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Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation
ARTICLE IN PRESS
Renewable Energy 31 (2006) 2492–2504 www.elsevier.com/locate/renene
Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation C.P. Jacovidesa,, F.S. Tymviosa,b, V.D. Assimakopoulosc, N.A. Kaltsounidesa a
Laboratory of Meteorology, Department of Applied Physics, University of Athens, University Campus, Builds PHYS-V, Athens 15784, Greece b Meteorological Service of Cyprus, Nicosia, Cyprus c Institute for Environmental Research and Sustainable Development, National Observatory of Athens, Athens 15236, Greece Received 25 January 2005; accepted 30 November 2005 Available online 24 January 2006
Abstract Proper design and performance predictions of solar energy systems require accurate information on the availability of solar radiation. The diffuse-to-global solar radiation correlation, originally developed by Liu and Jordan, has been extensively used as the technique providing accurate results, although it is latitude dependent. Thus, in the present study, empirical correlations of this type were developed to establish a relationship between the hourly diffuse fraction (kd) and the hourly clearness index (kt) using hourly global and diffuse irradiation measurements on a horizontal surface performed at Athalassa, Cyprus. The proposed correlations were compared against 10 models available in the literature in terms of the widely used statistical indicators, rmse, mbe and t test. From this analysis, it can be concluded that the proposed yearly correlation predicts diffuse values accurately, whereas all candidate models examined appear to be location-independent for diffuse irradiation predictions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Hourly diffuse fraction; Clearness index; Standard models; Statistical indicators; Athalassa (Cyprus)
Corresponding author. Tel.: +30 210 7276931; fax: +30 210 7295281.
E-mail address: [email protected] (C.P. Jacovides). 0960-1481/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2005.11.009
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1. Introduction The increasing global energy demands and increasing fossil fuel prices stimulate countries to downsize energy consumption and exploit renewable energy sources. In addition, environmental problems caused by mass consumption of fossil energy (e.g., global warming), are also reason for concern. For example, the European Union aims at achieving a 25 percent reduction in carbon dioxide emissions by the year 2005; there exists also an additional EU renewable energy obligation that purports to be one of the key mechanisms for enabling the Mediterranean partners to reach their renewable energy targets (20% by 2010). In general, solar and wind energy are thought to be good alternative energy sources for overcoming these problems due to their safety and positive contribution to the global environmental state because of their lack of emissions during operation [1]. Reliable solar radiation measurements for estimating the dynamic behavior of solar energy systems’ processes and for simulating long-term operations are required. For thermal analysis performance through transient simulation algorithms, a crucial input is the solar energy components incident on the collector surfaces. Usually, an hourly time step is used in these systems and thus hourly solar energy data are needed which in turn are seldom available at the site of interest. Given global solar irradiation measurements on a horizontal surface (the most widely available data for solar energy) direct and diffuse radiant components can be obtained from global solar energy data through various correlations. Thus, as early as the early 1960s, numerous models for evaluating the diffuse component based on the pioneer work of Liu and Jordan [2] appeared in the literature. These models are usually expressed in terms of 1st–4th-degree polynomials relating the diffuse fraction kd (ratio of the diffuse-to-global solar radiation) with the clearness index kt (ratio of the global-to-extraterrestrial solar radiation). Nevertheless, the range of the diffuse fraction, as reported in the literature, suggests the desirability for recalibration accounting for local climatic differences. This study investigates the applicability of various standard models correlating hourly diffuse fraction kd and kt, in the eastern Mediterranean basin. Of direct pertinence are the works of Orgill and Hollands [3], Reindl et al. [4], Boland et al. [5], Hawlader [6], Miguel et al. [7], Karatasou et al. [8], Erbs et al. [9], Chandrasekaran and Kumar [10], Oliveira et al. [11], and Soares et al. [12], who established hourly correlations between kd and kt under diverse climatic conditions. The models can be categorized as: firstorder (Eqs. (1)–(3)), second–third-order (Eqs. (6)–(8)), and fourth-order (Eqs. (9)–(12)) correlations that are briefly reviewed below. Orgill and Hollands [3] used data from one location in Toronto, Canada, derived the following correlation between kd and kt: kd ¼ 1:557 1:84kt
(1)
for 0.35pktp0.75; kd ¼ 1.00.249 kt for kto0.35, and kd ¼ 0.177 for kt40.75. Reindl et al. [4] studied the influence of climatic-geometric variables on the hourly diffuse fraction based on data from five European and US locations; they established the following expressions: kd ¼ 1:45 1:67kt for 0.3okto0.78; kd ¼ 1.020.248kt for ktp0.3, and kd ¼ 0.147 for ktX0.78.
(2)
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Boland et al. [5] used data from one location in Victoria, Australia, constructed a simple exponential correlation of the form, kd ¼
1 . 1 þ e7:997ðkt 0:586Þ
(3)
Hawlader [6] using data from a tropical site in Singapore derived the second-order polynomial correlation, kd ¼ 1:135 0:9422kt 0:3878k2t
(4)
for 0.225o kto0.775; kd ¼ 0.915 for ktp0.225 and kd ¼ 0.215 for ktX0.775. Miguel et al. [7] used an assembled data set from several countries in the North Mediterranean Belt area, and yielded a third-order polynomial for hourly diffuse fraction correlations, kd ¼ 0:724 þ 2:738kt 8:32k2t þ 4:967k3t
(5)
for 0.21oktp0.76; kd ¼ 0.9950.081kt for ktp0.21 and kd ¼ 0.18 for kt40.76. Karatasou et al. [8] based on data from Athens, Greece, proposed a third-order polynomial correlation, kd ¼ 0:9995 0:05kt 2:4156k2t þ 1:4926k3t
(6)
for 0.oktp0.78; and kd ¼ 0.20 for kt40.78. Erbs et al. [9] used an assembled data set from four US locations derived a fourth-order polynomial correlation for the hourly diffuse fraction, kd ¼ 0:951 0:1604kt þ 4:388k2t 16:638k3t þ 12:336k4t
(7)
for 0.22oktp0.80; kd ¼ 1.00.09kt for ktp0.22 and kd ¼ 0.165 for kt40.80. Chandrasekaran and Kumar [10] used data from a tropical environment in Madras, India, derived a fourth-order polynomial correlation, kd ¼ 0:9686 þ 0:1325kt þ 1:4183k2t 10:1862k3t þ 8:3733k4t
(8)
for 0.24oktp0.80; kd ¼ 1.00860.178kt for ktp0.24, and kd ¼ 0.197 for kt40.80. Oliveira et al. [11] using data from a tropical Sao Paolo site, Brazil, proposed a fourthorder polynomial correlation, kd ¼ 0:97 þ 0:8kt 3:0k2t 3:1k3t þ 5:2k4t
(9)
for 0.17okto0.75; kd ¼ 1.0 for ktp0.17, and kd ¼ 0.17 for kt40.75. Finally, Soares et al. [12] based on the same data set as [11], established a synthetic fourth-order polynomial correlation by means of a neural network technique, kd ¼ 0:90 þ 1:1kt 4:5k2t þ 0:01k3t þ 3:14k4t
(10)
for 0.17okto0.75; kd ¼ 1.0 for ktp0.17, and kd ¼ 0.17 for kt40.75 Furthermore, in the process of designing accurate solar energy projects in Cyprus, a country with abundant solar radiation and economics favorable to its utilization it was deemed important to investigate the accuracy of the above earlier correlations when applied locally. Using data from one location in Cyprus, the correlation between hourly diffuse fraction of global irradiation and the clearness index is established.
