ASRAdvances in Science and ResearchASRAdv. Sci. Res.1992-0636Copernicus PublicationsGöttingen, Germany10.5194/asr-13-1-2016On the effective solar zenith and azimuth angles to use with measurements of hourly irradiationBlancP.https://orcid.org/0000-0002-6345-0004WaldL.lucien.wald@mines-paristech.frhttps://orcid.org/0000-0002-2916-2391MINES ParisTech – PSL Research University, Sophia Antipolis, FranceL. Wald (lucien.wald@mines-paristech.fr)2February2016131630November201521January2016This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://asr.copernicus.org/articles/13/1/2016/asr-13-1-2016.htmlThe full text article is available as a PDF file from https://asr.copernicus.org/articles/13/1/2016/asr-13-1-2016.pdf
Several common practices are tested for assessing the effective solar zenith
angle that can be associated to each measurement in time-series of in situ
or satellite-derived measurements of hourly irradiation on horizontal
surface. High quality 1 min measurements of direct irradiation collected by
the BSRN stations in Carpentras in France and Payerne in Switzerland, are
aggregated to yield time series of hourly direct irradiation on both
horizontal and normal planes. Time series of hourly direct horizontal
irradiation are reconstructed from those of hourly direct normal irradiation
and estimates of the effective solar zenith angle by one of the six
practices. Differences between estimated and actual time series of the
direct horizontal irradiation indicate the performances of six practices.
Several of them yield satisfactory estimates of the effective solar angles.
The most accurate results are obtained if the effective angle is computed by
two time series of the direct horizontal and normal irradiations that should
be observed if the sky were cloud-free. If not possible, then the most
accurate results are obtained from using irradiation at the top of
atmosphere. Performances show a tendency to decrease during sunrise and
sunset hours. The effective solar azimuth angle is computed from the
effective solar zenith angle.
Introduction
Time-series of measurements of hourly irradiation on horizontal surface are
increasingly available from in-situ measurements, satellite retrievals,
meteorological numerical models, or combinations of these. They are useful
in many aspects in solar energy and other domains, e.g. architecture,
building management, agriculture, or biomass. Notably, hourly irradiations
are included in Typical Meteorological Year (TMY) data sets that are widely
used for simulation of solar conversion and building systems (Kalogirou,
2003; Hall et al., 1978).
In many cases, such data – whether in situ or satellite-derived – are
inputs to numerical procedures with different aims, ranging from quality
control and gap filling to assessment of the radiation impinging on a tilted
plane. Such computations can be performed only if a solar zenith angle θS,
and azimuth angle ΨS in some cases, can be
associated to each measurement. However, such angles are seldom given for
each hourly irradiation. Currently, only time stamp is given for each measurement.
If measurements are made with integration duration, also called
summarization, of 1 min, one may consider that the sun angles are
approximately constant, or more correctly that they vary approximately
linearly, and that they can be computed for the middle of the corresponding
minute. This is not the case for summarization of 15 min or 1 h. Angles are
greatly varying within such duration, especially at the beginning and end of
the day. In this context, there is a practical request from companies,
academics, or researchers: what is the best practice for computing these
angles? The article deals with this question. It does not intend to bring
definite answers which may be diverse if one considers the final goal of the
process requiring solar zenith angle as input. It presents a simple study
bringing practical answers to questions brought up by practitioners to the
attention of the International Energy Agency (IEA).
Several practices already exist. To the best of the knowledge of the
authors, there is no scientific publication supporting these practices and
comparing them. The work presented here compares the performances of a few
common practices and makes recommendations keeping in mind the practical
aspects faced by practitioners, companies, academics, and researchers.
Hourly values are dealt with for the sake of the simplicity but the work is
applicable to other summarizations.
Current practices and new ones
Let BN denote the direct irradiation received on a plane always normal
to the sun rays. Let note G, D and B respectively the global, diffuse and
direct irradiation received on a horizontal plane. The direct radiation is
also called the beam radiation. Practically, BN may be measured or B may
be deduced from the difference between G and D. In many cases, only G is known.
The following relationship holds:
BN=BcosθS
and θS is the angle to be used in further calculations. The
computation of BN is very sensitive to the solar zenith angle θS
which for this reason is the quantity dealt with in this work.
In case of summarization greater than 1 min, θS varies noticeably
and the application of Eq. (1) becomes a problem. Which value is the right one
to use? An effective angle θSeff must be used to handle
measured and modelled hourly irradiation whether global Gh, diffuse Dh,
or direct Bh or BNh irradiations where the
subscript “h” means hourly. Practically, Eq. (1) is rewritten with
BNh=BhcosθSeff.
Six practices to compute θSeff have been identified, named
from A0 to A5. Let t, expressed in h, define the time of the end of the
summarization Δt, equal to 1 h in this case, and assume that the
summarization is 1 h.
