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A detailed investigation of the variability of solar radiation can be proven useful towards more efficient and sustainable design of renewable resources systems. In this context, we analyze observations from Athens, Greece and we investigate the marginal distribution of the solar radiation process at a daily and hourly step, the long-term behavior based on the annual scale of the process, as well as the double periodicity (diurnal-seasonal) of the process. Finally, we apply a parsimonious double-cyclostationary stochastic model to generate hourly synthetic time series preserving the marginal statistical characteristics, the double periodicity and the dependence structure of the process.
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Available online at www.sciencedirect.com
Available online at www.sciencedirect.com
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling.
The 15th International Symposium on District Heating and Cooling
Assessing the feasibility of using the heat demand-outdoor
temperature function for a long-term district heat demand forecast
I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc
aIN+ Center for Innovation, Technology and Policy Research -Instituto Superior Técnico,Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal
bVeolia Recherche & Innovation,291 Avenue Dreyfous Daniel, 78520 Limay, France
cDépartement Systèmes Énergétiques et Environnement -IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France
Abstract
District heating networks are commonly addressed in the literature as one of the most effective solutions for decreasing the
greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heat
sales. Due to the changed climate conditions and building renovation policies, heat demand in the future could decrease,
prolonging the investment return period.
The main scope of this paper is to assess the feasibility of using the heat demand outdoor temperature function for heat demand
forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665
buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district
renovation scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were
compared with results from a dynamic heat demand model, previously developed and validated by the authors.
The results showed that when only weather change is considered, the margin of error could be acceptable for some applications
(the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation
scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered).
The value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the
decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and
renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the
coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and
improve the accuracy of heat demand estimations.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and
Cooling.
Keywords: Heat demand; Forecast; Climate change
Energy Procedia 125 (2017) 398–404
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientic committee of the European Geosciences Union (EGU) General Assembly 2017 – Division
Energy, Resources and the Environment (ERE).
10.1016/j.egypro.2017.08.076
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientic committee of the European Geosciences Union (EGU) General Assembly 2017 – Division
Energy, Resources and the Environment (ERE).
10.1016/j.egypro.2017.08.076
10.1016/j.egypro.2017.08.076 1876-6102
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientic committee of the European Geosciences Union (EGU) General Assembly
2017 – Division Energy, Resources and the Environment (ERE).
Available online at www.sciencedirect.com
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
European Geosciences Union General Assembly 2017, EGU
Division Energy, Resources & Environment, ERE
Investigation on the stochastic nature of the solar radiation process
Giannis Koudouris
a,
*, Panayiotis Dimitriadis
a
, Theano Iliopoulou
a
, Nikos Mamassis
a
,
Demetris Koutsoyiannis
a
a
Department of Water Resources & Environmental Engineering, School of Civil Engineering, National Technical University of Athens,
Heroon Polytechniou 9, Zografou15780, Greece
Abstract
A detailed investigation of the variability of solar radiation can be proven useful towards more efficient and sustainable design of
renewable resources systems. In this context, we analyze observations from Athens, Greece and we investigate the marginal
distribution of the solar radiation process at a daily and hourly step, the long-term behavior based on the annual scale of the
process, as well as the double periodicity (diurnal-seasonal) of the process. Finally, we apply a parsimonious double-
cyclostationary stochastic model to generate hourly synthetic time series preserving the marginal statistical characteristics, the
double periodicity and the dependence structure of the process.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
Keywords: synthetic hourly solar radiation; Kumaraswamy distribution; Hurst parameter; double periodicity
*Corresponding author. Tel.: +302107722831; fax: +302107722831
E-mail address: koudouris121212@gmail.com
1. Introduction
Several studies have been conducted to investigate the stochastic simulation of solar radiation for the purpose of
renewable energy simulation and management. For example, in the analysis of [1] the Beta distribution is suggested
* Corresponding author. Tel.: +30 210 77 22 831; fax: +30 210 77 22 831.
