Adv. Geosci., 45, 139–145, 2018https://doi.org/10.5194/adgeo-45-139-2018© Author(s) 2018. This work is distributed underthe Creative Commons Attribution 4.0 License.

A stochastic model for the hourly solar radiation process forapplication in renewable resources managementGiannis Koudouris, Panayiotis Dimitriadis, Theano Iliopoulou, Nikos Mamassis, and Demetris KoutsoyiannisDepartment of Water Resources and Environmental Engineering, School of Civil Engineering, NationalTechnical 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 renewableenergy sources, with solar radiation being one of the mostimportant. However, the variability of solar radiation has asignificant impact on solar energy conversion systems, suchas in photovoltaic systems, characterized by a fast and non-linear response to incident solar radiation. The performanceprediction of these systems is typically based on hourly ordaily data because those are usually available at these timescales. The aim of this work is to investigate the stochasticnature and time evolution of the solar radiation process fordaily and hourly scale, with the ultimate goal of creating anew cyclostationary stochastic model capable of reproduc-ing the dependence structure and the marginal distribution ofhourly solar radiation via the clearness index KT .

1 Introduction

Human activities are either explicitly or implicitly linkedwith 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 geophysicssuch as (Gamma, Pareto, Lognormal, Logistic, mixture oftwo Normal etc.) are suggested in the literature (Ayodeleand Ogunjuyigbe, 2015; Jurado et al., 1995; Aguiar andCollares-Pereira, 1992; Hollands and Huget, 1983) and mayexhibit a good fit, they cannot adequately fit the right tailof 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 rightbounded should not be applied for solar radiation. Koudouriset al. (2017) introduce a new marginal distribution (i.e. Ku-maraswamy distribution) for daily solar radiation processwhich 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 advantagesin modelling, e.g. invertible closed form of the cumulativedistribution function:

F (z;a,b)=

x∫0

f (ξ ;a,b)dξ = 1−(1− zα

)β⇔

Q(z)= F−1 (z)= {1− (1− z)1a }

1b , (1)

where z ∈ [0,1] is standardized according to z= z−zminzmax−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 theexamined process are being investigated with the ultimategoal to synthesize a preliminary cyclostationary stochasticmodel capable of generating synthetic series calibrated fromregional climate data. Furthermore, one the most commoncharacteristics of hydrometeorological processes is the dou-ble periodicity (i.e. the diurnal and seasonal variation of theprocess); 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 itsaxis of rotation.

In order to proceed to the hourly scale data analysis, thedouble periodicity must be deduced from the measured data.To achieve that, the clearness index KT is introduced Eq. (2)

KT =I

I0, KT ∈ [0,1] (2)

The index can be considered as a ratio of both deterministicand stochastic elements. The clearness index KT , describesall 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 I0 and 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 =12πGscd

[180π(ω2−ω1)sin(δ)sin(ϕ)

+ cos(δ)cos(ϕ)(sin(ω2)− sin(ω1))

], (3)

where d is the solar eccentricity (Eq. 4),Gsc is the solar con-stant, ω is the hour angle (Eq. 5), ϕ is the latitude and δ is thedeclination (Eq. 6); these quantities are given as

d = 1+ 0.0034cos(

2πJ365− 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 successivehours ω1−ω2 = 15◦;

δ =−0.49cos(

2πJ365+ 0.16

)(6)

Therefore, investigating the stochastic nature of clearness in-dex KT can lead to conclusions for the fluctuations in hourlysolar radiation.

2 Experimental data

The analysis of hourly solar radiation is conducted via theclearness index KT . In order to examine the process be-haviour on a world-wide scale, data from both United Statesof America and Greece are examined. Data for Greecewere retrieved from the Hydrological Observatory of Athens(HOA). The network consists of more than 10 stations lo-cated in Attica region measuring environmental variables ofhydrometeorological interest. Each station is equipped witha data logger which records with 10 min interval. These mea-surements were aggregated to mean hourly data throughthe open software “Hydrognomon” (http://hydrognomon.org, last access: 9 August 2018). For the Attica region, theKT timeseries 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 RadiationDatabase) was used, which contains more than 1500 stationswith hourly solar radiation, but only 40 of them include timeseries with more than 14 years of measurements of hourlyglobal solar radiation measured on horizontal surface.

