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1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer

-review under responsibility of the GFZ German Research Centre for Geosciences

doi: 10.1016/j.egypro.2015.07.851

Energy Procedia 76 ( 2015 ) 406 – 411

ScienceDirect

European Geosciences Union General Assembly 2015, EGU

Division Energy, Resources & Environment, ERE

Application of stochastic methods to double cyclostationary

processes for hourly wind speed simulation

Panayiotis Dimitriadis* and Demetris Koutsoyiannis

National Technical University of Athens, Heroon Polytechniou 9, 15780 Zografou, Greece

Abstract

In this paper, we present a methodology to analyze processes of double cyclostationarity (e.g. daily and seasonal). This method

preserves the marginal characteristics as well as the dependence structure of a process (through the use of climacogram). It

consists of a normalization scheme with two periodicities. Furthermore, we apply it to a meteorological station in Greece and

construct a stochastic model capable of preserving the Hurst-Kolmogorov behaviour. Finally, we produce synthetic time-series

(based on aggregated Markovian processes) for the purpose of wind speed and energy production simulation (based on a

proposed industrial wind turbine).

© 2015 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the GFZ German Research Centre for Geosciences.

Keywords: hourly wind speed; double cyclostationarity; stochastic modelling; Hurst-Kolmogorov dynamics; climacogram; uncertainty-bias; wind

turbine

1. Introduction

Several methods exist for dealing with processes of single periodicity, with most of them preserving the marginal

characteristics of the process and assuming a short-range dependence structure (cf. [1]). However, neglecting a

possible long-range dependence, i.e. Hurst-Kolmogorov (HK) behaviour, could lead to unrealistic predictions and

wind load situations, causing some impact on the energy production and management of renewable sources. Here,

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

E-mail address: pandim@itia.ntua.gr

Available online at www.sciencedirect.com

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the GFZ German Research Centre for Geosciences

Panayiotis Dimitriadis and Demetris Koutsoyiannis / Energy Procedia 76 ( 2015 ) 406 – 411

407

we focus on the stochastic nature of wind speed in an hourly scale. The most challenging problem of wind speed

simulation is the internal periodicities (e.g. daily and seasonal cycle), a common characteristic of

hydrometeorological processes. In this paper, we apply the methodology presented in [1], which involves the

analysis of a monthly-scale process, but with preserving both daily and seasonal periodicity. Particularly, assuming

that the process has a double cyclostationarity, we first normalize each cyclostationary variable, using a scheme of

double periodicity with three parameters. Then, we analyze the stochastic structure of the wind process and we

construct a model based on the climacogram, a stochastic tool with many advantages in stochastic interpretation and

model building [2,3]. Additionally, we produce synthetic time-series for the purpose of wind speed and energy

production simulation (based on a proposed industrial wind turbine). Finally, we apply the methodology to the

meteorological station of Larissa (www.hnms.gr) in the area of Thessaly (Greece), with latitude 22.417

o

, longitude

39.633

o

and elevation +74 m. This is one of the older stations in Greece and includes up to 75 years of measurements

in an hourly scale. Its marginal mean wind speed is estimated as 1.7 m/s and its standard deviation as 2.71 m/s (for

more information see in [2]).

In the next section, we describe the normalization method, we show how to analyze the stochastic structure of a

normalized process and how to generate synthetic time-series based on aggregated Markovian processes. Finally, we

produce a one week hourly wind speed time-series (that preserves the marginal characteristics as well as the

dependence structure of the examined process) and we estimate the hypothetically produced energy from a wind

turbine. Note that underlined symbols denote random variables and the overline symbol (^) denotes estimation.