E-mail address: papoulakoskon@gmail.com
Available online at www.sciencedirect.com
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
European Geosciences Union General Assembly 2017, EGU
Division Energy, Resources & Environment, ERE
Investigation on the stochastic nature of the solar radiation process
Giannis Koudouris
a,
*, Panayiotis Dimitriadis
a
, Theano Iliopoulou
a
, Nikos Mamassis
a
,
Demetris Koutsoyiannis
a
a
Department of Water Resources & Environmental Engineering, School of Civil Engineering, National Technical University of Athens,
Heroon Polytechniou 9, Zografou15780, Greece
Abstract
A detailed investigation of the variability of solar radiation can be proven useful towards more efficient and sustainable design of
renewable resources systems. In this context, we analyze observations from Athens, Greece and we investigate the marginal
distribution of the solar radiation process at a daily and hourly step, the long-term behavior based on the annual scale of the
process, as well as the double periodicity (diurnal-seasonal) of the process. Finally, we apply a parsimonious double-
cyclostationary stochastic model to generate hourly synthetic time series preserving the marginal statistical characteristics, the
double periodicity and the dependence structure of the process.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
Keywords: synthetic hourly solar radiation; Kumaraswamy distribution; Hurst parameter; double periodicity
*Corresponding author. Tel.: +302107722831; fax: +302107722831
E-mail address: koudouris121212@gmail.com
1. Introduction
Several studies have been conducted to investigate the stochastic simulation of solar radiation for the purpose of
renewable energy simulation and management. For example, in the analysis of [1] the Beta distribution is suggested
* Corresponding author. Tel.: +30 210 77 22 831; fax: +30 210 77 22 831.
E-mail address: papoulakoskon@gmail.com
2 Koudouris et al. / Energy Procedia 00 (2017) 000–000
0
100
200
300
400
500
600
700
800
900
1000
w/m2
hours
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
wh/m2
days
for the modelling of hourly solar radiation recorded in Algiers. However, little research has been done in comparing
different marginal distributions for the process of the hourly solar radiation. Here, we aim at investigating the
marginal distribution for each month at an hourly step (24 hours for 12 months) fitting two of the most suitable
distributions for this process. Preliminary analyses in a monthly scale (with a daily step) showed that popular
distributions used in geophysics (such as Gamma, Pareto, Lognormal, Pearson etc.), that were fitted through the
open-software Hydrognomon (hydrognomon.org), could not adequately fit the right tail of the empirical distribution.
This can be explained considering that the solar irradiation process is left and right bounded. Although the left
boundary is close to zero, the right boundary varies at a seasonal scale. Therefore, distributions like Gamma and
Pareto, although they may exhibit a good fit (based on the Kolmogorov–Smirnov test), they should not be applied
for the solar irradiation, since they are not right bounded.
After analysing both scales, hourly and daily, we conclude that the Kumaraswamy distribution [2] describes
adequately well the observed distributions of diurnal and monthly solar irradiation and also exhibits certain technical
advantages in model building and simulation, as discussed in Section 5.
2. Data
The study area is located in Athens, Greece. We analyze more than 12 year of hourly time series of solar irradiance,
that is equivalent to more than 102,920 hours (Fig. 1a) and daily data spanning more than 25 years (Fig. 1b). Hourly
data are obtained from the Hydrological Observatory of Athens (http://hoa.ntua.gr/) and daily data from the NASA
SSE -Surface meteorology Solar Energy- (http://www.soda-pro.com/web-services/radiation/nasa-sse). From the 288
hourly time series (24 hours × 12 months) we only consider the 170 time series of records of good quality and with a
mean solar radiation much larger than zero, i.e. excluding night hours.
a b
Fig.1. (a) One year of hourly time series of solar irradiance (Athens); (b) One year of daily time series of solar irradiation (Athens).
3. Marginal distribution
3.1 Double periodicity
One of the most common characteristic of atmospheric processes, such as the solar radiation process (Fig. 2), is
the double periodicity, i.e., the diurnal and seasonal variation of the process. Therefore, for a robust generation of a
synthetic time series we have to analyze the hourly and monthly statistical characteristic of solar radiation (such as
the double periodic statistical mean and standard deviation).
Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404 399
Available online at www.sciencedirect.com
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
European Geosciences Union General Assembly 2017, EGU
Division Energy, Resources & Environment, ERE
Investigation on the stochastic nature of the solar radiation process
Giannis Koudouris
a,
*, Panayiotis Dimitriadis
a
, Theano Iliopoulou
a
, Nikos Mamassis
a
,
Demetris Koutsoyiannis
a
a
Department of Water Resources & Environmental Engineering, School of Civil Engineering, National Technical University of Athens,
Heroon Polytechniou 9, Zografou15780, Greece
Abstract
A detailed investigation of the variability of solar radiation can be proven useful towards more efficient and sustainable design of
renewable resources systems. In this context, we analyze observations from Athens, Greece and we investigate the marginal
distribution of the solar radiation process at a daily and hourly step, the long-term behavior based on the annual scale of the
process, as well as the double periodicity (diurnal-seasonal) of the process. Finally, we apply a parsimonious double-
cyclostationary stochastic model to generate hourly synthetic time series preserving the marginal statistical characteristics, the
double periodicity and the dependence structure of the process.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
Keywords: synthetic hourly solar radiation; Kumaraswamy distribution; Hurst parameter; double periodicity
*Corresponding author. Tel.: +302107722831; fax: +302107722831
E-mail address: koudouris121212@gmail.com
1. Introduction
Several studies have been conducted to investigate the stochastic simulation of solar radiation for the purpose of
renewable energy simulation and management. For example, in the analysis of [1] the Beta distribution is suggested
* Corresponding author. Tel.: +30 210 77 22 831; fax: +30 210 77 22 831.
