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Global Investigation of Double Periodicity οf Hourly Wind Speed for Stochastic Simulation; Application in Greece

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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.
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
at
T
at
aaȝ¸
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¹
·
¨
¨
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§
+
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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 ܽ 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 ܽ 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 ܽ with latitude and (b) vrh with ܽ.
a
b
Fig. 3. Variation of (a) vrm and ܽ with latitude and (b) vrm with ܽ.
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 24:
β
+= alah (2)
where Į = 12/180 h/deg; l is the longitude varying from 180 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
¸
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¹
·
¨
¨
©
§+
¸
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¹
·
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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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... Finally, only the SAR model (which comprises of many AR(1) models) can handle explicitly non-Gaussian distributions through the preservation of moments higher than the second (e.g., the PGAR model of Fernandez and Salas, 1986), whereas this cannot be explicitly done by any other AR(q) or ARMA(q,p) model for q >1 (see also section 3.3.1). Note that the SAR has been already applied to several processes such as benchmark experiments (Dimitriadis and Koutsoyiannis, 2015a), the wind process (Deligiannis et al., 2016), the process of solar radiation (Koudouris et al., 2017) or the process of wave height and wave period . ...
... Note that this scheme has been applied to several (stationary and single/double cyclostationary as shown in the following section) processes, such as solar radiation (Koudouris et al., 2017), wave height and wind process for renewable energy production , as well as for the wind speed using a special case of the PBF distribution (Deligiannis et al., 2016) but also a generalized non-linear transformation (equivalent to a distribution function) based on the maximization of entropy when the distribution function is unknown (Dimitriadis and Koutsoyiannis, 2015b). ...
... Interestingly, we manage to also adequately preserve the cross-correlations between each cycle, without introducing a cyclostationary model (for more details on this method see Dimitriadis et al., 2018a). A useful remark is that the marginal characteristic of each period should follow a comprehensible periodic function (e.g., including sinus or cosines functions) as shown in Dimitriadis and Koutsoyiannis (2015b) for the wind process in Greece and through a global analysis in Deligiannis et al. (2016). In case where the periodic function of the parameters is not known or apparent it is advisable to use a parsimonious periodic function rather than use the empirical results that may be due to sample errors. ...
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Thesis
The high complexity and uncertainty of atmospheric dynamics has been long identified through the observation and analysis of hydroclimatic processes such as temperature, dew-point, humidity, atmospheric wind, precipitation, atmospheric pressure, river discharge and stage etc. Particularly, all these processes seem to exhibit high unpredictability due to the clustering of events, a behaviour first identified in Nature by H.E. Hurst in 1951 while working at the River Nile, although its mathematical description is attributed to A. N. Kolmogorov who developed it while studying turbulence in 1940. To give credits to both scientists this behaviour and dynamics is called Hurst-Kolmogorov (HK). In order to properly study the clustering of events as well as the stochastic behaviour of hydroclimatic processes in general we would require numerous of measurements in annual scale. Unfortunately, large lengths of high quality annual data are hardly available in observations of hydroclimatic processes. However, the microscopic processes driving and generating the hydroclimatic ones are governed by turbulent state. By studying turbulent phenomena in situ we may be able to understand certain aspects of the related macroscopic processes in field. Certain strong advantages of studying microscopic turbulent processes in situ is the recording of very long time series, the high resolution of records and the controlled environment of the laboratory. The analysis of these time series offers the opportunity of better comprehending, control and comparison of the two scientific methods through the deterministic and stochastic approach. In this thesis, we explore and further advance the second-order stochastic framework for the empirical as well as theoretical estimation of the marginal characteristic and dependence structure of a process (from small to extreme behaviour in time and state). Also, we develop and apply explicit and implicit algorithms for stochastic synthesis of mathematical processes as well as stochastic prediction of physical processes. Moreover, we analyze several turbulent processes and we estimate the Hurst parameter (H >> 0.5 for all cases) and the drop of variance with scale based on experiments in turbulent jets held at the laboratory. Additionally, we propose a stochastic model for the behaviour of a process from the micro to the macro scale that results from the maximization of entropy for both the marginal distribution and the dependence structure. Finally, we apply this model to microscale turbulent processes, as well as hydroclimatic ones extracted from thousands of