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Computing diffuse fraction of global horizontal solar radiation: A model comparison
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Solar Energy 86 (2012) 1796–1802 www.elsevier.com/locate/solener
Computing diffuse fraction of global horizontal solar radiation: A model comparison Sokol Dervishi ⇑, Ardeshir Mahdavi Department of Building Physics and Building Ecology, Vienna University of Technology, Vienna, Austria Received 3 August 2011; received in revised form 11 March 2012; accepted 12 March 2012 Available online 5 April 2012 Communicated by: Associate Editor Frank Vignola
Abstract For simulation-based prediction of buildings’ energy use or expected gains from building-integrated solar energy systems, information on both direct and diffuse component of solar radiation is necessary. Available measured data are, however, typically restricted to global horizontal irradiance. There have been thus many efforts in the past to develop algorithms for the derivation of the diffuse fraction of solar irradiance. In this context, the present paper compares eight models for estimating diffuse fraction of irradiance based on a database of measured irradiance from Vienna, Austria. These models generally involve mathematical formulations with multiple coefficients whose values are typically valid for a specific location. Subsequent to a first comparison of these eight models, three better performing models were selected for a more detailed analysis. Thereby, the coefficients of the models were modified to account for Vienna data. The results suggest that some models can provide relatively reliable estimations of the diffuse fractions of the global irradiance. The calibration procedure could only slightly improve the models’ performance. Ó 2012 Elsevier Ltd. All rights reserved. Keywords: Diffuse irradiance; Solar irradiance; Diffuse fraction models; Measurements
1. Introduction A crucial input required in the simulation of buildings’ energy performance is the availability of detailed information on the magnitudes of diffuse and direct irradiance data. Moreover, configuration and sizing of solar energy systems (e.g. photovoltaic cells, solar-thermal collectors) necessitates reliable solar radiation measurements. However, concurrent measured data of global and diffuse irradiance on horizontal surface or direct normal irradiance are available only for a limited number of locations. The measurement of global horizontal irradiance is rather simple and cost-effective. It can be, conceivably, an integral part of the sensory equipment of every building. Given global solar irradiation measurements on a horizontal surface ⇑ Corresponding author. Tel.: +43 1 58801 27003; fax: +43 1 58801 27093. E-mail address: [email protected] (S. Dervishi).
0038-092X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2012.03.008
(as the most widely available data) direct and diffuse solar radiation components can be obtained through various correlations. Starting from the early 1960s, numerous models for evaluating the diffuse component based on the pioneer work of Liu and Jordan (1960) appeared in the literature, such as Orgill and Hollands (1977), Erbs et al. (1982), Reindl et al. (1990), Lam and Li (1996), and Perez et al. (1990). These models are usually expressed in terms of first to fourth degree polynomial functions 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 on horizontal surface), as well as to other variables such as solar altitude, air temperature, relative humidity. Although these models are typically derived following sound approaches, their performance appears to lessen once they are applied to regions other than those, which provided the initial data for model development (Soler, 1990; LeBaron and Dirmhirn, 1983; Tuller, 1976).
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
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Nomenclature I Id Io Isc Ib kt kd a Ta /
global horizontal irradiance (W m2) diffuse horizontal irradiance (W m2) extraterrestrial solar irradiance (W m2) extraterrestrial solar constant irradiance (1367 W m2) direct irradiance on a horizontal surface (W m2) clearness index (dimensionless) diffuse fraction (dimensionless) sun altitude (°) outdoor air temperature (°C) relative humidity (%)
Thus, further studies would be necessary to better accommodate the influences of local climatic factors on model performance (Wong and Chow, 2001; Boland et al., 2001; Jacovides et al., 2006). This study investigates the applicability of various standard models correlating hourly diffuse fraction for Vienna– Austria, based on their prior reported performance and the availability of required measurement data for model comparison.
monthly average global radiation on a horizontal surface (W m2) tilt angle of a surface measured from the horizontal (°) air mass (dimensionless) at actual pressure air mass (dimensionless) at standard pressure (1013.25 mbar) the local air-pressure (mbar) day number in the year (No.) number of data zenith angle (°)
Gt f ma mr p N n hz
Interval : 0:22 6 k t 6 0:8 k d ¼ 0:9511 0:1604k t þ 4:39k 2t 16:64k 3t þ 12:34k 4t ð2Þ Interval : k t > 0:8; kt ¼
It I o sinðaÞ
k d ¼ 0:165
kd ¼
Id It
ð3Þ ð4Þ
where
360n cos hz I o ¼ I sc 1 þ 0:33 cos 365
ð5Þ
2. Approach For the purpose of the present study, eight models were considered for estimating diffuse fraction of irradiance as documented in Erbs et al. (1982), Orgill and Hollands (1977), Reindl et al. (1990), Lam and Li (1996), Skartveit and Olseth (1987), Louche et al. (1991), Maxwell (1987), and Vignola and McDaniels (1984). The comparison was based on measured irradiance data from Vienna, Austria. The selection of the models was influenced by their prior reported performance as well as the availability of required measurement data for model comparison. The models typically involve mathematical formulations with multiple coefficients whose values are generally valid for a specific location. Subsequent to a first comparison of the eight models, three better performing ones were selected for further analysis. Thereby, the original model versions were compared with modified versions with coefficients adjusted for a better match to Vienna data. Toward this end, polynomial curves fitting functions were applied using standard curve fitting toolbox in MATLAB (2010). A short summary of the selected models is provided in the following. 2.1. Models 2.1.1. Erbs model (ER) Erbs et al. (1982) used direct normal and global irradiance data on a horizontal surface from 5 stations in USA. Diffuse fraction kd is given by Interval : k t 6 0:22;
k d ¼ 1 0:09k t
ð1Þ
2.1.2. Reindl model (RE) Reindl et al. (1990) estimated diffuse fraction kd based on measured global and diffuse horizontal irradiance data from 5 locations in the USA and Europe. The algorithm considers three characteristic intervals using the following parameters: clearness index (kt), sun altitude (a), outdoor air temperature (Ta), and the relative humidity u. Depending on clearness index value, the diffuse fractions (Id/It) are calculated as per Eqs. (6)–(8). Interval : 0 6 k t 6 0:3 k d ¼ 1:0 0:232k t þ 0:0239 sin a 0:000682T a þ þ0:019/ ð6Þ Interval : 0:3 < k t 6 0:78 k d ¼ 1:329 1:716k t þ 0:267 sin a 0:00357T a þ þ0:106/ ð7Þ Interval : k t P 0:78 k d ¼ 0:426k t þ 0:256 sin a 0:00349T a þ þ0:0734/
ð8Þ
2.1.3. Orgill and Holands model (OH) Orgill and Hollands (1977) estimated diffuse fraction kd using the clearness index kt as the only variable. The model was based on measured global and diffuse irradiance data from Toronto. The relationship between diffuse fraction on a horizontal surface kd and clearness index kt is given as per Eqs. (9)–(12).
