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1876-6102 © 2016 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 organizing committee of the General Assembly of the European Geosciences Union (EGU)

doi: 10.1016/j.egypro.2016.10.001

Energy Procedia 97 ( 2016 ) 278 – 285

ScienceDirect

European Geosciences Union General Assembly 2016, EGU

Division Energy, Resources & Environment, ERE

Global investigation of double periodicity Ƞf hourly wind speed for

stochastic simulation; application in Greece

Ilias Deligiannis, Panayiotis Dimitriadis* , Olympia Daskalou, Yiannis Dimakos and

Demetris Koutsoyiannis

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

Abstract

The wind process is considered an important hydrometeorological process and one of the basic resources of renewable energy. In

this paper, we analyze the double periodicity of wind, i.e., daily and annual, for numerous wind stations with hourly data around

the globe and we develop a four-parameter model. Additionally, we apply this model to several stations in Greece and we

estimate their marginal characteristics and stochastic structure best described by an extended-Pareto marginal probability

function and a Hurst-Kolmogorov process, respectively.

© 2016 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the organizing committee of the General Assembly of the European Geosciences Union

(EGU).

Keywords: wind speed; double periodicity; marginal distribution; dependence strusture; stochastic simulation

1. Introduction

beware of the periodic double threat of the windmill maneuver,

dedicated to Bobby Fischer for the 1956 game of the century.

Several studies have been conducted for the stochastic simulation of hourly wind speed on the purpose of

renewable energy simulation and management [1]. However, the double periodicity of wind [2,3] is often

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

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

Available online at www.sciencedirect.com

© 2016 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 organizing committee of the General Assembly of the European Geosciences Union (EGU)

Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285 279

overlooked with most models focusing solely on the annual cycle and therefore, neglecting the contribution of daily

wind fluctuation in energy production and management. In this work, we present a methodology based on [3] for

wind speed simulation that includes a deterministic model for the double periodicity of wind as well as a stochastic

model for the probability and dependence structure of the process under a cyclostationary concept [4]. In section 2,

we describe the former model and we compare the hourly-monthly mean wind profiles with the corresponding

temperature ones in an attempt to provide a physical reasoning. We then test the double periodic model to

approximately 2000 stations around the globe with high quality and large quantity of records and we show several

statistical characteristics related to the model performance for each station and model parameter. Finally, in section 3

we estimate the parameters of the double cyclostationary model including the stochastic structure and marginal

characteristics of the most credible stations in Greece.

2. Double periodicity of wind

2.1. Data

From the original database of more than 15000 land-based stations around the globe downloaded from noaa

(www.ncdc.noaa.gov), we choose all stations that are still operational (7500) and we form two groups. The first

group includes stations with at least 105 observations in total and at least one observation per hour (1600 stations).

For the purpose of having as much as possible a uniform spatial distribution of stations around the globe, we add

250 stations located mostly at the Southern Hemisphere (group B). These stations have at least 1800 records per

year corresponding to one measurement per 3 hours and for at least 10 months per year. In Map 1, we depict the

selected stations for each group.

Map. 1. Spatial distribution of wind stations with hourly data.

0 5,000 10,000 15,000 20,0002,500 km

group A stations

group B stations

280 Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285

2.2. Correlation between temperature and wind speed

The kinetic state of air molecules is related to both their velocity and thermal energy [5]. Therefore, wind speed

and air temperature must have a strong correlation not only in microscale but also in macroscale, i.e. a difference in

temperature causing a difference in air pressure and as a consequence, in wind speed, similarly to the lake

stratification process. In Fig. 1, we estimate the correlation coefficient (denoted r) between hourly wind speed and

temperature and we plot the monthly average correlation (rav) for each station. It is notable that 90% of stations have

rav > 0.65 and 46% of stations have rav > 0.9.

Fig. 1. Correlation coefficient between hourly-monthly mean wind speed and temperatu re.