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2. Radiation data set The island of Cyprus is surrounded by the eastern Mediterranean between 34.61 and 35.61N latitude and 32–34.51E longitude; its climate is characterized by one wet season, from November to March, and a long dry season beginning in April and stretching through the month of October. This generalization is modified by the influence of marine factors giving cooler summers and warmer winters near much of the coastline and lowlying areas. Visibility is generally good, however, during spring and autumn months the atmosphere is very hazy with dust brought by prevailing winds (SW and SE) from the Saharan and Arabian deserts (desert depressions). The analysis is based on hourly radiometric data collected at the semi-rural Athalassa site, Cyprus (165 m a.m.s.l., 8 km from the center of Nicosia city) for a 5-year period (1 January 1998–31 December 2002). It is worth mentioning that the time interval 1998–1999 lies in a dry period observed in the eastern Mediterranean basin during the last decade, i.e., 1990–1999. Global solar irradiance (Gh) was measured using a Kipp and Zonen model CM6B (Delft, The Netherlands), while another Kipp&Zonen CM6B with a polar axis shadow band was used to measure diffuse irradiance (Gd). Diffuse irradiance measurements obtained by means of the shadow band has been corrected [13]. The hourly data were checked for inconsistencies to eliminate problems associated with shadow-band misalignments. Solar radiant flux measurements have an estimated experimental error of 2–3%. Further, the hourly irradiation values were checked against quality controls proposed by European Commission—Daylight I, 1993. Data were eliminated when: Gd41.1Gh, Gh41.2G0h, Gd40.8 G0h, Gho5 Wm2, and Gb4G0h. G0h is the hourly horizontal extraterrestrial solar irradiation and Gb is the hourly direct-beam horizontal solar irradiation, obtained as the difference GhGd. Two additional ‘‘extreme’’ limits [4] are applied to identify particular cases of: overcast skies Gd/Gho0.90 for kto0.20 and clear skies Gd/Gh40.90 for kt40.60. Less than 4% and 2.5% of the total data were deleted based on these limits, respectively. The final database consists of about 14,105 hourly values. 3. Results and discussion The accuracy of the different correlations was assessed by means of the widely used statistical indicators, mbe and rmse. These indicators, as a percentage of the averaged ðk¯ d Þ value, are defined as " # N 100 1 X ðkdp kdm Þ , (11) mbe ¼ k¯ d N 1 " #1=2 N 100 1 X 2 rmse ¼ ðkdp kdm Þ , k¯ d N 1
(12)
where N is the number of data points, kdp is the ith predicted value and kdm is the ith measured value. In this analysis, an additional statistical indicator, the t test, combining both mean bias error and root mean square error in their original form [14], is used.
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The t test is defined as
ðN 1ÞMBE2 t-stat ¼ RMSE2 MBE2
1=2 .
(13)
The t-statistic is used for a hypothesis that is defined as H0: there is no systematic difference of the means between measured and estimated kd samples, and Ha: indeed, there is a systematic difference between means. The application of a t test equation such as this, follows several steps: first, t values are calculated as in Eq. (13); second, if the calculated t value is greater than the tabulated one (tcrit), as determined from standard statistical texts, at the specified level of significance, the null hypothesis H0 is rejected and we can conclude that the difference between means is significant. Third, if it is not greater than the tabulated value at, for instance, 5% level of significance, the null hypothesis is accepted. It is noted that if two distributions provide very different variances, they may also be substantially different in shape; in that case, the difference of means may not be a particularly useful thing to know. Further, from individual hourly global and diffuse irradiation measurements, the diffuse fraction kd and the clearness index kt were calculated, through a type of moving average on the kt data; for that purpose moving a window of size 25 through the kt values is applied averaging the kd values as they go, shown in Fig. 1. Accordingly, for three different kt ranges, a resulting hourly polynomial correlation fitted the cloud points of Fig. 1, is expressed as kd ¼ 0:94 þ 0:937kt 5:01k2t þ 3:32k3t
(14)
1.0
diffuse fraction k d
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4 0.6 clearness index k t
0.8
1.0
Fig. 1. Scatter plot of the hourly diffuse fraction kd versus hourly clearness index kt, as determined through moving average procedure. The experimental data (squares) and the fitted curve (dashed line) are also shown.
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1.0
1.0
0.8
0.8
diffuse fraction kd
diffuse fraction kd
for 0.1oktp0.8; kd ¼ 0.987 for ktp0.1 and kd ¼ 0.177 for kt40.8. These intervals comprise 92%, 3% and 5% of the data, respectively. Fig. 1 reveals that Eq. (14) is very close to the experimental data; whereas for kt40.8 Eq. (14) it does not fit the data well [7,9,11]. Next, the earlier correlations reviewed here are shown in Figs. 2a–c, while their statistics are given in Table 1. The entries in Table 1 indicate that the models [6,8,11] provide the lower mbe and rmse values, while models [4,3,7,5,9,10,12] follow in that order. All these correlations predict diffuse irradiation values that are statistically significant at the particular confidence level 97.5%, since their t values are less than the critical one (tcrit). On these graphs, the resulting yearly curve (Eq. (14)) is plotted for comparison purposes, whereas its statistical performance is also included in Table 1. It is clear that the proposed correlation matches the statistical performances of the candidate correlations. Fig. 2a shows the first-order standard correlations examined. This figure indicates relatively good agreement between models [3–5] and the experimental kd values
0.6
0.4 present
0.6
present
0.4
Hawlader [6] Miguel et al. [7]
Orgill-Hollands [3] Reindl et al. [4]
0.2
Karatasou et al. [8]
0.2
Boland et al. [5]
0.0
0.0 0.0
(a)
0.2
0.4
0.6
0.8
1.0
0.0
0.2
(b)
clearness index kt
0.4
0.6
0.8
1.0
clearness index kt
1.0
diffuse fraction kd
0.8
0.6 present Erbs et al [9]
0.4
Chandrasekaran -Kumar [10] oliveira et al [11]
0.2
0.0
(c)
Soares et al [12]
0.0
0.2
0.4 0.6 clearness index kt
0.8
1.0
Fig. 2. (a)–(c): The hourly diffuse fraction kd versus clearness index kt, showing standard and proposed correlations: (a) first-order; (b) second–third order; and (c) fourth-order polynomial correlations.