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S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
Interval : k t < 0:35 k d ¼ 1 0:249k t Interval : 0:35 6 k t 6 0:75
k d ¼ 1:577 1:84k t
k d ¼ 0:177
Interval : k t > 0:75;
ð27Þ
ð10Þ
c4 0:5 c03 ¼ 0:15 1 þ sin p d3
ð11Þ
c04 ¼ 01:09c2 c1
ð28Þ
ð12Þ
2.1.6. Louche et al. model (LO) Louche et al. (1991) used the clearness index kt to estimate the direct irradiance kb as per the Eq. (29). The correlation includes global and direct irradiance data for Ajaccio (Corsica, France, 44.9°N latitude) between October 1983 and June 1985.
ð9Þ
The direct irradiance Ib is obtained by: Ib ¼
I t ð1 k d Þ sin a
2.1.4. Lam and Li model (LL) Lam and Li (1996) derived diffuse fraction kd based on measured direct and diffuse irradiance data for Hong Kong as per Eqs. (13)–(15). k d ¼ 0:977
Interval : k t 6 0:15;
Interval : 0:15 < k t 6 0:7; Interval : k t > 0:7;
k d ¼ 1:237 1:361k t
k d ¼ 0:273
ð13Þ ð14Þ ð15Þ
2.1.5. Skartveit and Olseth model (SO) Skartveit and Olseth (1987) derived the diffuse fraction kd, as a function of the clearness index kt, solar altitude, temperature, and relative humidity. The model was tested against data from 10 stations worldwide. Direct irradiance Ib is derived from global irradiance Gt using the following equations: Gt ð1 wÞ Ib ¼ sin a Interval
ð16Þ
k t < c1
w¼1
ð17Þ
0:059k t þ 0:02
ð29Þ
2.1.7. Maxwell model (MA) A quasi- physical model for converting hourly global horizontal to direct normal insolation is proposed by Maxwell in 1987. The model is a combination of a clear physical model with experimental fits for other conditions. The direct irradiance Ib is calculated as per Eqs. (30)–(39). I b ¼ Iofw ðd 4 þ d 5 ema d 6 Þg w ¼ 0:866 0:122ma þ þ
0:000653m4a
where ma ¼ mr
þ
ð30Þ
0:0121m2a
1=2
ð18Þ
where c2 ¼ 0:87 0:56e0:06a c4 c3 ¼ 0:5 1 þ sin pð 0:5Þ d3
ð19Þ
c4 ¼ k t c1
ð21Þ
d 1 ¼ 0:15 þ 0:43e
0:06a
ð31Þ
p 1013:25
ð32Þ
The air mass, at standard pressure mr, is approximated by Kasten’s formula (Iqbal, 1978) as per Eq. (33). 1:253 1
ð20Þ
Interval : k t 6 0:6 d 4 ¼ 0:512 1:56k t þ 2:286k 2t 2:222k 3t
ð34Þ
d 5 ¼ 0:37 þ 0:962k t
ð35Þ
d 6 ¼ 0:28 þ 0:923k t
2:048k 2t
ð36Þ
Interval : k t > 0:6 d 4 ¼ 5:743 þ 21:77k t 27:49k 2t þ 11:56k
ð22Þ
d 5 ¼ 41:4 118:5k t þ
d 2 ¼ 0:27
ð23Þ
d 6 ¼ 47:01 þ 184:2k t 222k 2t þ 73:81k 3t
d 3 ¼ c2 c1
ð24Þ
Interval : k t > 1:09c2 1n w ¼ 1 1:09c2 kt
ð25Þ
where 1 2 n ¼ 1 ð1 d 1 Þ d 2 c3 þ ð1 d 2 Þc3 2
ð33Þ
and d4, d5 and d6 are functions of the clearness index kt as given below:
c1 6 k t 6 1:09c2
w ¼ 1 ð1 d 1 Þ½d 2 c3 þ ð1 d 2 Þc23
0:000653m3a
0:000014m5a
mr ¼ ½cos hz þ 0:15ð93:885 hz Þ
c1 ¼ 0:2 Interval
k b ¼ 10:676k 5t þ 15:307k 4t 5:205k 3t þ 0:99k 2t
ð26Þ
66:05k 2t
þ
31:9k 3t
ð37Þ ð38Þ ð39Þ
2.1.8. Vignola and McDaniels model (VM) Vignola and McDaniels (1984) was created based on the measurements in seven sites in Oregon and Idaho, USA. The diffuse irradiance kd is calculated (see Eq. (40)) as a function of clearness index (kt) and the number of the day (N). k d ¼ 0:162 1:451k t þ 0:045 sin½2pðN 40Þ=365
ð40Þ
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
2.2. Measurements For the comparison of the diffuse fraction models, measured global and diffuse horizontal irradiance were obtained using the microclimatic monitoring station of the Department of Building Physics and Building Ecology at the Vienna University of Technology, Austria (location: 48°120 N, 16°220 E). Given the mounting height of the station (highest point of campus), the elevation of the effective horizon due to obstruction (e.g. surrounding buildings) is insignificant. Two sets of measured data were used for this study. One set that was collected over a 17-month period (from January 2009 to May 2010) was used to compare the models’ performance. A second set of measurements, which was used to derive the local (Vienna) version of the three better performing models, contained measured global irradiance data collected over a 2-year period (from January 2007 to December 2008). To arrive at the values of the adapted coefficients, polynomial curve fitting functions were applied using standard curve fitting toolbox in MATLAB (2010) software. Measurements of global horizontal and diffuse irradiance were performed every 5 min during the daylight hours, covering a variety of sky conditions, from sunny, to partly cloudy, to overcast. Subsequent to a comprehensive data quality check, 32,265 pairs of measured irradiance values in the first database and 47,087 pairs in the second database were included in the study. Global horizontal irradiance measurements under 50 W m2 and those at sun altitudes less than 5° were
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removed from the data sets, given the comparatively less accurate sensor performance at these ranges. Note that the sensor components of the monitoring station of the Building Physics and Building Ecology department are regularly calibrated via certified agencies. The technical specification of the applied pyranometer and the meteorological weather station is shown in Table 1. To obtain necessary input parameter for the eight diffuse fraction models (see Table 2), parallel to radiometric measurements, a weather station at the same location monitored other external environmental parameter such as air temperature and relative humidity. 2.3. Model comparison To compare the performance of the models, three common statistical indicators were used, namely the relative mean bias deviation MBD (Eq. (41)), the Relative Error (Eq. (42)), and the root mean square deviation RMSD (Eq. (43)). Pn I dmðiÞ I dcðiÞ MBD ¼
i¼1
I dmðiÞ
100 ð%Þ
n
ð41Þ
I dmðiÞ I dcðiÞ 100 ð%Þ I dmðiÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ½ðI dmðiÞ I dcðiÞ Þ=I dmðiÞ RMSD ¼ ðW m2 Þ n REi ¼
ð42Þ
ð43Þ
In these equations, Idm(i) denotes the measured diffuse irradiance, Idc(i) is the computed diffuse irradiance, and n is the total number of pairs of measured and computed values.
