Dependence of one-minute global irradiance probability density distributions on hourly irradiation

Energy 26 (21) 659 668 www.elsevier.com/locate/energy Dependence of one-minute global irradiance probability density distributions on hourly irradiation J. Tovar a, F.J. Olmo b, F.J. Batlles c, L. Alados-Arboledas b,* a Dpto. de Física Aplicada, Universidad de Jaén, 2371 Spain b Dpto. de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 1871 Spain c Dpto. de Física Aplicada, Facultad de Ciencias, Universidad de Almería, 412 Spain Abstract This paper analyzes the behavior of one minute global irradiance distributions as a function of hourly average solar global irradiance. For this purpose, we have used the clearness index k t which describes the atmospheric transmittance. Our interest is in characterizing the intrahourly variability of solar global irradiance and the behavior of the instantaneous values as a function of hourly values of solar global irradiance. The distributions are unimodal and show a marked symmetry around a central value that is close to the corresponding hourly average value. The probability density functions have been modeled using functions based on the Boltzmann statistic used in recent studies of the one minute distributions of k t conditioned to the optical air mass. These functions provide good fit of the distributions and are analytically integrable and can be inverted analytically. The one minute global irradiance data used in this study have been recorded during a three year period in a radiometric station located in south eastern Spain. 21 Elsevier Science Ltd. All rights reserved. 1. Introduction The analysis of the performance of the solar energy conversion systems, both natural and manmade, requires knowledge of the solar radiation temporal evolution. This is particularly important for those systems characterized by a fast and non-linear response to the incoming solar radiation. This is the situation, for example, for photosynthesis or photovoltaic systems. The non-linear response of some energy conversion devices implies that the performance cannot be estimated accurately using average values for the solar irradiance [1,2]. In these cases it is necessary to consider the variable nature of solar radiation. Thus, an appropriate treatment of this variability * Corresponding author. Tel.: 34 58 24424; fax: 34 58 243214. E-mail address: alados@ugr.es (L. Alados-Arboledas). 36-5442/1/$ - see front matter 21 Elsevier Science Ltd. All rights reserved. PII: S36-5442(1)24-X

66 J. Tovar et al. / Energy 26 (21) 659 668 may allow a better estimation of a natural process like plant photosynthesis, because the physiological process evolved a certain non-linear dependence on solar global irradiance and temperature [3]. These discrepancies are enhanced under cloudy conditions because of the strong influence of clouds over the solar radiation. Several authors, for example, Skartveit and Olseth [4], Suehrcke and McCormick [1,5,6], Gansler et al. [2], Jurado et al. [7] and Tovar et al. [8], have shown important differences between distributions of hourly and instantaneous values, represented by one or five minutes values. These differences are subsequently responsible for differences in the simulation of the photovoltaic systems. On the other hand, the characterization of the temporal variability of the solar radiation is important for the estimation of solar global irradiance using remote sensing information. Due to their global coverage, meteorological satellite imagery is an important information source that is used increasingly for the estimation of solar global irradiance. There is a great variety of algorithms that compute the instantaneous values of solar global irradiance using satellite imagery [9 11] to obtain the hourly, daily or monthly values required for different applications as the elaboration of solar radiation maps [12,13] or for the simulation of solar energy conversion systems [14]. The solar global irradiance obtained by processing the satellite images corresponds to instantaneous values. Computation of hourly or daily values of solar irradiation requires the appropriate processing of the instantaneous values acquired at regular intervals. There are algorithms such as HELIOSAT [15,16] or GISTEL [17], that use a limited number of images to obtain the daily solar global irradiation values, which is in the range of three to four per day [13,17]. In this sense, knowledge of the features of the instantaneous distribution of solar global irradiance could provide valuable information to determine the temporal frequency of the satellite images that must be processed. Previous studies that have analyzed solar global irradiance distributions have been centered on the influence of two variables: optical air mass [1,2,5,8] and solar global irradiance average values computed for greater temporal intervals [18,19]. Our purpose is to analyze the variability of the solar global irradiance values at a given location. For this reason, we have studied the distribution function of the one minute values of solar global irradiance as a function of the hourly values that correspond to the hourly period in which they are included. Liu and Jordan [18] first analyzed the behavior of daily values of solar global irradiation by means of the corresponding clearness index (k D t for a given monthly average value of the same index k M t. Following their study, these authors postulated the universality of the distribution function F=F(k D t k M t ). Liu and Jordan [18] did not provide a functional representation for these distributions, but they showed that the distributions are essentially dependent only on the monthly average value k M t. Later, Bendt et al. [19] performed an exhaustive analysis to confirm Liu and Jordan [18] results, studying the distributions F=F(k D t k M t )fork M t =.3,.4,.5,.6,.7. In this work we have analyzed the one minute values k t in order to develop a procedure that allows the estimation of the distribution functions of instantaneous values from knowledge of k t hourly averages. 2. Experimental data and procedure The radiometric and meteorological data set reported in this study was registered as one minute value at the radiometric station of Armilla (37.13 N, 3.63 W, 66 m m.a.s.l., near Granada,

