A stochastic model for the hourly solar radiation process for application in renewable resources management

Abstract

Since the beginning of the 21st century, the scientific community has made huge leaps to exploit renewable energy sources, with solar radiation being one of the most important. However, the variability of solar radiation has a significant impact on solar energy conversion systems, such as in photovoltaic systems, characterized by a fast and non-linear response to incident solar radiation. The performance prediction of these systems is typically based on hourly or daily data because those are usually available at these time scales. The aim of this work is to investigate the stochastic nature and time evolution of the solar radiation process for daily and hourly scale, with the ultimate goal of creating a new cyclostationary stochastic model capable of reproducing the dependence structure and the marginal distribution of hourly solar radiation via the clearness index KT.

Full text

Adv. Geosci., 45, 139–145, 2018
https://doi.org/10.5194/adgeo-45-139-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
A stochastic model for the hourly solar radiation process for
application in renewable resources management
Giannis Koudouris, Panayiotis Dimitriadis, Theano Iliopoulou , Nikos Mamassis, and Demetris Koutsoyiannis
Department of Water Resources and Environmental Engineering, School of Civil Engineering, National
Technical University of Athens (NTUA), Heroon Polytechneiou 5, 157 80, Zographou, Greece
Correspondence: Giannis Koudouris (koudourisgiannis@gmail.com)
Received: 31 May 2018 – Revised: 1 August 2018 – Accepted: 3 August 2018 – Published: 14 August 2018
Abstract. Since the beginning of the 21st century, the sci-
entific community has made huge leaps to exploit renewable
energy sources, with solar radiation being one of the most
important. However, the variability of solar radiation has a
significant impact on solar energy conversion systems, such
as in photovoltaic systems, characterized by a fast and non-
linear response to incident solar radiation. The performance
prediction of these systems is typically based on hourly or
daily data because those are usually available at these time
scales. The aim of this work is to investigate the stochastic
nature and time evolution of the solar radiation process for
daily and hourly scale, with the ultimate goal of creating a
new cyclostationary stochastic model capable of reproduc-
ing the dependence structure and the marginal distribution of
hourly solar radiation via the clearness index KT.
1 Introduction
Human activities are either explicitly or implicitly linked
with the dynamic behaviour of the solar radiation process.
As a result, during the last two decades extensive research
(Ettoumi et al., 2002; Tovar-Pescador, 2008; Reno et al.,
2012; Tsekouras and Koutsoyiannis, 2014) has been car-
ried out into the stochastic nature of the solar radiation pro-
cess (e.g. marginal distribution, dependence structure etc.).
Although many popular distributions, used in geophysics
such as (Gamma, Pareto, Lognormal, Logistic, mixture of
two Normal etc.) are suggested in the literature (Ayodele
and Ogunjuyigbe, 2015; Jurado et al., 1995; Aguiar and
Collares-Pereira, 1992; Hollands and Huget, 1983) and may
exhibit a good fit, they cannot adequately fit the right tail
of the distribution. This can be explained, considering that
the right boundary of the process varies at a seasonal scale.
Also, the maximum value of solar radiation that can be mea-
sured at the earth surface is the solar constant (i.e. Gsc =
1367 W m−2) and therefore, distributions which are not right
bounded should not be applied for solar radiation. Koudouris
et al. (2017) introduce a new marginal distribution (i.e. Ku-
maraswamy distribution) for daily solar radiation process
which is verified by three goodness of fit tests. The Ku-
maraswamy distribution may not be a very popular distri-
bution, but it originates from the Beta family of distribu-
tions (Jones, 2009) and exhibits some technical advantages
in modelling, e.g. invertible closed form of the cumulative
distribution function:
F(z;a, b)=
x
Z
0
f(ξ;a, b)dξ=1−1−zαβ⇔
Q(z)=F−1(z)= {1−(1−z) 1
a}1
b,(1)
where z∈[0,1]is standardized according to z=z−zmin
zmax−zmin ,
with zmin and zmax are minimum and maximum values deter-
mined from the empirical time series.
In this research framework a more extensive analysis is be-
ing conducted for hourly solar radiation. The marginal distri-
bution for hourly scale and the dependence structure of the
examined process are being investigated with the ultimate
goal to synthesize a preliminary cyclostationary stochastic
model capable of generating synthetic series calibrated from
regional climate data. Furthermore, one the most common
characteristics of hydrometeorological processes is the dou-
ble periodicity (i.e. the diurnal and seasonal variation of the
process); therefore, solar radiation exhibits same behaviour
(e.g. Fig. 1). The seasonality occurs due to the deterministic
Published by Copernicus Publications on behalf of the European Geosciences Union.