Table 1 Input parameters for the eight diffuse fraction models. Models Model codes
Orgill and Holland OH
Erbs ER
Reindl RE
Lam and Li LL
Skartveit and Olseth SO
Louche LO
Maxwell MA
Vignola and McDaniels VM
Global irradiance It Solar constant Io Solar altitude a Relative humidity U Air temperature Ta Local air pressure p
x x x
x x x
x x x x x
x x x
x x x
x x x
x x x
x x x
x
Table 2 Overview of the instrumentation specifications. Sensor
Information
Global and diffuse irradiance (sunshine pyranometer SPN1)
Overall accuracy: ±5% daily integrals, ±5% ± 10 W m2 hourly averages ±8% ± 10 W m2 individual readings Resolution: 0.6 W m2 = 0.6 mV, range: 0 to >2000 W m2, sunshine status threshold: 120 W m2 in the direct beam, temperature range: 20 to +70 °C, accuracy: Cosine Correction ±2% of incoming radiation over 0– 90” Zenith angle, accuracy: azimuth angle ±5% over 360° rotation, Response time <200 ms Outdoor temperature: Absolute Error: <0.3 K; Temperature range: 30 to +70 °C; Response time <20 s (P1.5 m s1) Outdoor relative humidity: Absolute Error: <±2%; Humidity range: 0–100%; Response time <10 s (P1.5 m s1) Wind speed: Absolute Error: <1%; Wind speed range 0–75 m s1
Monitoring weather station
1800
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
3. Results
Table 4 Comparison of eight models in terms of MBD (%) and RMSD (W m2).
To compare the performance of the eight diffuse fraction models (versions with original coefficients), Fig. 1 shows the cumulative distribution function (CDF) of the relative errors (in ±%) for the eight models with the original coefficients. Table 3 lists this information numerically for discrete values of Relative Error (from ±5% to ±40%). Table 4 compares the eight options in terms of RMSD and MBD. To further compare the three better performing models (in their versions with adapted coefficients), Fig. 2 shows the cumulative distribution function of the relative errors (in ±%) for the three models (ER, RE, OH) with the adapted coefficients. Table 5 shows, for both original and adapted versions, the same information numerically for discrete values of relative error (±5% to ±20%). Table 6 compares the three models in terms of RMSD and MBD for original and adapted coefficients. Fig. 3 shows MBD of the three models (ER, RE, OH) with the adapted coefficients as a function of solar altitude. For this illustration, discrete bins of solar altitude were considered as follows: 5–10°, 10–20°, 20–30°, 30–40°, 40–50°, 50–60°, <60°.
Models
MBD (%)
RMDS (W m2)
ER RE OH LL SO LO MA VM
9.2 10.5 13.3 11.9 98.3 19.5 21.1 60.38
37.4 41.6 43.1 45.7 199.9 29.6 33.2 50.4
Fig. 2. The cumulative distribution function (CDF) of the relative errors (in ±%) for the three models (ER, RE, OH) with the adapted coefficients.
100
Percentage of results (%)
90
4. Discussion
80 70 OH ER RE LL SO LO MA VM
60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100
Relative Error ( %) Fig. 1. The cumulative distribution function (CDF) of the relative errors (in ±%) for the eight models with the original coefficients.
A visual inspection of the results warrants a number of inferences. Diffuse fraction models do not “transport” well and display considerable errors. The comparison of the eight models (original coefficients) for the derivation of the diffuse fraction of horizontal irradiance (see Fig. 1 and Tables 3 and 4) suggests that Erbs et al. (1982), Reindl et al. (1990), and Orgill and Hollands (1977) algorithms provide, for the Vienna location, better results. About 62% of the results derived based on these three models display a Relative Error of less than ±20% (see Table 5). The cumulative error representation in Fig. 1 (and Table 4) implies that a higher fraction of the results have lower relative errors for Reindl model, followed by Erbs model and
Table 3 Percentage of results with corresponding maximum Relative Error (in ±%) for the original versions of the eight models as compared based on Vienna data. Models
±5%
±10%
±15%
±20%
±25%
±30%
±35%
±40%
ER RE OH LL SO LO MA VM
32.4 38.1 31.8 19.4 18.0 11.0 19.2 8.26
46.3 48.7 49.5 31.9 23.6 24.9 33.8 12.47
55.6 57.5 56.0 42.8 27.9 38.0 44.6 18.72
62.4 64.7 61.2 52.7 31.6 48.7 53.6 28.60
67.4 70.2 65.7 60.8 35.3 57.7 60.9 38.28
71.3 74.5 69.3 67.3 38.9 65.1 67.2 47.64
74.9 78.1 72.6 72.3 42.6 71.0 72.8 56.29
78.2 81.1 75.6 75.9 45.9 75.9 77.4 63.68
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802 Table 5 Percentage of results with corresponding maximum Relative Error for the original and adapted versions of the models OH, ER, and RE. Model
Original coefficients ±5%
±10%
±15%
±20%
ER RE OH
32.4 38.1 31.8
46.3 48.7 49.5
55.6 57.5 56.0
62.4 64.7 61.2
ER RE OH
Adapted coefficients 37.3 49.4 41.2 53.6 39.6 49.9
57.1 60.8 56.9
63.6 66.9 63.3
ER RE OH
Original coefficients
Adapted coefficients 2
rather modest improvement: The percentage of results with a Relative Error less than ±20% grew only by 2%. MBD was reduced to about 7%, and RMSD was reduced to about 35 W m2. A further analysis of the results suggests that model errors (as expressed in terms of MBD attributes) are generally higher for lower solar altitudes (see Fig. 3). 5. Conclusion
Table 6 Comparison of the models (OH, ER, RE) based on MBD (%) and RMSD (W m2). Models
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MBD (%)
RMSD (W m )
MBD (%)
RMSD (W m2)
9.2 10.5 13.3
37.4 41.6 43.1
4.7 6.5 9.2
33.2 35.1 37.6
Fig. 3. MBD results of the three models (ER, RE, OH) with the adapted coefficients as a function of solar altitude.
