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Stochastic analysis and simulation of hydrometeorological processes
associated with wind and solar energy
Georgios Tsekouras and Demetris Koutsoyiannis
Department of Water Resources and Environmental Engineering, Faculty of Civil
Engineering, National Technical University of Athens, Heroon Polytechneiou 5, GR157 80
Zographou, Greece(georgtsek@yahoo.gr)
Abstract
The current model for energy production, based on the intense use of fossil fuels, is both
unsustainable and environmentally harmful and consequently, a shift is needed in the
direction of integrating the renewable energy sources into the energy balance. However, these
energy sources are unpredictable and uncontrollable as they strongly depend on time varying
and uncertain hydrometeorological variables such as wind speed, sunshine duration and solar
radiation. To study the design and management of renewable energy systems we investigate
both the properties of marginal distributions and the dependence properties of these natural
processes, including possible longterm persistence by estimating and analyzing the Hurst
coefficient. To this aim we use time series of wind speed and sunshine duration retrieved from
European databases of daily records. We also study a stochastic simulation framework for
both wind and solar systems using the software system Castalia, which performs multivariate
and multitimescale stochastic simulation, in order to conduct simultaneous generation of
synthetic time series of wind speed and sunshine duration, on yearly, monthly and daily scale.
Keywords: wind speed, sunshine duration, long termpersistence, Hurst coefficient,
marginal distributions, multivariate stochastic simulation
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1. Introduction
The drawbacks of conventional energy sources including their negative environmental
impacts emphasize the need to integrate renewable energy sources, like wind and solar, into
the energy balance. However these energy sources, unlike fossil fuels on which the past
model was based on, are unpredictable, at both short and long run, and uncontrollable as they
are associated with relevant hydrometeorological processes which are characterized by high
variability and uncertainty [1]. As a result, the estimation of renewable energy sources
requires analyzing hydrometeorological data, especially wind speed, sunshine duration and
solar radiation time series.
Such hydrometeorological time series are typically modelled as stationary stochastic
processes. In this case their probability distribution does not change in time. It is thus
necessary, in order to characterize uncertainty, to assume a theoretical distribution function
that fits properly the data. Normal distribution is suitable for most hydrometeorological
processes at an annual or higher time scale due to the Central Limit Theorem. However,
variables of lower time scales, such as monthly or daily, are characterized by skewness and
thus, nonnormal distributions are more appropriate for their representation. In particular, the
sunshine duration is represented as a random variable bounded from both below and above,
and thus a theoretical distribution with these properties should be chosen for its
representation.
Various studies have been performed in order to examine theoretical distribution
functions fitting properly wind speed and relative sunshine duration data. The Weibull and
Gamma distributions have been found to be appropriate to represent wind speed data while
the Beta distribution has been assumed to be suitable for the relative sunshine duration. Carta
et al. [2] and Zhou et al. [3] proposed that both the Weibull and the Gamma distribution are
suitable for hourly wind speed representation in the Canary Islands and in North Dakota
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respectively. Garcia et al. [4] and Yilmaz and Celik [5] suggested that the Weibull
distribution fits satisfactorily hourly wind speed data in Spain and Turkey. Darbandi et al. [6]
noted that the Gamma distribution fits well the maximum annual wind speed data in Iran.
Concerning the relative sunshine duration, Sulaiman et al. [7] and Bashahu and Nsabimana
[8] examined the fit of the Beta distribution to daily data in Malaysia and Africa and
concluded that it fitted satisfactorily the data in several cases. Chia and Hutchinson [9]
examined the possibility of fitting the Beta distribution to daily relative cloud duration time
series in Australia.
In addition to the marginal distributional properties, the fact that climate is not static
but exhibits fluctuations at all time scales should also be taken into consideration for the
effective exploitation of renewable energy sources. In a stochastic framework, these
fluctuations are quantified through the dependence of the processes in time [10]. While in
several studies these processes have been regarded as independent identically distributed
random variables, particularly at coarse time scales such as monthly or annual, other studies
have verified the presence of dependence, sometimes longrange dependence, which has been
also termed as longterm persistence (LTP), selfsimilar behaviour, or the HurstKolmogorov
dynamics. This longrange dependence expresses the tendency of similar physical events to
cluster in time and its quantification is expressed through the Hurst coefficient, the
mathematics of which is discussed later. The HurstKolmogorov dynamics has been identified
by several researchers in various environmental variables [11]. The British engineer Hurst
[12] was the first to notice this behaviour while studying Nile’s runoff and, earlier, the
Russian mathematician Kolmogorov [13] had proposed a mathematical model consistent with
this behaviour when studying turbulence. Later, this behaviour was also observed in other
variables such as river runoff [14,15,16,17,18,19,20], temperature [21,22,23,24,25,17,26,27],
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climatic indices such as North Atlantic oscillation [28] and decadal Pacific oscillation [29]
and atmospheric pressure fields [30].