E-mail address: papoulakoskon@gmail.com
Available online at www.sciencedirect.com
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
European Geosciences Union General Assembly 2017, EGU
Division Energy, Resources & Environment, ERE
Investigation on the stochastic nature of the solar radiation process
Giannis Koudouris
a,
*, Panayiotis Dimitriadis
a
, Theano Iliopoulou
a
, Nikos Mamassis
a
,
Demetris Koutsoyiannis
a
a
Department of Water Resources & Environmental Engineering, School of Civil Engineering, National Technical University of Athens,
Heroon Polytechniou 9, Zografou15780, Greece
Abstract
A detailed investigation of the variability of solar radiation can be proven useful towards more efficient and sustainable design of
renewable resources systems. In this context, we analyze observations from Athens, Greece and we investigate the marginal
distribution of the solar radiation process at a daily and hourly step, the long-term behavior based on the annual scale of the
process, as well as the double periodicity (diurnal-seasonal) of the process. Finally, we apply a parsimonious double-
cyclostationary stochastic model to generate hourly synthetic time series preserving the marginal statistical characteristics, the
double periodicity and the dependence structure of the process.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the scientific committee of the European Geosciences Union (EGU) General Assembly 2017
– Division Energy, Resources and the Environment (ERE).
Keywords: synthetic hourly solar radiation; Kumaraswamy distribution; Hurst parameter; double periodicity
*Corresponding author. Tel.: +302107722831; fax: +302107722831
E-mail address: koudouris121212@gmail.com
1. Introduction
Several studies have been conducted to investigate the stochastic simulation of solar radiation for the purpose of
renewable energy simulation and management. For example, in the analysis of [1] the Beta distribution is suggested
* Corresponding author. Tel.: +30 210 77 22 831; fax: +30 210 77 22 831.
E-mail address: papoulakoskon@gmail.com
0
100
200
300
400
500
600
700
800
900
1000
w/m2
hours
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
wh/m2
days
for the modelling of hourly solar radiation recorded in Algiers. However, little research has been done in comparing
different marginal distributions for the process of the hourly solar radiation. Here, we aim at investigating the
marginal distribution for each month at an hourly step (24 hours for 12 months) fitting two of the most suitable
distributions for this process. Preliminary analyses in a monthly scale (with a daily step) showed that popular
distributions used in geophysics (such as Gamma, Pareto, Lognormal, Pearson etc.), that were fitted through the
open-software Hydrognomon (hydrognomon.org), could not adequately fit the right tail of the empirical distribution.
This can be explained considering that the solar irradiation process is left and right bounded. Although the left
boundary is close to zero, the right boundary varies at a seasonal scale. Therefore, distributions like Gamma and
Pareto, although they may exhibit a good fit (based on the Kolmogorov–Smirnov test), they should not be applied
for the solar irradiation, since they are not right bounded.
After analysing both scales, hourly and daily, we conclude that the Kumaraswamy distribution [2] describes
adequately well the observed distributions of diurnal and monthly solar irradiation and also exhibits certain technical
advantages in model building and simulation, as discussed in Section 5.
2. Data
The study area is located in Athens, Greece. We analyze more than 12 year of hourly time series of solar irradiance,
that is equivalent to more than 102,920 hours (Fig. 1a) and daily data spanning more than 25 years (Fig. 1b). Hourly
data are obtained from the Hydrological Observatory of Athens (http://hoa.ntua.gr/) and daily data from the NASA
SSE -Surface meteorology Solar Energy- (http://www.soda-pro.com/web-services/radiation/nasa-sse). From the 288
hourly time series (24 hours × 12 months) we only consider the 170 time series of records of good quality and with a
mean solar radiation much larger than zero, i.e. excluding night hours.
a b
Fig.1. (a) One year of hourly time series of solar irradiance (Athens); (b) One year of daily time series of solar irradiation (Athens).