140 G. Koudouris et al.: A stochastic model for the hourly solar radiation process
600.000
700.000
800.000
900.000
Τίτλος γραφήματος
0
100
200
300
400
500
600
700
800
900
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
W m-2
Hours
January February
March April
May June
July August
September October
November December
Figure 1. Double periodicity diagram of Solar radiation (i.e. averaged measured value for each hour) for Denver station.
movement of the earth in orbit around the sun and around its
axis of rotation.
In order to proceed to the hourly scale data analysis, the
double periodicity must be deduced from the measured data.
To achieve that, the clearness index KTis introduced Eq. (2)
KT=I
I0
, KT∈[0,1](2)
The index can be considered as a ratio of both deterministic
and stochastic elements. The clearness index KT, describes
all the meteorological stochastic influences (e.g. cloudiness,
dew point, temperature, atmospheric aerosols) as it is the ra-
tio of the actual solar radiation measured on the ground I,
to that available at the top of the atmosphere I0and so it ac-
counts for the transparency of the atmosphere.
The denominator of Eq. (2) is a deterministic process,
which can be determined from
I0=12
πGscd180
π(ω2−ω1)sin(δ)sin(ϕ)
+cos(δ)cos(ϕ)(sin(ω2)−sin(ω1)),(3)
where dis the solar eccentricity (Eq. 4), Gsc is the solar con-
stant, ωis the hour angle (Eq. 5), ϕis the latitude and δis the
declination (Eq. 6); these quantities are given as
d=1+0.0034cos2πJ
365 −0.05(4)
where J=1 and 365 refer to 1 January and 31 December,
respectively;
ω=(St −12)h
24 ·h360 ⇔ω=15(St −12),
ω∈ [−180◦,180◦](5)
where St is the solar time, St ∈[0,23]so between successive
hours ω1−ω2=15◦;
δ= −0.49cos2πJ
365 +0.16(6)
Therefore, investigating the stochastic nature of clearness in-
dex KTcan lead to conclusions for the fluctuations in hourly
solar radiation.
2 Experimental data
The analysis of hourly solar radiation is conducted via the
clearness index KT. In order to examine the process be-
haviour on a world-wide scale, data from both United States
of America and Greece are examined. Data for Greece
were retrieved from the Hydrological Observatory of Athens
(HOA). The network consists of more than 10 stations lo-
cated in Attica region measuring environmental variables of
hydrometeorological interest. Each station is equipped with
a data logger which records with 10 min interval. These mea-
surements were aggregated to mean hourly data through
the open software “Hydrognomon” (http://hydrognomon.
org, last access: 9 August 2018). For the Attica region, the
KTtimeseries were generated utilizing Eq. (3) by produc-
ing hourly solar top of atmosphere intensity at the local sta-
tion. For USA, the data base of NRLE (National Renew-
able Energy Laboratory)-NSRDB (National Solar Radiation
Database) was used, which contains more than 1500 stations
with hourly solar radiation, but only 40 of them include time
series with more than 14 years of measurements of hourly
global solar radiation measured on horizontal surface.
Adv. Geosci., 45, 139–145, 2018 www.adv-geosci.net/45/139/2018/

G. Koudouris et al.: A stochastic model for the hourly solar radiation process 141
3 Hourly solar radiation stochastic investigation
3.1 Marginal distribution
According to previous research (Koudouris et al., 2017),
daily solar radiation can be modelled by the Kumaraswamy
distribution Eq. (1). A new investigation for the hourly
marginal distribution of hourly solar radiation is being con-
ducted through the clearness index KT. Firstly, for every ex-
amined station, the hourly empirical data KTis divided into
288 times series (i.e. 24 hourly time series for each month
which are constructed with approximately 30 observations
during days for a certain number of years). This technique
is usually found to be sufficient allowing the construction of
a stationary model for the solar radiation time series. The
Kumaraswamy distribution is fitted to the empirical data of
the clearness index, to evaluate the statistical properties of
the solar data under study. Furthermore, three goodness of
fit tests (Marsaglia and Marsaglia, 2004; Csorgo and Far-
way, 1996; Burnham and Anderson, 2003) were applied (i.e.
Kolmogorov-Smirnov, Cramer von Misses and the Anderson
Darling) by setting the significance level at 5 %. For the Psyt-
talia station (Greece), from the 288 times series only 170 are
considered with a mean value of solar radiation much larger
than zero (0, 1367], as a result of the absence of the sun-
shine during the night period. We calculated that for only
44 of them the Kumaraswamy distribution is appropriate.
