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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−zαβ⇔

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

πGscd180

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+0.0034cos2πJ

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= −0.49cos2πJ

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−xa1b1i+(1−λ)h1−1−xa2b2i(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−xa1b1i

+(1−λ)hα2b2x(a2−1)1−xa2b2i(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.

www.adv-geosci.net/45/139/2018/ Adv. Geosci., 45, 139–145, 2018

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

Fxi,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-

www.adv-geosci.net/45/139/2018/ Adv. Geosci., 45, 139–145, 2018

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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