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Application of Stochastic Methods to Double Cyclostationary Processes for Hourly Wind Speed Simulation

DOI:10.1016/j.egypro.2015.07.851
Authors:
Panayiotis Dimitriadis at National Technical University of Athens
Panayiotis Dimitriadis
Demetris Koutsoyiannis at National Technical University of Athens
Demetris Koutsoyiannis
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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).
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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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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.
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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
[1] Koutsoyiannis, D., H. Yao, and A. Georgakakos, Medium-range flow prediction for the Nile: a comparison of stochastic and deterministic
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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... 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). ...
... distribution function (using an explicit or even an implicit scheme as shown in previous sections) and then, we transform implicitly each part of the time series (that corresponds to each periodic time series) to the one with the desired periodic marginal characteristics. In case where for the generation scheme we apply the SAR (or SARMA) or SMA models we abbreviate this framework as pCSAR (or pCSARMA; for such application to the wind process see Dimitriadis and Koutsoyiannis, 2015b;to solar radiation Koudouris et al., 2017; and to the wave height and wind speed Moschos et al., 2017) or pCSMA (see for such applications to precipitation in Dimitriadis and Koutsoyiannis, 2017;and in Dimitriadis et al., 2018a), respectively. For an application to a proper cyclo-stationary scheme using the SMA model (i.e., abbreviated as CSMA) see in Dimitriadis et al. (2018a). ...
... 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. ...
Hurst-Kolmogorov dynamics in hydrometeorological processes and in the microscale of turbulence
Thesis
Full-text available
  • Dec 2017
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).
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... In this implicit way, we manage to homogenize the timeseries x i,j ∼ F (x i,j ; p i,j ) to y ∼ N (0, 1). 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). ...
... It is noted that higher-order moments of processes with HK behaviour cannot be adequately preserved in an implicit manner (see an illustrative example in Dimitriadis and Koutsoyiannis, 2018, their Appendix D) and thus, for a more accurate preservation of the dependence structure an explicit algorithm is necessary . We may use a simple generation scheme, such as the sum of AR(1) models (SAR; Dimitriadis and Koutsoyiannis, 2015b), that can synthesize any N (0, 1) autoregressive- like dependence structure, which later it can be transformed back to the original distribution function F (x i,j ; p i,j ) and so in this way produce a double-periodic process with the desired marginal distribution for each diurnal-seasonal cycle as well as the desired dependence structure. Finally, we multiply each value of the synthetic K T with the deterministically determined value of the hourly intensity of solar radiation at the top of the atmosphere. ...
A stochastic model for the hourly solar radiation process for application in renewable resources management
Article
Full-text available
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.
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... Prior to the model identification and stochastic synthesis, we transform the timeseries with a nonlinear technique, so that we may approximately reduce the impacts of the double periodicities to the dependence structure. The applied technique is based on the fitted marginal distributions and their quantile expressions, i.e., first the probability of every record of the timeseries is estimated based on the fitted marginal distribution of the double periodic cycle it belongs to, and then is re-evaluated based on the quantile expression of the overall marginal distribution (Dimitriadis and Koutsoyiannis, 2015a). Note that other transformation techniques exist with the most common to be the standardization one (i.e., every record of the timeseries is subtracted from the mean and divided by the standard deviation of the double periodic cycle it belongs to) or the mixed standardization one, where instead of the mean and standard deviation of each cycle, model parameters are used that are estimated based on the least impact to the dependence structure evaluated through the climacogram (Vavoulogiannis et al., 2021). ...
Stochastic simulation of wind wave parameters for energy production
Article
A combination of stochastic and deterministic models is applied for the study of ocean wind waves. Timeseries of significant wave height and mean zero up-crossing period, obtained from globally scattered floating buoys, are analyzed in order to construct a double periodic model, and select an optimal marginal distribution and dependence function for the description of the stochastic structure of wind waves. It is concluded that wind waves, in contrast to the atmospheric wind speed process, are mostly governed by the seasonal periodicity rather than the diurnal periodicity, which is often weak and can be neglected. Also, the Pareto-Burr-Feller distribution is found to be a fair selection among other common three-parameter marginal distributions. The dependence function is simulated through the Hurst-Kolmogorov (HK) dynamics using the climacogram (i.e., variance of the averaged process in the scale domain), a stochastic tool that can robustly estimate both the short-term fractal and long-range dependence behaviors both apparent at the wind wave process. To test the validity of the model, a stochastic synthesis of the wind wave process is performed through the Symmetric Moving Average scheme, focused on the explicit preservation of the probabilistic and the dependence structures. Finally, the stochastic model is applied for simulation to an offshore station southeast of Australia having one of the largest record lengths. The energy potential is also estimated through the significant wave height and the mean zero up-crossing period of both the synthetic and observed timeseries, and the effectiveness of the model is further discussed.