The empirical probability distributions that were constructed
for these results, seem to indicate that the clearness index
and therefore the solar radiation exhibit a bimodal behaviour.
As a result, from the analysis of all HOA network stations,
the Kumaraswamy distribution cannot describe the empiri-
cal data sufficiently well and thus it is an insufficient dis-
tribution for modelling hourly solar radiation in the Attica
region. Nevertheless, the empirical probability distributions
that were constructed from NSRDB stations indicate that
hourly solar radiation does not always exhibit a bimodal be-
haviour at any geographic location. This is due to the fact that
hourly solar radiation is extremely correlated with the cloudi-
ness process. Consequently, in regions where the cloudiness
process does not exhibit fluctuations in small time scales, the
Kumaraswamy distribution seem to adequately fit the hourly
empirical data. This assumption is confirmed after investi-
gating the marginal distribution of the clearness index at the
Barrow station in Alaska where cloudiness does not exhibit a
highly varying behaviour in contrast to the Attica region. For
a better representation of the multivariate analysis scenarios
of clearness index, a linear combination of Kumaraswamy
distributions is proposed. A new distribution is constructed
considering the sum of two Kumaraswamy distributions con-
ventionally chosen and weighted by a parameter λ∈ [0,1].
The probability distribution and density functions of the pro-
posed distribution are:
F(x;λ, a1, b1, a2, b2)=
λF x;a1,b1+(1−λ)F(x;a2,b2)
F(x;λ, a1, b1, a2, b2)=
λ
χ
Z
0
f(ξ;α1,b1)dξ+(1−λ)
χ
Z
0
f(ξ;α2,b2)dξ
F(x;λ, a1, b1, a2, b2)=
λh1−1−xa1b1i+(1−λ)h1−1−xa2b2i(7)
f=λf1+(1−λ)f2⇔f(x;λ, a1, b1, a2, b2)=
λf x;a1,b1+(1−λ)f(x;a2, b2)
f(x;λ, a1, b1, a2, b2)=
λhα1b1x(a1−1)1−xa1b1i
+(1−λ)hα2b2x(a2−1)1−xa2b2i(8)
The parameters are calculated via the least square estimator
method or the maximum likelihood estimation. The resulting
f (x) and F (x) exhibit good agreement with the empirical
ones (e.g. Fig. 2),notably when solar radiation exhibits a bi-
modal behaviour (e.g. Fig. 2c, b).
3.2 Dependence structure
It is well known from the literature that hydrometeorolog-
ical processes show a large variability (often linked to the
maximum entropy; Koutsoyiannis, 2011) at different time
scales, exhibiting so-called long-term persistence (LTP) or
else the Hurst-Kolmogorov dynamics (Koutsoyiannis, 2002).
In order to investigate if solar radiation exhibits an LTP be-
haviour, the Hurst parameter (representing the dependence
structure of the process) is estimated via the climacogram
tool (i.e. double logarithmic plot of variance of the averaged
process versus averaging time scale k; Koutsoyiannis, 2010).
The Hurst parameter is identified only at large scales. Thus,
in Fig. 3, it is identified for time scales >1 year (8760 h),
where the seasonal variation is averaged out and equals the
half slope of the climacogram, as scale tends to infinity, plus
1. The climacogram has some important statistical advan-
tages if compared to the autocovariance and power spec-
trum (Dimitriadis and Koutsoyiannis, 2015a; Harrouni et al.,
2005). Exploiting the Hurst parameter, the persistence be-
haviour of the process is quantified and examined. The anal-
ysis of both NSRDB and HOA stations estimates the Hurst
parameter larger than 0.5 (where the latter value corresponds
to a white-noise behaviour); consequently, the examined pro-
cess indicates long-term persistence and cannot be consid-
ered as white noise nor a Markov process.
