is the global irradiance in the surface normal to the sun beam, we investigate the ratio Gzs/Gn, which suggests a dependence in terms of angular distance to the sun’s position (sz). An additional analysis shown in Fig. 4 indicates that this ratio depends also on the clearness index, kt (global to extraterrestrial horizontal irradiance ratio). This is especially true for inclined planes far from the sun’s position (greater sz). In our case, we have taken the global irradiance at normal incidence to the sun, Gn, as maximum (Fig. 3). Then we have modelled the global irradiance distribution in the solar zenith plane in terms of this maximum value and an exponential function that includes the clearness index, kt, and the angular distance to the sun’s position. The exponential function is adopted by convenience, as it will be explained later, even though we could also model the global irradiance in the solar zenith plane as a function of cosines with similar behaviours. In this procedure, we have only used 8% of the data set, i.e. data lying exactly in the solar zenith plane. Thus, we propose the following expression: Gzs ⫽ Gn exp( ⫺ kt2zs)

(1)

where zs is expressed in radians. The kt 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

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

for inclined and oriented surfaces have been studied. It has been shown that the influence of turbidity 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 geometrically this procedure to the entire hemisphere (by means of the experimental function adopted). For this purpose the angular distance can be evaluated as follows: cos ⫽ sin sin s ⫹ cos cos s cos(␣s ⫺ ␣)

(2)

where represents the zenith angle and ␣ the azimuth. The subscript s refers to the sun’s position. We should point out that the angular distance is the so-called scattering angle. The scheme developed provides a good representation of the global irradiance distribution on the complete hemisphere. Nevertheless, the global irradiance on a surface normal to the sun is not measured in most radiometric networks. Therefore, considering that horizontal global irradiance is the usually available term, we have modified the proposed scheme in order to use as input the horizontal global irradiance. For horizontal global irradiance our model reads: GH ⫽ Gn exp( ⫺ kt2H)

(3)

where H denotes the angular distance between the normal direction to the horizontal plane and the sun’s position, that is, H reduces to the solar zenith angle s.

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

From Eq. (3) and extending Eq. (1) to the whole hemisphere, we can obtain: G ⫽ GH exp( ⫺ kt(2 ⫺ 2H))

(4)

where and H 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 development 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 cases with the sun’s elevations in the range 60–80° at Granada. We must take into account that this subset includes data for different ranges of clearness index and the fact that we represent experimental instantaneous values. Nevertheless, it must be pointed out that in the present formulation the model does not allow

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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 factor that considers the effect of anisotropic reflections. The multiplying factor that we propose is a modified version of that proposed by Temps and Coulson [34]: Fc ⫽ 1 ⫹ sin2(/2)

(5)

where is the albedo of the underlying surface. In our case, for uncoloured concrete, ⫽ 0.35 [31]. Finally, our model reads: G ⫽ GH exp( ⫺ kt(2 ⫺ 2H))Fc

(6)

Taking into account the expression for Fc, the correction is stronger for surfaces at 180° azimuth angle from the sun, that present the greater contribution of ground reflected radiation. The use of the 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 data base (excluding data in the solar zenith plane). In Table 1 we show the model’s statistical results for six different solar elevation ranges, where a is the slope, b the intercept and r the correlation coefficient of the experimental versus calculated values [35]. The correlation coefficient gives an evaluation of the experimental data variance explained by the model, while the other two provide information about the tendency to over- or under-estimate in a particular range. Moreover, the model performance was evaluated using the root mean square deviation (RMSD) and the mean

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

bias deviation (MBD) [35]. These statistics allow for the detection of differences between the experimental data and the model estimates and the existence of systematic over- or under-estimation tendencies, respectively. The slopes and the intercept variability can be explained because of the influence of almost vertical surfaces. These surfaces present anisotropic ground reflectance effects. 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 obstructed by buildings, which reaches in the worse case elevations of 20° (over the terrace level), especially in 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 is lower than 40°. In these cases, the reflection from the buildings around is very important, and we have not introduced these effects on the model. The departure of the slope from the ideal value 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 in Eq. (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. During this time, the global irradiance variation can reach 20% for mean solar elevations. These circumstances can explain the RMSD between model and experimental data, which increases when the solar elevation decreases. Nevertheless, as suggested by Table 1 and Figs. 6 and 7, the model