So far, the concept of LTP has been incorporated in the modelling of runoff time
series applied to reservoir sizing design studies, where it has been found that neglect of LTP
usually leads to undersizing of the reservoir storage [3134]. However, the HurstKolmogorov
behaviour has been found to be omnipresent in the most natural processes [11]. Consequently,
it will strongly affect the design variables of any engineering project related to the
exploitation of natural phenomena, like a large scale hybrid renewable energy system
comprising wind and solar energy [35], as it expresses a general natural variability,
manifested by the aforementioned fluctuations in multiple time scales [36,10]. Such large
scale renewable energy projects aim not only at maximizing the profit but also the reliability
in satisfying energy demand or energy target via storage of the excess energy in each time
step of the project’s operation, usually by means of pumped storage hydropower facilities.
Therefore, neglect of this clustering of natural events at the design phase may lead to
undersizing of the facilities for energy storage or overestimation of the reliability of produced
energy.
A prerequisite for the investigation of HurstKolmogorov dynamics is the analysis of
time series having long time period records so that their fluctuations on multiple time scales
can be detected. For this reason a sufficient time length (which is generally regarded 100
years or more and no less than 7080 years) is required. However, hydrometeorological data
records, relevant to wind and sun, started in the beginning of the previous century or later and
as a consequence time series of such a length are not often available.
Once the marginal distribution and the dependence structure of the wind speed and
sunshine duration processes are identified, another relevant issue is to construct a simulation
model that can simulate such processes for arbitrarily long times, respecting the identified
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stochastic properties of each process, as well as the crosscorrelation between the two
processes. This issue is not trivial as typical stochastic generators may not be able to generate
processes with nonnormal distributions and with autocorrelations departing from a Markov
process.
This study aims to perform a comprehensive analysis of existing data with sufficient
length of observation, concerning wind speed and sunshine duration, retrieved from
measuring stations all over Europe, in order to investigate the properties of their marginal
distributions and mainly to detect the possible presence of LTP by estimating and analyzing
the Hurst coefficient values of both processes.
At the same time, it also aims to construct a general stochastic simulation scheme, able
to reproduce the statistical characteristics of the natural series including their long term
variations. By this, the unpredictable fluctuations characterizing the natural resources (wind
and solar radiation) can be taken into account effectively in the design and management of
renewable energy systems that comprise both wind and solar systems, and possibly
hydropower systems with pumped storage, which provide means for energy storage. The
framework is particularly useful to study the reliability in fulfilling the energy demand or the
energy production target. This is attempted by using the multivariate stochastic simulation
system, Castalia, for the simultaneous generation of long synthetic time series, on the concept
of the steadystate simulation, of both variables, being statistically equivalent to the historical
ones at all time scales including the large scale, annual and overannual, in which the Hurst
Kolmogorov behaviour dominates.
2. Description of data
Wind speed and sunshine duration records used in the analysis were retrieved from measuring
stations whose data are available online. The criteria for station selection were:
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a) A minimum acceptable record length of 70 years so as the Hurst coefficient to be
estimated with some accuracy.
b) The existence of metadata related to wind speed data and especially information about
the measurement height above ground.
c) A maximum number of three homogeneous periods, in each of which the
measurement height above ground is constant (stations with too frequent changes may
not be reliable).
However, records of this time length are rare worldwide. In particular, in some
continents (Asia, Africa) there is lack of data series of this length. Even in Europe and North
America most of the free access records refer to a period of 50 years or less. Furthermore,
there is total lack of long solar radiation records. However, this variable can be indirectly
estimated by sunshine duration [37].
After an extensive search, wind speed and sunshine duration records fulfilling these
criteria were retrieved exclusively from European databases, the KNMI Climate Explorer, the
European Climate Assessment & Data (ECA&D) and the Deutscher Wetterdienst databases.
A total number of 20 wind speed records and 21 sunshine duration records were found. In
addition, wind speed time series from Dublin Airport and Valentia stations were found from
the website of the Irish Meteorological Service in chart form and were digitized using the
Engauge software. The data retrieved consist of daily records except for the ones of the Irish
stations whose records are annual averages. In Tables 1 and 2 stations details are shown.
As known, wind speed at a certain site increases with height and thus, the analysis of
wind speed time series requires a single measurement height above ground during the time
period of operation. However, the collected daily time series do not refer to a single height
above ground due to changes of the observation height at certain time periods in measuring
stations. Based on the assumption of process stationarity, this problem was overcome by
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modifying the data of all the time periods except for the last period of records for each station
so as to refer to a unified altitude (data homogenization). Specifically the following procedure
was applied:
a) The daily time series were grouped to time periods where the measurement height
above ground is constant.
b) The average value of each homogeneous period was estimated.
c) The daily values of each period were multiplied with the ratio of the last period
average value to each period average value.
Thus, all records refer to the last measurement height above ground. Neglect of this
modification would lead to incorrect analysis, especially Hurst coefficient overestimation,
because, as exemplified in Figure 1, the annual raw time series at De Bilt station present a
sudden shift owing to the change of measurement height above ground. It is noted though that
the procedure followed may result in negative bias for the Hurst coefficient, because the
assumption that time averages of different periods are equal, which lies behind step (c), is not
true in general and may result in too stable time series.