J. Tovar et al. / Energy 26 (21) 659 668 661 Spain) by a Kipp and Zonen CM-11 pyranometer. The radiometer measurements are registered using digital data logger at five second intervals and are averaged to provide one minute solar global irradiance. Measurements used in this work started in December 1993 and finished at the end of 1996, including approximately 1.5 1 6 one minute values of solar global irradiance. The periods covered by the data set guarantee that a complete range of seasonal conditions and solar angles is included among the samples taken. To avoid cosine response errors we have used data to solar zenith angle less than 85. Measurements of global solar irradiance have an estimated experimental error of about 2 3%. The pyranometer is intercompared yearly against a reference CM-11, reserved for this purpose, and exposed to solar radiation only during the intercomparison campaigns. Degradation lower than 1% per year can be quoted. We checked the calibration constants of the radiometric devices periodically. Degradation of less than a few tenths per cent per year were observed in the CM-11 pyranometer. The solar global irradiance behavior was studied by means of the clearness index k t which represents the ratio of solar global horizontal irradiance incoming at surface level to that received at the top of the atmosphere. This variable is studied using the distribution function, F(k t,t), which represents the probability that the event k t (t) at a time t is smaller than the given value k t. F(k t,t) P[k t (t) k t ] (1) For stochastic sequences, this quantity also represents the fraction of the time that the stochastic variable is below a given value. This second interpretation is more appropriate for our study. We have divided the possible range of clearness index ( 1) into 5 intervals,.2 in width, denoted (.2 1). The values in parenthesis indicate the beginning of the first interval, the width of each bin and the final interval value. The distributions are also characterized by the probability density function f(k t ), where we omit the time dependence for clarity, given by: f(k t ) F(k t) (2) k t and the normalization condition: k tmax 1 f(k t )dk t f(k t )dk t 1 (3) considering 1 the maximum possible value of k t. We have analyzed the k t probability density functions conditioned to a particular value of the k t hourly average value f(k t k H t ). In this sense F(k t k H t ) and f(k t k H t ) refer respectively to cumulative distributions and density distributions conditioned to the k t hourly average. 3. Probability distributions of one-minute k t values conditioned to the hourly average values k H t In order to obtain the one minute probability distributions of k t we have computed the hourly average values of solar global irradiance corresponding to the three years period of data available.

662 J. Tovar et al. / Energy 26 (21) 659 668 Following that evaluation we have classified the data in bins of.1 width centered at k H t = (.3,,35,.4,.45,.5,.55,.6,.65,.7,.75). The one minute data have been classified according to these criteria. These bins are grouped into two groups, one including the bins with hourly average values centered in.3,.4,.5,.6 and.7 including 91575 values. A second one corresponds to distribution of values around.35,.45,.55,.65 y.75 that includes 98113 values. The second group has been reserved for validation purposes. Fig. 1 shows the density probability distributions of k t for given k H t values (.3.7). These Fig. 1. Probability density distributions of k t for given k Ht values (.3,.4,.5,.6 and.7), obtained from data recorded with one minute resolution.

J. Tovar et al. / Energy 26 (21) 659 668 663 distributions show an evident unimodality that contrasts with the bimodality that characterizes the distributions conditioned to the optical air mass [8]. This fact can be explained in terms of the reduced range of the k t values that corresponds to a given k H t. On the contrary, when the bins are defined as a function of the optical air mass f(k t m a ) the distributions tend to be bimodal. This bimodality can be associated with the existence of two levels of irradiation corresponding to two extreme atmospheric situations, namely cloudless and cloudy conditions. The distributions present a marked symmetry around a central value that is close to the corresponding k H t. This feature is more evident for k H t in the range.45.65, while the distributions corresponding to k H t out of this range show a slight asymmetry. For values of k H t below.45 there is an asymmetry toward the higher values indicating that k H t in this range can be the result of the combination of very low and very high instantaneous values of k t. This can be connected with transient situations under partial cloud cover with clouds close to the sun position, that in a short time can block the sun or enhance the sun direct beam due to reflections in cloud border. Another relevant feature is the range of k t instantaneous values associated to a given k H t. Excluding the higher values of k H t the range of k t instantaneous values is rather wide. This indicates that for these categories we include partially covered skies characterized by a great variability of instantaneous k t, especially, if the clouds are close to the sun. For higher k H t the k t instantaneous value range is reduced, indicating that these higher hourly values are associated to cloudless sky conditions. A rather narrow range of k t values characterizes these distributions. In the distributions corresponding to intermediate values of k t, associated to partially cloudy skies, we register the highest k t values. This is a result of multiple reflections in the clouds located close to the sun position. Under these conditions the reflections in the clouds sides leads to an increment of global irradiance due to the enhancement of the diffuse component. 4. Functional representation of the distributions Considering the shape of the curves we have approximated the probability distributions using a function based on Boltzmann s statistic. This function has been used previously for modeling one minute distributions conditioned to the air mass [8,2]. This function reads as follows: Ale(k t k t)l f(k t k H t ) (4) [1+e (k t k t )l ] 2 where A, k t and l are constants for a given distribution, with the normalization condition 1 f(k t k H t )dk t 1 (5) Eq. (4) yields unimodal curves that are symmetric around k t, that represent the location of the maximum. The product Al determines the maximum of function and l is related to the width of the distribution f(k t k H t ). This equation has the advantage of being analytically integrable.