A0: θSeff is taken as θS at half-hour,
i.e. (t- 0.5)
θSeff,A0=θS(t-0.5).
A1: θSeff is taken as the average of θS
over the hour
θSeff,A1=1Δt∫t-1tθS(u)du.
A2: θSeff is taken as the average of θS
over the hour provided θS<π/2θSeff,A2=1Δt∫t-1θS<π/2tθS(u)du.
A3: θSeff is taken as the average of θS
over the hour but limited to the daylight period in the astronomical sense
θSeff,A3=∫t-1θS<π/2tθS(u)du∫t-1θS<π/2tdu.
A4: θSeff is computed from hourly irradiations
BhTOA and BNhTOA received at the top of atmosphere. Note
that at top of atmosphere, there is no downwelling diffuse component and
that the direct irradiation is equal to the global irradiation.
θSeff,A4=cos-1BhTOABNhTOA
A5: θSeff is computed from hourly irradiations given
by a clear-sky model
θSeff,A5=cos-1BhclearBNhclear
where a clear-sky model is a model providing estimates of Bhclear
and BNhclear that would be observed if the sky were clear at this
instant and location. The McClear model (Lefèvre et al., 2013) is such a
model and is used here.
Practices A0 to A4 were discussed during a meeting of the Task #46 of the
Solar Heating and Cooling Implementing Agreement of the IEA held in Almeria,
Spain, in January 2015. The practice A5 was used by Korany et al. (2015).
Methodology for assessing the performances of each practice
Time-series of measurements of the BSRN stations at Carpentras, France, and
Payerne, Switzerland, were collected that span 2008 to 2010. Carpentras is
located in Provence, in the Southeast of France (Table 1) and experiences
Mediterranean climate, i.e. warm temperate climate with dry and hot summer
with many days of cloud-free skies throughout the year. Payerne experiences
oceanic climate, i.e. warm temperate, fully humid, and warm summer; many
small cumulus clouds can be observed during summer days. Measurements are
acquired every 1 min for BN as well as for G and D. Uncertainty
requirements for BSRN data are 5 W m-2 for global irradiance and
2 W m-2 for direct irradiance (Ohmura et al., 1998). Only measurements
passing the quality check procedures described by Roesch et al. (2011) has
been considered here.
Geographical coordinates of the two BSRN stations.
Correlogram between BNh (horizontal axis) and BNh*
computed with θSeff,A5 (vertical axis) for all solar zenith
angles (SZA) for Carpentras.
For each station, every 1 min, the actual θS was accurately
computed by the means of the SG2 algorithm (Blanc and Wald, 2012). The actual
direct irradiance on horizontal surface BN can be computed using Eq. (1).
Then, hourly measurements are simulated by aggregating BN and B over
1 h, yielding hourly irradiations BNh and Bh. Only hours with no
missing nor invalid data have been selected. Given these hourly time-series,
θSeff is computed with the six proposed practices. Then, using
Eq. (2), an estimated time series BNh* is computed from the actual
Bh time series:
BNh*=BhcosθSeff.
Finally, the actual BNh and estimated BNh* time series are
compared. The deviations: BNh*-BNh are computed and then
summarized by the bias, root mean square error and correlation coefficient.
The smaller the discrepancies, the more accurate the practice.
Results
A subset of the data, called “low sun”, has been created to better study
the cases of sun low above horizon, i.e. θS> 75∘. Tables 2
and 3 report the results at Carpentras and Payerne for daylight time (all
angles) and for the subset “low sun” for the six different practices.
Correlograms between BNh (horizontal axis) and BNh* (vertical axis) at
Carpentras and Payerne for practice A5 are shown in Figs. 1–4.
One observes that the error depends on the range of θS. For large
θS, errors are much greater than for smaller θS. One
may also observe that errors are far from being negligible for most practices.
Errors are the greatest for practices A0, A1 and A2. The bias ranges from
5 to 6 Wh m-2 for all θS, and from 16 to 21 Wh m-2
for θS> 75∘. In the latter case, it means a relative bias
of 16–17 % which is quite large. The RMSE ranges from 16 to 25 Wh m-2
for all θS, and from 31 to 48 Wh m-2 for
θS> 75∘ – relative values are 30–40 %. Correlation
coefficients are very large as a whole. The minima are observed for large
θS and are greater than 0.974.
Better results are attained for practices A3 and A4. The bias is 4–5 Wh m-2
for all θS, and 13–15 Wh m-2 for θS> 75∘.
It corresponds to respectively 1 and 12–13 % in
relative values. The RMSE is 13–15 Wh m-2 (relative RMSE is 4–5 %)
for all θS and ranges from 24 to 28 Wh m-2 for
θS> 75∘ – relative RMSE is 23–24 %. Correlation
coefficients are very large as a whole. The minima are observed for large
θS and are greater than 0.986.