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

Table 1 Statistical model results for different solar elevations in Granada data set; where a is the slope, b the intercept, r the correlation coefficient of the experimental vs. calculated values, MBD the mean bias deviation and RMSD the root mean square deviation Solar elevation

a

b (Wm−2)

r

MBD (%)

10–20° 20–30° 30–40° 40–50° 50–60° 60–80°

0.888 1.009 1.104 1.041 1.081 0.998

19.5 ⫺ 1.0 ⫺ 20.9 ⫺ 22.5 ⫺ 53.0 9.3

0.887 0.936 0.969 0.963 0.972 0.982

⫺ 1.2 0.6 5.3 0.7 ⫺ 2.1 1.3

38.0 26.9 17.8 16.0 13.0 8.1

10.1

0.966

0.2

17.8

Complete data set 1.027

RMSD (%)

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provides a good estimation for solar elevations greater than 40°. The relatively marked overestimation 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 data set [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 reliable existing methodologies. It is important to point out that our model can be used with either instantaneous measurements of global horizontal irradiance or any kind of averaged values. In contrast, the Perez model needs hourly values. If we don’t know the direct and diffuse horizontal irradiance values, they have to be calculated by means of the model described in [36]. In order for the Perez method to calculate the direct and diffuse irradiance on a horizontal surface, knowledge of the hourly global irradiance in the previous, next and actual hour of the considered period is required. As the horizontal global irradiance has to be the only radiometric input, several previous steps had to be carried out. Using the 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 magnitudes were available in the data set. We also had to carry out the proper averaging process in the Skyscan’834 data set that the Perez model demands. After this it was possible to compute the direct irradiance projection and the diffuse irradiance on a tilted surface, the latter by means of the method described in [3], to get the global irradiance on a tilted surface by adding the other two and having the same initial conditions for both models, that is, global horizontal irradiance as the only input. In Figs. 8 and 9, we can observe the performance of the Perez model and our model for G hourly values, respectively. We have not used the factor Fc when testing the model against the Skyscan’834 data set, because of the very well shielded pyranometer as mentioned above. As the data set covers a full year period, a variety of solar elevation angles as well as cloudy and turbidity conditions are included. Just one thing has to be pointed out before we analyse the statistics for both models. The Skyscan’834 data set is composed of 5845 instantaneous measurements for the sloping 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 for both models. As can be observed in Table 2, together with Figs. 8 and 9, our model offers an over-estimation of 22.6 W/m2 (4.8% of the averaged measured slope irradiance) in contrast to an under-estimation of the Perez model of only 3.9%. The RMSD follows a similar behaviour, being 1% greater for our model than for the Perez model. But considering the statistic as a whole, our model gives good results as well, with a slope close to 1 and a general performance that, together with 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 of 5.2%, a correlation coefficient greater than 0.99 and a slope close to 1. Thus, we can conclude that our model offers a reliable tool for use when solar radiation data are scarce or limited to global horizontal irradiance. Additionally, a fast and simple estimation is often necessary and preferred. Compared with the Perez model, our model gives a similar performance but has the advantage of its simpler formulation.

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

5. Conclusions As our study shows, it is possible to obtain the global irradiance angular distribution from knowledge of the global horizontal irradiance and the astronomic parameters, by means of a model that has been proposed for our latitude. This model depends on local atmospheric conditions by means of kt, and avoids the classical partitioning of the global solar irradiance into direct and diffuse components. This fact gives a general applicability character to the model, which could be used in sites where only horizontal global irradiance is measured. This model has been developed taking into account the exponential pattern shown by the global irradiance angular distribution and its symmetry using polar diagrams. Our model takes into account the ground reflection effects by means of an anisotropic factor, although the effects of an artificial horizon are not taken into account. This provides a good estimation for instantaneous global irradiance values on inclined surfaces. In order to establish its applicability, the model had to be validated against other experimental data 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 turbidity conditions. The results corroborate our predictions and show that when used with either instantaneous or averaged measurements, the new model performs well. Owing to the appropriate performance of the model for different solar elevations and inclined