664 J. Tovar et al. / Energy 26 (21) 659 668 1 )l F(k t ) A 1 1+e (k t k t (6) In addition, its primitive can be analytically inverted k t k t 1 l ln F(k t) A F(k t ) (7) These characteristics simplify the computation required for synthetic data generation [8]. To account for the asymmetry of the analyzed distributions, we now introduce a parameter (b) in the above function. This parameter has been previously used to analyze the diffuse component distributions of the radiation with the air mass f(k b m a ). The modified equation reads as follows Ale(k t k t)l f(k t k H t ) (8) [1+e (k t k t )(l+b) ] 2 This equation satisfies the normalization condition 1 1 f(k t k H t )dk t A le (k t k t )l [1+e (k t k t )(l+b) ] 2dk t 1 (9) This function is flexible and it provides very good fits, even for distributions that exhibit a high degree of asymmetry. The degree of asymmetry depends on the relationship between b and l. The sign of b determines the sign of the asymmetry. The dependence of the coefficients k t, l and b with k H t may be formulated by means of polynomials functions: k t.6 1.1k H t with r 2 (k t ).999 (1) l 11.284 115.37(k H t ) 7.25 with r 2 (l).935 (11) b.293 6.93k H t 15.643(k H t ) 2 with r 2 (b).984 (12) where r 2 is the correlation coefficient of the polynomial fit. Fig. 2 shows the performance of these functions depending on k H t. Fig. 2 also shows the ratio b/l that provides information about the distribution asymmetry. It can be seen that this ratio presents a sign change about k H t =.45. The values of the parameter A must satisfy the normalization condition (Eq. (9)) and the computation of 1 k t f(k t k H t )dk t must return the corresponding value of k H t. The values for the A parameter that

J. Tovar et al. / Energy 26 (21) 659 668 665 Fig. 2. Polynomial fit of the coefficients of the distributions versus k H t. (a) k t ; (b) l; (c) b. (d) shows the asymmetry of the function characterized by the ratio b/l. correspond to the distributions in the range k H t (,1) in.5 steps are included in Table 1. In this table we have excluded the situations k H t =, corresponding to the absence of radiation, and k H t =1, corresponding to situations when all the instantaneous values of k t are equal to unity. It is worth to note that the A parameter can be fitted by a polynomial function: A 2.8137 237859k H t 143.72282(k H t ) 2 456.5793(k H t ) 3 771.8157(k H t ) 4 (13) 656.1663(k H t ) 5 22.62361(k H t ) 6 for the range to 1; although the experimental values that we found for k H t are comprised in the.3 to.8 range. This equation has an associated value of R 2 about.999 when we consider the range.5 k H t.95. Fig. 3 shows this fit. The developed probability density functions have been checked using the data set reserved for validation purposes. Fig. 4 shows the degree of accordance between the experimental distributions and the corresponding functions with coefficients depending of k H t as shown by Eqs. (1) (12). 5. Conclusions Our analysis of the k t distributions of data with one minute resolution conditioned to the hourly average value, f(k t k H t ), shows that the distributions are unimodal in contrast to those conditioned

666 J. Tovar et al. / Energy 26 (21) 659 668 Table 1 Values for the A parameter that correspond to the distributions in the range k H t (,1) k H t A.5 1.934.1 1.471.15 1.298.2 1.198.25 1.136.3 1.93.35 1.56.4 1.22.45.99.5.966.55.954.6.953.65.959.7.966.75.973.8.977.85.988.9 1.153.95 1.575 Fig. 3. Polynomial fit equation corresponding to the parameter A. to the optical air mass which exhibit a marked bimodality. The range of k t instantaneous values covered by these distributions is rather high, specially for medium and low values of k H t. In the case of higher k H t values the distributions are grouped in a narrow interval around the central value. It is interesting to note that for the distributions associated to intermediate values of k H t the instantaneous clearness index can read values close to the unity due to enhancement of the diffuse components by reflections in the clouds borders. The k t instantaneous distributions have been modeled using a function based on Boltzmann