Orgill and Holland model. Comparisons based on MBD and RMSD show that Erbs model displays the lowest MBD and RMSD values (see Table 5) followed by Reindl model and Origll and Hollands model. However, the differences between the three algorithms are not highly pronounced. As it could be expected, these models perform better when their coefficients are modified according to local data (see Fig. 2 as well as Tables 5 and 6). Comparison of the adapted models based on RE, MBD and RMSD show the same trend as with model versions involving the original coefficients. Reindl model shows better results in terms of relative error, whereas Erbs model shows better results in terms of MBD and RMSD. However, the calibration of the models via the adapted coefficients resulted in a
We compared eight diffuse fraction models to derive horizontal diffuse irradiance values from the more widely available measured global horizontal irradiance values. These algorithms can be used to provide necessary input data for the generation of sky radiance maps. The comparison was conducted using measurement data from Vienna, Austria. It revealed that three models (RE, ER, and OH) reproduce measurement results more accurately. About 62% of the results derived based on these three models display a Relative Error of less than ±20%, a MBD in the order of 11%, and a RMSD in the order of 40 W m2. We further explored the potential for the performance improvement of these three models by calibration of their respective algorithms based on Vienna data. The model calibration (via derivation of new values for coefficients) resulted only in a modest improvement of the models’ predictive performance. These findings are in general agreement with previous studies. Wong and Chow (2001), Elminir (2005), Cucumo et al. (2007), Jacovides et al. (2006), suggest that models by Orgill and Hollands (1977), Erbs et al. (1982), and Reindl et al. (1990) show in general better performance as compared to other models. The differences between the predictive performance of these models does not appear to be significant. Various studies also agree in general concerning the influence of the solar altitude. Lower solar altitudes appear to be associated with larger errors. On the other hand, higher values of the clearness index (i.e. cloudless skies, and high solar altitude) are associated with better model performance. Building simulation specialists and designers of solar energy systems for architectural applications must thus consider such order of magnitude in potential errors while estimating the dynamic behavior of solar energy systems’ processes and for simulating long-term operations. The magnitude of the observed uncertainty suggests that proper “factor of safety” assumptions should be made while designing solar-thermal systems or selecting shading strategies for buildings. These concerns will be addressed in detail in ongoing and future studies, together with the extent of model errors for other locations toward developing more accurate and robust (globally applicable) diffuse fraction models. Likewise, the potential influences of other microclimatic parameters on the error levels will be addressed in detail. Moreover, the potential for alternative sky model generation schemes are being explored.
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S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
Acknowledgment The authors would like to acknowledge the contribution of Ms. Katka Zavodnikova toward collecting and structuring the data used in this paper. References Boland, J., McArthur, L.C., Luther, M., 2001. Modelling the diffuse fraction of global solar radiation on a horizontal surface. Environmetrics 12, 103–116. Cucumo, M., De Rossa, A., Ferraro, V., Kaliakatkos, D., Marinelli, V., 2007. Experimental testing of models for the estimation of hourly solar radiation on vertical surfaces at Arcavacata di Rende. Solar Energy 81, 692–695. Elminir, H., 2005. Experimental and theoretical investigation of diffuse solar radiation: data and models quality tested for Egyptian sites. Energy 32, 73–82. Erbs, D.G., Klein, S.A., Duffie, J.A., 1982. Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation. Solar Energy 28 (4), 293–302. Iqbal, M., 1978. Estimation of the monthly average of the diffuse component of total insolation on a horizontal surface. Solar Energy 20 (1), 101–105. Jacovides, C.P., Tymvios, F.S., Assimakopoulos, V.D., 2006. Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation. Renewable Energy 31 (15), 2492– 2504. Lam, J.C., Li, D.H.W., 1996. Correlation between global solar radiation and its direct and diffuse components. Building and Environment 31 (6), 527–535.
LeBaron, B., Dirmhirn, I., 1983. Strengths and limitations of the Liu and Jordan model to determine diffuse from global irradiance. Solar Energy 31, 167–172. Liu, B.Y.H., Jordan, R.C., 1960. The inter-relationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4 (3), 1–19. Louche, A., Notton, G., Poggi, P., Simonnot, G., 1991. Correlations for direct normal and global horizontal irradiations on a French Mediterranean site. Solar Energy 46 (4), 261–266. MATLAB, 2010. MATLAB Release 2010a. The MathWorks, Inc.. Maxwell, E.L., 1987. A quasi-physical model for converting hourly global horizontal to direct normal insolation. Report SERI/TR-215-3087. Solar Energy Institute, Golden, CO. Orgill, J.F., Hollands, K.G.T., 1977. Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19 (4), 357–359. Perez, R., Ineichen, P., Seals, R., Michalsky, J., Stewart, R., 1990. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 44 (5), 271–289. Reindl, D.T., Beckman, W.A., Duffie, J.A., 1990. Diffuse fraction corrections. Solar Energy 45 (1), 1–7. Skartveit, A., Olseth, J.A., 1987. A model for the diffuse fraction of hourly global radiation. Solar Energy 38 (4), 271–274. Soler, A., 1990. Dependence on latitude of the relation between the diffuse fraction of solar radiation and the ratio of global to extraterrestrial radiation for monthly average daily values. Solar Energy 1990 (44), 297–302. Tuller, S.F., 1976. The relationship between diffuse, total and extraterrestrial solar radiation. Solar Energy 18, 259–263. Vignola, F., McDaniels, D.K., 1984. Correlations between diffuse and global insolation for the Pacific Northwest. Solar Energy 32, 161. Wong, L.T., Chow, W.K., 2001. Solar radiation model. Applied Energy 69, 191–224.