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Table 1 Statistical performance of existing hourly correlations against Cyprus data, 1998–2002 Key references First-order models Orgill and Hollands [3] Reindl et al. [4] Exponential model Boland et al. [5]a Second– third-order models Hawlader [6] Miguel et al. [7] Karatasou et al. [8] Fourth-order models Erbs et al. [9] Chandrasekaran and Kumar [10] Oliveira et al. [11] Soares et al. [12] This analysis Yearly model (Eq. 14) Wet (Eq. (15)) Dry (Eq. (16)) a
rmse(%)
R2
t-stat.
tcrit.
4.47 3.93
30.2 29.5
0.902 0.901
0.638 0.880
1.96 1.96 1.96
6.95
30.5
0.901
0.923
1.96
1.88 4.52 1.38
29.2 29.8 29.2
0.902 0.901 0.903
0.697 0.893 0.719
1.96 1.96 1.96
3.5 6.15 2.82 8.21
30.9 30.6 29.4 30.9
0.899 0.901 0.898 0.896
0.664 1.214 1.005 1.263
1.22 7.6 5.47
28.4 29.8 31.6
0.926 0.907 0.892
0.756 0.835 0.582
1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96
mbe(%)
Range of clear sky data not mentioned; assigned in the current analysis at kd ¼ 0:153.
represented by the polynomial curve (Eq. (14)), being marginally above for 0.25okto0.60, coinciding for kto0.25 and crossing or being below for larger kt (40.65) values, tending to underestimate diffuse values on that interval. On the other hand, second- and third-order models [6–8] allow closer correspondence with the experimental curve as illustrated in Fig. 2b. However the Miguel et al. [7] model, although developed for a Mediterranean environment as Cyprus, clearly disagrees with the experimental curve in the range 0.25okt o0.60. This disagreement may result from the combined insolation data used over diverse Mediterranean sites that in turn even out the kdkt dependency on local weather. Interestingly, Karatasou et al. [8] and Hawlader [6] models are equally accurate with the proposed one in respect to mbe, rmse, R2 and t test. However, both models tend to underestimate marginally diffuse fractions for kto0.45; whereas for larger kt values they coincide with the experimental curve. Next, Fig. 2c shows the fourth-order standard correlations. It is clear that both Erbs et al. [9] and Chandrasekaran–Kumar [10] models exhibit a tendency to systematically overestimate diffuse irradiation values compared to the experimental curve (Eq (14)) for 0.20okto0.60; while for kt40.60 both models cross to being below the proposed curve. Oliveira et al. [11] model developed for a tropical environment allows a closer correspondence with the experimental polynomial curve, mainly for kto0.5; for larger kT values a tendency to underestimate is observed. On the other hand, Soares et al. [12] model that was established through a neural network technique, underestimates diffuse irradiation values for kt40.40, while for lower kt values it coincides with the experimental curve. Interestingly, the higher-order standard correlations exhibit higher rmse and mbe values; exception to this is the Oliveira et al. [11] model which is in line with the experimental one from the point of view of its statistical performance. A detailed inspection of Fig. 2 in conjunction with Table 1, reveals that most of the polynomial
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correlations are equally accurate for hourly diffuse irradiation predictions at least for the island of Cyprus. On the other hand, several regression models [4,7,9] that are based on combined insolation data from diverse locations over the globe, disagree with the experimental data. It is believed that the main reason for this disagreement is that these models using averaged kd values over finite kt intervals minimize the dependence of kdkt correlation on local weather. Further, considering that Cyprus has two distinct seasons, wet and dry, polynomial regressions similar to Eq. (14) are derived accordingly: For the wet season (November– March): kd ¼ 0:94 þ 0:938kt 4:48k2t þ 2:79k3t
(15)
for 0.1oktp0.85; kd ¼ 0.992 for ktp0.1 and kd ¼ 0.189 for kt40.85. For the dry season (April–October): kd ¼ 0:943 þ 0:586kt 4:31k2t þ 2:97k3t
(16)
for 0.1oktp0.8; kd ¼ 0.962 for ktp0.1 and kd ¼ 0.158 for kt40.8. In Fig. 3 the above experimental seasonal curves (Eqs. (15) and (16)) have been drawn together with the resulting yearly one (Eq. (14)) for comparison purposes. It is clear that the yearly curve lies between the seasonal curves, thus strengthening the assertion of various workers on seasonal dependency [5,9–11]. This clearly implies that the hourly diffuse fraction exhibits a more pronounced seasonal dependence as opposed to daily
1.0
diffuse fraction kd
0.8
0.6
0.4 yearly wet season
0.2
dry season
0.0 0.0
0.2
0.4 0.6 clearness index kt
0.8
1.0
Fig. 3. The hourly diffuse fraction kd as a function of clearness index kt, showing the proposed yearly and seasonal curves separately, for Athalassa, Cyprus.