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Fig. 9. The new model for the estimation of the global slope irradiance vs. Skyscan’834 hourly values for all weather conditions. Table 2 Statistical results obtained from testing the Perez model and the new model against Skyscan’834 hourly and instantaneous values Model

a

b (W/m2)

r

MBD (%)

RMSD (%)

Hourly values Perez model New model

0.975 1.01

⫺ 6.0 19.5

0.994 0.993

⫺ 3.9 4.8

8.3 9.3

Instantaneous values New 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 solar irradiance on inclined surfaces, using the horizontal global irradiance as unique input. The estimations provided by the model can be used for the estimation of the energy balance, in technological applications or in local and large-scale climate studies. On the other hand, this work represents a previous stage in the study of topographic effects on the solar irradiance field, with the goal of mapping solar radiation on local and large scales.

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

Acknowledgements This work was supported by La Direccio´n General de Ciencia y Tecnologı´a from the Education and Research Spanish Ministry through the project No. CLI98-0912-C02-01. We would like to thank sincerely Dr. Alfred Brunger for lending us the Skyscan’834 data set and for the permission to 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.

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Prediction of global irradiance on inclined surfaces from horizontal global irradiance F.J. Olmo, J. Vida, I. Foyo, Y. Castro-Diez, L. Alados-Arboledas* Dpto. Fı´sica 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. Experimental data of global irradiance on inclined planes recorded at Granada (Spain, 37.08°N, 3.57°W) have been used in order to study the pattern of the angular distribution of global irradiance. We have modelled the global irradiance angular distribution, employing horizontal global irradiance as the only radiometric input, 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. The Skyscan’834 data set has also been used in order to test the model under completely different conditions from those in Granada, with respect to the amount of cloud, local peculiarities, experimental design and instrumentation. 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 global horizontal irradiance. 1999 Elsevier Science Ltd. All rights reserved.

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

* Corresponding author. Tel.: ⫹ 34-958-244024; fax: ⫹ 34-958-243214; e-mail: [email protected] 0360-5442/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 5 4 4 2 ( 9 9 ) 0 0 0 2 5 - 0

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been developed, such as Refs. [1–6]. Nevertheless, calculations using these kinds of models are not simple, since they require information of the direct or diffuse irradiance. This information can 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 oriented surface suggests that the parameters involved in these models must depend on the atmospheric state, 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 et al. model [3] with empirical correlations to estimate the direct component from horizontal global irradiance, can yield accurate predictions for locations where only global horizontal data are available. There has been growing interest in recent decades about the use of satellite data to obtain solar irradiance at a surface. This method provides the possibility of continuous and global monitoring of this radiation flux [16–19]. Although this technique seems powerful, the use of remote sensing data in the case of complex topography terrains requires additional efforts [20,21]. The algorithms that deal with the influence of complex topography in the solar irradiance field require the consideration of solar irradiance impinging on non-horizontal surfaces. This information must be acquired 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’s elevation and azimuth as input parameters. The added value of this model lies in its applicability to sites where only horizontal global irradiance is measured, as is the case at most conventional meteorological stations, or satellite-derived global irradiance data. In addition, this method allows for the estimation of the global irradiance distribution by means of the horizontal global irradiance in 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 diffuse irradiance from the horizontal global irradiance. This work represents a previous stage in the study of topographic effects on the solar irradiance field, with the goal of mapping solar radiation on local and large scales. At present we only intend to test the validity of the inclined global irradiance model that we propose in contrast to other type 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 direct irradiance, respectively [23]. 2. Data base Measurements of incoming global irradiance on surfaces with different slopes and orientations have been carried out at Granada (Spain). The measurement system consists of a pyranometer Kipp-Zonen CM-5, mounted on a device with the ability to vary both the elevation (0, the horizontal, to 90°, at 15° intervals) and azimuth (0, the south meridian, to 360°, at 45° intervals) of the inclined surface. This system has been installed on the terrace of the Science Faculty of the University of Granada (650 m a.s.l.) [24,25]. The stability of the radiometer calibration factor has been periodically tested against a reference pyranometer used as substandard and not exposed to the sun except during the intercomparison trials.