Solar Energy 86 (2012) 1796–1802 www.elsevier.com/locate/solener
Computing diffuse fraction of global horizontal solar radiation: A model comparison Sokol Dervishi ⇑, Ardeshir Mahdavi Department of Building Physics and Building Ecology, Vienna University of Technology, Vienna, Austria Received 3 August 2011; received in revised form 11 March 2012; accepted 12 March 2012 Available online 5 April 2012 Communicated by: Associate Editor Frank Vignola
Abstract For simulation-based prediction of buildings’ energy use or expected gains from building-integrated solar energy systems, information on both direct and diffuse component of solar radiation is necessary. Available measured data are, however, typically restricted to global horizontal irradiance. There have been thus many efforts in the past to develop algorithms for the derivation of the diffuse fraction of solar irradiance. In this context, the present paper compares eight models for estimating diffuse fraction of irradiance based on a database of measured irradiance from Vienna, Austria. These models generally involve mathematical formulations with multiple coefficients whose values are typically valid for a specific location. Subsequent to a first comparison of these eight models, three better performing models were selected for a more detailed analysis. Thereby, the coefficients of the models were modified to account for Vienna data. The results suggest that some models can provide relatively reliable estimations of the diffuse fractions of the global irradiance. The calibration procedure could only slightly improve the models’ performance. Ó 2012 Elsevier Ltd. All rights reserved. Keywords: Diffuse irradiance; Solar irradiance; Diffuse fraction models; Measurements
1. Introduction A crucial input required in the simulation of buildings’ energy performance is the availability of detailed information on the magnitudes of diffuse and direct irradiance data. Moreover, configuration and sizing of solar energy systems (e.g. photovoltaic cells, solar-thermal collectors) necessitates reliable solar radiation measurements. However, concurrent measured data of global and diffuse irradiance on horizontal surface or direct normal irradiance are available only for a limited number of locations. The measurement of global horizontal irradiance is rather simple and cost-effective. It can be, conceivably, an integral part of the sensory equipment of every building. Given global solar irradiation measurements on a horizontal surface ⇑ Corresponding author. Tel.: +43 1 58801 27003; fax: +43 1 58801 27093. E-mail address: [email protected] (S. Dervishi).
0038-092X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2012.03.008
(as the most widely available data) direct and diffuse solar radiation components can be obtained through various correlations. Starting from the early 1960s, numerous models for evaluating the diffuse component based on the pioneer work of Liu and Jordan (1960) appeared in the literature, such as Orgill and Hollands (1977), Erbs et al. (1982), Reindl et al. (1990), Lam and Li (1996), and Perez et al. (1990). These models are usually expressed in terms of first to fourth degree polynomial functions 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 on horizontal surface), as well as to other variables such as solar altitude, air temperature, relative humidity. Although these models are typically derived following sound approaches, their performance appears to lessen once they are applied to regions other than those, which provided the initial data for model development (Soler, 1990; LeBaron and Dirmhirn, 1983; Tuller, 1976).
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
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Nomenclature I Id Io Isc Ib kt kd a Ta /
global horizontal irradiance (W m2) diffuse horizontal irradiance (W m2) extraterrestrial solar irradiance (W m2) extraterrestrial solar constant irradiance (1367 W m2) direct irradiance on a horizontal surface (W m2) clearness index (dimensionless) diffuse fraction (dimensionless) sun altitude (°) outdoor air temperature (°C) relative humidity (%)
Thus, further studies would be necessary to better accommodate the influences of local climatic factors on model performance (Wong and Chow, 2001; Boland et al., 2001; Jacovides et al., 2006). This study investigates the applicability of various standard models correlating hourly diffuse fraction for Vienna– Austria, based on their prior reported performance and the availability of required measurement data for model comparison.
monthly average global radiation on a horizontal surface (W m2) tilt angle of a surface measured from the horizontal (°) air mass (dimensionless) at actual pressure air mass (dimensionless) at standard pressure (1013.25 mbar) the local air-pressure (mbar) day number in the year (No.) number of data zenith angle (°)
Gt f ma mr p N n hz
Interval : 0:22 6 k t 6 0:8 k d ¼ 0:9511 0:1604k t þ 4:39k 2t 16:64k 3t þ 12:34k 4t ð2Þ Interval : k t > 0:8; kt ¼
It I o sinðaÞ
k d ¼ 0:165
kd ¼
Id It
ð3Þ ð4Þ
where
360n cos hz I o ¼ I sc 1 þ 0:33 cos 365
ð5Þ
2. Approach For the purpose of the present study, eight models were considered for estimating diffuse fraction of irradiance as documented in Erbs et al. (1982), Orgill and Hollands (1977), Reindl et al. (1990), Lam and Li (1996), Skartveit and Olseth (1987), Louche et al. (1991), Maxwell (1987), and Vignola and McDaniels (1984). The comparison was based on measured irradiance data from Vienna, Austria. The selection of the models was influenced by their prior reported performance as well as the availability of required measurement data for model comparison. The models typically involve mathematical formulations with multiple coefficients whose values are generally valid for a specific location. Subsequent to a first comparison of the eight models, three better performing ones were selected for further analysis. Thereby, the original model versions were compared with modified versions with coefficients adjusted for a better match to Vienna data. Toward this end, polynomial curves fitting functions were applied using standard curve fitting toolbox in MATLAB (2010). A short summary of the selected models is provided in the following. 2.1. Models 2.1.1. Erbs model (ER) Erbs et al. (1982) used direct normal and global irradiance data on a horizontal surface from 5 stations in USA. Diffuse fraction kd is given by Interval : k t 6 0:22;
k d ¼ 1 0:09k t
ð1Þ
2.1.2. Reindl model (RE) Reindl et al. (1990) estimated diffuse fraction kd based on measured global and diffuse horizontal irradiance data from 5 locations in the USA and Europe. The algorithm considers three characteristic intervals using the following parameters: clearness index (kt), sun altitude (a), outdoor air temperature (Ta), and the relative humidity u. Depending on clearness index value, the diffuse fractions (Id/It) are calculated as per Eqs. (6)–(8). Interval : 0 6 k t 6 0:3 k d ¼ 1:0 0:232k t þ 0:0239 sin a 0:000682T a þ þ0:019/ ð6Þ Interval : 0:3 < k t 6 0:78 k d ¼ 1:329 1:716k t þ 0:267 sin a 0:00357T a þ þ0:106/ ð7Þ Interval : k t P 0:78 k d ¼ 0:426k t þ 0:256 sin a 0:00349T a þ þ0:0734/
ð8Þ
2.1.3. Orgill and Holands model (OH) Orgill and Hollands (1977) estimated diffuse fraction kd using the clearness index kt as the only variable. The model was based on measured global and diffuse irradiance data from Toronto. The relationship between diffuse fraction on a horizontal surface kd and clearness index kt is given as per Eqs. (9)–(12).