2.3. Double periodic model

Several models exist for simulating the deterministic behaviour of hourly-monthly air temperature with the most

popular ones to be a combination of periodical and exponential functions [6,7]. Since the correlation coefficient

between wind speed and temperature is high enough, it is only reasonable to adopt similar models for describing the

double periodicity of wind. Here, we expand the model presented in [3] for the hourly-monthly mean wind speed of

the form A(t) eB(t) + C(t), where A, B and C are periodic functions describing the annual variability and with the

exponential function corresponding to the daily variability of the process:

h4

m

mm

3

h

hh

m

mm

21c ʌ2cosʌ2cosexpʌ2cos ȝa

T

at

a

T

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+= (1)

where ȝc (m/s) is the mean for the specific hour of the day and month of the year (24×12 different values in total); ȝh

(m/s) is the overall mean of the process (one value); th is the continuous time in hours and tm the continuous time in

months; Th = 24 h; Tm = 12 months; a1, a2, a3 are dimensionless parameters; a4 equals

()

³

−ʌ2

0

1dxcosexp1xa

1

266.11 a−≈ , in order to exactly preserve the mean of the process; am is a parameter depicting the month of

maximum wind speed and varies from 0 to 12 months; and ah is considered a coefficient depicting the hour of

maximum wind speed varying from 0 h to 24 h (see at the end of section for justification).

The four parameters are calculated through the minimization of the average squared error between the observed

and modeled values. Parameter ܽଵ is closely related to daily fluctuation of wind speed. Furthermore, we estimate the

average of the daily velocity ratios, i.e., vmax/vmin in order to evaluate the temporal variation of wind speed. The

monthly-average ratio vrh describes the weighting factor of the temporal variation. We estimate that the 82% of

stations have vrh > 1.5 and 26% of stations have vrh > 2.5.

Likewise, parameter ܽଶ is closely related to the annual periodicity of wind. To evaluate the monthly variation of

annual wind speed, the ratio vrm = vM/vm is calculated, where vM, vm are the maximum and minimum monthly wind

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Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285 281

speed. This ratio is evaluated quite larger than unity for most stations indicating a significant annual variation, with

64% of stations having vrm > 1.5.

a

b

Fig. 2. Variation of (a) vrh and ܽଵ with latitude and (b) vrh with ܽଵ.

a

b

Fig. 3. Variation of (a) vrm and ܽଷ with latitude and (b) vrm with ܽଷ.

Fig. 4. Variation of ܽ with longitude.

Parameter a2 in combination with a3 can capture the most commonly met profiles of wind speed (see Fig. 7 in

section 3). There are three profiles exhibiting hourly-monthly means: (1) almost parallel to each other, i.e., a2 = 0;

(2) with similar low values and different peak values for each month, i.e., a3 = 0; and (3) with similar peak values

and different low values for each month, i.e., a2 a3 0.

Coefficients ah and parameter am determine the peak hour and month, respectively. The variation of ah with

longitude is linear with r2 around 0.7, meaning that the maximum velocity seems to appear at the same

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282 Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285

approximately local hour (14h00) for all examined stations around the globe (all observations are recorded in

Greenwich Time). As a result of this, ah can be calculated as follows (note that if ah > 24 then ah = ah ᄙ 24:

β

+= alah (2)

where Į = 12/180 h/deg; l is the longitude varying from ᄙ180 to +180 deg; ȕ = 14.2 h.

2.4. Model performance from global analysis

Coefficient r and nrmse (abbreviation for the normalized root mean square error) between observed and modeled

values are in most cases remarkably high and low, respectively. In Fig. 5, we plot the monthly average r and nrmse

(denoted rav and nrmseav) and we observe that 90% of stations indicate rav > 0.65 and 75% of stations rav > 0.9. In

addition, 34% of stations have nrmseav < 0.1 and 90% of stations nrmseav < 0.2.

a

b

Fig. 5. Variationof (a) rav with latitude and (b) nrmseav with latitude.