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correlations [7,15]. Another important feature in Fig. 3 is that for kto0.30 cloud effects are the regulating factor so that seasonal trends between wet and yearly curves are not discernible. Comparisons between earlier correlations and both experimental yearly and wet curves clearly indicate that the diffuse fraction kd at Athalassa, Cyprus, is relatively large at high clearness index values ðkt 40:80Þ, as demonstrated in Table 2. From this table it is obvious that upper kt limits of the present data set are in line with most reported results in the literature concerning both temperate and tropical climates. Viewing Table 2 further one can notice the high kt limits imposed by [6,8,10] models and the proposed curves. It is accepted that seasonal, location differences and moisture content incur substantial influences on the diffuse fraction. It is also known that winter months attain clearer atmospheric conditions as opposed to summer months for most locations. Such conditions may affect directly the atmospheric turbidity which increases more in the presence of air pollutants and aerosols-induced scattering processes that in turn increase the diffuse irradiation ratios for high kt values. It is believed that in the Cyprus environment the dominant factors are the increased moisture content and air pollutants-dust loads, that in turn result differently on kdkt shapes [7,15]. In order to better realize the seasonal evolutions of standards and proposed correlations the data were averaged on a monthly basis by means of rmse and mbe indicators. Figs. 4a and b show monthly average values of these indicators for all models examined. The results clearly indicate a pronounced seasonal dependence. The rmse values increase in summer months implying substantial models deviations at this time interval. Also, it is clear that the models tend to underestimate diffuse irradiation ratios in winter, spring and autumn, while during summer they clearly overestimate [7,8]. Viewing further the overall results, the departures from the experimental data of the various correlations examined may not be due only to differences in local atmospheric and geographic characteristics under which these models have been established. As underlined by several researchers [5,7,9,16] the hourly correlation kdkt is not completely correct due to several errors. As Fig. 4 reveals, the error increases during summer, which in turn implies the necessity of taking into account the effect of solar altitude. In this way, by
Table 2 kt and kd upper limits for various correlations examined Models
kt
kd
Orgill and Hollands [3] Reindl et al. [4] Boland et al. [5] Hawlader [6] Miguel et al. [7] Karatasou et al. [8] Erbs et al. [9] Chandrasekaran and Kumar [10] Oliveira et al. [11] Soares et al. [12] Proposed yearly (Eq. (14)) Proposed wet (Eq. (15))
40.75 40.78 40.78 40.775 40.76 40.78 40.80 40.80 40.75 40.75 40.80 40.85
0.177 0.147 0.153 0.215 0.180 0.200 0.165 0.197 0.180 0.180 0.177 0.189
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Erbs et al [9] Chandrasekaran-Kumar [10] Miguel et al [7]
60
Oliveira et al [11] Soares et al [12] Present
40 (%)
rmse
20 mbe 0 − 20
j
f
m
a
m
(a)
j
j
a
s
o
n
d
months 80
Karatasou et al [8] Hawlader [6] Orgill-Hollands [3] Reindl et al [4] Boland et al [5] Present
60
rmse
(%)
40
20
0 mbe − 20
(b)
j
f
m
a
m
j
j
a
s
o
n
d
months
Fig. 4. (a) and (b): Comparison of the monthly rmse and mbe errors of hourly diffuse irradiation correlations examined here and the proposed yearly polynomial curve, for Athalassa, Cyprus.
applying the methodology of [7], the differences between experimental and predicted diffuse values as a function of solar altitude (sin h), is given in Fig. 5, assuming two kt intervals: 0.35okto0.80 and 0.80okto1.0. Fig. 5 reveals that the difference increases with solar altitude in the first kt range, while in the second range the difference is maximized at lower solar altitudes. Further, an extensive set of experimental data is considered an essential part of this research in order to provide a basis for the verification of the accuracy of candidate correlations tested. For this purpose, another data set of diffuse irradiation values obtained at Athalassa, Cyprus, during the time period 1988–1992 is considered. As discussed above, some problems concerning solar altitude usually have a detrimental effect on the diffuse irradiation predictions. It is worth noting that Skartveit and Olseth [17] underlined that under partly cloudy and clear skies, solar altitudes lower than 301 have a marked effect on the diffuse fraction, while this effect is minimal at higher solar elevations.
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0.4 0.35 < kt < 0.80
difference (Gdp - Gdm)
0.3
0.80 < kt < 1.0
0.2 0.1 0.0 − 0.1 − 0.2 0.2
0.4
0.6 Sin (h)
0.8
1.0
Fig. 5. Differences between experimental and predicted diffuse irradiation values as a function of the solar altitude angle.
In addition, Perez et al. [18] testing several diffuse fraction models pointed out that the algorithms employing solar elevation-dependent kt boundaries perform better than the others, which do not use solar elevation as an active variable. Therefore, by omitting the data collected round sunrise–sunset hours, approximately when ho101 [5,7,16–18], predicted diffuse irradiation values through each model are compared with experimental ones. It is found that most of the standard correlations predict accurately the experimental results; however, with regard to statistical performances the original models [5,7,8] and the proposed curve are the best in reproducing the mean diffuse irradiation value, mbe, rmse and R2. Fig. 6 shows experimental diffuse irradiation values versus predicted ones through these models and the proposed one. It is worth noting that these findings are in line with the results reported by Miguel et al. [7] and Notton et al. [16] in respect to statistical indicators. 4. Concluding remarks The analyses of hourly global and diffuse solar irradiance measurements covering the period between 1 January 1998 and 31 December 2002 in Athalssa, Cyprus, are briefly summarized as follows: The proposed hourly correlations relating kd with kt are based on a yearly, and on seasonal basis (wet and dry periods). It is found that the experimental correlation matches the statistical performances of the existing models. All earlier correlations and the suggested one yield diffuse irradiation values accurately, while their predictions are statistically significant. The disparity of earlier correlations from the suggested one for high kt 40:7 values may be attributed not only to different climatic conditions but also to several other reasons such as, atmospheric moisture content, clear skies and dust-induced high aerosol loads that in turn impact differently on the kdkt shape. From the overall analysis it can be concluded that the standard correlations tested are location-independent and can also be applied for other locations in the eastern Mediterranean region having the same climatic and geographic characteristics as Cyprus.
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0
0
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0
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(c)
(a) 800
800
600
600
400
400
200
200
0
0 0
(b)
200
400
600
0
800
(d)
Fig. 6. Validation of various correlations tested: (a) proposed Eq. (14); (b) Karatasou et al. [8]; (c) Boland et al. [5]; and (d) Miguel et al. [7]; the diagonal lines are given.
References [1] Jacovides CP, Theophilou K, Tymvios FS, Pashiardes S. Wind statistics for coastal station in Cyprus. Theor Appl Climatol 2002;72:259–63. [2] Liu BYH, Jordan RC. The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Sol Energy 1960;4:1–19. [3] Orgill JF, Hollands KGT. Correlation equation for hourly diffuse radiation on a horizontal surface. Sol Energy 1977;19:357–9. [4] Reindl DT, Beckman WA, Duffie JA. Diffuse fraction correlations. Sol Energy 1990;45:1–7. [5] Boland J, Scott L, Luther M. Modeling the diffuse fraction of global solar radiation on a horizontal surface. Environmetrics 2001;12:103–16. [6] Hawlader MNA. Diffuse, global and extraterrestrial solar radiation for Singapore. Int J Ambient Energy 1984;5:31–8.