Correlogram between BNh (horizontal axis) and BNh*
computed with θSeff,A5 (vertical axis) for the subset “low
sun” (SZA: solar zenith angle) for Carpentras.
Performance of each practice for Carpentras for all angles and the
subset “low sun”. Relative values are computed relative to the mean value
of BNh. Best results are in bold.
Performance of each practice for Payerne for all angles and the
subset “low sun”. Relative values are computed relative to the mean value
of BNh. Best results are in bold.
The best results are attained for practice A5. The bias is very small:
0 Wh m-2 or close to, for all θS and 1–3 Wh m-2
for θS> 75∘. The RMSE is 6–7 Wh m-2 – relative
RMSE is 2 % – for all SZA and 9–10 Wh m-2 for
θS> 75∘ – relative RMSE is 8–9 %. Correlation coefficients
are very large as a whole and greater than 0.998. One may observe in Figs. 1
to 4 that the points are well aligned along the y=x line with a very small scattering.
Correlogram between BNh (horizontal axis) and BNh*
computed with θSeff,A5 (vertical axis) for all solar zenith
angles (SZA) for Payerne.
The performances decrease with large θS. It should be noted
that the decrease is much less pronounced with practice A5 than with the
others. For example, the RMSE for A5 increases from 7 to 10 Wh m-2
for Carpentras, and from 6 to 9 Wh m-2 for Payerne, while it doubles
for the other practices, e.g. from 25 to 48 Wh m-2 for Carpentras and A0.
Correlogram between BNh (horizontal axis) and BNh*
computed with θSeff,A5 (vertical axis) for the subset “low
sun” (SZA: solar zenith angle) for Payerne.
The azimuth of the sun ΨS is defined as the angle between the
projection of the direction of the sun on the horizontal plane and a
reference direction. The ISO convention is to count ΨS
clockwise from North where its value is 0. Thus, it is π2 for
East, π for South and 3π2 for West. The effective solar
azimuth ΨSeff may be computed with the following
equations (ESRA, 2000):
ΨSeff=π-cos-1sinΦcosθSeff-sinδcosΦsinθSeffbeforenoonΨSeff=π+cos-1sinΦcosθSeff-sinδcosΦsinθSeffafternoon
where Φ is the latitude and δ the declination angle.
Conclusions
Several practices may be used to compute the effective solar angles. Though
dealing with a limited number of cases, this study has shown that errors are
far from being negligible for most practices. One must be careful in selecting the practice.
The practice A5 produces the best results by far with no bias and small
RMSE. It is followed by A4 and A3. The worst ones are A0, A1 and A2. For all
methods, performances show a tendency to decrease during sunrise and sunset
hours. The errors may double, except for A5 which shows little degradation
in performance for large θS.
Practically, how to implement these practices? Practices A0 to A4 needs a
library to compute θS every 1 min. Among several solutions, such
as the Python code PyEphem available at http://rhodesmill.org/pyephem/, one
may use the equations in ESRA (2000) which form the Solar Geometry 1 library (SG1)
or the more accurate SG2 library (Blanc and Wald, 2012); both are
available at http://www.oie.mines-paristech.fr/Valorisation/Outils/Solar-Geometry/. If
one does not possess software for computing this angle, the web site SoDa
Service for professionals in solar radiation (www.soda-pro.com) offer
free-of-charge efficient services that implement SG2 and deliver time series
of solar zenith and azimuth angles and declination angle.
Implementing practice A4 requests in addition the computation of the hourly
irradiation at the top of atmosphere on horizontal and normal-to-sun
surfaces. The above mentioned tools, including the SoDa Service, may be used
in that purpose.
Implementing practice A5 may be costly if one does not possess software for
the estimation of hourly – or better resolution – direct irradiation on
both normal and horizontal surfaces under cloud-free skies. The ESRA
clear-sky model (Rigollier et al., 2004) is easy to implement (code
available at http://www.oie.mines-paristech.fr/Valorisation/Outils/Clear-Sky-Library/)
but several others are, too. If one does not possess a clear-sky model, the
SoDa Service offer free-of-charge a service that implements the McClear
model and that delivers in one click time series of hourly irradiation on
horizontal and normal-to-sun surfaces for any place in the world.
Acknowledgements
The authors thank the two reviewers whose comments help in improving this
text. The research leading to these results has been undertaken within the
Task #46 of the Solar Heating and Cooling Implementing Agreement of the
International Energy Agency, and has been partly funded by the French Agency
ADEME, research grant no. 1105C0028. The authors thank the operators of the
Carpentras and Payerne stations for their valuable measurements and the
Alfred Wegener Institute for hosting the BSRN website from which data may be
downloaded. The McClear service in the SoDa Service is made available to
anyone by the MINES ParisTech and Transvalor within the Copernicus
Atmosphere Monitoring Service implemented by the ECMWF on behalf of the
European Commission.
Edited by: S.-E. Gryning
Reviewed by: two anonymous referees
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