In addition to wind speed gauge data, the potential of using Reanalysis data for further
investigation of LTP in other sites was also examined. Data were obtained from the
NCEP/NCAR database in grid mode and their reliability was checked by comparing their
records with the stationbased ones. This was achieved by selecting the same longitude and
latitude with this of measuring stations. As can be seen in Appendix A, Reanalysis data are
unreliable as their annual average values are higher than the stationbased ones even though
they refer to the same or a lower altitude and consequently they are not used any further.
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3. Analysis
3.1. Properties of marginal distributions
While wind speed is a nonnegative variable not bounded from above, sunshine duration has a
domain bounded from both below and above and its upper bound varies in different sites.
Thus, the relative sunshine duration is used instead, which varies in the interval [0,1] as it is
the ratio of the sunshine duration, n, to the astronomical day length, N. The latter depends on
the latitude and time of the year.
In this study, it was verified that both the Weibull and the Gamma distributions are
suitable for wind speed data (as summarized in the Introduction), mainly using graphical
depictions like that in Figure 2. Wind speed on daily scale is characterized by positive
skewness as shown in Figure 2, representing the frequency histogram of the time series at the
Eelde station. The probability density functions of the Gamma and the Weibull distributions
are nearly indistinguishable from each other and fit satisfactorily the data. Thus, both
distribution functions can be regarded as appropriate for the representation of wind speed.
Considering the fact that the Gamma distribution is used in several software systems
(including the software Castalia used in this study as described below) for generation of
synthetic time series, it is useful to also associate the distribution of the relative sunshine
duration with the Gamma distribution by performing a nonlinear transformation so that the
transformed variable takes values in the interval [0,+∞). Specifically, denoting the relative
sunshine duration variable as X, the logarithmic transformation Y= –ln(1 – X) is defined,
whose values belong indeed to [0,+∞), the value X = 0 corresponds to Y = 0 and X
approaching to 1 corresponds to Y approaching to +∞. As shown in Figure 3, the Gamma
distribution fits satisfactorily the empirical distribution of the transformed variable Y for the
relative sunshine duration data of the Eelde station.
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Thus, assuming that Y has Gamma distribution with shape parameter κ and scale
parameter λ, it is proved (see Appendix B) that the probability density function of the variable
X is:
(1)
The probability density functions of both the theoretical distribution (1) and the Beta
distribution are fitted to the frequency histogram of the historical time series of the relative
sunshine duration as shown in Figure 3. These two distributions are almost indistinguishable
from each other and they fit very well the historical data.
3.2. Longterm persistence
3.2.1. The presence of the LTP in wind speed and sunshine duration
The basic condition for the investigation of LTP is the elimination of seasonality so that the
Hurst coefficient estimation be reliable and not affected by periodic fluctuations. As a result,
it is preferable to analyze annual time series. Bakker and Bart [38] analyzed annual
geostrophic wind speed, deriving from sea level pressure, in Northwestern Europe, and
identified LTP using the maximum likelihood method for the simultaneous estimation of
standard deviation and the Hurst coefficient, according to Tyralis and Koutsoyiannis [39].
However, in most relevant studies, time series of lower time scales were used,
especially daily and hourly ones. Haslett and Raftery [40] proposed the existence of long term
persistence in daily wind speed time series, which were firstly deseasonalized and then
adjusted to an ARFIMA model. Bouette et al. [41], based on the former study, adjusted a
GARMA model using the periodogram and Whittle methods and analyzing both seasonality
and long term persistence. The results of the methods indicated that seasonality is the
dominant phenomenon in these time series.
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Furthermore, several researchers tried to investigate LTP using the socalled
Detrended Fluctuation Analysis [42]. Feng et al. [43] and Kavasseri and Nagarajan [44] used
MultiFractal Detrended Fluctuation Analysis to analyze daily and hourly wind speed time
series in China and North Dakota respectively, having previously deseasonalized the data.
Feng et al. concluded that no specific behaviour can be determined while Kavasseri and
Nagarajan noted that wind speed time series are multifractal processes, as suggested by the
range of their multifractal spectrum. Kocak [45] noted, after analyzing hourly wind speed
time series in Turkey using Detrended Fluctuation Analysis, that the scaling region is divided
into two parts. In the first one, time series fluctuate randomly while in the second one they are
characterized by long term correlations.
Additionally, Rehman and Siddiqi [46] estimated the Hurst coefficient in daily wind
speed time series in Saudi Arabia using the wavelet method, but no exact result was extracted
as these processes presented either persistence or antipersistence. Finally Benth and Saltyre
[47] claimed that daily wind speed data in Lithuania can be modelled using autoregression
models of third and fourth order which indicates short term memory.
As far as sunshine duration is concerned, Liu et al. [48] noted intense selfsimilar
behaviour in annual time series in China using the original Hurst’s range analysis which led to
remarkably high Hurst coefficient values. Tsekov [49] also used the Detrended Fluctuation
Analysis in order to examine the existence of LTP in daily sunshine duration time series in
Bulgaria and concluded that data do not indicate long term correlations. Finally, Harrouni and
Guessoum [50] investigated LTP in both daily and annual global solar radiation time series in
Panama and USA using the fractal dimension method and suggested that this process is
characterized by antipersistence.
