Correlation of global solar radiation values estimated and measured on an inclined surface for clear days in Bogotá

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Renewable Energy 32 (2007) 2590–2602 www.elsevier.com/locate/renene

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Correlation of global solar radiation values estimated and measured on an inclined surface for clear days in Bogota´ N.L. Foreroa,, L.M. Caicedob, G. Gordillob a

Licenciatura en Fı´sica, Universidad Distrital, Bogota´, Colombia Departamento de Fı´sica, Universidad Nacional de Colombia, Bogota´, Colombia

b

Received 22 September 2006; accepted 23 December 2006 Available online 13 March 2007

Abstract An empirical expression developed to estimate global solar radiation in clear days, on inclined surfaces located at any geographical position, is presented. This expression allows determining the global solar radiation in a specific day of the year, considering the attenuation of radiation in the atmosphere, the air mass factor, astronomic geometric and geographic parameters, and in particular, the altitude. Data calculated with this expression were correlated with those obtained experimentally in Bogota´, Colombia (74140 W, 41350 N and 2580 m altitude). The correlation of the calculated with the experimental data yielded a coefficient of 0.9980, which indicates the reliability of the former and that the developed expression facilitates the construction of data bases with information on solar radiation potential in ample regions characterized by their locations at different altitudes above sea level. These data bases will supply preliminary information on sites adequate for the installation of photovoltaic systems. r 2007 Elsevier Ltd. All rights reserved. Keywords: Global irradiance; Clearness index; Beer’s law; Virtual instrumentation

Corresponding author. Tel.: +57 1 3165000x13019; fax: +57 1 3165135.

E-mail address: [email protected] (N.L. Forero). 0960-1481/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2006.12.012

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Nomenclature G G0 B0 e0d d dn yzs ys f o os b a gs Hl AM KT tal tsl A

global solar radiation (W/m2) extraterrestrial global solar radiation (W/m2) solar constant (W/m2) eccentricity correction factor solar declination day number, starting on 1st January (28 and 29 February, 1 day) solar zenith angle. Angle between the vertical and the incident solar ray incidence angle between sun rays and the normal to the surface location’s latitude, positive north of the equator solar time or true solar time. Positive in the morning, negative in the afternoon dawn angle. Always negative by sign convention surface angle with respect to the horizontal azimuth of the normal to the surface solar altitude local time air mass factor clearness index atmospheric absorption index atmospheric dispersion index local altitude

1. Introduction It is of great importance to know the global solar radiation at a specific geographic region for several purposes, especially for designing and sizing photovoltaic systems [1]. Several works have been developed to estimate the global solar radiation for a site, from the number of hours of standard irradiance [2] and from sunshine hours [3]. Other authors predict global radiation through calculations carried out with expressions built from theoretical models [4–6], which include fitting meteorological parameters obtained from the correlation with experimental data [7]. In other cases geographic and meteorological parameters have been taken into account to predict it [8,9]. Some other authors have estimated global radiation from the clearness index KT and the air mass factor, without explicitly considering astronomic and geographic parameters [10,11]. Yet, few studies have taken into account of the local altitude for the calculation of global solar radiation. Within the structure of this work experimental measurements of global solar radiation were made in the city of Bogota´, Colombia (74140 W, 41350 N and 2580 m altitude), which were correlated initially with the calculated one using different empirical models, reported in literature, to foretell profiles of global solar radiation under diverse meteorological and astronomical conditions [12,13]. This correlation showed significant differences between the experimental and theoretical profiles, so, a new expression was developed including explicitly the effect of altitude, which was not considered in models mentioned before.

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The solar radiation profile obtained with the expression developed in this work better reproduced the experimental results in Bogota´ than those obtained with expressions not including the altitude effect. The global solar radiation calculations were performed for inclined surfaces to facilitate the sizing of photovoltaic systems in open fields or on roofs of houses and buildings, for which it is necessary to consider the latitude of the place and architectonic restrictions [14,15]. 2. Development of the expression to calculate the global solar radiation The amount of solar radiation received by a location varies with the relative position of the sun. The expression allowing the calculation of the global solar radiation G falling upon an inclined surface at any moment of a day of a year dn, depends mainly on astronomic, geographic, geometric as well as atmospheric parameters: 0 1 Astronom::Param:   B C G @ Geom: and Geog: Param: A ¼ G0 ð0; d n Þ  SðyS ; f; b; :::;Þ  x tal ; tsl ; AM; A; :::; . Atmospher:Param (1) The term G0 (0,dn) ¼ B0e0d represents the astronomic effects on the extraterrestrial global solar radiation upon a surface perpendicular to the incident solar rays and can be calculated as a function of the solar constant B0 and the terrestrial eccentricity correction factor e0d. B0 corresponds to the amount of solar radiation received by the external layers of the earth’s atmosphere at an average earth–sun distance of one astronomical unit (1 AU ¼ 1.496  1011 m); its average value on a surface perpendicular to the sun’s rays is 1367 W/m2 (118.108 mJ/m2/day), obtained from annual averages of satellite measurements [16]. The eccentricity e0 is expressed as  2   r 360 d n 0d ¼ 0 ðd n Þ ¼ ¼ 1 þ 0:033 Cos , (2) r0 365 where dn is the day number, counting from 1st January on. The second term S (ys,f,b) is equal to Cos ys, where ys is the angle between the incident solar rays and the normal to the surface, taking into account that the surface is inclined at an angle b with the horizontal and it is oriented through an angle a, which corresponds to the azimuth with respect to the astronomic south. Cosys is given by (see Fig. 1) Cos ys ¼ Sind Sinf Cos b  ½sign ðfÞ Sin d Cos f Sin b Cos a þ Cos d Cos f Cos b Cos o þ ½signðfÞ Cos d Sin f Sin b Cos a Cos o þ Cos d Sin a Sin o Sin b. Where the solar declination d is given by [17]    d n þ 284 dðd n Þ ¼ 23:45 Sin 360 , 365