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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 in Fig. 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 the day distributed uniformly around solar noon. Thus, a complete range of solar azimuth (from 247.5° 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 (clear skies only) and possible local dependencies from Granada in our results, we have used the Skyscan’834 data set [26,27]. The measurement system consists of an Eppley Precision Spectral Pyranometer (PSP) installed on the Mechanical Engineering Building rooftop (University of Toronto, 43.7°N, 79.4°W, 111 m a.s.l.). The pyranometer was calibrated by the National Atmospheric Radiation Centre in July 1982. This is a well-known data set, widely used and referenced in this type of works. Skyscan’834 covers a full year of measurements in a completely different environment to that of our trials in Granada. The main limitation is that the Skyscan’834 data set only contains slope irradiance measurements for the case of surfaces oriented to the south with an elevation angle of 44°. On the other hand, the system has the advantage that the pyranometer is shielded from the radiation reflected from the ground [26]. We have characterised the different atmospheric conditions by means of the clearness index, kt, defined as the global horizontal irradiance to extraterrestrial horizontal irradiance ratio. kt ranges from 0.4 to 0.8 in Granada and from 0 to 1 in the Skyscan’834 data set. In the latter, we observe certain values for which an increase of the diffuse fraction comes together with an increase of kt. This fact is associated with partly cloudy conditions with some clouds located near the sun. Under these circumstances, cloud borders reflect the direct irradiance during periods when the sun is unshaded by the surrounding clouds, thus enhancing both global and diffuse irradiance, and these situations are associated with changing sky conditions [28,29]. 3. Model formulation In order to represent graphically the measurements of global irradiance obtained, we have used two types of diagram: polar and three dimensional. Fig. 1 presents a three-dimensional and polar global irradiance diagram for one of our experiments. In the figure caption, specific features of this experimental series are explained. In short, this diagram facilitates the visual information of each experience. Both diagrams have been obtained from experimental values, using the kriging interpolation method [30]. The complete Granada data set has been analysed by means of this type of diagram. In this sense, Fig. 1 can be considered as an example of the general performance of the irradiance distribution. These diagrams show the existence of a dependence between global irradiance values and the angular distance subtended by the normal to the inclined plane and the sun (). To evidence this correlation we have restricted our analysis to the solar zenith plane. This is the plane that contains the zenith and the sun’s position (Fig. 2). Fig. 3 shows the global irradiance values in the solar zenith plane (Gsz) versus the angular distance to the sun’s position (sz), which, for this plane, is the difference between the surface zenith angle and the sun’s zenith angle. In this way, we eliminate the azimuth dependence. In this figure, we have plotted only the values that

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

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Fig. 2. Sky dome showing the solar zenith plane geometry: sz, angular distance, in the zenith plane, of the normal to the considered surface (n⬘) with respect to the sun’s position; , angular distance, for an arbitrary plane, of the normal to the considered surface (n) with respect to the sun’s position; and s, zenith angles for the normal (n⬘) and the sun, respectively; ␣ and ␣s, azimuth angles for the normal (n⬘) and the sun, respectively.

correspond to the intervals 60–80 and 20–30° of solar elevation in the Granada data set. It is well known that direct irradiance on an inclined plane can be written as direct irradiance on a horizontal surface multiplied by a geometrical factor [31]. This geometrical factor is a ratio of cosines. In order to investigate if a similar ratio could be applied when dealing with global irradiance let us refer to Fig. 3. In this figure, in which Gn is the global irradiance in the surface normal to the sun beam, we investigate the ratio Gzs/Gn, which suggests a dependence in terms of angular distance to the sun’s position (sz). An additional analysis shown in Fig. 4 indicates that this ratio depends also on the clearness index, kt (global to extraterrestrial horizontal irradiance ratio). This is especially true for inclined planes far from the sun’s position (greater sz). In our case, we have taken the global irradiance at normal incidence to the sun, Gn, as maximum (Fig. 3). Then we have modelled the global irradiance distribution in the solar zenith plane in terms of this maximum value and an exponential function that includes the clearness index, kt, and the angular distance to the sun’s position. The exponential function is adopted by convenience, as it will be explained later, even though we could also model the global irradiance in the solar zenith plane as a function of cosines with similar behaviours. In this procedure, we have only used 8% of the data set, i.e. data lying exactly in the solar zenith plane. Thus, we propose the following expression: Gzs ⫽ Gn exp( ⫺ kt2zs)