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3.2.2. Stochastic representation of LTP and estimation of the Hurst coefficient
Let Xi denote a hydrometeorological process with i=1,2,…., denoting discrete time (years).
Furthermore, let its mean be μ :=E[Xi], its autocovariance γj :=cov[Xi,Xi+j], its autocorrelation
ρj :=corr[Χi.Χi+j] = γj/γ0 (j=0,±1, ±2,…) and its standard deviation σ := γ01/2. Let k be a positive
integer representing time scale of aggregation for the Xi process. The mean aggregated
stochastic process on that time scale is:
()
( 1) 1
1
: ik
k
ii
l i k
XX
k
(2)
Hydrometeorological processes characterized by LTP can be represented by a
stochastic process known as simple scaling stochastic (SSS) process or a HurstKolmogorov
(HK) process [17,51]. By definition of the HK process, the variance of the mean aggregated
stochastic process is a power law of k with exponent 2H –2, where H is the Hurst coefficient:
( ) 2 2 0
Var[ ] ,
kH
i
Xk
(3)
The value of H varies from 0 to 1. A value in the interval (0,0.5) indicates anti
persistence, which denotes that an increase in the values of time series is followed by a
subsequent decrease, a behaviour that is not normally observed in nature. A value 0.5
corresponds to a purely random process known as white noise. A value in the interval (0.5,1)
indicates persistence and implies positive autocorrelation. This behaviour is visualized as
multiyear fluctuations and apparent trends, as can be seen in Figure 4, which depicts a
sunshine duration time series characterized by a high value of Hurst coefficient (H=0.9) and
for comparison, a time series of a white noise with the same mean and standard deviation.
From equation (3) it is observed that if standard deviation (square root of variance) is
plotted against time scale on a logarithmic graph, known as the climacogram, a straight line is
formed with slope H – 1. Wind speed and sunshine duration data support this behaviour, as an
approximation, as can be seen in the climacograms of Figure 5. Based on this scaling property
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Koutsoyiannis [17] proposed a method for the simultaneous estimation of standard deviation
σ and the Hurst coefficient H using the least square error of sample standard deviation
estimates (LSSD). The method was later examined by Tyralis and Koutsoyiannis [39] and
Sheng et al. [52] and was found to perform very satisfactorily. In general, the method can be
perceived as a fitting of a straight line in a climacogram like that of Figure 5 using linear
regression of the mean aggregated standard deviation σ(k) on time scale k. The Hurst
coefficient is estimated from the slope of the climacogram using sample standard deviation
estimates on all time scales. The only difference in the LSDD method is that it explicitly
considers the bias of the standard statistical estimator of standard deviation on all time scales.
3.2.3. Results of the LSSD method
The results derived from the LSSD method are summarized in Figure 6 in terms of frequency
histograms of the Hurst coefficient values of the historical time series. Concerning both
variables, markedly high Hurst coefficient values are estimated and it is remarkable that no
value lies in the interval (0,0.5) i.e., no time series is characterized by antipersistence. The
majority of the values of both variables lie in the interval (0.6,0.9) and it is notable that nearly
15% of the values lie in the interval (0.9,1) which indicates very strong selfsimilar behaviour.
In general, the results of the method imply that both wind speed and sunshine duration time
series are characterized by HurstKolmogorov behaviour.
3.2.4. Representative value of Hurst coefficient
In order to assume a representative value of the Hurst coefficient for the historical time series
of both variables, the same number of synthetic time series is generated, each one of the same
mean value, standard deviation and record length with the respective historical one, but to
define the autocorrelation function a unique value of the Hurst coefficient is used, the same
for all time series. For this generation the multiple timescale fluctuation approach [16] is
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used. This method implies that a weighted sum of three exponential functions of time lag can
approximate satisfactorily the autocorrelation function of a simple scaling process on the
basic time scale. The generated process is the sum of three independent AR(1) processes.
In the end of the generation process the sample value of the Hurst coefficient is
estimated for all synthetic time series using the LSSD method. The frequency histogram of
these H values is then compared to the histogram of the historical H values. By trying
consecutive initial theoretical values of H until a convergence between the histogram of
historical and synthetic timeseries is achieved, it is concluded that the value H = 0.84 can be
regarded as representative for both variables as can be seen in Figure 6.
4. Modelling
4.1. The software system Castalia
Castalia is a software system that was initially introduced for the simulation of
hydrometeorological processes, like rainfall, runoff and evaporation. It performs multivariate
stochastic simulation on annual, monthly and daily scale using the threeparameter Gamma
distribution function in order to generate synthetic time series. The program preserves the
essential marginal statistics, specifically mean value, standard deviation and skewness, as well
as the joint second order statistics, namely auto and crosscorrelation, on these three time
scales. To estimate these statistics, historical daily time series are used as input data and
subsequently, historical monthly and annual time series are created by aggregating daily data
inside the system. Furthermore, Castalia reproduces the LTP considering it as a special
instance of a parametrically defined generalized autocovariance function [53].