2. Stochastic analysis of the wind speed process

2.1. Cyclostationarity

One of the most common characteristics of hydrometeorological processes (in a sub-climatic scale) is the double

periodicity, i.e. the continuous change of the process’ statistical properties in both daily and seasonal scales. Several

techniques have been developed to model this behaviour (a brief description can be seen in [1]). However, most of

them can capture the marginal characteristics of the process assuming a short-range dependence structure between

daily and seasonal variables. A method to model a single periodicity with any type of internal dependence structure

is presented in [1], where the process is assumed to be cyclostationary in seasonal scale (e.g. monthly scale). The

main feature of this method is the application of a normalization scheme (derived from the principle of maximum

entropy) to all seasonal variables, capturing in this way both the marginal properties as well as the dependence

structure of the process (zero values are excluded from the analysis since the wind process cannot exhibit zero

speeds). Here, we apply this scheme but with also including the daily periodicity since we are interested in sub-daily

(e.g. hourly) scale simulation. The normalization scheme is the following:

¸

¸

¹

·

¨

¨

©

§

¸

¸

¹

·

¨

¨

©

§

−

+

¸

¸

¹

·

¨

¨

©

§

+

¸

¸

¹

·

¨

¨

©

§

−

=

2

c

c

c

cc

c

1ln

1

1sign

σ

μ

σ

μ

X

g

g

X

Z

(1)

where ǽ

~N(0,1) is the transformed process of X, ȝ

c

and ı

c

are the mean and standard deviation for each

cyclostationary variable (i.e. one for each hour and month), and g

c

is a parameter related to the distribution tail of the

cyclostationary process.

From Fig. 1, we observe that the cyclostationary mean value of the process can be well described by a periodic

exponential function for the daily scale and with a simple cosine function for the monthly scale (performance of

these models to the Larissa station can be also seen in [2]). Also, we observe that the standard deviation can be well

modeled by two simple periodic functions and that g

c

significantly varies only within the daily scale and thus, can be

described by a single cosine function:

408 Panayiotis Dimitriadis and Demetris Koutsoyiannis / Energy Procedia 76 ( 2015 ) 406 – 411

h3

2ʌ

cos-

2

h

1

d

e

2ʌ

cos

μμ

aa

T

t

a

T

t

c

+

¸

¸

¹

·

¨

¨

©

§

+

¸

¸

¹

·

¨

¨

©

§

=

¸

¸

¹

·

¨

¨

©

§

(2)

h6

d

5

h

4

2ʌ

sin

2ʌ

cos ıa

T

t

a

T

t

a

c

+

¸

¸

¹

·

¨

¨

©

§

¸

¸

¹

·

¨

¨

©

§

+

¸

¸

¹

·

¨

¨

©

§

=

σ

(3)

2ʌ

cos

8

d

7

a

T

t

ag

c

+

¸

¸

¹

·

¨

¨

©

§

=

(4)

where t denotes time (h), Į

i

are dimensionless coefficients, T

h

equals the annual time duration in hours and T

d

=24

h. For the Larissa station the coefficients Į

i

are calculated (with fitting R

2

coefficient around 95% for all cases) as:

Į

1

=0.463, Į

2

=0.177, Į

3

=0.6, Į

4

=0.07, Į

5

=-0.1, Į

6

=0.738, Į

7

=0.217 and Į

8

=0.541.

ab

cd

Fig. 1. (a) fluctuation of hourly mean wind speed for each month; (b) fluctuation of hourly wind speed standard deviation for each month; (c)

fluctuation in a monthly scale of both mean and standard deviation of hourly wind speed (hourly-averaged); (d) fluctuation in a hourly scale of

parameter g

c

(monthly-averaged).

2.2. Stochastic structure

By normalizing the process, we have no longer effects of the internal periodicities to the stochastic structure of

the process and thus, we can now proceed to the estimation of the latter. There are several stochastic tools available

for the analysis of the dependence structure of a process (e.g. autocovariance, power spectrum, variogram). Based on

the analysis of [3], we choose to use the climacogram (i.e. plot of variance of the mean aggregated process vs. scale,