stations around the globe including countless of data. The most important innovation of this thesis is that, to the Author’s knowledge, a unique framework (through modelling of common expression of both the marginal density distribution function and the second-order dependence structure) is presented that can include the simulation of the discretization effect, the statistical bias, certain aspects of the turbulent intermittent (or else fractal) behaviour (at the microscale of the dependence structure) and the long-term behaviour (at the macroscale of the dependence structure), the extreme events (at the left and right tail of the marginal distribution), as well as applications to 13 turbulent and hydroclimatic processes including experimentation and global analyses of surface stations (overall, several billions of observations). A summary of the major innovations of the thesis are: (a) the further development, and extensive application to numerous processes, of the classical second-order stochastic framework including innovative approaches to account for intermittency, discretization effects and statistical bias; (b) the further development of stochastic generation schemes such as the Sum of Autoregressive (SAR) models, e.g. AR(1) or ARMA(1,1), the Symmetric-Moving-Average (SMA) scheme in many dimensions (that can generate any process second-order dependence structure, approximate any marginal distribution to the desired level of accuracy and simulate certain aspects of the intermittent behaviour) and an explicit and implicit (pseudo) cyclo-stationary (pCSAR and pCSMA) schemes for simulating the deterministic periodicities of a process such as seasonal and diurnal; and (c) the introduction and application of an extended stochastic model (with an innovative identical expression of a four-parameter marginal distribution density function and correlation structure, i.e. g(x;C)=λ/[(1+|x/a+b|^c )]^d, with C=[λ,a,b,c,d]), that encloses a large variety of distributions (ranging from Gaussian to powered-exponential and Pareto) as well as dependence structures (such as white noise, Markov and HK), and is in agreement (in this form or through more simplified versions) with an interestingly large variety of turbulent (such as horizontal and vertical thermal jet of positively buoyancy processes using laser-induced-fluorescence techniques as well as grid-turbulence generated within a wind-tunnel), geostatistical (such as 2d rock formations), and hydroclimatic processes (such as temperature, atmospheric wind, dew-point and thus, humidity, precipitation, atmospheric pressure, river discharges and solar radiation, in a global scale, as well as a very long time series of river stage, and wave height and period). Amazingly, all examined physical processes (overall 13) exhibited long-range dependence and in particular, most (if treated properly within a robust physical and statistical framework, e.g. by adjusting the process for sampling errors as well as discretization and bias effects) with a mean long-term persistence parameter equal to H ≈ 5/6 (as in the case of isotropic grid-turbulence), and (for the processes examined in the microscale such atmospheric wind, surface temperature and dew-point, in a global scale, and a long duration discharge time series and storm event in terms of precipitation and wind) a powered-exponential behaviour with a fractal parameter close to M ≈ 1/3 (as in the case of isotropic grid-turbulence).
... In case where the marginal distribution is unknown or difficult to estimate, we may use non-linear transformation schemes based on the maximization of entropy (Koutsoyiannis et al., 2008;Dimitriadis and Koutsoyiannis, 2015b). It is noted that a more robust approach to reduce the 12 × 24 set of parameters would be to employ an analytical expression for the double solar periodicity (as done for the wind process in Deligiannis et al., 2016). This homogenization scheme has been applied to several processes such as wind (Deligiannis et al., 2016), solar radiation (Koudouris et al., 2017), wave height, wave period and wind for renewable energy production (Moschos et al., 2017), river discharge (Pizarro et al., 2018) and precipitation . ...
... It is noted that a more robust approach to reduce the 12 × 24 set of parameters would be to employ an analytical expression for the double solar periodicity (as done for the wind process in Deligiannis et al., 2016). This homogenization scheme has been applied to several processes such as wind (Deligiannis et al., 2016), solar radiation (Koudouris et al., 2017), wave height, wave period and wind for renewable energy production (Moschos et al., 2017), river discharge (Pizarro et al., 2018) and precipitation . However, it is noted that this scheme assumes stationary in the dependence structure rather cyclostationary (for such analyses see Koutsoyiannis et al., 2008, and references therein). ...
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Since the beginning of the 21st century, the scientific community has made huge leaps to exploit renewable energy sources, with solar radiation being one of the most important. However, the variability of solar radiation has a significant impact on solar energy conversion systems, such as in photovoltaic systems, characterized by a fast and non-linear response to incident solar radiation. The performance prediction of these systems is typically based on hourly or daily data because those are usually available at these time scales. The aim of this work is to investigate the stochastic nature and time evolution of the solar radiation process for daily and hourly scale, with the ultimate goal of creating a new cyclostationary stochastic model capable of reproducing the dependence structure and the marginal distribution of hourly solar radiation via the clearness index KT.