The generation of the synthetic time series is performed in three timeaggregation
levels. Initially, a theoretical autocovariance function is defined for each annual variable,
which can include LTP. The dependence structure is reproduced through the Symmetric
Moving Average (SMA) model, whose parameters are estimated using the annual marginal
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and joint second order statistics and the autocovariance function [53]. The SMA model is
used for the generation of the synthetic annual time series. Following this, auxiliary monthly
time series are generated using the Periodic Autoregressive (PAR(1)) model, whose
parameters are estimated using the monthly marginal and joint second order statistics. A
disaggregation process (called a coupling transformation) modifies the auxiliary monthly time
series in order to become consistent with the annual ones [54]. The arising deviations in the
statistics that must be preserved, which may be caused by the disaggregation process, are
overcome through a repetitive MonteCarlo process that accompanies the disaggregation
process. By this, a statistically independent sequence of monthly variables is found which
approximates the annual value. The synthetic daily time series are then generated in a similar
manner by disaggregation of monthly values into daily.
4.2. Preservation of probability of zero values
Sunshine duration can take on a zero value with nonzero probability and thus the preservation
of the historical probability of zero values is important. This is similar to rainfall, which is an
intermittent process and the Castalia program, which was originally developed to simulate
rainfall, can handle the intermittent behaviour on the daily time scale [55]. This can be
achieved by appropriate choice of the parameters that are used for this aim into the system
after the generation of daily auxiliary time series. These include:
a) The parameters λ1 and λ2 that are used to adjust the probabilities k1 and k2 of a Markov
chain model. These probabilities are adjusted by the parameters λ1 and λ2, respectively,
of the historical probability of zero values of each variable. Specifically, k1 expresses
the probability of a zero value occurring in the current time step if there is also a zero
value in the previous time step while k2 expresses the probability of a zero value
occurring in the current time step if there is a nonzero value in the previous time step.
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b) A parameter k3 which expresses the probability of the values of all variables in the
current time step being zero if one of them is zero.
c) The parameters of a roundoff rule and specifically a threshold l0 and a percentage π0,
where the rule implies that a percentage π0 of values below this threshold are
converted into zero values.
Note that all three procedures described above generate zero values and thus they all
contribute to the final frequency of zero values.
4.3. Results
4.3.1. First application – Untransformed variables
In the first application of the program 1000year synthetic time series are generated in 8
measuring stations having both wind speed and sunshine duration records (16 variables in
total). The length of the simulation period (1000 years) was determined as explained in
Appendix C. Although the domain of the sunshine duration is bounded from above and below
and thus, the Gamma distribution is not appropriate to represent this variable, data are initially
imported into the system without any transformation to test the program performance in a fast
application without preprocessing of data. For this reason possible deviations concerning the
skewness of the sunshine duration variable are expected.
The model parameters are calculated automatically by the program except for those
controlling the preservation of the probability of zero values, which are determined by a trial
and error procedure. For the latter, the following values were finally adopted π0 = 0.9 for
l0=0.4; λ1 = 0.3, λ2 = 0.1; k3= 0.
Application results are presented in Figures 710 for one of the stations (Eelde). It is
observed that the statistical characteristics, the Hurst coefficient as well as the cross
correlation coefficient of both processes are preserved on annual and monthly scale. On daily
scale wind speed statistics and crosscorrelation coefficient are also satisfactorily preserved.
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However, the skewness of the sunshine duration variable is overestimated. This divergence is
caused, as expected, by the fact that the domain of the sunshine duration variable differs from
the domain of the Gamma distribution, used for the generation of the time series. As a
consequence, several synthetic sunshine duration values, being higher than the upper bound of
the variable domain, are generated due to the fact that the domain of Gamma distribution is
the [0,+∞). These values are obviously unrealistic and lead to skewness overestimation.
4.3.2. Second application – transformed sunshine duration
In order to overcome the overestimation of the skewness which was encountered in the first
application, a second application is performed using the previous wind speed data and the
proposed logarithmic transformation of sunshine duration data in each of the 8 stations. After
the generation of the synthetic time series, the Y daily time series are manually subject to the
reverse transformation so as the synthetic daily time series of the sunshine duration to arise.
Then these time series are compared with the historical ones.
The model parameters controlling the preservation of the probability of zero values are
also determined by a trial and error procedure. For the latter, the following values were finally
adopted π0 = 0.9 for l0 =0; λ1 = 0.3, λ2 = 0.1; k3 = 0.