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Ŗ Ś Ş ŗŘŗŜŘŖŘŚ

ΐ

ǻȦǼ

ǻǼ

ŗ Ř ř Ś ś Ŝ ŝ Ş ş ŗŖ ŗŗ ŗŘ

ŗǯŖ

ŗǯś

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řǯŖ

řǯś

Ŗ Ś Ş ŗŘŗŜŘŖŘŚ

Η

ǻȦǼ

ǻǼ

ŗ Ř ř Ś ś Ŝ

ŝ Ş ş ŗŖ ŗŗ ŗŘ

ŗǯś

ŘǯŖ

Řǯś

Ȭŗ

Ŗ

ŗ

ŗŘřŚśŜŝŞşŗŖŗŗŗŘ

Η

ǻȦǼ

ΐ

ǻȦǼ

ǻǼ

ΐ Η

ΐ

Η

ŖǯŖ

ŖǯŘ

ŖǯŚ

ŖǯŜ

ŖǯŞ

ŗǯŖ

Ŗ Ś Ş ŗŘ ŗŜ ŘŖ ŘŚ

ǻȬǼ

ǻǼ

Panayiotis Dimitriadis and Demetris Koutsoyiannis / Energy Procedia 76 ( 2015 ) 406 – 411

409

cf. [4]). It has been shown that for simple processes, such as Markovian, HK and combinations thereof, the latter

stochastic tool often outperforms the aforementioned tools in terms of smaller statistical uncertainty. Furthermore, it

has a plethora of advantages in terms of stochastic analysis (e.g. in determining the Hurst coefficient) and model

building (e.g. it has simple and analytical expressions for the expected value of the process). The climacogram

definition, classical estimator and expected value are shown in the equations below.

()

2

0

/dVar)( mXmȖ

m

»

¼

º

«

¬

ª

=

³

ξξ

(5)

()

¦¦¦

==+−=

¸

¸

¹

·

¨

¨

©

§

−

−−

=

n

i

n

l

ǻ

ki

ikl

ǻ

ll

X

n

X

kn

ǻkȖ

1

2

1

)(

11

)(

^

1

1

1

1

1

)(

(6)

)(

/1

)()/(1

)(E

^

ǻkȖ

nk

ǻkȖǻnȖ

ǻkȖ

−

−

=

»

¼

º

«

¬

ª

(7)

where Ȗ is the continuous-time climacogram (in m

2

/s

2

), m is the continuous-time scale (in h), ǻ is the sampling

time interval (in our analysis equals 1 h), n is the total number of observations and k is the discrete-time scale

(dimensionless).

In Fig. 2, we observe that the empirical (from the normalized process) climacogram exhibits a Markovian decay

at small scales and an HK behaviour at large ones (similar observations in the wind process are derived in [3]). Here,

we choose to fit a Markovian model (to control the small scales) and an HK one for the larger scales (shown in the

equation below), by assuming that the empirical climacogram represents the expected value of the process. The best

fitted parameters are estimated as: Ȝ

M

=6 m

2

/s

2

, q=0.05 h, Ȝ

HK

=0.1 m

2

/s

2

and H=0.75:

()

()

()

Ǿ

qkǻ

kǻ

qkǻ

qkǻ

kǻȖ

22

HK

/

2

M

1e/

/

2

)(

−

−

+−+=

λ

λ

(8)

410 Panayiotis Dimitriadis and Demetris Koutsoyiannis / Energy Procedia 76 ( 2015 ) 406 – 411

ab

Fig. 2. (a) qq-plot of standardized and normalized time-series of the 1

st

hour of the day of the 1

st

month (where w denotes wind speed); (b)

continuous-time climacograms for a random (H=0.5) process, empirical (standardized and normalized) climacograms from the analysis of the

Larissa station, the adapted for bias climacogram of the HK and Markovian fitting model to the empirical normalized climacogram as well as the

continuous-time model used for the stochastic generation based on the aggregated Markovian process (described in section 2.3).

2.3. Stochastic generation and application in energy production simulation

For the stochastic generation we choose the methodology presented in [3]. We produce synthetic HK Gaussian

distributed time series based on an aggregation of Markovian processes:

()

()

()

1e/

/

2

/

2

−+=

−

l

qkǻ

l

l

l

l

qkǻ

qkǻ

kǻ

λ

γ

(9)

whose parameters q

l

are connected to each other in a pre-defined way (parameters Ȝ

l

can be calculated analytically

following the analysis of [3]), particularly:

1

21

−

=

l

l

ppq

(10)

where p

1

and p

2

are parameters, which can be calculated by minimizing the residouble between the modeled and

aggregated-Markovian processes. For the chosen HK process and for n§10

6

, we choose to generate four Markovian

processes, with the best fit corresponding to p

1

=0.113 and p

2

=0.099 (Fig. 2).