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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 Kumaraswamydistribution Eq. (1). A new investigation for the hourlymarginal distribution of hourly solar radiation is being con-ducted through the clearness index KT . Firstly, for every ex-amined station, the hourly empirical data KT is divided into288 times series (i.e. 24 hourly time series for each monthwhich are constructed with approximately 30 observationsduring days for a certain number of years). This techniqueis usually found to be sufficient allowing the construction ofa stationary model for the solar radiation time series. TheKumaraswamy distribution is fitted to the empirical data ofthe clearness index, to evaluate the statistical properties ofthe solar data under study. Furthermore, three goodness offit 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 AndersonDarling) by setting the significance level at 5 %. For the Psyt-talia station (Greece), from the 288 times series only 170 areconsidered with a mean value of solar radiation much largerthan zero (0, 1367], as a result of the absence of the sun-shine during the night period. We calculated that for only44 of them the Kumaraswamy distribution is appropriate.The empirical probability distributions that were constructedfor these results, seem to indicate that the clearness indexand 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 Atticaregion. Nevertheless, the empirical probability distributionsthat were constructed from NSRDB stations indicate thathourly solar radiation does not always exhibit a bimodal be-haviour at any geographic location. This is due to the fact thathourly solar radiation is extremely correlated with the cloudi-ness process. Consequently, in regions where the cloudinessprocess does not exhibit fluctuations in small time scales, theKumaraswamy distribution seem to adequately fit the hourlyempirical data. This assumption is confirmed after investi-gating the marginal distribution of the clearness index at theBarrow station in Alaska where cloudiness does not exhibit ahighly varying behaviour in contrast to the Attica region. Fora better representation of the multivariate analysis scenariosof clearness index, a linear combination of Kumaraswamydistributions is proposed. A new distribution is constructedconsidering 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∫0

f (ξ ;α1,b1)dξ + (1− λ)

χ∫0

f (ξ ;α2,b2)dξ

F (x;λ,a1,b1,a2,b2)=

λ[1−

(1− xa1

)b1]+ (1− λ)

[1−

(1− xa2

)b2]

(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)=

λ[α1b1x

(a1−1)(1− xa1)b1]

+ (1− λ)[α2b2x

(a2−1)(1− xa2)b2]

(8)

The parameters are calculated via the least square estimatormethod or the maximum likelihood estimation. The resultingf (x) and F(x) exhibit good agreement with the empiricalones (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 themaximum entropy; Koutsoyiannis, 2011) at different timescales, exhibiting so-called long-term persistence (LTP) orelse the Hurst-Kolmogorov dynamics (Koutsoyiannis, 2002).In order to investigate if solar radiation exhibits an LTP be-haviour, the Hurst parameter (representing the dependencestructure of the process) is estimated via the climacogramtool (i.e. double logarithmic plot of variance of the averagedprocess 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 thehalf slope of the climacogram, as scale tends to infinity, plus1. 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 Hurstparameter larger than 0.5 (where the latter value correspondsto 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

Cd

f

KT

Month: June - Hour: 09:00

Empirical Cdf

Bimodal Kumaraswamy Cdf

a1=2.3b1=25.4a2=9.82b2=4,1λ=0.56

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

Pd

f

KT

Month: June - hour: 09:00

Empirical pdf

Bimodal Kumaraswamy Pdf

a1=2.3b1=25.4a2=9.82b2=4,1λ=0.56

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

Cd

f

KT

Month: February - hour: 19:00

Empirical Cdf

Bimodal Kumaraswamy Cdf

a1=15.0b1=14.4a2=1.8b2=1.7λ=0.002

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2P

df

KT

Month: February - hour: 19:00

Empirical pdf

Bimodal Kumaraswamy Pdf

a1=15.0b1=14.4a2=1.8b2=1.7λ=0.002

(a) (b)

(c) (d)

Figure 2. Examples of comparison between empirical and theoretical distribution, where the latter is a linear combination of Kumaraswamydistributions. (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

1 10 100 1000 10 000 100 000

Var

ien

e 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), weestimate 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 functionF(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 (orany 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 proposedhomogenization is equivalent to the standard normalizationscheme, where for each process the mean is subtracted andthe 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 themarginal distribution is unknown or difficult to estimate, wemay use non-linear transformation schemes based on themaximization 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 wouldbe to employ an analytical expression for the double solarperiodicity (as done for the wind process in Deligiannis etal., 2016). This homogenization scheme has been applied toseveral 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 etal., 2017), river discharge (Pizarro et al., 2018) and precipi-tation (Dimitriadis and Koutsoyiannis, 2018). However, it isnoted that this scheme assumes stationary in the dependencestructure rather cyclostationary (for such analyses see Kout-soyiannis et al., 2008, and references therein).