3. Marginal distribution
3.1 Double periodicity
One of the most common characteristic of atmospheric processes, such as the solar radiation process (Fig. 2), is
the double periodicity, i.e., the diurnal and seasonal variation of the process. Therefore, for a robust generation of a
synthetic time series we have to analyze the hourly and monthly statistical characteristic of solar radiation (such as
the double periodic statistical mean and standard deviation).
400 Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404
Koudouris et al. / Energy Procedia 00 (2017) 000–000 3
0
100
200
300
400
500
600
700
800
900
1000
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/M2
HOURS
Jan ua ry February
March Apri l
May June
July A ugust
September October
Nove mber December
Fig.2. Solar radiation (in W/m
2
) for each month and daily hour (Athens).
3.2 Kumaraswamy and Beta distributions
The Kumaraswamy distribution is a probability distribution suitable for double bounded random processes. It is
very familiar to the Beta distribution, allowing us to generate a large variety of probability distribution shapes of
processes. Moreover, the Kumaraswamy distribution, has the advantage of an invertible closed form of the
cumulative distribution function [3], as shown in Eqn. 1. Therefore, using Kumaraswamy rather than Beta for
simulation purposes may be proven less computationally intensive.
In particular, the Kumaraswamy cumulative density function can be expressed as:
; , =; ,  =1−1−
 (1)
where ∈0,1 is standardized according to =





, with

and

are the minimum and maximum
values of the empirical time series.
Also, the Beta Cumulative density function is given by:
;, =
;,
,
(2)
where ; , is the incomplete Beta function ;, =

1−

d and , is the Beta function
, =

1−

d.
3.3 Comparison between the Kumaraswamy and Beta distributions for the monthly scale at the daily and hourly step
In order to examine whether the marginal distribution of solar irradiance can be adequately fitted from the
Kumaraswamy or the Beta distribution, we apply three tests of goodness of fit. Also, we employ one model
selection criterion, i.e., the Akaike information criterion [4] which is a function of the number of parameters in the
model and the resulting log-likelihood value. However, since the Kumaraswamy and the Beta distributions have
4 Koudouris et al. / Energy Procedia 00 (2017) 000–000
0
1
2
3
4
5
6
7
8
9
Akaike informat ion
criteri on
Kolm ogorov Sm irnov Crammer von Miss es Aders on Darl ing
Months
Beta Kumaraswamy
0
1
2
3
4
5
6
7
8
9
Kolm ogorov Sm irnov Cramm er von Mis ses Aders on Darl ing
Months
Beta Kumaraswamy
0
0.2
0.4
0.6
0.8
1
0 2000 4000 6000 8000 10000
WH/M
2
MAY
Obsr ved- cdf Kumaraswamy-Cdf
0
0.2
0.4
0.6
0.8
1
0 2000 4000 6000 8000 10000
WH/M
2
AUGUST
Observ ed-cdf Ku ma ra swa my- cdf
only two parameters, the above test compares only the likelihood value of each distribution. For the goodness of fit
we use the Kolmogorov-Smirnov [5,6], Cramer von Misses [7] and the Anderson Darling tests [8]. Computations
are carried out in the R statistical environment [14].
After applying all the tests, we conclude that at the monthly scale (using time series of daily irradiation), for the
AIC test (which does not provide information on the goodness of fit of the model) the Kumaraswamy distribution
performs better than the Beta distribution. Note that, we adopt the suggestion of [9] that for a difference below 2
points between the AIC values of the two models, both models have good support. Considering the latter, our results
show that the Kumaraswamy distribution is always selected by the test contrariwise to the Beta distribution.
However, for three time series (for April, May and December) both distributions are selected by the test (Fig. 3a).
For the goodness of fit, we set a 5% confidence level. According to the Kolmogorov-Smirnov test, Crammer von
Misses and the Anderson & Darling test, the Kumaraswamy distribution is rejected in fewer months than the Beta
distribution (Fig. 3b). Nevertheless, all hypothesis tests reject both the Kumaraswamy and the Beta distributions for
the summer months (see the difference between Fig. 3c and 3d). For the hourly step, according to AIC criterion, the
Kumaraswamy distribution is again preferred to the Beta distribution. However, based on the goodness of fit tests,
the Kumaraswamy and the Beta distributions are not rejected only for the 44 out of the 170 time series (mostly at the
midday hours). This inability of both distributions to adequately fit mainly the hours with the potential highest solar
radiation within the day (e.g., 15:00 during the summer months), may be due to the variability induced by the
clearness index process K
T
(a measure of the ratio of measured irradiation in a locale relative to the extraterrestrial
irradiation calculated at the given locale i.e. for K
T
1: atmosphere is clear and for K
T
0: atmosphere is cloudy),
which highly affects the behaviour of the marginal distribution.
a b
c d
Fig.3. (a) Selected model for marginal distribution of the monthly scale based on the AIC, KS, CvM and AD tests; (b) results from the goodness
of fit of the monthly scale of the marginal distribution base on the KS, CvM and AD tests; (c) plot of the Kumaraswamy distribution drawn from
the August time series which is rejected from the above tests; (d) plot of the Kumaraswamy distribution from a non-rejected time series in May.
Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404 401
Koudouris et al. / Energy Procedia 00 (2017) 000–000 3
0
100
200
300
400
500
600
700
800
900
1000
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/M2
HOURS
Jan ua ry February
March Apri l
May June
July A ugust
September October
Nove mber December
Fig.2. Solar radiation (in W/m
2
) for each month and daily hour (Athens).
3.2 Kumaraswamy and Beta distributions
The Kumaraswamy distribution is a probability distribution suitable for double bounded random processes. It is
very familiar to the Beta distribution, allowing us to generate a large variety of probability distribution shapes of
processes. Moreover, the Kumaraswamy distribution, has the advantage of an invertible closed form of the
cumulative distribution function [3], as shown in Eqn. 1. Therefore, using Kumaraswamy rather than Beta for
simulation purposes may be proven less computationally intensive.
In particular, the Kumaraswamy cumulative density function can be expressed as:
; , =; ,  =1−1−
 (1)
where ∈0,1 is standardized according to =





, with

and

are the minimum and maximum
values of the empirical time series.
Also, the Beta Cumulative density function is given by:
;, =
;,
,
(2)
where ; , is the incomplete Beta function ;, =

1−

d and , is the Beta function
, =

1−

d.
3.3 Comparison between the Kumaraswamy and Beta distributions for the monthly scale at the daily and hourly step
In order to examine whether the marginal distribution of solar irradiance can be adequately fitted from the
Kumaraswamy or the Beta distribution, we apply three tests of goodness of fit. Also, we employ one model
selection criterion, i.e., the Akaike information criterion [4] which is a function of the number of parameters in the
model and the resulting log-likelihood value. However, since the Kumaraswamy and the Beta distributions have
4 Koudouris et al. / Energy Procedia 00 (2017) 000–000
0
1
2
3
4
5
6
7
8
9
Akaike informat ion
criteri on
Kolm ogorov Sm irnov Crammer von Miss es Aders on Darl ing
Months
Beta Kumaraswamy
0
1
2
3
4
5
6
7
8
9
Kolm ogorov Sm irnov Cramm er von Mis ses Aders on Darl ing
Months
Beta Kumaraswamy
0
0.2
0.4
0.6
0.8
1
0 2000 4000 6000 8000 10000
WH/M
2
MAY
Obsr ved- cdf Kumaraswamy-Cdf
0
0.2
0.4
0.6
0.8
1
0 2000 4000 6000 8000 10000
WH/M
2
AUGUST
Observ ed-cdf Ku ma ra swa my- cdf
only two parameters, the above test compares only the likelihood value of each distribution. For the goodness of fit
we use the Kolmogorov-Smirnov [5,6], Cramer von Misses [7] and the Anderson Darling tests [8]. Computations
are carried out in the R statistical environment [14].
After applying all the tests, we conclude that at the monthly scale (using time series of daily irradiation), for the
AIC test (which does not provide information on the goodness of fit of the model) the Kumaraswamy distribution
performs better than the Beta distribution. Note that, we adopt the suggestion of [9] that for a difference below 2
points between the AIC values of the two models, both models have good support. Considering the latter, our results
show that the Kumaraswamy distribution is always selected by the test contrariwise to the Beta distribution.
However, for three time series (for April, May and December) both distributions are selected by the test (Fig. 3a).
For the goodness of fit, we set a 5% confidence level. According to the Kolmogorov-Smirnov test, Crammer von
Misses and the Anderson & Darling test, the Kumaraswamy distribution is rejected in fewer months than the Beta
distribution (Fig. 3b). Nevertheless, all hypothesis tests reject both the Kumaraswamy and the Beta distributions for
the summer months (see the difference between Fig. 3c and 3d). For the hourly step, according to AIC criterion, the
Kumaraswamy distribution is again preferred to the Beta distribution. However, based on the goodness of fit tests,
the Kumaraswamy and the Beta distributions are not rejected only for the 44 out of the 170 time series (mostly at the
midday hours). This inability of both distributions to adequately fit mainly the hours with the potential highest solar
radiation within the day (e.g., 15:00 during the summer months), may be due to the variability induced by the
clearness index process K
T
(a measure of the ratio of measured irradiation in a locale relative to the extraterrestrial
irradiation calculated at the given locale i.e. for K
T
1: atmosphere is clear and for K
T
0: atmosphere is cloudy),
which highly affects the behaviour of the marginal distribution.
a b
c d
Fig.3. (a) Selected model for marginal distribution of the monthly scale based on the AIC, KS, CvM and AD tests; (b) results from the goodness
of fit of the monthly scale of the marginal distribution base on the KS, CvM and AD tests; (c) plot of the Kumaraswamy distribution drawn from
the August time series which is rejected from the above tests; (d) plot of the Kumaraswamy distribution from a non-rejected time series in May.