J. Tovar et al. / Energy 26 (21) 659 668 667 Fig. 4. Probability density distributions for k H t values (.35,.45,.55,.65, and.75) reserved for testing purposes and the corresponding adjust functions of the model proposed. statistics. The parameters included in these functions are related to different features of the distributions and present a dependence on k H t, which can be modeled by means of polynomial functions. The performance of the proposed functions has been checked using the data set reserved for validation purposes. The generality of the model must be confirmed using data set acquired in locations with different climates that were encountered at Granada. The agreement between the experimental distributions and that modeled seems very appropriate.

668 J. Tovar et al. / Energy 26 (21) 659 668 Acknowledgements This work was supported by La Dirección General de Ciencia y Tecnología from the Education and Research Spanish Ministry through Project No. CLI99-835-C2-1. We are very grateful to the Armilla Air Base Meteorological Office Staff and specially to the meteorologist Guillermo Ballester for the maintenance of the radiometric devices. We are also indebted to Dr Noam Lior and the anonymous reviewers for their comments and suggestions witch improved the presentation of the results. References [1] Suehrcke H, McCormick PG. Solar radiation utilizability. Solar Energy 1989;43:339 45. [2] Gansler RA, Klein SA, Beckman WA. Investigation of minute solar radiation data. Solar Energy 1995;55:21 7. [3] Berninger F. Simulated irradiance and temperatures estimates as a possible source of bias in the simulation of photosynthesis. Agric For Meteorol 1994;71:19 32. [4] Skartveit A, Olseth JA. The probability density and autocorrelation of short-term global and beam irradiance. Solar Energy 1992;49:477 87. [5] Suehrcke H, McCormick PG. A performance prediction method for solar energy systems. Solar Energy 1992;48:169 75. [6] Suehrcke H, McCormick PG. The frequency distributions of instantaneous insolation values. Solar Energy 1988;4:413 22. [7] Jurado M, Caridad JM, Ruiz V. Statistical distribution of the clearness index with radiation data integrated over five minute intervals. Solar Energy 1995;55:469 73. [8] Tovar J, Olmo FJ, Alados-Arboledas L. One-minute global irradiance probability density distributions conditioned to the optical air mass. Solar Energy 1998;62:387 93. [9] Stuhlmann R, Rieland M, Raschke E. An improvement of the IGMK model to derive total and diffuse solar radiation at the surface from satellite data. Journal of Applied Meteorology 199;29:586 63. [1] Noia M, Ratto CF, Festa R. Solar irradiance estimation from geostationary satellite data:i. Statistical Models. Solar Energy 1993;51:449 56. [11] Noia M, Ratto CF, Festa R. Solar irradiance estimation from geostationary satellite data:ii. Physical Models. Solar Energy 1993;51:457 65. [12] D Agostino V, Zelenka A. Supplementing solar radiation network data by co-kriging with satellite images. International Journal of Climatology 1992;12:749 61. [13] Möser W, Raschke E. Incident solar radiation over Europe estimated from METEOSAT data. Journal of Climate and Applied Meteorology 1984;23:166 7. [14] Perez R, Seals R, Stewart R, Zelenka A, Estrada-Cagigal V. Using satellite derived insolation data for the site/time specific simulation of solar energy systems. Solar Energy 1994;53:491 5. [15] Beyer HG, Constanzo C, Reise CH. Multiresolution analysis of satellite derived irradiance maps. An evaluation of a new tool for the spatial characterization of hourly irradiance fields. Solar Energy 1995;55:9 2. [16] Beyer HG, Constanzo C, Heinemann D. Modifications of the Heliosat procedure for irradiance estimates from satellite images. Solar Energy 1996;56:27 13. [17] Delorme C, Gallo A, Olivieri J. Quick use of WEFAX images from METEOSAT to determine daily solar radiation in France. Solar Energy 1992;49:191 7. [18] Liu BYH, Jordan RC. The interrelationship and the characteristics distribution of direct, diffuse and total solar radiation. Solar Energy 1963;4:1 19. [19] Bendt P, Collares-Pereira M, Rabl A. The frequency distribution of daily insolation values. Solar Energy 1981;27:1 5. [2] Tovar J, Olmo FJ, Batlles FJ, Alados-Arboledas L. One-minute k b and k d probability density distributions conditioned to the optical air mass. Solar Energy 1999;65:297 34.