Hence, we can generate a N(0,1) process with the desired stochastic structure and then, by applying the inverse

normalization scheme described in section 2.1, we can produce a time-series with the same statistical characteristics

as the original one, for the purpose of simulation (note that we set all negative synthetic values to zero). In Fig. 3, we

illustrate a weekly time-window of generated hourly wind speed with the same stochastic structure and seasonality

properties of the Larissa station. Furthermore and for illustration purposes, we assume a reference wind speed (i.e.

10 min mean wind speed at hub height with a 50-year return period) equal to 42.5 m/s and a larger annual average

wind speed of 10 m/s. Based on the latter specifications and on the IEC-61400 standards [5], we can install a wind

turbine generator of class II, with an industrial solution of ENERCON E-82 (cf. [2]). Finally, we show in Fig. 3 the

simulation of the energy production based on the turbine’s power curve.

ȬŘ

Ȭŗ

Ŗ

ŗ

Ř

ř

Ś

ȬřǯŖ ȬŘǯŖ ȬŗǯŖ ŖǯŖ ŗǯŖ ŘǯŖ řǯŖ

ȱȱǻȦǼ

ǻŖǰŗǼȱȱǻȦǼ

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Panayiotis Dimitriadis and Demetris Koutsoyiannis / Energy Procedia 76 ( 2015 ) 406 – 411

411

ab

Fig. 3. (a) wind turbine power curve of ENERCON E-82 (enercon.de); (b) a weekly-window of hourly wind speed simulation and the

corresponding energy production from the installed wind turbine (where w denotes wind speed).

3. Conclusions

In this paper, we present a methodology for dealing with processes of double cyclostationarity (e.g. daily and

seasonal). Most existing methodologies preserve the marginal characteristics and assume a process with a short-

range dependence structure. The present method is based on a normalization scheme with two periodicities and it is

more appropriate for the wind speed process. Furthermore, we describe how to analyze the stochastic structure of a

normalized process with the use of climacogram, a stochastic tool with many advantages in stochastic interpretation

and model building. Also, we construct a stochastic model capable of preserving an HK behaviour and we produce

synthetic time-series (based on aggregated Markovian processes) for the purpose of simulation. Finally, we apply the

above to a meteorological station in Greece and we illustrate an example of simulation of wind speed and energy

production (based on a proposed industrial wind turbine).

Acknowledgements

This paper was partly funded by the Greek General Secretariat for Research and Technology through the research

project “Combined REnewable Systems for Sustainable ENergy DevelOpment” (CRESSENDO; programme

ARISTEIA II; grant number 5145).

References

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methods, Hydrological Sciences Journal, 2008; 53(1):142–164.

[2] Dimitriadis, P., L. Lappas, ȅ. Daskalou, A. M. Filippidou, M. Giannakou, Ǽ. Gkova, R. Ioannidis, ǹ. Polydera, Ǽ. Polymerou, Ǽ. Psarrou, A.

Vyrini, S.M. Papalexiou, and D. Koutsoyiannis, Application of stochastic methods for wind speed forecasting and wind turbines design at the

area of Thessaly, Greece, European Geosciences Union General Assembly 2015, Geophysical Research Abstracts, Vol. 17, Vienna,

EGU2015-13810, European Geosciences Union, 2015.

[3] Dimitriadis, P., and D. Koutsoyiannis, Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and

Hurst–Kolmogorov processes, Stochastic Environmental Research & Risk Assessment, doi:10.1007/s00477-015-1023-7, 2015.

[4] Koutsoyiannis, D., Generic and parsimonious stochastic modelling for hydrology and beyond, Hydrological Sciences Journal,

doi:10.1080/02626667.2015.1016950, 2015.

[5] Burton T., Sharpe D., Jenkins N. and Bossanyi E., Wind Energy Handbook, John Wiley & Sons, New York, 2001.

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