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[7] Miguel A, Bilbao J, Aguiar R, Kambezidis H, Negro E. Diffuse solar irradiation model evaluation in the north Mediterranean belt area. Sol Energy 2001;70:143–53. [8] Karatasou S, Santamouris M, Geros V. Analysis of experimental data on diffuse solar radiation in Athens, Greece, for building applications. Int J Sustain Energy 2003;23(1–2):1–11. [9] Erbs DG, Klein SA, Duffie JA. Estimation of the diffuse radiation fraction for hourly, daily and monthly average global radiation. Sol Energy 1982;28:293–302. [10] Chandrasekaran J, Kumar S. Hourly diffuse fraction correlation at a tropical location. Sol Energy 1994;53:505–10. [11] Oliveira AP, Escobedo JF, Machado AJ, Soares J. Correlation models of diffuse solar radiation applied to the city of Sao Paulo, Brazil. Appl Energy 2002;71:59–73. [12] Soares J, Oliveira AP, Boznar MZ, Mlakar P, Escobedo JF, Machado AJ. Modeling hourly diffuse solar radiation in the city of Sao Paulo using a neural-network technique. Appl Energy 2004;79:201–14. [13] Batlles FJ, Olmo FJ, Alados-Arboledas L. On shadowband correction methods for diffuse irradiance measurements. Sol Energy 1995;54:105–14. [14] Jacovides CP, Kontogiannis H. Statistical procedure for the evaluation of evapotranspiration computing models. Agric Water Manage 1995;27:365–71. [15] Jacovides CP, Hadjioannou L, Pashiardis S, Stefanou L. On the diffuse fraction of daily and monthly global radiation for the island of Cyprus. Sol Energy 1996;56:565–72. [16] Notton G, Cristofari C, Muselli M, Poggi P. Calculation on an hourly basis of solar diffuse irradiations from global data for horizontal surfaces in Ajaccio. Energy Convers Manage 2004;45:2849–66. [17] Skartveit A, Olseth JA. A model for the diffuse fraction of hourly global radiation. Sol Energy 1987;38:271–4. [18] Perez R, Seals R, Zelenka A, Ineichen P. Climatic evaluation of models that predict hourly direct irradiance from hourly global irradiance: prospects for performance improvements. Sol Energy 1990;44:99–108.
Renewable Energy 31 (2006) 2492–2504 www.elsevier.com/locate/renene
Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation C.P. Jacovidesa,, F.S. Tymviosa,b, V.D. Assimakopoulosc, N.A. Kaltsounidesa a
Laboratory of Meteorology, Department of Applied Physics, University of Athens, University Campus, Builds PHYS-V, Athens 15784, Greece b Meteorological Service of Cyprus, Nicosia, Cyprus c Institute for Environmental Research and Sustainable Development, National Observatory of Athens, Athens 15236, Greece Received 25 January 2005; accepted 30 November 2005 Available online 24 January 2006
Abstract Proper design and performance predictions of solar energy systems require accurate information on the availability of solar radiation. The diffuse-to-global solar radiation correlation, originally developed by Liu and Jordan, has been extensively used as the technique providing accurate results, although it is latitude dependent. Thus, in the present study, empirical correlations of this type were developed to establish a relationship between the hourly diffuse fraction (kd) and the hourly clearness index (kt) using hourly global and diffuse irradiation measurements on a horizontal surface performed at Athalassa, Cyprus. The proposed correlations were compared against 10 models available in the literature in terms of the widely used statistical indicators, rmse, mbe and t test. From this analysis, it can be concluded that the proposed yearly correlation predicts diffuse values accurately, whereas all candidate models examined appear to be location-independent for diffuse irradiation predictions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Hourly diffuse fraction; Clearness index; Standard models; Statistical indicators; Athalassa (Cyprus)
Corresponding author. Tel.: +30 210 7276931; fax: +30 210 7295281.
E-mail address: [email protected] (C.P. Jacovides). 0960-1481/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2005.11.009
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1. Introduction The increasing global energy demands and increasing fossil fuel prices stimulate countries to downsize energy consumption and exploit renewable energy sources. In addition, environmental problems caused by mass consumption of fossil energy (e.g., global warming), are also reason for concern. For example, the European Union aims at achieving a 25 percent reduction in carbon dioxide emissions by the year 2005; there exists also an additional EU renewable energy obligation that purports to be one of the key mechanisms for enabling the Mediterranean partners to reach their renewable energy targets (20% by 2010). In general, solar and wind energy are thought to be good alternative energy sources for overcoming these problems due to their safety and positive contribution to the global environmental state because of their lack of emissions during operation [1]. Reliable solar radiation measurements for estimating the dynamic behavior of solar energy systems’ processes and for simulating long-term operations are required. For thermal analysis performance through transient simulation algorithms, a crucial input is the solar energy components incident on the collector surfaces. Usually, an hourly time step is used in these systems and thus hourly solar energy data are needed which in turn are seldom available at the site of interest. Given global solar irradiation measurements on a horizontal surface (the most widely available data for solar energy) direct and diffuse radiant components can be obtained from global solar energy data through various correlations. Thus, as early as the early 1960s, numerous models for evaluating the diffuse component based on the pioneer work of Liu and Jordan [2] appeared in the literature. These models are usually expressed in terms of 1st–4th-degree polynomials relating the diffuse fraction kd (ratio of the diffuse-to-global solar radiation) with the clearness index kt (ratio of the global-to-extraterrestrial solar radiation). Nevertheless, the range of the diffuse fraction, as reported in the literature, suggests the desirability for recalibration accounting for local climatic differences. This study investigates the applicability of various standard models correlating hourly diffuse fraction kd and kt, in the eastern Mediterranean basin. Of direct pertinence are the works of Orgill and Hollands [3], Reindl et al. [4], Boland et al. [5], Hawlader [6], Miguel et al. [7], Karatasou et al. [8], Erbs et al. [9], Chandrasekaran and Kumar [10], Oliveira et al. [11], and Soares et al. [12], who established hourly correlations between kd and kt under diverse climatic conditions. The models can be categorized as: firstorder (Eqs. (1)–(3)), second–third-order (Eqs. (6)–(8)), and fourth-order (Eqs. (9)–(12)) correlations that are briefly reviewed below. Orgill and Hollands [3] used data from one location in Toronto, Canada, derived the following correlation between kd and kt: kd ¼ 1:557 1:84kt
(1)
for 0.35pktp0.75; kd ¼ 1.00.249 kt for kto0.35, and kd ¼ 0.177 for kt40.75. Reindl et al. [4] studied the influence of climatic-geometric variables on the hourly diffuse fraction based on data from five European and US locations; they established the following expressions: kd ¼ 1:45 1:67kt for 0.3okto0.78; kd ¼ 1.020.248kt for ktp0.3, and kd ¼ 0.147 for ktX0.78.