402 Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404
Koudouris et al. / Energy Procedia 00 (2017) 000–000 5
0
1
2
3
4
5
6
mle Lm om Ls So lver
0
1
2
3
4
5
6
Mle Lmom Ls Sol ver-excel
10
100
1000
1 10 100 1000 10000
St.Devi a ti on
K(h)
3.4 Kumaraswamy distribution parameters for the monthly scale in a daily step.
We estimate the parameters of the Kumaraswamy distribution applying four methods, i.e., the maximum
likelihood, the L-moments and two least square methods (one based on quartiles and the other on the cumulative
distribution) (Fig 4.a, b).
a b
Fig. 4. (a) Plot of Kumaraswamy’s a parameter calculated from four different methods; (b) plot of Kumaraswamy’s {b} parameter calculated
from four different.
4. Dependence structure at the hourly scale
For the estimation of the dependence structure of the process, we analyze a time series of more than 17 years of
hourly solar radiation values. To quantify the persistence behavior of the process, we estimate the Hurst parameter
(H = 0.83) via the climacogram introduced in [10] (i.e. plot of standard deviation σ (k) vs. averaging scale k), as
shown in Fig. 5. We justify the use of climacogram to estimate the stochastic structure of the process instead of the
commonly used autocorrelation functions and power spectrums as explained in [11], where it is shown that the
climacogram has always a smaller statistical uncertainty from the other tools for common processes such as Markov
and Hurst-Kolmogorov (HK). Since H > 0.5, we conclude that the examined process follows a HK behavior and the
annual solar radiation is strongly correlated and cannot be considered as a white noise process (H = 0.5).
Fig. 5. Standardized climacogram (i.e., standard deviation of the scaled process).
6 Koudouris et al. / Energy Procedia 00 (2017) 000–000
0
200
400
600
800
1000
1200
solar irradiance (W/m
2
)
time
mode l
observed
y = 1.0932x
R² = 0.9663
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 1000 2000 3000 4000 5000 6000 70 00 8000 9000
WH/ M2
WH /M2
5. Generation of synthetic time series
In this section, we apply a double periodic model, to generate synthetic hourly solar irradiance time series. The
mean hourly synthetic time series is produced using the methodology of [12] suitable for double cyclostationary
processes such as the ones examined in this study. Particularly, this methodology preserves the double periodicity
(i.e., diurnal and seasonal) of a process through the hourly-monthly marginal distributions, including intermittent
characteristics such as probability of zero values (i.e. during night time), as well as the dependence structure of the
process through the climacogram. For the dependence structure, we apply an HK model based on the empirical
climacogram of the solar irradiance as estimated in the previous section. Finally, for the generation scheme we use
the CSAR algorithm (Cyclostationary Sum of finite independent AR(1) processes, [13]) capable of generating any
length of time series following an HK, or various other processes, and with arbitrary distributions of each internal
stationary process of the double cyclostationary process. In Figures 6 we compare the synthetic time series with the
observed one.
a b
Fig.6. (a) Synthetic and observed hourly synthetic time series of daily irradiation; (b) Yearly average of daily observed and synthetic values.
6. Conclusions
In this study, we investigate the statistical properties of the solar radiation process at a monthly scale for both a
daily and an hourly step. Regarding the marginal distribution, we conclude that the Kumaraswamy distribution can
adequately describe the (daily step) monthly solar radiation and is generally preferred to the Beta distribution based
on the three proposed tests of goodness of fit and on one model selection criterion. However, further research needs
to be conducted in order to investigate the impact of the clearness index to the hourly process. Also, we calculate the
parameters of the marginal distribution of the monthly solar radiation in Athens according to four different statistical
methods (maximum likelihood, L-moments and two least square methods). An important result is that, solar
radiation is found to exhibit a strong Hurst-Kolmogorov behaviour since the Hurst parameter is estimated as high as
0.83, that implies high correlation between successive years. Finally, we present a double periodicity model for
generating hourly solar radiation time series which reproduces exceptionally well all the above statistical
characteristics of the examined process.