The above homogenization enables the estimation andmodelling of the dependence structure after the effect of thedouble periodicity has been approximately removed. Thishomogenization 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 hourlysolar 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

Dis

trib

uti

on

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

Den

sity

KT

Modelled

Empirical

0

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0.8

1

0 0.2 0.4 0.6 0.8 1

Den

sity

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 thesecond-order climacogram and we fit a generalized-HurstKolmogorov (GHK) model (Dimitriadis and Koutsoyiannis,2018):

γ (k)= 1/(1+ k/q)2−2H (9)

where γ is the standardized variance, k the scale (h), q ascale-parameter and H the Hurst parameter (for the exam-ined process we estimate q = 2 and H = 0.83, consideringthe bias effect; Fig. 7).

In this implicit manner, the marginal characteristics ofeach periodic part are exactly preserved (since the marginaldistribution functions are implicitly handled through the pro-

posed homogenization) and the expectation of the second-order dependence structure (e.g. correlation function) is alsoexactly preserved after properly adjusting for bias throughthe mode or expected value of the estimator (Dimitriadis,2017). It is noted that higher-order moments of processeswith HK behaviour cannot be adequately preserved in animplicit manner (see an illustrative example in Dimitriadisand Koutsoyiannis, 2018, their Appendix D) and thus, for amore accurate preservation of the dependence structure anexplicit algorithm is necessary (Dimitriadis and Koutsoyian-nis, 2018). We may use a simple generation scheme, such asthe 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

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0.9

1

7 9 11 13 15 17 19

Co

rrel

atio

n f

un

ctio

n

Lag 1 (hour)

January February March April May June

July August September October November December

0

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

7 9 11 13 15 17 19

Co

rrel

atio

n f

un

ctio

n

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

1 10 100 1000 10 000 100 000

Stan

dard

ized

varia

ne 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 KT vs. whitenoise and a homogenized process without periodicities.

like dependence structure, which later it can be transformedback to the original distribution function F(xi,j ;pi,j ) and soin this way produce a double-periodic process with the de-sired marginal distribution for each diurnal-seasonal cycle aswell as the desired dependence structure. Finally, we multi-ply each value of the synthetic KT with the deterministicallydetermined value of the hourly intensity of solar radiation atthe top of the atmosphere.

5 Conclusions

The hourly marginal distribution of solar radiation is inves-tigated via the clearness index KT . Regarding the marginaldistribution, 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 sumof two Kumaraswamy distributions This distribution is fullycharacterized by five parameters and can adequately fit theempirical hourly data of the clearness index in in the regionsinvestigated in this study. Also, the dependence structure ofsolar radiation process is investigated via the climacogramtool. 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 noisenor a Markov process. Finally, a new preliminary stochas-tic model is proposed, capable of reproducing the clearnesssky index KT and so the hourly solar radiation. The modelcan maintain and preserve the probability density functionby means of the first four central moments and also the Hurstparameter which represents the correlation and the persistentbehaviour.

Data availability. The datasets generated during the current studyare 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 Renvironment statistical tests. PD verified the analytical methods andencouraged GK to investigate the LTP behaviour of the solar radi-ation process. NM contributed to sample preparation. DK and PDsupervised the findings of this work. GK took the lead in writingthe manuscript. All authors provided critical feedback and helpedshape the research, analysis and manuscript.

Competing interests. The authors declare that they have no conflictof interest.

Special issue statement. This article is part of the special issue “Eu-ropean Geosciences Union General Assembly 2018, EGU DivisionEnergy, Resources & Environment (ERE)”. It is a result of the EGUGeneral Assembly 2018, Vienna, Austria, 8–13 April 2018.

Acknowledgements. The authors thank warmly the anonymousreviewers for their most helpful comments. The statistical analyseswere 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 MartensReviewed by: two anonymous referees

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