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Interval : k t < 0:35 k d ¼ 1 0:249k t Interval : 0:35 6 k t 6 0:75
k d ¼ 1:577 1:84k t
k d ¼ 0:177
Interval : k t > 0:75;
ð27Þ
ð10Þ
c4 0:5 c03 ¼ 0:15 1 þ sin p d3
ð11Þ
c04 ¼ 01:09c2 c1
ð28Þ
ð12Þ
2.1.6. Louche et al. model (LO) Louche et al. (1991) used the clearness index kt to estimate the direct irradiance kb as per the Eq. (29). The correlation includes global and direct irradiance data for Ajaccio (Corsica, France, 44.9°N latitude) between October 1983 and June 1985.
ð9Þ
The direct irradiance Ib is obtained by: Ib ¼
I t ð1 k d Þ sin a
2.1.4. Lam and Li model (LL) Lam and Li (1996) derived diffuse fraction kd based on measured direct and diffuse irradiance data for Hong Kong as per Eqs. (13)–(15). k d ¼ 0:977
Interval : k t 6 0:15;
Interval : 0:15 < k t 6 0:7; Interval : k t > 0:7;
k d ¼ 1:237 1:361k t
k d ¼ 0:273
ð13Þ ð14Þ ð15Þ
2.1.5. Skartveit and Olseth model (SO) Skartveit and Olseth (1987) derived the diffuse fraction kd, as a function of the clearness index kt, solar altitude, temperature, and relative humidity. The model was tested against data from 10 stations worldwide. Direct irradiance Ib is derived from global irradiance Gt using the following equations: Gt ð1 wÞ Ib ¼ sin a Interval
ð16Þ
k t < c1
w¼1
ð17Þ
0:059k t þ 0:02
ð29Þ
2.1.7. Maxwell model (MA) A quasi- physical model for converting hourly global horizontal to direct normal insolation is proposed by Maxwell in 1987. The model is a combination of a clear physical model with experimental fits for other conditions. The direct irradiance Ib is calculated as per Eqs. (30)–(39). I b ¼ Iofw ðd 4 þ d 5 ema d 6 Þg w ¼ 0:866 0:122ma þ þ
0:000653m4a
where ma ¼ mr
þ
ð30Þ
0:0121m2a
1=2
ð18Þ
where c2 ¼ 0:87 0:56e0:06a c4 c3 ¼ 0:5 1 þ sin pð 0:5Þ d3
ð19Þ
c4 ¼ k t c1
ð21Þ
d 1 ¼ 0:15 þ 0:43e
0:06a
ð31Þ
p 1013:25
ð32Þ
The air mass, at standard pressure mr, is approximated by Kasten’s formula (Iqbal, 1978) as per Eq. (33). 1:253 1
ð20Þ
Interval : k t 6 0:6 d 4 ¼ 0:512 1:56k t þ 2:286k 2t 2:222k 3t
ð34Þ
d 5 ¼ 0:37 þ 0:962k t
ð35Þ
d 6 ¼ 0:28 þ 0:923k t
2:048k 2t
ð36Þ
Interval : k t > 0:6 d 4 ¼ 5:743 þ 21:77k t 27:49k 2t þ 11:56k
ð22Þ
d 5 ¼ 41:4 118:5k t þ
d 2 ¼ 0:27
ð23Þ
d 6 ¼ 47:01 þ 184:2k t 222k 2t þ 73:81k 3t
d 3 ¼ c2 c1
ð24Þ
Interval : k t > 1:09c2 1n w ¼ 1 1:09c2 kt
ð25Þ
where 1 2 n ¼ 1 ð1 d 1 Þ d 2 c3 þ ð1 d 2 Þc3 2
ð33Þ
and d4, d5 and d6 are functions of the clearness index kt as given below:
c1 6 k t 6 1:09c2
w ¼ 1 ð1 d 1 Þ½d 2 c3 þ ð1 d 2 Þc23
0:000653m3a
0:000014m5a
mr ¼ ½cos hz þ 0:15ð93:885 hz Þ
c1 ¼ 0:2 Interval
k b ¼ 10:676k 5t þ 15:307k 4t 5:205k 3t þ 0:99k 2t
ð26Þ
66:05k 2t
þ
31:9k 3t
ð37Þ ð38Þ ð39Þ
2.1.8. Vignola and McDaniels model (VM) Vignola and McDaniels (1984) was created based on the measurements in seven sites in Oregon and Idaho, USA. The diffuse irradiance kd is calculated (see Eq. (40)) as a function of clearness index (kt) and the number of the day (N). k d ¼ 0:162 1:451k t þ 0:045 sin½2pðN 40Þ=365
ð40Þ
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802
2.2. Measurements For the comparison of the diffuse fraction models, measured global and diffuse horizontal irradiance were obtained using the microclimatic monitoring station of the Department of Building Physics and Building Ecology at the Vienna University of Technology, Austria (location: 48°120 N, 16°220 E). Given the mounting height of the station (highest point of campus), the elevation of the effective horizon due to obstruction (e.g. surrounding buildings) is insignificant. Two sets of measured data were used for this study. One set that was collected over a 17-month period (from January 2009 to May 2010) was used to compare the models’ performance. A second set of measurements, which was used to derive the local (Vienna) version of the three better performing models, contained measured global irradiance data collected over a 2-year period (from January 2007 to December 2008). To arrive at the values of the adapted coefficients, polynomial curve fitting functions were applied using standard curve fitting toolbox in MATLAB (2010) software. Measurements of global horizontal and diffuse irradiance were performed every 5 min during the daylight hours, covering a variety of sky conditions, from sunny, to partly cloudy, to overcast. Subsequent to a comprehensive data quality check, 32,265 pairs of measured irradiance values in the first database and 47,087 pairs in the second database were included in the study. Global horizontal irradiance measurements under 50 W m2 and those at sun altitudes less than 5° were