Acknowledgment
Τhe statistical analyses were performed in the R statistical environment [14] by also using the contributed
packages VGAM [15], fitdistrplus [16], goftest [17] and lmomco [18]
References
[1] F. Youcef Ettoumi, A. Mefti, A. Adane and M. Y. Bouroubi. (2002) "Statistical analysis of solar measurements in Algeria using beta
distributions", Renew. Energy, vol. 26, no. 1 May (2002): 47-67.
Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404 403
Koudouris et al. / Energy Procedia 00 (2017) 000–000 5
0
1
2
3
4
5
6
mle Lm om Ls So lver
0
1
2
3
4
5
6
Mle Lmom Ls Sol ver-excel
10
100
1000
1 10 100 1000 10000
St.Devi a ti on
K(h)
3.4 Kumaraswamy distribution parameters for the monthly scale in a daily step.
We estimate the parameters of the Kumaraswamy distribution applying four methods, i.e., the maximum
likelihood, the L-moments and two least square methods (one based on quartiles and the other on the cumulative
distribution) (Fig 4.a, b).
a b
Fig. 4. (a) Plot of Kumaraswamy’s a parameter calculated from four different methods; (b) plot of Kumaraswamy’s {b} parameter calculated
from four different.
4. Dependence structure at the hourly scale
For the estimation of the dependence structure of the process, we analyze a time series of more than 17 years of
hourly solar radiation values. To quantify the persistence behavior of the process, we estimate the Hurst parameter
(H = 0.83) via the climacogram introduced in [10] (i.e. plot of standard deviation σ (k) vs. averaging scale k), as
shown in Fig. 5. We justify the use of climacogram to estimate the stochastic structure of the process instead of the
commonly used autocorrelation functions and power spectrums as explained in [11], where it is shown that the
climacogram has always a smaller statistical uncertainty from the other tools for common processes such as Markov
and Hurst-Kolmogorov (HK). Since H > 0.5, we conclude that the examined process follows a HK behavior and the
annual solar radiation is strongly correlated and cannot be considered as a white noise process (H = 0.5).
Fig. 5. Standardized climacogram (i.e., standard deviation of the scaled process).
6 Koudouris et al. / Energy Procedia 00 (2017) 000–000
0
200
400
600
800
1000
1200
solar irradiance (W/m
2
)
time
mode l
observed
y = 1.0932x
R² = 0.9663
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 1000 2000 3000 4000 5000 6000 70 00 8000 9000
WH/ M2
WH /M2
5. Generation of synthetic time series
In this section, we apply a double periodic model, to generate synthetic hourly solar irradiance time series. The
mean hourly synthetic time series is produced using the methodology of [12] suitable for double cyclostationary
processes such as the ones examined in this study. Particularly, this methodology preserves the double periodicity
(i.e., diurnal and seasonal) of a process through the hourly-monthly marginal distributions, including intermittent
characteristics such as probability of zero values (i.e. during night time), as well as the dependence structure of the
process through the climacogram. For the dependence structure, we apply an HK model based on the empirical
climacogram of the solar irradiance as estimated in the previous section. Finally, for the generation scheme we use
the CSAR algorithm (Cyclostationary Sum of finite independent AR(1) processes, [13]) capable of generating any
length of time series following an HK, or various other processes, and with arbitrary distributions of each internal
stationary process of the double cyclostationary process. In Figures 6 we compare the synthetic time series with the
observed one.
a b
Fig.6. (a) Synthetic and observed hourly synthetic time series of daily irradiation; (b) Yearly average of daily observed and synthetic values.
6. Conclusions
In this study, we investigate the statistical properties of the solar radiation process at a monthly scale for both a
daily and an hourly step. Regarding the marginal distribution, we conclude that the Kumaraswamy distribution can
adequately describe the (daily step) monthly solar radiation and is generally preferred to the Beta distribution based
on the three proposed tests of goodness of fit and on one model selection criterion. However, further research needs
to be conducted in order to investigate the impact of the clearness index to the hourly process. Also, we calculate the
parameters of the marginal distribution of the monthly solar radiation in Athens according to four different statistical
methods (maximum likelihood, L-moments and two least square methods). An important result is that, solar
radiation is found to exhibit a strong Hurst-Kolmogorov behaviour since the Hurst parameter is estimated as high as
0.83, that implies high correlation between successive years. Finally, we present a double periodicity model for
generating hourly solar radiation time series which reproduces exceptionally well all the above statistical
characteristics of the examined process.