(2)
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Boland et al. [5] used data from one location in Victoria, Australia, constructed a simple exponential correlation of the form, kd ¼
1 . 1 þ e7:997ðkt 0:586Þ
(3)
Hawlader [6] using data from a tropical site in Singapore derived the second-order polynomial correlation, kd ¼ 1:135 0:9422kt 0:3878k2t
(4)
for 0.225o kto0.775; kd ¼ 0.915 for ktp0.225 and kd ¼ 0.215 for ktX0.775. Miguel et al. [7] used an assembled data set from several countries in the North Mediterranean Belt area, and yielded a third-order polynomial for hourly diffuse fraction correlations, kd ¼ 0:724 þ 2:738kt 8:32k2t þ 4:967k3t
(5)
for 0.21oktp0.76; kd ¼ 0.9950.081kt for ktp0.21 and kd ¼ 0.18 for kt40.76. Karatasou et al. [8] based on data from Athens, Greece, proposed a third-order polynomial correlation, kd ¼ 0:9995 0:05kt 2:4156k2t þ 1:4926k3t
(6)
for 0.oktp0.78; and kd ¼ 0.20 for kt40.78. Erbs et al. [9] used an assembled data set from four US locations derived a fourth-order polynomial correlation for the hourly diffuse fraction, kd ¼ 0:951 0:1604kt þ 4:388k2t 16:638k3t þ 12:336k4t
(7)
for 0.22oktp0.80; kd ¼ 1.00.09kt for ktp0.22 and kd ¼ 0.165 for kt40.80. Chandrasekaran and Kumar [10] used data from a tropical environment in Madras, India, derived a fourth-order polynomial correlation, kd ¼ 0:9686 þ 0:1325kt þ 1:4183k2t 10:1862k3t þ 8:3733k4t
(8)
for 0.24oktp0.80; kd ¼ 1.00860.178kt for ktp0.24, and kd ¼ 0.197 for kt40.80. Oliveira et al. [11] using data from a tropical Sao Paolo site, Brazil, proposed a fourthorder polynomial correlation, kd ¼ 0:97 þ 0:8kt 3:0k2t 3:1k3t þ 5:2k4t
(9)
for 0.17okto0.75; kd ¼ 1.0 for ktp0.17, and kd ¼ 0.17 for kt40.75. Finally, Soares et al. [12] based on the same data set as [11], established a synthetic fourth-order polynomial correlation by means of a neural network technique, kd ¼ 0:90 þ 1:1kt 4:5k2t þ 0:01k3t þ 3:14k4t
(10)
for 0.17okto0.75; kd ¼ 1.0 for ktp0.17, and kd ¼ 0.17 for kt40.75 Furthermore, in the process of designing accurate solar energy projects in Cyprus, a country with abundant solar radiation and economics favorable to its utilization it was deemed important to investigate the accuracy of the above earlier correlations when applied locally. Using data from one location in Cyprus, the correlation between hourly diffuse fraction of global irradiation and the clearness index is established.
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2. Radiation data set The island of Cyprus is surrounded by the eastern Mediterranean between 34.61 and 35.61N latitude and 32–34.51E longitude; its climate is characterized by one wet season, from November to March, and a long dry season beginning in April and stretching through the month of October. This generalization is modified by the influence of marine factors giving cooler summers and warmer winters near much of the coastline and lowlying areas. Visibility is generally good, however, during spring and autumn months the atmosphere is very hazy with dust brought by prevailing winds (SW and SE) from the Saharan and Arabian deserts (desert depressions). The analysis is based on hourly radiometric data collected at the semi-rural Athalassa site, Cyprus (165 m a.m.s.l., 8 km from the center of Nicosia city) for a 5-year period (1 January 1998–31 December 2002). It is worth mentioning that the time interval 1998–1999 lies in a dry period observed in the eastern Mediterranean basin during the last decade, i.e., 1990–1999. Global solar irradiance (Gh) was measured using a Kipp and Zonen model CM6B (Delft, The Netherlands), while another Kipp&Zonen CM6B with a polar axis shadow band was used to measure diffuse irradiance (Gd). Diffuse irradiance measurements obtained by means of the shadow band has been corrected [13]. The hourly data were checked for inconsistencies to eliminate problems associated with shadow-band misalignments. Solar radiant flux measurements have an estimated experimental error of 2–3%. Further, the hourly irradiation values were checked against quality controls proposed by European Commission—Daylight I, 1993. Data were eliminated when: Gd41.1Gh, Gh41.2G0h, Gd40.8 G0h, Gho5 Wm2, and Gb4G0h. G0h is the hourly horizontal extraterrestrial solar irradiation and Gb is the hourly direct-beam horizontal solar irradiation, obtained as the difference GhGd. Two additional ‘‘extreme’’ limits [4] are applied to identify particular cases of: overcast skies Gd/Gho0.90 for kto0.20 and clear skies Gd/Gh40.90 for kt40.60. Less than 4% and 2.5% of the total data were deleted based on these limits, respectively. The final database consists of about 14,105 hourly values. 3. Results and discussion The accuracy of the different correlations was assessed by means of the widely used statistical indicators, mbe and rmse. These indicators, as a percentage of the averaged ðk¯ d Þ value, are defined as " # N 100 1 X ðkdp kdm Þ , (11) mbe ¼ k¯ d N 1 " #1=2 N 100 1 X 2 rmse ¼ ðkdp kdm Þ , k¯ d N 1
(12)
where N is the number of data points, kdp is the ith predicted value and kdm is the ith measured value. In this analysis, an additional statistical indicator, the t test, combining both mean bias error and root mean square error in their original form [14], is used.
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The t test is defined as
ðN 1ÞMBE2 t-stat ¼ RMSE2 MBE2
1=2 .
(13)
The t-statistic is used for a hypothesis that is defined as H0: there is no systematic difference of the means between measured and estimated kd samples, and Ha: indeed, there is a systematic difference between means. The application of a t test equation such as this, follows several steps: first, t values are calculated as in Eq. (13); second, if the calculated t value is greater than the tabulated one (tcrit), as determined from standard statistical texts, at the specified level of significance, the null hypothesis H0 is rejected and we can conclude that the difference between means is significant. Third, if it is not greater than the tabulated value at, for instance, 5% level of significance, the null hypothesis is accepted. It is noted that if two distributions provide very different variances, they may also be substantially different in shape; in that case, the difference of means may not be a particularly useful thing to know. Further, from individual hourly global and diffuse irradiation measurements, the diffuse fraction kd and the clearness index kt were calculated, through a type of moving average on the kt data; for that purpose moving a window of size 25 through the kt values is applied averaging the kd values as they go, shown in Fig. 1. Accordingly, for three different kt ranges, a resulting hourly polynomial correlation fitted the cloud points of Fig. 1, is expressed as kd ¼ 0:94 þ 0:937kt 5:01k2t þ 3:32k3t
(14)
1.0
diffuse fraction k d
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4 0.6 clearness index k t
0.8
1.0
Fig. 1. Scatter plot of the hourly diffuse fraction kd versus hourly clearness index kt, as determined through moving average procedure. The experimental data (squares) and the fitted curve (dashed line) are also shown.
