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Content may be subject to copyright.

ScienceDirect

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 scientic 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 scientic 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 scientic 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.

– 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.

– 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.

– 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.

– 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

Heroon Polytechniou 9, Zografou15780, Greece

Abstract

double periodicity and the dependence structure of the process.

© 2017 The Authors. Published by Elsevier Ltd.

– Division Energy, Resources and the Environment (ERE).

*Corresponding author. Tel.: +302107722831; fax: +302107722831

E-mail address: koudouris121212@gmail.com

1. Introduction

* 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).

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):

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[7] Csörgo, S. and Faraway, J.J. (1996) "The exact and asymptotic distributions of Cramér-von Mises statistics." Journal of the Royal Statistical

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[8] Marsaglia, G. and Marsaglia, J. (2004) "Evaluating the Anderson-Darling Distribution." Journal of Statistical Software 9.2 February (2004):

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[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.