ð3Þ

(4)

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Fig. 1. Position of the receiving surface inclined an angle b respect to the horizontal, and incident solar rays forming an angle ys with the normal to the surface.

f is the site’s latitude (positive sign(f) north of the equator and negative otherwise); o is the standard local time, determined through the expression   360 o¼ ðH l  12Þ ¼ 15 ðH l  12Þ, (5) 24 o is also called true solar time and is expressed in degrees. The last term of Eq. (1) represents the atmospheric effects attenuating solar radiation and might also include meteorological and weather parameters, which in turn might depend of each other and of geographic and astronomic parameters. When solar radiation crosses the atmosphere its intensity is attenuated because of absorption by the different atmospheric components. The absorption effect is determined by the absorption index tal that is given by the sum of the absorption indexes of all the absorption centers present (ozone, oxygen, carbon dioxide, water vapor, etc.); additionally, the radiation is scattered by gases conforming the atmosphere and it is described by the scattering index tsl . These phenomena are the reason why the incident solar radiation on the earth’s surface is formed by a direct radiation component, a diffuse one and another reflected by the obstacles around the incidence site. The total incident radiation on a surface is given then by these three components and is called global radiation. The law allowing determining the effect of absorption processes on the radiation intensity when crossing a medium is known as Beer’s Law [18] (also named Beer–Bouguer–Lambert’s Law) and can be expressed as a

a

GðlÞ ¼ G 0 ðlÞetl ð1=Cos ys Þ ¼ G0 ðlÞetl AM ,

(6)

where AM is the air mass factor, which is a gauge for the radiation’s optical path length to the geographic point where the global solar radiation is measured. An analogous form of Beer’s Law for the case of dispersion effects can be written as follows: S

S

GðlÞ ¼ G 0 ðlÞetl ð1=Cos ys Þ ¼ G 0 ðlÞetl AM , tsl

(7)

where is the spectral dispersion index. It is very difficult to include absorption and dispersion effects in the calculation of global solar radiation for inhomogeneous media, so, from now on, the calculated global solar

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radiation will be assumed to correspond to clear days, in which the atmosphere is considered homogeneous, and therefore, with constant absorption and dispersion indexes. This work includes, in the expression for x, the effect of the altitude A on the measurement of clear day global solar radiation, because it determines the length of the air mass column the radiation has to travel through before reaching a given geographical point, as shown in Fig. 2, where it is observed that the distance travelled by rays incident on places above sea level is shorter than for those at sea level. The case in which the radiation falls perpendicularly on the receiving surface the angle y is zero and hence S (ys,f,b) ¼ Cos y ¼ 1; in this case we have:   (8) G d ð0Þ  B0d 0d x A; f; b; y; o; a; AM; tal ; tsl ; :::; . The following expressions have often been used to calculate Gd [19–21]: Q

G d ¼ B0d 0d f ðyz ÞPAM ,

(9)

where P and Q could be variables or coefficients to be determined and f (yz) represents the orientation and geographic position of the surface receiving the global solar radiation with respect to earth. Meinel and Meinel [22] developed the following empirical relation to calculate the global radiation and the clearness index KT : G ¼ B0 0 0:7AM

0:678

;

KT ¼

0:678 G ¼ 0:7AM . G0

(10)

The G and KT values calculated with Eq. (10) reproduce well the values obtained experimentally at sea level. However, in this work global radiation data measured experimentally in Bogota´ (located at 2580 m above sea level) were correlated with values acquired using that equation, producing low correlation coefficients in most cases. It was decided, then, to develop an empirical expression to improve the correlation between the experimental and theoretical data, by including explicitly the effect of the measurement site’s altitude. This equation maintains the form of Beer’s law and the empirical expression developed by Meinel and Meinel, adding the term representing the effect of the altitude on

Fig. 2. Diagram showing the different distances travelled by solar rays incident upon places at sea level and at a height A, respectively.