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1.0
1.0
0.8
0.8
diffuse fraction kd
diffuse fraction kd
for 0.1oktp0.8; kd ¼ 0.987 for ktp0.1 and kd ¼ 0.177 for kt40.8. These intervals comprise 92%, 3% and 5% of the data, respectively. Fig. 1 reveals that Eq. (14) is very close to the experimental data; whereas for kt40.8 Eq. (14) it does not fit the data well [7,9,11]. Next, the earlier correlations reviewed here are shown in Figs. 2a–c, while their statistics are given in Table 1. The entries in Table 1 indicate that the models [6,8,11] provide the lower mbe and rmse values, while models [4,3,7,5,9,10,12] follow in that order. All these correlations predict diffuse irradiation values that are statistically significant at the particular confidence level 97.5%, since their t values are less than the critical one (tcrit). On these graphs, the resulting yearly curve (Eq. (14)) is plotted for comparison purposes, whereas its statistical performance is also included in Table 1. It is clear that the proposed correlation matches the statistical performances of the candidate correlations. Fig. 2a shows the first-order standard correlations examined. This figure indicates relatively good agreement between models [3–5] and the experimental kd values
0.6
0.4 present
0.6
present
0.4
Hawlader [6] Miguel et al. [7]
Orgill-Hollands [3] Reindl et al. [4]
0.2
Karatasou et al. [8]
0.2
Boland et al. [5]
0.0
0.0 0.0
(a)
0.2
0.4
0.6
0.8
1.0
0.0
0.2
(b)
clearness index kt
0.4
0.6
0.8
1.0
clearness index kt
1.0
diffuse fraction kd
0.8
0.6 present Erbs et al [9]
0.4
Chandrasekaran -Kumar [10] oliveira et al [11]
0.2
0.0
(c)
Soares et al [12]
0.0
0.2
0.4 0.6 clearness index kt
0.8
1.0
Fig. 2. (a)–(c): The hourly diffuse fraction kd versus clearness index kt, showing standard and proposed correlations: (a) first-order; (b) second–third order; and (c) fourth-order polynomial correlations.
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Table 1 Statistical performance of existing hourly correlations against Cyprus data, 1998–2002 Key references First-order models Orgill and Hollands [3] Reindl et al. [4] Exponential model Boland et al. [5]a Second– third-order models Hawlader [6] Miguel et al. [7] Karatasou et al. [8] Fourth-order models Erbs et al. [9] Chandrasekaran and Kumar [10] Oliveira et al. [11] Soares et al. [12] This analysis Yearly model (Eq. 14) Wet (Eq. (15)) Dry (Eq. (16)) a
rmse(%)
R2
t-stat.
tcrit.
4.47 3.93
30.2 29.5
0.902 0.901
0.638 0.880
1.96 1.96 1.96
6.95
30.5
0.901
0.923
1.96
1.88 4.52 1.38
29.2 29.8 29.2
0.902 0.901 0.903
0.697 0.893 0.719
1.96 1.96 1.96
3.5 6.15 2.82 8.21
30.9 30.6 29.4 30.9
0.899 0.901 0.898 0.896
0.664 1.214 1.005 1.263
1.22 7.6 5.47
28.4 29.8 31.6
0.926 0.907 0.892
0.756 0.835 0.582
1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96
mbe(%)
Range of clear sky data not mentioned; assigned in the current analysis at kd ¼ 0:153.
represented by the polynomial curve (Eq. (14)), being marginally above for 0.25okto0.60, coinciding for kto0.25 and crossing or being below for larger kt (40.65) values, tending to underestimate diffuse values on that interval. On the other hand, second- and third-order models [6–8] allow closer correspondence with the experimental curve as illustrated in Fig. 2b. However the Miguel et al. [7] model, although developed for a Mediterranean environment as Cyprus, clearly disagrees with the experimental curve in the range 0.25okt o0.60. This disagreement may result from the combined insolation data used over diverse Mediterranean sites that in turn even out the kdkt dependency on local weather. Interestingly, Karatasou et al. [8] and Hawlader [6] models are equally accurate with the proposed one in respect to mbe, rmse, R2 and t test. However, both models tend to underestimate marginally diffuse fractions for kto0.45; whereas for larger kt values they coincide with the experimental curve. Next, Fig. 2c shows the fourth-order standard correlations. It is clear that both Erbs et al. [9] and Chandrasekaran–Kumar [10] models exhibit a tendency to systematically overestimate diffuse irradiation values compared to the experimental curve (Eq (14)) for 0.20okto0.60; while for kt40.60 both models cross to being below the proposed curve. Oliveira et al. [11] model developed for a tropical environment allows a closer correspondence with the experimental polynomial curve, mainly for kto0.5; for larger kT values a tendency to underestimate is observed. On the other hand, Soares et al. [12] model that was established through a neural network technique, underestimates diffuse irradiation values for kt40.40, while for lower kt values it coincides with the experimental curve. Interestingly, the higher-order standard correlations exhibit higher rmse and mbe values; exception to this is the Oliveira et al. [11] model which is in line with the experimental one from the point of view of its statistical performance. A detailed inspection of Fig. 2 in conjunction with Table 1, reveals that most of the polynomial
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correlations are equally accurate for hourly diffuse irradiation predictions at least for the island of Cyprus. On the other hand, several regression models [4,7,9] that are based on combined insolation data from diverse locations over the globe, disagree with the experimental data. It is believed that the main reason for this disagreement is that these models using averaged kd values over finite kt intervals minimize the dependence of kdkt correlation on local weather. Further, considering that Cyprus has two distinct seasons, wet and dry, polynomial regressions similar to Eq. (14) are derived accordingly: For the wet season (November– March): kd ¼ 0:94 þ 0:938kt 4:48k2t þ 2:79k3t
(15)
for 0.1oktp0.85; kd ¼ 0.992 for ktp0.1 and kd ¼ 0.189 for kt40.85. For the dry season (April–October): kd ¼ 0:943 þ 0:586kt 4:31k2t þ 2:97k3t
(16)
for 0.1oktp0.8; kd ¼ 0.962 for ktp0.1 and kd ¼ 0.158 for kt40.8. In Fig. 3 the above experimental seasonal curves (Eqs. (15) and (16)) have been drawn together with the resulting yearly one (Eq. (14)) for comparison purposes. It is clear that the yearly curve lies between the seasonal curves, thus strengthening the assertion of various workers on seasonal dependency [5,9–11]. This clearly implies that the hourly diffuse fraction exhibits a more pronounced seasonal dependence as opposed to daily
1.0
diffuse fraction kd
0.8
0.6
0.4 yearly wet season
0.2
dry season
0.0 0.0
0.2
0.4 0.6 clearness index kt
0.8
1.0
Fig. 3. The hourly diffuse fraction kd as a function of clearness index kt, showing the proposed yearly and seasonal curves separately, for Athalassa, Cyprus.