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removed from the data sets, given the comparatively less accurate sensor performance at these ranges. Note that the sensor components of the monitoring station of the Building Physics and Building Ecology department are regularly calibrated via certified agencies. The technical specification of the applied pyranometer and the meteorological weather station is shown in Table 1. To obtain necessary input parameter for the eight diffuse fraction models (see Table 2), parallel to radiometric measurements, a weather station at the same location monitored other external environmental parameter such as air temperature and relative humidity. 2.3. Model comparison To compare the performance of the models, three common statistical indicators were used, namely the relative mean bias deviation MBD (Eq. (41)), the Relative Error (Eq. (42)), and the root mean square deviation RMSD (Eq. (43)). Pn I dmðiÞ I dcðiÞ MBD ¼
i¼1
I dmðiÞ
100 ð%Þ
n
ð41Þ
I dmðiÞ I dcðiÞ 100 ð%Þ I dmðiÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ½ðI dmðiÞ I dcðiÞ Þ=I dmðiÞ RMSD ¼ ðW m2 Þ n REi ¼
ð42Þ
ð43Þ
In these equations, Idm(i) denotes the measured diffuse irradiance, Idc(i) is the computed diffuse irradiance, and n is the total number of pairs of measured and computed values.
Table 1 Input parameters for the eight diffuse fraction models. Models Model codes
Orgill and Holland OH
Erbs ER
Reindl RE
Lam and Li LL
Skartveit and Olseth SO
Louche LO
Maxwell MA
Vignola and McDaniels VM
Global irradiance It Solar constant Io Solar altitude a Relative humidity U Air temperature Ta Local air pressure p
x x x
x x x
x x x x x
x x x
x x x
x x x
x x x
x x x
x
Table 2 Overview of the instrumentation specifications. Sensor
Information
Global and diffuse irradiance (sunshine pyranometer SPN1)
Overall accuracy: ±5% daily integrals, ±5% ± 10 W m2 hourly averages ±8% ± 10 W m2 individual readings Resolution: 0.6 W m2 = 0.6 mV, range: 0 to >2000 W m2, sunshine status threshold: 120 W m2 in the direct beam, temperature range: 20 to +70 °C, accuracy: Cosine Correction ±2% of incoming radiation over 0– 90” Zenith angle, accuracy: azimuth angle ±5% over 360° rotation, Response time <200 ms Outdoor temperature: Absolute Error: <0.3 K; Temperature range: 30 to +70 °C; Response time <20 s (P1.5 m s1) Outdoor relative humidity: Absolute Error: <±2%; Humidity range: 0–100%; Response time <10 s (P1.5 m s1) Wind speed: Absolute Error: <1%; Wind speed range 0–75 m s1
Monitoring weather station
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3. Results
Table 4 Comparison of eight models in terms of MBD (%) and RMSD (W m2).
To compare the performance of the eight diffuse fraction models (versions with original coefficients), Fig. 1 shows the cumulative distribution function (CDF) of the relative errors (in ±%) for the eight models with the original coefficients. Table 3 lists this information numerically for discrete values of Relative Error (from ±5% to ±40%). Table 4 compares the eight options in terms of RMSD and MBD. To further compare the three better performing models (in their versions with adapted coefficients), Fig. 2 shows the cumulative distribution function of the relative errors (in ±%) for the three models (ER, RE, OH) with the adapted coefficients. Table 5 shows, for both original and adapted versions, the same information numerically for discrete values of relative error (±5% to ±20%). Table 6 compares the three models in terms of RMSD and MBD for original and adapted coefficients. Fig. 3 shows MBD of the three models (ER, RE, OH) with the adapted coefficients as a function of solar altitude. For this illustration, discrete bins of solar altitude were considered as follows: 5–10°, 10–20°, 20–30°, 30–40°, 40–50°, 50–60°, <60°.
Models
MBD (%)
RMDS (W m2)
ER RE OH LL SO LO MA VM
9.2 10.5 13.3 11.9 98.3 19.5 21.1 60.38
37.4 41.6 43.1 45.7 199.9 29.6 33.2 50.4
Fig. 2. The cumulative distribution function (CDF) of the relative errors (in ±%) for the three models (ER, RE, OH) with the adapted coefficients.
100
Percentage of results (%)
90
4. Discussion
80 70 OH ER RE LL SO LO MA VM
60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100
Relative Error ( %) Fig. 1. The cumulative distribution function (CDF) of the relative errors (in ±%) for the eight models with the original coefficients.