... The empirical and modelled probability of standardized wind speed less than or equal to 1 are both around 50%. Note that r and k should approximate unity but they are slightly larger due to the double cyclostationary effects of the daily and seasonal periodicities of the wind process (Deligiannis et al. 2016). These effects cause the small increase of climacogram around daily and annual scales (Fig. 5) but here, for simplicity, we apply a stationary rather than a cyclostationary model through the non-linear transformation of the probability function of Deligiannis et al. (2016). ...
... Note that r and k should approximate unity but they are slightly larger due to the double cyclostationary effects of the daily and seasonal periodicities of the wind process (Deligiannis et al. 2016). These effects cause the small increase of climacogram around daily and annual scales (Fig. 5) but here, for simplicity, we apply a stationary rather than a cyclostationary model through the non-linear transformation of the probability function of Deligiannis et al. (2016). Again, due to the weak periodicities of the examined process the double cyclostationarity can be generated through the inverse transformation or through the CSMA framework. ...
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Article
An extension of the symmetric-moving-average (SMA) scheme is presented for stochastic synthesis of a stationary process for approximating any dependence structure and marginal distribution. The extended SMA model can exactly preserve an arbitrary second-order structure as well as the high order moments of a process, thus enabling a better approximation of any type of dependence (through the second-order statistics) and marginal distribution function (through statistical moments), respectively. Interestingly, by explicitly preserving the coefficient of kurtosis, it can also simulate certain aspects of intermittency, often characterizing the geophysical processes. Several applications with alternative hypothetical marginal distributions, as well as with real world processes, such as precipitation, wind speed and grid-turbulence, highlight the scheme’s wide range of applicability in stochastic generation and Monte-Carlo analysis. Particular emphasis is given on turbulence, in an attempt to simulate in a simple way several of its characteristics regarded as puzzles.
... One of the main features of the hydrometeorological processes (such as wind speed [73] and solar irradiance [74]) is the double periodicity, interpreting the diurnal and seasonal variation of these uncertain variables. Note that the seasonality occurs considering the deterministic movement of the earth in orbit around the sun and around its axis of rotation [74]. ...
Article
This paper presents a stochastic planning algorithm to plan an operation of a multi-microgrid (MMG) in an electricity market considering the integration of stochastic renewable energy resources (RERs). The proposed planning algorithm investigates the optimal operation of resources (i.e., wind turbine (WT), fuel cell (FC), Electrolyzer, photovoltaic (PV) panel, and microturbine (MT)) and energy storage (ES). Various uncertainties (e.g., the power production of WT, the power production of PV, the departure time of electric vehicle (EV), the arrival time of EV, and the traveled distance of EV) are initially forecasted according to the observed data. The prediction error is estimated by fitting the forecasted data and observed data using a Copula method. A Cournot equilibrium and game theory (GT) are applied to model the real-time electricity market and its interactions with the MMG. The proposed algorithm is examined in a sample MMG to determine the operation of uncertain resources and ES. The obtained results are compared with a baseline and the other conventional optimization methods to verify the effectiveness of the proposed algorithm. The obtained results authenticate the importance of modeling the interaction between the MMG and electricity market, especially under the high integration of uncertain RERs, resulting in above 8% cost reduction in the MMG.
... A simple sine-exponential method for describing diurnal wind characteristics is proposed by Ref. [73], however, the double periodic effects are not considered. Thus, the proposed models are based on the statistical analysis performed by Deligiannis et al. [74], but here, further advancements are proposed for the expressions of the standard deviation and the skewness coefficient. So, the double periodicity of the mean, the standard deviation and the skewness coefficient of the wind process can be simulated by the functions shown in (Eqs. ...