However, r can sometimes underestimate the goodness of fit, especially if vrh is close to unity. In that case, nrmse

is close to zero and a smooth hourly-monthly mean profile can be easily fitted. Reasonably, when both nrmse and vrh

have large values then so will r. In general, both r and nrmse show adequate results with 80% of stations having rav

> 0.7 and nrmseav < 0.2 (Fig. 6).

a

b

Fig. 6. Variation of (a) vrh and nrmseav with rav and (b) vrh with nrmseav.

3. Application

In this section, we apply the double periodic model to 17 stations of high quality and large quantity of records in

Greece (Table 1-2 and Fig. 7). Additionally, we model the standard deviation of the process by a single periodic

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Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285 283

function corresponding solely to annual fluctuation since daily fluctuation is minimal for all stations:

h

m

mm

c1ʌ2cos

σσ

¸

¸

¹

·

¨

¨

©

§+

¸

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·

¨

¨

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§−

=T

bt

b (3)

where ıc (m/s) is the standard deviation for each month, ıh (m/s) is the hourly standard deviation of the process, b is

a dimensionless parameter related to the magnitude of the monthly fluctuation and bm is a coefficient depicting the

month of maximum wind speed standard deviation and varying from 0 to 12 months.

Furthermore, we estimate the dependence structure of the wind process all over Greece by combining the

climacogram (i.e., the variance of the mean process vs. scale, denoted as Ȗ (m2/s2) and introduced in [8]) of the 17

stations (Fig. 8). The justification for the use of climacogram to estimate the stochastic structure of the process

instead of the commonly used autocorrelation function or power spectrum can be seen in [9]. There, it is illustrated

that the climacogram has always smaller statistical uncertainty from the other two stochastic tools for common

processes such as Markov and Hurst-Kolmogorov (HK) as well as combinations thereof. In Fig. 8, we conclude that

the wind process in Greece follows an HK process:

H

k22

/−

=

λγ

(4)

where Ȝ = 2 m2/s2 is the standardized variance of the discretized stationary process and Ǿ = 0.9 is the Hurst

coefficient.

This behaviour is somehow expected based on the analysis of [10,11], where the HK behaviour is detected in an

annual scale and in approximately 4000 stations around the globe. Also, we estimate the average marginal

probability function for the standardized process and we fit a two-parameter extended Pareto-type cumulative

probability function that shows good agreement with data in a global scale [12]:

()

()

()

p

2

p

/1/11

β

avvF +−= (5)

with Įp § 10 and ȕp § 8.5.

Table 1. General characteristics of the 17 stations in Greece downloaded from noaa (www.ncdc.noaa.gov).

station name longitude

(deg)

latitude

(deg)

elevation

(m)

years

of

records

mean wind

speed (m/s)

std wind

speed (m/s)

Karpathos 27.13 35.42 20 17 7.6 4.1

Santorini 25.47 36.40 39 24 5.7 3.2

Syros 24.95 37.42 72 17 5.1 3.0

Samos 26.92 37.70 7 37 4.4 3.1

El. Venizelos 23.95 37.93 94 11 4.0 3.1

Chios 26.13 38.33 4 24 3.7 2.8

Limnos 25.23 39.92 4 38 4.4 3.5

Paros 25.13 37.02 36 11 5.5 3.3

Kavala 24.60 40.98 5 24 2.4 2.1

Meganisi 20.77 38.62 4 42 3.6 2.7

Zakynthos 20.88 37.75 5 24 2.5 2.6

Kos 27.07 36.78 129 81 4.8 2.6

N. Aghialos 22.80 39.22 15 62 3.3 2.3

Larissa 22.42 39.63 74 32 1.7 2.7

Aleksandroupoli 25.92 40.85 3 80 3.6 3.1

Herakleio 25.18 35.33 39 41 4.6 2.9

Araksos 21.42 38.15 12 17 2.6 2.1

284 Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285

Table 2. Estimation of the four parameters and the one coefficient of the hourly-monthly mean and standard deviation model for the 17 stations in

Greece along with the model performance.

mean model (parameters) mean model

(coefficients) rav nrmseav

stdev model

(parameter)

stdev model

(coefficient) r nrmseav

a1 a2 a3 ah (h) am

(months) b bm (months)