(1)

where zs is expressed in radians. The kt 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

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

for inclined and oriented surfaces have been studied. It has been shown that the influence of turbidity 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 geometrically this procedure to the entire hemisphere (by means of the experimental function adopted). For this purpose the angular distance can be evaluated as follows: cos ⫽ sin sin s ⫹ cos cos s cos(␣s ⫺ ␣)

(2)

where represents the zenith angle and ␣ the azimuth. The subscript s refers to the sun’s position. We should point out that the angular distance is the so-called scattering angle. The scheme developed provides a good representation of the global irradiance distribution on the complete hemisphere. Nevertheless, the global irradiance on a surface normal to the sun is not measured in most radiometric networks. Therefore, considering that horizontal global irradiance is the usually available term, we have modified the proposed scheme in order to use as input the horizontal global irradiance. For horizontal global irradiance our model reads: GH ⫽ Gn exp( ⫺ kt2H)

(3)

where H denotes the angular distance between the normal direction to the horizontal plane and the sun’s position, that is, H reduces to the solar zenith angle s.

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

From Eq. (3) and extending Eq. (1) to the whole hemisphere, we can obtain: G ⫽ GH exp( ⫺ kt(2 ⫺ 2H))

(4)

where and H 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 development 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 cases with the sun’s elevations in the range 60–80° at Granada. We must take into account that this subset includes data for different ranges of clearness index and the fact that we represent experimental instantaneous values. Nevertheless, it must be pointed out that in the present formulation the model does not allow

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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 factor that considers the effect of anisotropic reflections. The multiplying factor that we propose is a modified version of that proposed by Temps and Coulson [34]: Fc ⫽ 1 ⫹ sin2(/2)

(5)

where is the albedo of the underlying surface. In our case, for uncoloured concrete, ⫽ 0.35 [31]. Finally, our model reads: G ⫽ GH exp( ⫺ kt(2 ⫺ 2H))Fc

(6)

Taking into account the expression for Fc, the correction is stronger for surfaces at 180° azimuth angle from the sun, that present the greater contribution of ground reflected radiation. The use of the 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 data base (excluding data in the solar zenith plane). In Table 1 we show the model’s statistical results for six different solar elevation ranges, where a is the slope, b the intercept and r the correlation coefficient of the experimental versus calculated values [35]. The correlation coefficient gives an evaluation of the experimental data variance explained by the model, while the other two provide information about the tendency to over- or under-estimate in a particular range. Moreover, the model performance was evaluated using the root mean square deviation (RMSD) and the mean

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

bias deviation (MBD) [35]. These statistics allow for the detection of differences between the experimental data and the model estimates and the existence of systematic over- or under-estimation tendencies, respectively. The slopes and the intercept variability can be explained because of the influence of almost vertical surfaces. These surfaces present anisotropic ground reflectance effects. 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 obstructed by buildings, which reaches in the worse case elevations of 20° (over the terrace level), especially in 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 is lower than 40°. In these cases, the reflection from the buildings around is very important, and we have not introduced these effects on the model. The departure of the slope from the ideal value 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 in Eq. (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. During this time, the global irradiance variation can reach 20% for mean solar elevations. These circumstances can explain the RMSD between model and experimental data, which increases when the solar elevation decreases. Nevertheless, as suggested by Table 1 and Figs. 6 and 7, the model

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

Table 1 Statistical model results for different solar elevations in Granada data set; where a is the slope, b the intercept, r the correlation coefficient of the experimental vs. calculated values, MBD the mean bias deviation and RMSD the root mean square deviation Solar elevation

a

b (Wm−2)

r

MBD (%)

10–20° 20–30° 30–40° 40–50° 50–60° 60–80°

0.888 1.009 1.104 1.041 1.081 0.998

19.5 ⫺ 1.0 ⫺ 20.9 ⫺ 22.5 ⫺ 53.0 9.3

0.887 0.936 0.969 0.963 0.972 0.982

⫺ 1.2 0.6 5.3 0.7 ⫺ 2.1 1.3

38.0 26.9 17.8 16.0 13.0 8.1

10.1

0.966

0.2

17.8

Complete data set 1.027

RMSD (%)