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correlations [7,15]. Another important feature in Fig. 3 is that for kto0.30 cloud effects are the regulating factor so that seasonal trends between wet and yearly curves are not discernible. Comparisons between earlier correlations and both experimental yearly and wet curves clearly indicate that the diffuse fraction kd at Athalassa, Cyprus, is relatively large at high clearness index values ðkt 40:80Þ, as demonstrated in Table 2. From this table it is obvious that upper kt limits of the present data set are in line with most reported results in the literature concerning both temperate and tropical climates. Viewing Table 2 further one can notice the high kt limits imposed by [6,8,10] models and the proposed curves. It is accepted that seasonal, location differences and moisture content incur substantial influences on the diffuse fraction. It is also known that winter months attain clearer atmospheric conditions as opposed to summer months for most locations. Such conditions may affect directly the atmospheric turbidity which increases more in the presence of air pollutants and aerosols-induced scattering processes that in turn increase the diffuse irradiation ratios for high kt values. It is believed that in the Cyprus environment the dominant factors are the increased moisture content and air pollutants-dust loads, that in turn result differently on kdkt shapes [7,15]. In order to better realize the seasonal evolutions of standards and proposed correlations the data were averaged on a monthly basis by means of rmse and mbe indicators. Figs. 4a and b show monthly average values of these indicators for all models examined. The results clearly indicate a pronounced seasonal dependence. The rmse values increase in summer months implying substantial models deviations at this time interval. Also, it is clear that the models tend to underestimate diffuse irradiation ratios in winter, spring and autumn, while during summer they clearly overestimate [7,8]. Viewing further the overall results, the departures from the experimental data of the various correlations examined may not be due only to differences in local atmospheric and geographic characteristics under which these models have been established. As underlined by several researchers [5,7,9,16] the hourly correlation kdkt is not completely correct due to several errors. As Fig. 4 reveals, the error increases during summer, which in turn implies the necessity of taking into account the effect of solar altitude. In this way, by
Table 2 kt and kd upper limits for various correlations examined Models
kt
kd
Orgill and Hollands [3] Reindl et al. [4] Boland et al. [5] Hawlader [6] Miguel et al. [7] Karatasou et al. [8] Erbs et al. [9] Chandrasekaran and Kumar [10] Oliveira et al. [11] Soares et al. [12] Proposed yearly (Eq. (14)) Proposed wet (Eq. (15))
40.75 40.78 40.78 40.775 40.76 40.78 40.80 40.80 40.75 40.75 40.80 40.85
0.177 0.147 0.153 0.215 0.180 0.200 0.165 0.197 0.180 0.180 0.177 0.189
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Erbs et al [9] Chandrasekaran-Kumar [10] Miguel et al [7]
60
Oliveira et al [11] Soares et al [12] Present
40 (%)
rmse
20 mbe 0 − 20
j
f
m
a
m
(a)
j
j
a
s
o
n
d
months 80
Karatasou et al [8] Hawlader [6] Orgill-Hollands [3] Reindl et al [4] Boland et al [5] Present
60
rmse
(%)
40
20
0 mbe − 20
(b)
j
f
m
a
m
j
j
a
s
o
n
d
months
Fig. 4. (a) and (b): Comparison of the monthly rmse and mbe errors of hourly diffuse irradiation correlations examined here and the proposed yearly polynomial curve, for Athalassa, Cyprus.
applying the methodology of [7], the differences between experimental and predicted diffuse values as a function of solar altitude (sin h), is given in Fig. 5, assuming two kt intervals: 0.35okto0.80 and 0.80okto1.0. Fig. 5 reveals that the difference increases with solar altitude in the first kt range, while in the second range the difference is maximized at lower solar altitudes. Further, an extensive set of experimental data is considered an essential part of this research in order to provide a basis for the verification of the accuracy of candidate correlations tested. For this purpose, another data set of diffuse irradiation values obtained at Athalassa, Cyprus, during the time period 1988–1992 is considered. As discussed above, some problems concerning solar altitude usually have a detrimental effect on the diffuse irradiation predictions. It is worth noting that Skartveit and Olseth [17] underlined that under partly cloudy and clear skies, solar altitudes lower than 301 have a marked effect on the diffuse fraction, while this effect is minimal at higher solar elevations.
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0.4 0.35 < kt < 0.80
difference (Gdp - Gdm)
0.3
0.80 < kt < 1.0
0.2 0.1 0.0 − 0.1 − 0.2 0.2
0.4
0.6 Sin (h)
0.8
1.0
Fig. 5. Differences between experimental and predicted diffuse irradiation values as a function of the solar altitude angle.
In addition, Perez et al. [18] testing several diffuse fraction models pointed out that the algorithms employing solar elevation-dependent kt boundaries perform better than the others, which do not use solar elevation as an active variable. Therefore, by omitting the data collected round sunrise–sunset hours, approximately when ho101 [5,7,16–18], predicted diffuse irradiation values through each model are compared with experimental ones. It is found that most of the standard correlations predict accurately the experimental results; however, with regard to statistical performances the original models [5,7,8] and the proposed curve are the best in reproducing the mean diffuse irradiation value, mbe, rmse and R2. Fig. 6 shows experimental diffuse irradiation values versus predicted ones through these models and the proposed one. It is worth noting that these findings are in line with the results reported by Miguel et al. [7] and Notton et al. [16] in respect to statistical indicators. 4. Concluding remarks The analyses of hourly global and diffuse solar irradiance measurements covering the period between 1 January 1998 and 31 December 2002 in Athalssa, Cyprus, are briefly summarized as follows: The proposed hourly correlations relating kd with kt are based on a yearly, and on seasonal basis (wet and dry periods). It is found that the experimental correlation matches the statistical performances of the existing models. All earlier correlations and the suggested one yield diffuse irradiation values accurately, while their predictions are statistically significant. The disparity of earlier correlations from the suggested one for high kt 40:7 values may be attributed not only to different climatic conditions but also to several other reasons such as, atmospheric moisture content, clear skies and dust-induced high aerosol loads that in turn impact differently on the kdkt shape. From the overall analysis it can be concluded that the standard correlations tested are location-independent and can also be applied for other locations in the eastern Mediterranean region having the same climatic and geographic characteristics as Cyprus.
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800
600
600
400
400
200
200
0
0
200
400
600
0
800
0
2503
200
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800
200
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(c)
(a) 800
800
600
600
400
400
200
200
0
0 0
(b)
200
400
600
0
800
(d)
Fig. 6. Validation of various correlations tested: (a) proposed Eq. (14); (b) Karatasou et al. [8]; (c) Boland et al. [5]; and (d) Miguel et al. [7]; the diagonal lines are given.
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