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142 G. Koudouris et al.: A stochastic model for the hourly solar radiation process
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cdf
KT
Month: June - Hour: 09:00
Empirical Cdf
Bimodal Kumaraswamy Cdf
a1=2.3
b1=25.4
a2=9.82
b2=4,1
λ=0.56
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1 1.2
Pdf
KT
Month: June - hour: 09:00
Empirical pdf
Bimodal Kumaraswamy Pdf
a1=2.3
b1=25.4
a2=9.82
b2=4,1
λ=0.56
0
0.2
0.4
0.6
0.8
1
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cdf
KT
Month: February - hour: 19:00
Empirical Cdf
Bimodal Kumaraswamy Cdf
a1=15.0
b1=14.4
a2=1.8
b2=1.7
λ=0.002
0
1
2
3
4
5
6
7
0 0.2 0.4 0.6 0.8 1 1.2
Pdf
KT
Month: February - hour: 19:00
Empirical pdf
Bimodal Kumaraswamy Pdf
a1=15.0
b1=14.4
a2=1.8
b2=1.7
λ=0.002
(a) (b)
(c) (d)
Figure 2. Examples of comparison between empirical and theoretical distribution, where the latter is a linear combination of Kumaraswamy
distributions. (a, b) Psitallia station, Greece (bimodal); (c, d) Barrow station, Alaska (unimodal).
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
110 100 1000 10 000 100 000
Variene γ(κ)
Time scale k (h)
H = 0.65
H = 0.78
H = 0.75
Figure 3. Climacogram of three investigated stations: Denver, H=
0.78, Elizabeth city, H=0.75, Bluefield Virginia, H=0.64.
4 Methodology and application of the model
In this section, we describe a simple methodology to pro-
duce synthetic hourly solar radiation. After we cautiously se-
lect a marginal distribution model (e.g., Eq. 8, Fig. 5), we
estimate the distribution parameters pi,j for each diurnal-
seasonal process xi,j (e.g., 12 ×24 different set of pa-
rameters; Fig. 4). Then we homogenize the timeseries
(Dimitriadis, 2017) by applying the distribution function
Fxi,j ;pi,j to each one of the diurnal-seasonal processes
(with the estimated, e.g. i=12 ×j=24, set of parame-
ters pi,j ) and then, by employing the standard Gaussian (or
any other) distribution function to each diurnal-seasonal pro-
cess, i.e. y=N−1(F (xi,j ;pi,j );0,1). Note that in case the
diurnal-seasonal processes are all Gaussian, the proposed
homogenization is equivalent to the standard normalization
scheme, where for each process the mean is subtracted and
the residual is divided by the standard deviation.
In this implicit way, we manage to homogenize the time-
series xi,j ∼F (xi,j ;pi,j )to y∼N(0,1). In case where the
marginal distribution is unknown or difficult to estimate, we
may use non-linear transformation schemes based on the
maximization of entropy (Koutsoyiannis et al., 2008; Dim-
itriadis and Koutsoyiannis, 2015b). It is noted that a more ro-
bust approach to reduce the 12 ×24 set of parameters would
be to employ an analytical expression for the double solar
periodicity (as done for the wind process in Deligiannis et
al., 2016). This homogenization scheme has been applied to
several processes such as wind (Deligiannis et al., 2016), so-
lar radiation (Koudouris et al., 2017), wave height, wave pe-
riod and wind for renewable energy production (Moschos et
al., 2017), river discharge (Pizarro et al., 2018) and precipi-
tation (Dimitriadis and Koutsoyiannis, 2018). However, it is
noted that this scheme assumes stationary in the dependence
structure rather cyclostationary (for such analyses see Kout-
soyiannis et al., 2008, and references therein).
The above homogenization enables the estimation and
modelling of the dependence structure after the effect of the
double periodicity has been approximately removed. This
homogenization scheme also enables approximating the cor-
relations among the diurnal processes for the same scale
(Fig. 6). After the estimation of the N(0,1)homogenized
Adv. Geosci., 45, 139–145, 2018 www.adv-geosci.net/45/139/2018/

G. Koudouris et al.: A stochastic model for the hourly solar radiation process 143
Figure 4. Results of the simulation model for the Denver Station: (a) 2-year synthetic timeseries of hourly KT;(b) yearly average of hourly
solar radiation observations vs. synthetic values of 15 years simulation.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Distribution
KT
Modelled
Empirical
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1
Density
KT
Modelled
Empirical
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Density
KT
Modelled
Empirical
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Distribution
KT
Modelled
Empirical
(a) (b)
(c) (d)
Figure 5. Comparison of distribution (a, c) and density (b, d) function of simulated and observed data, for hour 13:00: (a, b) May;
(c, d) September.
timeseries, we estimate the dependence structure through the
second-order climacogram and we fit a generalized-Hurst
Kolmogorov (GHK) model (Dimitriadis and Koutsoyiannis,
2018):
γ(k)=1/(1+k/q)2−2H(9)
where γis the standardized variance, kthe scale (h), qa
scale-parameter and Hthe Hurst parameter (for the exam-
ined process we estimate q=2 and H=0.83, considering
the bias effect; Fig. 7).