A visual inspection of the results warrants a number of inferences. Diffuse fraction models do not “transport” well and display considerable errors. The comparison of the eight models (original coefficients) for the derivation of the diffuse fraction of horizontal irradiance (see Fig. 1 and Tables 3 and 4) suggests that Erbs et al. (1982), Reindl et al. (1990), and Orgill and Hollands (1977) algorithms provide, for the Vienna location, better results. About 62% of the results derived based on these three models display a Relative Error of less than ±20% (see Table 5). The cumulative error representation in Fig. 1 (and Table 4) implies that a higher fraction of the results have lower relative errors for Reindl model, followed by Erbs model and
Table 3 Percentage of results with corresponding maximum Relative Error (in ±%) for the original versions of the eight models as compared based on Vienna data. Models
±5%
±10%
±15%
±20%
±25%
±30%
±35%
±40%
ER RE OH LL SO LO MA VM
32.4 38.1 31.8 19.4 18.0 11.0 19.2 8.26
46.3 48.7 49.5 31.9 23.6 24.9 33.8 12.47
55.6 57.5 56.0 42.8 27.9 38.0 44.6 18.72
62.4 64.7 61.2 52.7 31.6 48.7 53.6 28.60
67.4 70.2 65.7 60.8 35.3 57.7 60.9 38.28
71.3 74.5 69.3 67.3 38.9 65.1 67.2 47.64
74.9 78.1 72.6 72.3 42.6 71.0 72.8 56.29
78.2 81.1 75.6 75.9 45.9 75.9 77.4 63.68
S. Dervishi, A. Mahdavi / Solar Energy 86 (2012) 1796–1802 Table 5 Percentage of results with corresponding maximum Relative Error for the original and adapted versions of the models OH, ER, and RE. Model
Original coefficients ±5%
±10%
±15%
±20%
ER RE OH
32.4 38.1 31.8
46.3 48.7 49.5
55.6 57.5 56.0
62.4 64.7 61.2
ER RE OH
Adapted coefficients 37.3 49.4 41.2 53.6 39.6 49.9
57.1 60.8 56.9
63.6 66.9 63.3
ER RE OH
Original coefficients
Adapted coefficients 2
rather modest improvement: The percentage of results with a Relative Error less than ±20% grew only by 2%. MBD was reduced to about 7%, and RMSD was reduced to about 35 W m2. A further analysis of the results suggests that model errors (as expressed in terms of MBD attributes) are generally higher for lower solar altitudes (see Fig. 3). 5. Conclusion
Table 6 Comparison of the models (OH, ER, RE) based on MBD (%) and RMSD (W m2). Models
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MBD (%)
RMSD (W m )
MBD (%)
RMSD (W m2)
9.2 10.5 13.3
37.4 41.6 43.1
4.7 6.5 9.2
33.2 35.1 37.6
Fig. 3. MBD results of the three models (ER, RE, OH) with the adapted coefficients as a function of solar altitude.
Orgill and Holland model. Comparisons based on MBD and RMSD show that Erbs model displays the lowest MBD and RMSD values (see Table 5) followed by Reindl model and Origll and Hollands model. However, the differences between the three algorithms are not highly pronounced. As it could be expected, these models perform better when their coefficients are modified according to local data (see Fig. 2 as well as Tables 5 and 6). Comparison of the adapted models based on RE, MBD and RMSD show the same trend as with model versions involving the original coefficients. Reindl model shows better results in terms of relative error, whereas Erbs model shows better results in terms of MBD and RMSD. However, the calibration of the models via the adapted coefficients resulted in a
We compared eight diffuse fraction models to derive horizontal diffuse irradiance values from the more widely available measured global horizontal irradiance values. These algorithms can be used to provide necessary input data for the generation of sky radiance maps. The comparison was conducted using measurement data from Vienna, Austria. It revealed that three models (RE, ER, and OH) reproduce measurement results more accurately. About 62% of the results derived based on these three models display a Relative Error of less than ±20%, a MBD in the order of 11%, and a RMSD in the order of 40 W m2. We further explored the potential for the performance improvement of these three models by calibration of their respective algorithms based on Vienna data. The model calibration (via derivation of new values for coefficients) resulted only in a modest improvement of the models’ predictive performance. These findings are in general agreement with previous studies. Wong and Chow (2001), Elminir (2005), Cucumo et al. (2007), Jacovides et al. (2006), suggest that models by Orgill and Hollands (1977), Erbs et al. (1982), and Reindl et al. (1990) show in general better performance as compared to other models. The differences between the predictive performance of these models does not appear to be significant. Various studies also agree in general concerning the influence of the solar altitude. Lower solar altitudes appear to be associated with larger errors. On the other hand, higher values of the clearness index (i.e. cloudless skies, and high solar altitude) are associated with better model performance. Building simulation specialists and designers of solar energy systems for architectural applications must thus consider such order of magnitude in potential errors while estimating the dynamic behavior of solar energy systems’ processes and for simulating long-term operations. The magnitude of the observed uncertainty suggests that proper “factor of safety” assumptions should be made while designing solar-thermal systems or selecting shading strategies for buildings. These concerns will be addressed in detail in ongoing and future studies, together with the extent of model errors for other locations toward developing more accurate and robust (globally applicable) diffuse fraction models. Likewise, the potential influences of other microclimatic parameters on the error levels will be addressed in detail. Moreover, the potential for alternative sky model generation schemes are being explored.
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Acknowledgment The authors would like to acknowledge the contribution of Ms. Katka Zavodnikova toward collecting and structuring the data used in this paper. References Boland, J., McArthur, L.C., Luther, M., 2001. Modelling the diffuse fraction of global solar radiation on a horizontal surface. Environmetrics 12, 103–116. Cucumo, M., De Rossa, A., Ferraro, V., Kaliakatkos, D., Marinelli, V., 2007. Experimental testing of models for the estimation of hourly solar radiation on vertical surfaces at Arcavacata di Rende. Solar Energy 81, 692–695. Elminir, H., 2005. Experimental and theoretical investigation of diffuse solar radiation: data and models quality tested for Egyptian sites. Energy 32, 73–82. Erbs, D.G., Klein, S.A., Duffie, J.A., 1982. Estimation of the diffuse radiation fraction for hourly, daily and monthly-average global radiation. Solar Energy 28 (4), 293–302. Iqbal, M., 1978. Estimation of the monthly average of the diffuse component of total insolation on a horizontal surface. Solar Energy 20 (1), 101–105. Jacovides, C.P., Tymvios, F.S., Assimakopoulos, V.D., 2006. Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation. Renewable Energy 31 (15), 2492– 2504. Lam, J.C., Li, D.H.W., 1996. Correlation between global solar radiation and its direct and diffuse components. Building and Environment 31 (6), 527–535.
LeBaron, B., Dirmhirn, I., 1983. Strengths and limitations of the Liu and Jordan model to determine diffuse from global irradiance. Solar Energy 31, 167–172. Liu, B.Y.H., Jordan, R.C., 1960. The inter-relationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4 (3), 1–19. Louche, A., Notton, G., Poggi, P., Simonnot, G., 1991. Correlations for direct normal and global horizontal irradiations on a French Mediterranean site. Solar Energy 46 (4), 261–266. MATLAB, 2010. MATLAB Release 2010a. The MathWorks, Inc.