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the global radiation. Therefore, the global solar radiation for any hour of the day, on a surface inclined an angle b respect to the horizontal and oriented with an azimuth angle a, could be determine as follows:  AM0:678 a s G ðb; aÞ ¼ B0 0 Cos yS 1  eðc1 Aþtl þtl Þ . (11) Assuming that absorption and dispersion are indistinguishable in clear days, when the atmosphere is homogeneous, Eq. (11) can be written as  AM0:678 G ðb; aÞ ¼ B0 0 Cos yS 1  eðc1 Aþc2 Þ , (12) where c1 and c2 are constants to be determined and represent the altitude, absorption and dispersion effects on the global radiation. The following section describes the procedure to find constants c1 and c2. 3. Results and discussion 3.1. Measurement of global radiation The global solar radiation data were obtained with a Kipp and Zonen SP LITE pyrometer fulfilling the ISO 9060 specifications, with a 72 mV/W m2 sensibility, response time shorter than a second, placed on a meteorological station located at the Universidad Nacional de Colombia in Bogota´ (74140 W, 41350 N, altitude 2580 m). The acquisition of data was carried out with an automatic system developed and implemented with virtual instrumentation using a National Instruments FP 2000 module and a personal computer. The acquisition, storage and statistical processing of the global solar radiation data were performed through a virtual instrument (VI) developed with the 7.1 LabVIEW package. This VI has additional tools to watch in real time, on the screen of a PC, the data itself and the graphs showing the time evolution of the global solar radiation. The VI acquires one datum per second and executes averages every minute, hour and day. Fig. 3 shows the mean daily irradiance per month, measured in Bogota´ during 2004 and 2005. It is observed that January and February are the months with the highest solar radiation in Bogota´ with values of around 400 W/m2; April and May present the lowest average daily radiation of around 250 W/m2. These results also revealed that the annual daily averages of solar radiation in Bogota´, during 2004 and 2005, were 335.6 and 297.7 W/m2, respectively, which are values relatively low for a region near the equator. This behavior of solar radiation is attributed to Bogota´’s position on the Andean Range, characterized by sudden atmospheric changes and a high and intermittent cloud cover. The data shown in Fig. 3 were also analyzed to determine the number of clear, cloudy and partially cloudy days in Bogota´ during 2004 and 2005. These data were obtained through the calculation of the daily atmospheric clearness index KT, given by the ratio between experimental global radiation measured in Bogota´ on a horizontal surface and the extraterrestrial irradiance G0, calculated using the expression G0 ¼ B0e0d. Days with KT between 0 and 0.3 are classified as cloudy, days with KT between 0.3 and 0.55 are classified as partially cloudy and those with KT between 0.55 and 0.8 are considered with clear skies. The results of the classification mentioned above and shown in Table 1 reveal that April–June were the months with higher number of cloudy days and almost no clear days.

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Fig. 3. Daily per month averages of global solar radiation measured in Bogota´ during 2004 and 2005.

Table 1 Presents a relation of the number of clear, cloudy and partially cloudy days occurring in Bogota´ each month during 2004 and 2005 2004

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

2005

Cloudy days 0oKo0.3

Semi-cloudy days 0.31oKo0.55

Clear days K40.56

Cloudy days 0oKo0.3

Semi-cloudy days 0.31oKo0.55

Clear days K40.56

9 5 12 15 23 7 11 7 11 18 19 11

11 12 13 15 8 23 19 22 17 12 11 19

11 12 6 0 0 0 1 2 2 1 0 1

8 8 7 18 12 10 6 14 21 11 12 16

16 17 20 12 19 19 23 16 9 19 18 15

7 4 4 0 0 1 2 1 0 1 0 0

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On the contrary, January and February had the highest number of clear days. On the other hand, and from those criteria, it is worth emphasizing that 2004 presented 36 clear days whilst during 2005 only 20 clear days occurred, which explains the low average solar radiation level Bogota´ shows. Another aspect worth mentioning is the 2004 behavior of solar radiation that was significantly different from 2005 probably caused by meteorological changes occurring worldwide and regionally associated to climatic fluctuations caused by the El Nin˜o equatorial phenomenon (and its negative phase, La Nin˜a) in the Pacific littoral of South America, during 2004–2005. 3.2. Correlation of global solar radiation experimental and theoretical data The procedure to determine constants c1 and c2 of Eq. (12) consisted in the initial calculation of c2 assuming that at sea level, when A ¼ 0, Eq. (12) is the same expression developed by Meinel and Meinel to calculate the global solar radiation (Eq. (10)); in this way c2 ¼ 1.2039. To find c1 the global solar radiation values obtained with Eq. (12) for a particular day in Bogota´, on an inclined surface, were fitted with the least-squares method to the corresponding experimental data. Hence, c1 is obtained from the application of such method, through the following equation: d

