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

### Giannis Koudouris

### Panayiotis Dimitriadis

### Theano Iliopoulou

### Nikos Mamassis

### Demetris Koutsoyiannis

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 *K*_{T}.

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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 carried out into
the stochastic nature of the solar radiation process (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
measured at the earth surface is the solar constant (i.e.
*G*_{sc}=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. Kumaraswamy
distribution) for daily solar radiation process which is verified by three
goodness of fit tests. The Kumaraswamy distribution may not be a very popular
distribution, but it originates from the Beta family of distributions (Jones,
2009) and exhibits some technical advantages in modelling, e.g. invertible
closed form of the cumulative distribution function:

where $z\in \left[\mathrm{0},\mathrm{1}\right]$ is standardized according to $z=\frac{z-{z}_{min}}{{z}_{max}-{z}_{min}}$, with *z*_{min} and *z*_{max} are
minimum and maximum values determined from the empirical time series.

In this research framework a more extensive analysis is being conducted for hourly solar radiation. The marginal distribution 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 double 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 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 *K*_{T} is introduced Eq. (2)

The index can be considered as a ratio of both deterministic and stochastic
elements. The clearness index *K*_{T}, describes all the meteorological
stochastic influences (e.g. cloudiness, dew point, temperature, atmospheric
aerosols) as it is the ratio of the actual solar radiation measured on the
ground *I*, to that available at the top of the atmosphere *I*_{0}
and so it accounts for the transparency of the atmosphere.

The denominator of Eq. (2) is a deterministic process, which can be determined from

where *d* is the solar eccentricity (Eq. 4), *G*_{sc} is the solar constant,
*ω* is the hour angle (Eq. 5), *φ* is the latitude and
*δ* is the declination (Eq. 6); these quantities are given as

where *J*=1 and 365 refer to 1 January and 31 December,
respectively;

where St is the solar time, $\mathrm{St}\in \left[\mathrm{0},\mathrm{23}\right]$ so between successive hours ${\mathit{\omega}}_{\mathrm{1}}-{\mathit{\omega}}_{\mathrm{2}}=\mathrm{15}{}^{\circ}$;

Therefore, investigating the stochastic nature of clearness index *K*_{T} can
lead to conclusions for the fluctuations in hourly solar radiation.

The analysis of hourly solar radiation is conducted via the clearness index
*K*_{T}. In order to examine the process behaviour 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 located in Attica region measuring
environmental variables of hydrometeorological interest. Each station is
equipped with a data logger which records with 10 min interval. These
measurements were aggregated to mean hourly data through the open software
“Hydrognomon” (http://hydrognomon.org, last access: 9 August 2018).
For the Attica region, the *K*_{T} timeseries were generated utilizing
Eq. (3) by producing hourly solar top of atmosphere intensity at the local
station. For USA, the data base of NRLE (National Renewable 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.

## 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 conducted through the clearness index *K*_{T}. Firstly, for every
examined station, the hourly empirical data *K*_{T} is 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 Farway, 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 Psyttalia 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
sunshine 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 empirical data sufficiently well
and thus it is an insufficient distribution 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 behaviour at any geographic
location. This is due to the fact that hourly solar radiation is extremely
correlated with the cloudiness 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 investigating 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 conventionally chosen and weighted by a parameter $\mathit{\lambda}\in [\mathrm{0},\mathrm{1}]$. The probability distribution and density functions of the proposed
distribution are:

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
bimodal behaviour (e.g. Fig. 2c, b).

## 3.2 Dependence structure

It is well known from the literature that hydrometeorological 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 behaviour, 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
advantages if compared to the autocovariance and power spectrum (Dimitriadis
and Koutsoyiannis, 2015a; Harrouni et al., 2005).
Exploiting the Hurst parameter, the persistence behaviour of the process is
quantified and examined. The analysis 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 process
indicates long-term persistence and cannot be considered as white noise nor a
Markov process.

In this section, we describe a simple methodology to produce synthetic
hourly solar radiation. After we cautiously select a marginal distribution
model (e.g., Eq. 8, Fig. 5), we estimate the distribution parameters
**p**_{i,j} for each diurnal-seasonal process *x*_{i,j} (e.g.,
12×24 different set of parameters; Fig. 4). Then we homogenize
the timeseries (Dimitriadis, 2017) by applying the distribution function
$F\left({x}_{i,j};{\mathbf{p}}_{i,j}\right)$ to each one of the
diurnal-seasonal processes (with the estimated, e.g. $i=\mathrm{12}\times j=\mathrm{24}$, set of parameters **p**_{i,j}) and then, by employing the
standard Gaussian (or any other) distribution function to each
diurnal-seasonal process, i.e. $y={N}^{-\mathrm{1}}\left(F\right({x}_{i,j};{\mathbf{p}}_{i,j});\mathrm{0},\mathrm{1})$. 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 timeseries ${x}_{i,j}\sim F({x}_{i,j};{\mathbf{p}}_{i,j})$ to $y\sim N(\mathrm{0},\mathrm{1})$. 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; Dimitriadis and Koutsoyiannis, 2015b). It is noted that a more robust 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), solar radiation (Koudouris et al., 2017), wave height, wave period and wind for renewable energy production (Moschos et al., 2017), river discharge (Pizarro et al., 2018) and precipitation (Dimitriadis and Koutsoyiannis, 2018). However, it is noted that this scheme assumes stationary in the dependence structure rather cyclostationary (for such analyses see Koutsoyiannis 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 correlations among the diurnal processes for the same scale (Fig. 6).
After the estimation of the *N*(0,1) homogenized timeseries, we estimate the
dependence structure through the second-order climacogram and we fit a
generalized-Hurst Kolmogorov (GHK) model (Dimitriadis and Koutsoyiannis,
2018):

where *γ* is the standardized variance, *k* the scale (h), *q* a
scale-parameter and *H* the Hurst parameter (for the examined 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 proposed 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 Koutsoyiannis, 2018). We may use a simple generation scheme,
such as the sum of AR(1) models (SAR; Dimitriadis and Koutsoyiannis,
2015b), that can synthesize any *N*(0,1)
autoregressive-like dependence structure, which later it can be transformed
back to the original distribution function $F({x}_{i,j};{\mathbf{p}}_{i,j})$ and
so in this way produce a double-periodic process with the desired marginal
distribution for each diurnal-seasonal cycle as well as the desired
dependence structure. Finally, we multiply each value of the synthetic
*K*_{T} with the deterministically determined value of the hourly intensity
of solar radiation at the top of the atmosphere.

The hourly marginal distribution of solar radiation is investigated via the
clearness index *K*_{T}. Regarding the marginal distribution, after analysing
a variety of stations with different regional climate data, it is concluded
that the Kumaraswamy distribution cannot adequately describe the empirical
data of hourly solar radiation. Therefore, a new bimodal 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 estimated 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 stochastic model is
proposed, capable of reproducing the clearness sky index *K*_{T} and 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.

The datasets generated during the current study are available from the corresponding author on reasonable request.

GK conceived of the presented idea, developed 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 radiation 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.

The authors declare that they have no conflict of interest.

This article is part of the special issue “European 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.

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
(Asquith, 2018).

Edited by: Sonja Martens

Reviewed by: two anonymous referees

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*K*

_{T}.