Acknowledgment
Τhe statistical analyses were performed in the R statistical environment [14] by also using the contributed
packages VGAM [15], fitdistrplus [16], goftest [17] and lmomco [18]
References
[1] F. Youcef Ettoumi, A. Mefti, A. Adane and M. Y. Bouroubi. (2002) "Statistical analysis of solar measurements in Algeria using beta
distributions", Renew. Energy, vol. 26, no. 1 May (2002): 47-67.
404 Giannis Koudouris et al. / Energy Procedia 125 (2017) 398–404
Koudouris et al. / Energy Procedia 00 (2017) 000–000 7
[2] Giannis Koudouris, Panayiotis Dimitriadis, Theano Iliopoulou, Nikos Mamassis and Demetris Koutsoyiannis. Investigation of the stochastic
nature of solar radiation for renewable resources management. At: Vienna, Austria, Ordinal: Geophysical Research Abstracts, Vol. 19,
EGU2017-10189-4 04/2017
[3] Mitnik, P.A. (2013). "New properties of the Kumaraswamy distribution." Commun. Stat. – Theory Methods 42.5 (2013):741–755
[4] Sakamoto, Y., Ishiguro, M. and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.
[5] Durbin, J. (1973), Distribution theory for tests based on the sample distribution function. SIAM
[6] George Marsaglia, Wai Wan Tsang and Jingbo Wang. (2003) "Evaluating Kolmogorov's distribution" Journal of Statistical Software (2013):
8/18
[7] Csörgo, S. and Faraway, J.J. (1996) "The exact and asymptotic distributions of Cramér-von Mises statistics." Journal of the Royal Statistical
Society, Series B 58 (1996): 221–234
[8] Marsaglia, G. and Marsaglia, J. (2004) "Evaluating the Anderson-Darling Distribution." Journal of Statistical Software 9.2 February (2004):
1–5.
[9] Burnham, K.P. and Anderson, D.R. (2002). Information and likelihood theory: a basis for model selection and inference. Model selection and
multimodel inference: a practical information-theoretic approach (2002): 49–97.
[10] Koutsoyiannis D. (2010) "A random walk on water." Hydrology and Earth System Sciences 14 (2010): 585–601
[11] Dimitriadis, P. and Koutsoyiannis, D. (2015). "Climacogram versus autocovariance and power spectrum in stochastic modelling for
Markovian and Hurst–Kolmogorov processes." Stochastic Environmental Research and Risk Assessment 29.6 (2015): 1649–1669.
[12] P. Dimitriadis, and D. Koutsoyiannis. (2015) "Application of stochastic methods to double cyclostationary processes for hourly wind speed
simulation." Energy Procedia 76 (2015): 406–411
[13] E. Deligiannis, P. Dimitriadis, Ο. Daskalou, Y. Dimakos, and D. Koutsoyiannis. (2016) "Global investigation of double periodicity οf hourly
wind speed for stochastic simulation; application in Greece." Energy Procedia 97 (2016): 278–285.
[14] R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
URL https://www.R-project.org/.
[15] Thomas W. Yee (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: Springer.
[16] Marie Laure Delignette-Muller and Christophe Dutang (2015). fitdistrplus: "An R Package for Fitting Distributions." Journal of Statistical
Software 64.4 (2015): 1-34
[17] Adrian Baddeley (2017). Cramer-Von Mises and Anderson-Darling tests of goodness-of-fit for continuous univariate distributions, using
efficient algorithms. R package 1.1-1
[18] Asquith, W.H., 2017, lmomco---L-moments, censored L-moments, trimmed L-moments, L-comoments, and many distributions. R package
version 2.2.7, Texas Tech University, Lubbock, Texas.
... no diurnal, seasonal or annual fluctuations) and with uniform spatial distribution over the area of the campus. We consider the clearness index equal to 0.6 as estimated by the expected hourly value of the clearness index during daylight over the campus (Koudouris et al., 2017). ...
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... A proper measure to express the strength of the long-term persistence of a stochastic process is the Hurst parameter, (Hurst, 1951;Koutsoyiannis, 2010). Interestingly, the Hurst parameter for the clearness index over the area of interest is estimated around 0.8 (Koudouris et al., 2017) indi- cating strong long-term persistence of the process and thus, a relevant model will be developed in future research to assess the degree of uncertainty induced from the variability of the clearness index based on synthetic solar radiation timeseries longer than the historical data used here. However, even the single simulation of the system based on historical data is reflective of the effect of the scale of the system on both reliability, pertaining to minimizing the energy required from the grid, and profit, estimated from the process of buying and selling energy to the grid and assuming the present energy prices. ...
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