N X

ðE i  T i Þ2 ¼ 0,

(13)

i¼1

where Ei is the ith average datum obtained experimentally for global solar radiation and Ti the corresponding datum obtained theoretically from Eq. (12). The difference between the experimental and theoretical daily solar radiation profiles is minimized through the successive iteration up to 100 times of the process. A significant number of solar radiation spectral profiles were chosen corresponding to clear days (KT40.56) of 2004 and 2005 and each one was fitted to those obtained

Fig. 4. Effect of the altitude on solar radiation profiles calculated using Eq. (14).

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Fig. 5. Comparison of global irradiance profiles obtained experimentally on an inclined surface during 2004 for clear days in Bogota´, with theoretical ones obtained using Eq. (14).

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Fig. 6. Comparison of global irradiance profiles obtained experimentally on an inclined surface during 2005 for clear days in Bogota´, with theoretical ones obtained using Eq. (14).

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theoretically using the least-squares method. Each fitting yielded a value for c1 and all together gave an average value of c1 ¼ 5.87  105 m1. Eq. (12) with c1 and c2 included becomes  AM0:678 4 1 G ðb; aÞ ¼ B0 0 Cos yS 1  eðð5:8710 m ÞAþ1:2039Þ . (14) The clear day global solar radiation for locations at sea level (A ¼ 0) is reduced to G ðb; aÞ ¼ B0 0 0:7AM

0:678

Cos yS .

(15)

Corresponding to the expression originally proposed by Meinel and Meinel. Fig. 4 shows daily solar radiation spectral profiles calculated with Eq. (14), varying the altitude from sea level (A ¼ 0) up to values near the atmospheric boundary (80 km) assuming geographic parameters corresponding to Bogota´ and astronomic parameters corresponding to 24th February, a clear day. Figs. 5 and 6 compare several daily global solar radiation profiles measured on clear days in Bogota´ during different months of 2004 and 2005, with the corresponding ones calculated using Eq. (14) taking into account the astronomic, geographic and atmospheric parameters for the location. From the results shown in Figs. 5 and 6 it is worth stressing the following facts:







The results show that many days, classified as clear because their KT’s were greater than 0.56, presented abrupt fluctuations in the solar radiation intensities due to the continuous passing of low clouds over the city, driven by strong winds proper of July and August. Moreover, only during January and February of 2004 there were clear days with very little cloud cover. The theoretical global solar radiation profile fits very well the experimental one for these types of days. The correlation factor for these cases was around 0.9980. February of 2005 also showed days with low cloud density but with asymmetric morning radiation profiles (see the profile for 24th February ), due to the presence of low clouds formed by water vapor condensation produced by the low temperature proper in this part of the year. There were days at the end of July and during August for both years presenting frequent irradiance peaks greater than 1200 W/m2, followed by strong reductions. This behavior is caused by the presence of strong winds during this part of the year which drags elements of the atmosphere (CO2, pollution particles suspended in the atmosphere, etc.) that contribute to the absorption and scattering of solar radiation.

4. Conclusions The building of a global solar radiation data base in Bogota´ located at 74140 W, 41350 N and 2580 m altitude, has began. The statistical evaluation of the data obtained since 2004 indicated that the months with the highest daily global solar radiation averages are January and February, April and May the months with lowest radiation. It was also found that the annual daily averages of global solar radiation in Bogota´ were 335.6 and 297.7 W/m2, respectively, which are quite low to consider Bogota´ as a good place to install photovoltaic electricity generator plants. The low intensity of solar radiation in Bogota´ is

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caused by the high frequency of cloudy days in the city due to its position in the Andes Range. An empirical expression was developed based on Beer’s Law to estimate the global solar radiation for clear days, on inclined surfaces at any geographic position. This equation allows determining the global solar radiation considering atmospheric, astronomic, geometric and geographic parameters, in particular the altitude. It was established that the theoretical solar radiation values reproduce very well the experimental ones, through the comparison of daily global solar radiation spectral profiles, obtained with this equation, with those measured in Bogota´, Colombia. The correlation coefficient found for both types of solar radiation profiles is 0.9805, which indicates the good reliability of the global solar radiation data obtained using that expression. This enables building in a practical and simple manner data bases with information on solar radiation potential for ample regions located at different altitudes, which is valuable to determine adequate places where to install photovoltaic systems.

Acknowledgments This work was carried out due to the financial support from COLCIENCIAS and the Universidad Nacional de Colombia.

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