Both the Hurst coefficient and the marginal statistical characteristics of the wind speed
process are preserved satisfactorily on all time scales as in the first application. Application
results for the sunshine duration are presented in Figures 1114 for the same station as before
(Eelde). The Hurst coefficient and the statistical characteristics on annual and monthly scale
are also generally preserved. The only deviations observed are related to monthly skewness,
which is underestimated, and monthly autocorrelation, which is overestimated. However,
these features are not of great importance as the design focuses on the daily scale. It is
remarkable that the crosscorrelation of the variables is preserved even though the logarithmic
transformation is used in the simulation. On daily time scale, skewness is now satisfactorily
17
reproduced and the only existing deviation refers to autocorrelation which is slightly
overestimated. The crosscorrelation of the variables, which is high as the time scale of the
design is daily, as well as the probability of zero values are also preserved. The deviations
mentioned above on monthly and daily time scales are caused due to the fact that the iterative
Monte Carlo process and the adjustingdisaggregation procedures are performed inside the
program, so they are applied to the logarithmic transformation of the variable instead of the
actual one.
5. Conclusions
Simulation of wind speed and sunshine duration gains interest because these processes are
associated with renewable energy production. Knowing the marginal distribution as well as
the stochastic structure of these processes, including LTP, is important as the solar and wind
energy strongly depend on them.
The results derived from the analysis using data from Europe indicate that the Gamma
and the Weibull distributions are suitable for representing wind speed. For the relative
sunshine duration a theoretical distribution function, which is a transformation of the Gamma
distribution, is derived. This is practically indistinguishable from the Beta distribution and
they both fit satisfactorily to relative sunshine duration data.
Both processes are found to be characterized by strong selfsimilar behaviour with
values of the Hurst coefficient, H, (estimated using the LSSD method) markedly higher than
the value H = 0.5, corresponding to white noise, for the majority of the time series. After
implementing a Monte Carlo approach, a unique value H = 0.84 is found to be representative
for both variables. However, it should be noted that, due to the small amount of samples,
further investigation is required. Also, as the observations have started in their majority, all
over the world, after 1940, the record lengths are not quite sufficient to support a safe
estimation of the Hurst coefficient.
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The detailed empirical analysis of the stochastic properties allows stochastic
simulation of the two processes which is particularly useful for the rational design of a
renewable energy system comprising wind and solar energy. Specifically, synthetic time
series of wind speed and sunshine duration data are simultaneously generated on annual,
monthly and daily scale using the software system Castalia.
Castalia performs this stochastic simulation satisfactorily as the marginal and joint
second order statistics as well as the LTP of the historical data retrieved from measuring
stations with common observations are well reproduced. An overestimation of the skewness
of daily sunshine duration which arises when raw data are used is attributed to the bounded
domain of the variable and it can be overcome by using the proposed logarithmic
transformation. Some deviations which arise in the monthly skewness and autocorrelation
when the transformed variable is used are not of great importance due to the fact that the time
scale of the design is daily while the overestimation of the daily autocorrelation is slight.
Thus, considering that Castalia was initially developed for the simulation of rainfall, runoff
and evaporation, it is also confirmed that this program can conduct stochastic simulations of a
wide spectrum of hydrometeorological variables on three important time scales, annual,
monthly and daily.
Acknowledgements: We thank two anonymous reviewers for the constructive review
comments on an earlier version of this manuscript as well as Andreas Efstratiadis, Stefanos
Kozanis, Yannis Dialynas and Panagiotis Dimitriadis for their help during the preparation of
the study.
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25
Appendix A: Assessment of Reanalysis data
Figure A.1 compares stationbased and reanalysis data for the location of the De Bild station
at an annual basis. It can be seen that the two series do not correlate well. In addition the
reanalysis data series has annual average values higher than the stationbased ones even
though the former refer to lower altitude than the latter.
Figure A.1: Mean annual wind speed time series from stationbased (20m above ground) and
Reanalysis data (10m above ground) at De Bilt station.
Appendix B: Proof of equation (1)
Let X, Y, where 0< X <1 be random variables so that:
(B.1)
(B.2)
The event {Χ ≤ x} is identical to the event {Υ ≤ –1(x)}. As a result, the distribution functions
of Υ and Χ are linked through the equation:
(B.3)
The variables are continuous, and the function g is differentiable with derivative equal to:
(B.4)
Τhe probability density function of the variable X is a function of the density of the variable Y
[56]:
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
1940 1960 1980 2000 2020
Average annual wind
speed (m/s)
Year
Reanalysis
Station based
26
=
=
]
(B.5)
As the variable Y for the continuous part of the distribution (Y > 0) is represented by the
Gamma distribution function with shape parameter κ > 0 and scale parameter λ > 0, the
probability density function of the variable X is:
(B.6)
or
=
(B.7)
and finally
=
(B.8)
Appendix C: Determination of the length of simulation period
In the estimation, through stochastic simulation, of a probability p, from a sample with
relatively large size N of independent identically distributed random variables, it is known
that the length of the confidence interval of the estimate is [31,57]:
2 z(1+γ)/2 [p (1– p) / N]0.5 = 2 c p
(1)
where γ is the confidence coefficient, za is the aquantile of the standard normal distribution,
and c is the acceptable relative error. Solving for N, the minimum sample size Nmin that is
required for estimating the probability p is:
Nmin= (z2(1 + γ) / 2 / c2) (1/p – 1)
(2)
For a daily step of design, a confidence coefficient γ = 0.95, an acceptable relative error c =
10% and probability of failure p = 1‰, the required sample size (number of days of
simulation) needed to obtain an accurate estimate of p on a daily basis is Nmin ≈ 384000
(days), which is rounded off to a simulation period of 1000 years. It is noted though that, as
the hydrometeorological processes are not independent in time, the required period should be
greater than 1000 years.