Article
Lacking coastal and offshore wind speed time series of sufficient length, reanalysis data and wind speed models serve as the primary sources of valuable information for wind power management. In this study, long-length observational records and modelled data from Uncertainties in Ensembles of Regional Re-Analyses system are collected, analyzed and modelled. The first stage refers to the statistical analysis of the time series marginal structure in terms of the fitting accuracy, the distributions’ tails behavior, extremes response and the power output errors, using Weibull distribution and three parameter Weibull-related distributions (Burr Type III and XII, Generalized Gamma). In the second stage, the co-located samples in time and space are compared in order to investigate the reanalysis data performance. In the last stage, the stochastic generation mathematical framework is applied based on a Generalized Hurst-Kolmogorov process embedded in a Symmetric-Moving-Average scheme, which is used for the simulation of a wind process while preserving explicitly the marginal moments, wind’s intermittency and long-term persistence. Results indicate that Burr and Generalized Gamma distribution could be successfully used for wind resource assessment, although, the latter emerged enhanced performance in most of the statistical tests. Moreover, the credibility of the reanalysis data is questionable due to increased bias and root mean squared errors, however, high-order statistics along with the long-term persistence are thoroughly preserved. Eventually, the simplicity and the flexibility of the stochastic generation scheme to reproduce the seasonal and diurnal wind characteristics by preserving the long-term dependence structure are highlighted.
... To mitigate the effect that the periodicity of hydrological-cycle processes, prominent both in the diurnal and seasonal cycles (e.g., [125][126][127][128][129][130][131][132]), exerts on their modelling, we apply a double standardization on the processes with hourly resolution and a seasonal standardization on the ones with daily resolution. In particular, we subtract the mean from each periodicity cycle, and we divide with its standard deviation. ...
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Article
To seek stochastic analogies in key processes related to the hydrological cycle, an extended collection of several billions of data values from hundred thousands of worldwide stations is used in this work. The examined processes are the near-surface hourly temperature, dew point, relative humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow and precipitation. Through the use of robust stochastic metrics such as the K-moments and a secondorder climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic similarities exist in both the marginal structure, in terms of the first four moments, and in the secondorder dependence structure. Stochastic similarities are also detected among the examined processes, forming a specific hierarchy among their marginal and dependence structures, similar to the one in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can be also justified by the principle of maximum entropy. Therefore, they allow for the development of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics, with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence at large scales, and a transient behavior at intermediate scales.
... The literature offers a plethora of models that allow for representing important statistical characteristics of the process of interest, such as its marginal distribution structure and its second (and higher) order dependence structure. By robustly simulating both structures, several important behaviours of the process of interest can be preserved, such as the marginal distribution function along with the diurnal and seasonal periodicities, for example, through marginal transformations (Deligiannis et al. 2016), entropic transformations (Dimitriadis and Koutsoyiannis 2015), or copula-based schemes preserving different distribution functions and autocorrelation structures across seasons and scales (Tsoukalas et al. 2019), as well as the intermittency and the persistence on a wide range of scales (Dimitriadis and Koutsoyiannis 2018). ...
Chapter
The fundamental concepts in the field of water-energy systems and their historical evolution with emphasis on recent developments are reviewed. Initially, a brief history of the relation of water and energy is presented and the concept of the water-energy nexus in the 21th century is introduced. The investigation of the relationship between water and energy shows that this relationship comprises both conflicting and synergistic elements. Hydropower is identified as the major industry of the sector and its role in addressing modern energy challenges by means of integrated water-energy management is highlighted. Thus, the modelling steps of designing and operating a hydropower system are reviewed, followed by an analysis of theory and physics behind energy hydraulics. The key concept of uncertainty, which characterises all types of renewable energy, is also presented in the context of the design and management of water-energy systems. Subsequently, environmental considerations and impacts of using water for energy generation are discussed, followed by a summary of the developments in the emerging field of maritime energy. Finally, present challenges and possible future directions are presented.
... 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 ...
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Article
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.
... For the dependence structure we apply an HK model based on the empirical climacogram of each process. Finally, for the generation scheme we use the CSAR algorithm (cyclostationary sum of finite independent AR(1) processes [5]) 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. process of the double cyclostationary process. ...