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... Particularly, to account for the double periodicity, heavy-tail distribution, and long-term persistence, evident in hydrometeorological processes (Dimitriadis, 2017), the stochastic algorithm proposed by Dimitriadis and Koutsoyiannis (2015) is applied in offshore wind speed. An outline of the appropriate steps to be followed during the stochastic generation scheme is presented in Figure 5 ...
Spatial Decision Support System for Offshore Wind Farm siting using Geographic Information Systems, Spatial Analysis and Optimization
Thesis
Full-text available
  • Apr 2022
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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... Each step of the analysis already described in the previous sections is used in order to simulate the marginal structure, the seasonal characteristics and the second order dependence structure. Particularly, to account for the aforementioned characteristics, evident in hydrometeorological processes [71], the stochastic algorithm proposed by Dimitriadis and Koutsoyiannis [72] is applied. An outline of the appropriate steps to be followed during the stochastic generation scheme is presented in Fig. 10 including: ...
A stochastic simulation scheme for the long-term persistence, heavy-tailed and double periodic behavior of observational and reanalysis wind time-series
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.
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... 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. ...
A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes
Article
Full-text available
  • Mar 2021
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.
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... , n}, each with length T and the total difference in profit will be positive. There exists empirical evidence that wind velocity and solar radiation can be modeled as doubly cyclostationary random processes [13], [14] with daily and yearly cycles. By choosing n = 24 , we can consider that the extra profit that is accumulated to the aggregator is ...
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... The climacogram-implicit scheme has been applied to several (stationary and single/double cyclostationary) processes, such as solar radiation (Koudouris et al. 2017), wave height and wind process for renewable energy production (Moschos et al. 2017), 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). Note that in all the above applications the same dependence structure is used for the original and the transformed process, since a small deviation between then is noticed and therefore, additional trials are considered not necessary. ...
Stochastic synthesis approximating any process dependence and distribution
Article
Full-text available
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.
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... The internal periodicities are a common characteristic of hydrometeorological processes. For the generation of the synthetic time-series we implement the methodology of Dimitriadis and Koutsoyiannis [14], which is suitable for double periodic processes, such as the ones examined in this study. In particular, the applied methodology preserves the double cyclostationarity (diurnal and seasonal) of a process through the hourly-monthly marginal distributions, including intermittent characteristics, such as probability of zero values, as well as the dependence structure of the processes through the climacogram, which represents a plot of the variance of the scaled-averaged process as a function of scale [15]. ...
Harnessing wind and wave resources for a Hybrid Renewable Energy System in remote islands: a combined stochastic and deterministic approach
Article
Full-text available
  • Sep 2017
Wind and wave resources enclose an important portion of the planet’s energy potential. While wind energy has been effectively harnessed through the last decades to substitute other forms of energy production, the utilization of the synergy between wind and wave resource has not yet been adequately investigated. Such a hybrid energy system could prove efficient in covering the needs of non-connected remote islands. A combined deterministic and stochastic methodology is presented in a case study of a remote Aegean island, by assessing a 100-year climate scenario incorporating uncertainty parameters and exploring the possibilities of fully covering its energy demands.
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Transitioning remote Arctic settlements to renewable energy systems – A modelling study of Longyearbyen, Svalbard
Article
Full-text available
As transitioning away from fossil fuels to renewable energy sources comes on the agenda for a range of energy systems, energy modelling tools can provide useful insights. If large parts of the energy system turns out to be based on variable renewables, an accurate representation of their short-term variability in such models is crucial. In this paper, we have developed a stochastic long-term energy model and applied it to an isolated Arctic settlement as a challenging and realistic test case. Our findings suggest that the stochastic modelling approach is critical in particular for studies of remote Arctic energy systems. Furthermore, the results from a case study of the Norwegian settlement of Longyearbyen, suggest that transitioning to a system based on renewable energy sources is feasible. We recommend that a solution based mainly on renewable power generation, but also including energy storage, import of hydrogen and adequate back-up capacity is taken into consideration when planning the future of remote Arctic settlements.