27
Tables
Station
Country
Longitude(ο)
Latitude (ο)
Measurement height
above ground (m)
Altitude
(m)
Source
Period of
record
Vlissingen
Netherlands
3.60 E
51.45N
15
8
ΚΝΜΙ
19092011
Regenburg
Germany
12.10 E
49.00 N
15
377
ECA&D
19462011
Potsdam
Germany
13.10 E
52.40 N
37.7
81
ECA&D
19112011
Maastricht
Netherlands
5.78 E
50.92 N
10
114
ΚΝΜΙ
19062011
Karlsruhe
Germany
8.40 E
49.00 N
47.4
113
ECA&D
19452011
Hohenpeissenberg
Germany
11.01 E
47.48 N
15
980
ECA&D
19392011
Hof
Germany
11.90 E
50.30 N
12
624
ECA&D
19462011
Giessen
Wettenberg
Germany
8.38 E
50.36 N
10
203
ECA&D
19392011
Eelde
Netherlands
6.58 E
53.13 N
10
3.5
ΚΝΜΙ
19042011
Dresden
Germany
13.80 E
51.10 N
10
220
ΚΝΜΙ
19412011
De Kooy
Netherlands
4.78 E
52.92 N
10
0.5
ΚΝΜΙ
19082011
De Bilt
Netherlands
5.18 E
52.10 N
20
1.9
ΚΝΜΙ
19042011
Tarifa
Spain
5.35 W
36.00 N
10
32
ECA&D
19452012
Bjoernoeya
Norway
19.0 E
74.50 N
10
16
ECA&D
19202011
San Sebastian
Spain
2.54 W
43.18 N
10
251
ECA&D
19332011
Dublin Airport
Ireland
6.30 W
53.42 N

85
MET
EIREANN
19442010
Valentia
Ireland
10.24 W
51.94 N

9
MET
EIREANN
19402010
Fokstua
Norway
9.17 E
62.07 N
10
975
ECA&D
19232011
Karasjok
Norway
25.52 E
69.47 N
10
133
ECA&D
19392011
Sula
Norway
8.45 E
63.85 N
10
6
ECA&D
19382011
Table 1: Stations with wind speed measurements
28
Station
Country
Longitude(ο)
Latitude (ο)
Altitude (m)
Source
Period of record
Alicante
Spain
0.60 W
38.30 N
82
ECA&D
19392011
Vlissingen
Netherlands
3.60 E
51.45N
8
KNMI
19072011
Lindenberg
Germany
14.10 E
52.20 N
112
ECA&D
19072011
Potsdam
Germany
13.10 E
52.40 N
81
ECA&D
19012011
Maastricht
Netherlands
5.78 E
50.92 N
114
KNMI
19062011
Karlsruhe
Germany
8.40 E
49.00 N
113
ECA&D
19362011
Hohenpeissenberg
Germany
11.01 E
47.48 N
980
ECA&D
19372011
Zugspitze
Germany
10.98 E
47.42 N
2960
ECA&D
19012011
Zuerichfluntern
Switzerland
8.57 E
47.38 N
556
ECA&D
19012011
Eelde
Netherlands
6.58 E
53.13 N
3.5
KNMI
19062011
Geneve
Switzerland
6.10 E
46.20 N
238
ECA&D
19012011
De Kooy
Netherlands
4.78 E
52.92 N
0.5
KNMI
19092011
De Bilt
Netherlands
5.18 E
52.10 N
1.9
KNMI
19012011
Granada
Spain
3.59 W
37.17 N
680
ECA&D
19422010
Aachen
Germany
6.10 E
50.80 N
264
ECA&D
19352010
Basel
Switzerland
7.61 E
47.56 N
265
ECA&D
19012011
Lugano
Switzerland
8.97 E
46.00 N
276
ECA&D
19012011
Madrid
Spain
3.65 W
40.40 N
687
ECA&D
19202011
Saentis
Switzerland
9.40 E
47.20 N
2500
ECA&D
19012011
Valencia
Spain
0.38 W
39.48 N
35
ECA&D
19382011
Zagreb
Croatia
16.03 E
45.82 N
123
ECA&D
19002011
Table 2: Stations with sunshine duration measurements
29
Figures
Figure 1: Annual averages of wind speed time series at De Bilt station before and after
modification; three different periods were identified: 19041960 (37 m above ground), 1961
1992 (10 m above ground), 19932011 (20 m above ground).
Figure 2: Frequency histograms and probability density functions of historical daily wind
speed time series at Eelde station in June.