In this implicit manner, the marginal characteristics of
each periodic part are exactly preserved (since the marginal
distribution functions are implicitly handled through the pro-
posed homogenization) and the expectation of the second-
order dependence structure (e.g. correlation function) is also
exactly preserved after properly adjusting for bias through
the mode or expected value of the estimator (Dimitriadis,
2017). It is noted that higher-order moments of processes
with HK behaviour cannot be adequately preserved in an
implicit manner (see an illustrative example in Dimitriadis
and Koutsoyiannis, 2018, their Appendix D) and thus, for a
more accurate preservation of the dependence structure an
explicit algorithm is necessary (Dimitriadis and Koutsoyian-
nis, 2018). We may use a simple generation scheme, such as
the sum of AR(1) models (SAR; Dimitriadis and Koutsoyian-
nis, 2015b), that can synthesize any N(0,1)autoregressive-
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144 G. Koudouris et al.: A stochastic model for the hourly solar radiation process
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
7 9 11 13 15 17 19
Correlation function
Lag 1 (hour)
January February March April May June
July August September October November December
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
7 9 11 13 15 17 19
Correlation function
Lag 1 (hour)
January February March April May June
July August September October November December
(a) (b)
Figure 6. Lag-1 autocorrelation of hourly time series for each month (a) empirical; (b) simulated.
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
110 100 1000 10 000 100 000
Standardized variane γ(κ)
Time scale k (hours)
Observed
Modelled including double periodicities
Homogenized (without double periodicities)
White noise (uncorrelated)
Figure 7. Climacogram of observed and simulated KTvs. white
noise and a homogenized process without periodicities.
like dependence structure, which later it can be transformed
back to the original distribution function F (xi,j ;pi,j )and so
in this way produce a double-periodic process with the de-
sired marginal distribution for each diurnal-seasonal cycle as
well as the desired dependence structure. Finally, we multi-
ply each value of the synthetic KTwith the deterministically
determined value of the hourly intensity of solar radiation at
the top of the atmosphere.
5 Conclusions
The hourly marginal distribution of solar radiation is inves-
tigated via the clearness index KT. Regarding the marginal
distribution, after analysing a variety of stations with dif-
ferent regional climate data, it is concluded that the Ku-
maraswamy distribution cannot adequately describe the em-
pirical data of hourly solar radiation. Therefore, a new bi-
modal distribution is constructed which is a weighted sum
of two Kumaraswamy distributions This distribution is fully
characterized by five parameters and can adequately fit the
empirical hourly data of the clearness index in in the regions
investigated in this study. Also, the dependence structure of
solar radiation process is investigated via the climacogram
tool. It is concluded that since the Hurst parameter is esti-
mated higher than 0.5, the examined process exhibits long-
term persistence and cannot be considered as a white noise
nor a Markov process. Finally, a new preliminary stochas-
tic model is proposed, capable of reproducing the clearness
sky index KTand so the hourly solar radiation. The model
can maintain and preserve the probability density function
by means of the first four central moments and also the Hurst
parameter which represents the correlation and the persistent
behaviour.
Data availability. The datasets generated during the current study
are available from the corresponding author on reasonable request.
Author contributions. GK conceived of the presented idea, devel-
oped the theory and performed the computations. TI helped with R
environment statistical tests. PD verified the analytical methods and
encouraged GK to investigate the LTP behaviour of the solar radi-
ation process. NM contributed to sample preparation. DK and PD
supervised the findings of this work. GK took the lead in writing
the manuscript. All authors provided critical feedback and helped
shape the research, analysis and manuscript.
Competing interests. The authors declare that they have no conflict
of interest.
Special issue statement. This article is part of the special issue “Eu-
ropean Geosciences Union General Assembly 2018, EGU Division
Energy, Resources & Environment (ERE)”. It is a result of the EGU
General Assembly 2018, Vienna, Austria, 8–13 April 2018.
Acknowledgements. The authors thank warmly the anonymous
reviewers for their most helpful comments. The statistical analyses
were performed in the R statistical environment (RDC Team,
2006) by also using the contributed packages VGAM (Yee, 2015),
fitdistrplus (Delignette-Muller and Dutang, 2015) and lmomco
Adv. Geosci., 45, 139–145, 2018 www.adv-geosci.net/45/139/2018/

G. Koudouris et al.: A stochastic model for the hourly solar radiation process 145
(Asquith, 2018).
Edited by: Sonja Martens
Reviewed by: two anonymous referees
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