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Generic and parsimonious stochastic modelling for hydrology and beyond
Article
Full-text available
  • Jan 2016
The old principle of parsimonious modelling of natural processes has regained its importance in the last few years. The inevitability of uncertainty and risk, and the value of stochastic modelling in dealing with them, are also again appreciated, after a period of growing hopes for radical reduction of uncertainty. Yet in stochastic modelling of natural processes several families of models are used which are often non-parsimonious, unnatural or artificial, theoretically unjustified and, eventually, unnecessary. Here we develop a general methodology for more theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations. The discrete-time properties thereof are theoretically derived from the continuous-time ones and a general simulation methodology in discrete time is built, which explicitly handles the effects of discretization and truncation. Some additional modelling issues are discussed with focus on model identification and fitting, which are often made using inappropriate methods.
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Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes
Article
Full-text available
  • Jan 2015
Three common stochastic tools, the climacogram i.e. variance of the time averaged process over averaging time scale, the autocovariance function and the power spectrum are compared to each other to assess each one’s advantages and disadvantages in stochastic modelling and statistical inference. Although in theory, all three are equivalent to each other (transformations one another expressing second order stochastic properties), in practical application their ability to characterize a geophysical process and their utility as statistical estimators may vary. In the analysis both Markovian and non Markovian stochastic processes, which have exponential and power-type autocovariances, respectively, are used. It is shown that, due to high bias in autocovariance estimation, as well as effects of process discretization and finite sample size, the power spectrum is also prone to bias and discretization errors as well as high uncertainty, which may misrepresent the process behaviour (e.g. Hurst phenomenon) if not taken into account. Moreover, it is shown that the classical climacogram estimator has small error as well as an expected value always positive, well-behaved and close to its mode (most probable value), all of which are important advantages in stochastic model building. In contrast, the power spectrum and the autocovariance do not have some of these properties. Therefore, when building a stochastic model, it seems beneficial to start from the climacogram, rather than the power spectrum or the autocovariance. The results are illustrated by a real world application based on the analysis of a long time series of high-frequency turbulent flow measurements.
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Medium-range flow prediction for the Nile: a comparison of stochastic and deterministic methods / Prévision du débit du Nil à moyen terme: une comparaison de méthodes stochastiques et déterministes
Article
Full-text available
  • Feb 2008
Due to its great importance, the availability of long flow records, contemporary as well as older, and the additional historical information of its behaviour, the Nile is an ideal test case for identifying and understanding hydrological behaviours, and for model development. Such behaviours include the long-term persistence, which historically has motivated the discovery of the Hurst phenomenon and has put into question classical statistical results and typical stochastic models. Based on the empirical evidence from the exploration of the Nile flows and on the theoretical insights provided by the principle of maximum entropy, a concept newly employed in hydrological stochastic modelling, an advanced yet simple stochastic methodology is developed. The approach is focused on the prediction of the Nile flow a month ahead, but the methodology is general and can be applied to any type of stochastic prediction. The stochastic methodology is also compared with deterministic approaches, specifically an analogue (local nonlinear chaotic) model and a connectionist (artificial neural network) model based on the same flow record. All models have good performance with the stochastic model outperforming in prediction skills and the analogue model in simplicity. In addition, the stochastic model has other elements of superiority such as the ability to provide long-term simulations and to improve understanding of natural behaviours.
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Wind Energy Handbook
Chapter
  • Apr 2002
Project development. Initial site selection, project feasibility assessment including the measure-correlate-predict technique for estimating energy yields from wind farm sites, micro-siting of turbines, the importance of public consultation and an overview of the preparation of environmental impact assessments.Visual and landscape assessment. Wind farm design and mitigation measures, assessment of visual impact and the use of Zones of Visual Impact (ZVI), wire-frame representations and photomontages.Noise. Terminology and basic concepts, sources of noise from a wind turbine, measurement and prediction of wind farm noise.Electro-magnetic interference. Impact of wind farms on various types of communication signals, modelling and prediction of electro-magnetic interference from wind turbines.Ecological assessment. Impact on birds.Financing wind farm developments. Project appraisal using discounted cash flow techniques, project finance and support mechanisms for wind energy development.
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Application of stochastic methods for wind speed forecasting and wind turbines design at the area of Thessaly
  • Jan 2015
  • 2015-13810
  • P Dimitriadis
  • Lappas
  • A M Daskalou
  • M Filippidou
  • Giannakou
  • R Gkova
  • Ioannidis
  • Polydera
  • Polymerou
  • A Psarrou
  • S M Vyrini
  • D Papalexiou
  • Koutsoyiannis
Dimitriadis, P., L. Lappas,. Daskalou, A. M. Filippidou, M. Giannakou,. Gkova, R. Ioannidis,. Polydera,. Polymerou,. 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.