2.5
3
3.5
4
4.5
5
5.5
6
1900 1920 1940 1960 1980 2000 2020
Average annual
wind speed (m/s)
Year
Raw time
series
Modified time
series
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.25
0.75
1.25
1.75
2.25
2.75
3.25
3.75
4.25
4.75
5.25
5.75
6.25
6.75
7.25
7.75
8.25
8.75
9.25
9.75
10.25
Probability density
Wind speed (m/s)
Wind speed
frequency
histograms
Gamma
distribution
Weibull
distribution
30
Figure 3: Frequency histograms and probability density functions of the logarithmic
transformation of historical daily sunshine duration time series (above) and the relative
sunshine duration time series (below) at Eelde station in June.
0
0.25
0.5
0.75
1
1.25
1.5
0.125
0.375
0.625
0.875
1.125
1.375
1.625
1.875
2.125
2.375
2.625
2.875
3.125
3.375
3.625
Probability density
Transformed relative sunshine duration
Transformed
sunshine duration
frequency
histograms
Gamma
distribution
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
Probability density
Relative sunshine duration
Relative sunshine
duration
frequency
histograms
Theoretical
distribution (1)
Beta distribution
31
Figure 4: Annual and 10year averages of sunshine duration time series at Vlissingen station
(above) and a series of white noise with the same mean and standard deviation (below).
Figure 5: Climacograms of wind speed (Valentia, G.Wettenberg) and sunshine duration
(Vlissingen, De Bilt) time series.
2.5
3
3.5
4
4.5
5
5.5
6
1900 1920 1940 1960 1980 2000 2020
Sunshine duration (h)
Year
Average
annual time
series
10 year
average
2.5
3
3.5
4
4.5
5
5.5
6
1900 1920 1940 1960 1980 2000 2020
Year
Average
annual time
series
10 year
average
0.5
0.4
0.3
0.2
0.1
0
0.1 00.2 0.4 0.6 0.8 1
log (σ(k) /σ)
log (k)
G.Wettenberg
(Η=0.515)
Valentia
(Η=0.915)
Vlissingen
(Η=0.895)
De Bilt (Η=0.667)
32
Figure 6: Comparison of Hurst coefficient frequency histograms of historical and synthetic,
with a theoretical H = 0.84, time series for wind speed (above) and sunshine duration (below).
Figure 7: Comparison of Hurst coefficient of historical and synthetic time series at the Eelde
station; first application.
0%
5%
10%
15%
20%
25%
30%
0.50.6 0.60.7 0.70.8 0.80.9 0.91
Percentage of stations
Hurst coefficient
Historical
time series
Synthetic
time series
[3AR(1)]
0
0.1
0.2
0.3
0.4
0.5
0.5  0.6 0.6  0.7 0.7  0.8 0.8  0.9 0.9 1
Percentage of stations
Hurst coefficient
Historical
time series
Synthetic
time series
[3AR(1)]
0
0.2
0.4
0.6
0.8
1
Wind speed Sunshine duration
Hurst coefficient
Historical
Synthetic
33
Figure 8: Comparison of annual, monthly and daily standard deviations and skewness
coefficients of historical and synthetic wind speed and sunshine duration time series at the
Eelde station; first application.
0
0.5
1
1.5
2
2.5
3
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Standard deviations
(m/s)
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0
1
2
3
4
5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Standard deviations
(h)
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Skewness coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0.5
0
0.5
1
1.5
2
2.5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Skewness coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
34
Figure 9: Comparison of annual, monthly and daily auto and crosscorrelation coefficient of
historical and synthetic wind speed time series at the Eelde station; first application.
Figure 10: Comparison of the probability of zero sunshine duration of the historical and
synthetic time series at the Eelde station; first application.
0.05
0.05
0.15
0.25
0.35
0.45
0.55
0.65
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Autocorrelation
coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0
0.1
0.2
0.3
0.4
0.5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Autocorrelation
coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.05
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Crosscorrelation
factor
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0
0.1
0.2
0.3
0.4
0.5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Probability of zero
values
Historical (daily)
Synthetic (daily)
35
Figure 11: Comparison of Hurst coefficient of historical and synthetic time series at the Eelde
station; second application.
Figure 12: Comparison of annual, monthly and daily standard deviations and skewness
coefficients of historical and synthetic sunshine duration time series at the Eelde station;
second application.
0
0.2
0.4
0.6
0.8
1
Wind speed Sunshine duration
Hurst coefficient
Historical
Synthetic
0
1
2
3
4
5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Standard deviations
(h)
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0.5
0.1
0.3
0.7
1.1
1.5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Skewness
coefficioent
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
36
Figure 13: Comparison of annual, monthly and daily auto and crosscorrelation coefficient of
historical and synthetic sunshine duration time series at the Eelde station; second application.
Figure 14: Comparison of the probability of zero sunshine duration of the historical and
synthetic time series at the Eelde station; second application.
0
0.1
0.2
0.3
0.4
0.5
0.6
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Autocorrelation
coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0.35
0.25
0.15
0.05
0.05
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Crosscorrelation
coefficient
Historical (monthly)
Synthetic (monthly)
Historical (daily)
Synthetic (daily)
0
0.1
0.2
0.3
0.4
0.5
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Year
Probability of zero
values
Historical (daily)
Synthetic (daily)