0.130 0.042 0.213 11.88 7.12 0.94 0.13 0.019 5.6 0.43 0.05

0.144 0.000 0.051 11.89 2.00 0.96 0.08 0.164 1.2 0.98 0.06

0.185 0.000 0.102 12.07 0.34 0.96 0.08 0.122 1.3 0.95 0.08

0.165 0.000 0.036 12.09 11.00 0.81 0.15 0.098 0.6 0.96 0.16

0.416 0.188 -0.163 12.51 6.94 0.95 0.13 0.106 0.2 0.79 0.12

0.291 0.064 -0.140 11.67 5.51 0.96 0.13 0.207 0.6 0.97 0.12

0.264 0.000 0.139 11.28 0.35 0.95 0.12 0.280 0.6 0.98 0.12

0.250 0.000 0.005 12.02 11.00 0.95 0.12 0.051 1.2 0.56 0.13

0.401 0.189 -0.306 12.44 6.67 0.96 0.12 0.210 1.1 0.98 0.07

0.305 0.000 0.060 13.73 1.80 0.66 0.22 0.217 1.2 0.99 0.10

0.477 0.220 -0.400 12.75 6.71 0.91 0.16 0.346 1.1 0.98 0.15

0.251 0.016 0.007 13.23 3.47 0.96 0.07 0.231 1.2 0.98 0.07

0.314 0.290 -0.318 13.23 6.45 0.80 0.12 0.145 1.8 0.97 0.10

0.674 0.489 -0.295 15.35 6.36 0.97 0.15 0.121 2.2 0.89 0.16

0.427 0.164 -0.323 12.27 6.50 0.97 0.11 0.270 0.7 0.99 0.10

0.171 0.119 -0.225 11.95 6.50 0.89 0.10 0.163 1.5 0.97 0.06

0.527 0.265 -0.496 13.74 6.95 0.97 0.11 0.244 1.1 0.96 0.13

a

b

c

d

Fig. 7. Hourly-monthly mean velocities for the (a) Larissa, (b) Alexandroupoli and (c) Kos stations and monthly standard deviation of mean and

standard deviation for Larissa (continuous line), Alexandroupoli (dashed line) and Kos (dot dashed line) stations.

Finally, we describe a methodology to produce synthetic hourly wind timeseries with double periodicity as well

as preferable marginal characteristics and stochastic structure. Particularly, after we estimate the parameters for the

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Ilias Deligiannis et al. / Energy Procedia 97 ( 2016 ) 278 – 285 285

hourly-monthly mean wind speed (Eq. 1), the parameters for the standard deviation (Eq. 3), the dependence

structure of the process (Eq. 4) and the marginal probability function (Eq. 5), we can use the scheme described in [3]

to produce hourly wind speed timeseries approximating the desired distribution and with the desired dependence

structure generated by the sum of multiple Markov processes. The generation algorithm used in [3] is introduced in

[9] and although it includes only two parameters, it is capable of generating any length of timeseries following an

HK or various other processes. By applying this method we assume stationarity in autocorrelation rather than

cyclostationarity. Although this assumption can be cruel for certain hydrometeorological processes, it can be applied

as an approximation for the wind process, due to the small fluctuation of the autocorrelations of wind for the same

lag in different months.

a

b

Fig. 8. (a) Climacograms for all stations, best fitted HK, Markov and white noise processes and model; (b) empirical tail functions for all stations

and model.

4. Conclusions

In this paper, we investigate the double periodicity of wind and we present a model for the hourly-monthly

mean comprising four parameters. We further test our model against approximately 2000 stations around the globe

with 75% of stations having correlation coefficients with the observed values above 0.9. Finally, we apply our

model to several stations in Greece by also suggesting a deterministic model for the hourly-monthly standard

deviation and an HK stochastic model for the dependence structure with a Pareto-type marginal probability function,

all showing excellent agreement with data.

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microseconds to megayears. Orlob Second International Symposium on Theoretical Hydrology, University of California Davis 2016.

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