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Thesis
Planning an offshore wind project is considered as a highly complex and multivariable task since it involves a large number of parameters, controversial objectives and constraints to be considered. During the pre-feasibility and pre-planning stages for offshore wind farm site-prospecting, the current manual and sequential design approaches are not always sufficient to guarantee optimal solutions because inherent interactions and trade-offs are most of the times disregarded. Most of the already existing wind energy design tools are specifically built either for onshore environments or for specific offshore activities; hence most of them ignore many relevant key design aspects extended in both space and time. In addition, with the rapid evolution of the Geographic Information Systems (GIS) during the last two decades, numerous research studies, spatial modelling and spatial optimization approaches in the field of the Renewable Energy Sources (RES) gained attention. Highlighting the promising results occurred, considering the planning and designing procedures of such projects, in the near future, geospatial technology with its numerous services and fields can effectively be utilized for timely analysis and future planning assessments. Considering the aforementioned challenges, this Ph.D. thesis proposes the development of a set of tools, as a Spatial Decision Support System (SDSS) entitled SpOWNED-Opt (Spatial Optimization for Offshore WiNd Energy Development), in order to model, map, evaluate and identify continuous space for future OWF siting, towards the mathematical programming approach, based on GIS data structures and algorithms. Thus, the proposed tool can be defined as a more integrated GIS-based framework for the pre-feasibility assessment as also for parts of the Front and Engineering Stage of the Design (FEED) for offshore wind farm site-prospecting procedures in the North and Central Aegean Sea in Greece. In particular, the SpOWNED-Opt approach proposes a multi-level methodological framework for integrating different spatial modelling tools separated at four stages of development. The first stage consists of all preparative steps considering data acquisition pre-processing along with the screening analysis module, based on the Maritime Spatial Planning (MSP) guidelines and the national legislative regulations. Vector and raster data 10 are used expressing existing potential conflicts among human activities combined with socio-economic and environmental factors affecting the selection procedures. The second stage is linked to the cost assessment modules for the capital, operation and maintenance and decommissioning expenses (CAPEX, O&M and DECEX) approximation. An extensive review of all sub-cost components is carried out in order to formulate analytical expressions embedded in the SDSS. Moreover, graph-based optimization techniques are applied, based on Least Cost Path (LCP) algorithms upon raster surfaces in order to extract distance-based costs (transmission lines, installation, decommissioning and O&M costs). The third stage focuses on the energy yield estimation and wind power output variability based on the UERRA Regional Reanalysis data. Different probabilistic models (Weibull, Burr Type II and XII, Gen. Gamma), reanalysis data errors quantification, wind speed intermittent characteristics and the second-order dependence structure are examined, analyzed and modelled in order to stochastically generate wind power output time series that are served as inputs to the last stage of the SDSS. The final module refers to a multi-objective integer non-linear programming (INLP) algorithm; as a unified framework that allows exploring in a rigorous and systematic mode numerous alternatives for offshore wind farm site-prospecting. The economic viability and the performance of the proposed wind farms are assessed along with the optimality of the different scenarios, from which the best ones are finally identified and mapped. The novelty of this research lies both on the integrated nature of the SDSS and on the models used in the spatial modelling field. A critical advantage of the SDSS is that it addresses existing gaps on OWFs siting and overall, in RES location-allocation issues, by: i) introducing a holistic, step-by-step, spatial modelling framework, ii) providing a long-term planning approach, iii) implemented in a user-friendly graphical user interface (GUI), giving the opportunity to national and local authorities and stakeholders to delineate systematic assessment strategies in order to succeed an effective and sustainable renewable energy sources penetration.
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Reicosky, D.C., Winkelman, L.J., Baker, J.M. and Baker, D.G., 1989. Accuracy of hourly air tem- peratures calculated from daily minima and maxima. Agric. For. Meteorol., 46: 193-209. Temperature is one of the critical variables that drives biological systems and is of fundamental importance in crop growth models. The objective of this work was to determine the accuracy of several methods for calculating hourly air temperatures from daily maxima and minima. Methods that have as inputs daily minimum and maximum temperature were selected from the literature based on their use in existing growth models and simplicity. Four years of hourly air temperature data collected during the growing season at 2 m over well-watered grass were used to test the various methods. Six days from each growing season were randomly selected for detailed analysis, and an additional 9 days were selected to cover a range of daily maximum temperatures and solar radiation. The absolute mean error within a 24-h period ranged from 0.5 to 9.3 ° C for the 6 ran- domly selected days for all 4 years of the data. All methods worked reasonably well on clear days but with limited success on overcast days. Daily maximum temperature did not appear to affect the accuracy of any of the methods. If accurate timing of temperature input to models is critical, the results indicate direct measurement of hourly temperature may be necessary.
Investigation of the stochastic properties of wind
  • P Dimitriadis
  • D Koutsoyiannis
  • P Papanicolaou
Dimitriadis P, Koutsoyiannis D, Papanicolaou P. Investigation of the stochastic properties of wind. European Geosciences Union General Assembly 2016; 18 EGU2016-12434-1.