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699

provides a good estimation for solar elevations greater than 40°. The relatively marked overestimation 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 data set [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 reliable existing methodologies. It is important to point out that our model can be used with either instantaneous measurements of global horizontal irradiance or any kind of averaged values. In contrast, the Perez model needs hourly values. If we don’t know the direct and diffuse horizontal irradiance values, they have to be calculated by means of the model described in [36]. In order for the Perez method to calculate the direct and diffuse irradiance on a horizontal surface, knowledge of the hourly global irradiance in the previous, next and actual hour of the considered period is required. As the horizontal global irradiance has to be the only radiometric input, several previous steps had to be carried out. Using the 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 magnitudes were available in the data set. We also had to carry out the proper averaging process in the Skyscan’834 data set that the Perez model demands. After this it was possible to compute the direct irradiance projection and the diffuse irradiance on a tilted surface, the latter by means of the method described in [3], to get the global irradiance on a tilted surface by adding the other two and having the same initial conditions for both models, that is, global horizontal irradiance as the only input. In Figs. 8 and 9, we can observe the performance of the Perez model and our model for G hourly values, respectively. We have not used the factor Fc when testing the model against the Skyscan’834 data set, because of the very well shielded pyranometer as mentioned above. As the data set covers a full year period, a variety of solar elevation angles as well as cloudy and turbidity conditions are included. Just one thing has to be pointed out before we analyse the statistics for both models. The Skyscan’834 data set is composed of 5845 instantaneous measurements for the sloping 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 for both models. As can be observed in Table 2, together with Figs. 8 and 9, our model offers an over-estimation of 22.6 W/m2 (4.8% of the averaged measured slope irradiance) in contrast to an under-estimation of the Perez model of only 3.9%. The RMSD follows a similar behaviour, being 1% greater for our model than for the Perez model. But considering the statistic as a whole, our model gives good results as well, with a slope close to 1 and a general performance that, together with 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 of 5.2%, a correlation coefficient greater than 0.99 and a slope close to 1. Thus, we can conclude that our model offers a reliable tool for use when solar radiation data are scarce or limited to global horizontal irradiance. Additionally, a fast and simple estimation is often necessary and preferred. Compared with the Perez model, our model gives a similar performance but has the advantage of its simpler formulation.

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

5. Conclusions As our study shows, it is possible to obtain the global irradiance angular distribution from knowledge of the global horizontal irradiance and the astronomic parameters, by means of a model that has been proposed for our latitude. This model depends on local atmospheric conditions by means of kt, and avoids the classical partitioning of the global solar irradiance into direct and diffuse components. This fact gives a general applicability character to the model, which could be used in sites where only horizontal global irradiance is measured. This model has been developed taking into account the exponential pattern shown by the global irradiance angular distribution and its symmetry using polar diagrams. Our model takes into account the ground reflection effects by means of an anisotropic factor, although the effects of an artificial horizon are not taken into account. This provides a good estimation for instantaneous global irradiance values on inclined surfaces. In order to establish its applicability, the model had to be validated against other experimental data 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 turbidity conditions. The results corroborate our predictions and show that when used with either instantaneous or averaged measurements, the new model performs well. Owing to the appropriate performance of the model for different solar elevations and inclined

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Fig. 9. The new model for the estimation of the global slope irradiance vs. Skyscan’834 hourly values for all weather conditions. Table 2 Statistical results obtained from testing the Perez model and the new model against Skyscan’834 hourly and instantaneous values Model

a

b (W/m2)

r

MBD (%)

RMSD (%)

Hourly values Perez model New model

0.975 1.01

⫺ 6.0 19.5

0.994 0.993

⫺ 3.9 4.8

8.3 9.3

Instantaneous values New 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 solar irradiance on inclined surfaces, using the horizontal global irradiance as unique input. The estimations provided by the model can be used for the estimation of the energy balance, in technological applications or in local and large-scale climate studies. On the other hand, this work represents a previous stage in the study of topographic effects on the solar irradiance field, with the goal of mapping solar radiation on local and large scales.

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

Acknowledgements This work was supported by La Direccio´n General de Ciencia y Tecnologı´a from the Education and Research Spanish Ministry through the project No. CLI98-0912-C02-01. We would like to thank sincerely Dr. Alfred Brunger for lending us the Skyscan’834 data set and for the permission to 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.

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