A Fokker–Planck equation based approach for modelling wind speed and its

Energy Conversion and Management 222 (2020) 113152
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A Fokker–Planck equation based approach for modelling wind speed and its
power output
T
J. Pablo Arenas-López, Mohamed Badaoui
⁎
Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Av. Luis Enrique Erro s/n, 07738 Ciudad de México, Mexico
ARTICLE INFO
ABSTRACT
Keywords:
Wind speed
Power output
Probability density function
Stochastic differential equation
Fokker–Planck equation
The main purpose of this article is to estimate the wind power generation in Juchitan de Zaragoza-Mexico based
on stochastic analysis of the average wind speed. This estimation is carried out by simulating numerically a
stochastic differential equation (SDE) for a set of randomly generated trajectories of the Wiener process and a
particular probability density function (PDF). Due to the fact that wind power is formulated as an explicit
function of wind speed, then a good selection of the appropriate PDF can significantly reduce the wind power
estimation error. Moreover, and in order to get more insights into the statistical behavior of the wind power
output, we propose eight different PDFs to evaluate its suitability to characterize wind speed and wind power as
well. The challenging step is to find the drift and volatility of the SDE, which in this work are determined in
closed form via the solution of the stationary Fokker–Planck equation. To assess the practical value and competitiveness of the proposed approach, computational complexity, stability analysis and different measures of
accuracy have been addressed. Finally, the results show that among the eight PDFs evaluated in this particular
study, the three-parameter Beta distribution has the best performance in estimating wind resources in the shortterm, while for long-term closed results are obtained between the three-parameter generalized Gamma, threeparameter Beta and Log-Pearson 3 distributions.
1. Introduction
Over the past few years, many countries around the world have
recognized that the increase in demand for energy has reached a very
high level, which requires the consideration of new sources of energy to
achieve sustainability and the reliability of the power grid. Global
warming has been a red flag that has motivated governments to enforce
new policies that aim to create consciousness about the danger that may
be created if electricity is still generated by means of fossil fuel.
Motivated by the foregoing, it is worth noting that many countries
around the world have focused their attention on energy produced
through renewable energy sources such as wind energy, which has been
given great incentives to achieve a high percentage of clean energy
produced by 2030. However, the variability and natural uncertainty of
renewable energy sources has become a serious challenge because of its
volatility and the lack of accurate forecasting. This challenge has attracted the attention of both academia and system operator, because
deterministic models are not suitable to capture the uncertainty of renewable sources due to intermittence, which can compromise the security and reliability of the power grid. This reason thus, some studies
have focused on the implementation of probabilistic methodologies and
statistic analysis to characterize renewable sources and estimate its
power output. In Mexico, and due to climatic and geographical conditions wind energy is one of the most important sources of clean energy,
which can be appreciated by many statistical studies of wind characteristics that have been carried out from data obtained by meteorological stations. An analysis of wind power in Mexico using data from
133 weather stations throughout the country shows that Mexico has
great wind potential in much of the country [1]. The first wind farm in
Mexico was built in 1994 in La Venta, Oaxaca. The La Venta wind farm,
originally a prototype project, has seven 225 kW wind turbines and was
the first of its kind in Latin America [2]. Mexico had recorded 929 MW
of new installed capacity in 2018, reaching a total of almost 5 GW and
is expected to reach its goal of generating 35 percent of its energy
through renewable sources by the year 2024 [3]. Moreover, the recent
deregulation of the Mexican electricity market encouraged the appearance of clean energy certificates as one of the instruments of the
electricity market aimed at promoting investment in this type of energy.
However, the scientific research and development of new methodologies for the characterization of renewable sources and the production of
⁎
Corresponding author. Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Av. Luis Enrique Erro s/n, 07738 Ciudad de México,
Mexico.
E-mail address: mbadaoui@ipn.mx (M. Badaoui).
https://doi.org/10.1016/j.enconman.2020.113152
Received 22 April 2020; Received in revised form 24 June 2020; Accepted 25 June 2020
0196-8904/ © 2020 Elsevier Ltd. All rights reserved.
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
energy obtained through them, had already started since the year 1969
[4] which can be demonstrated by the large amount of published research on wind energy by Mexican institutions that has grown significantly since then. Many published works have focused on the statistical analysis of the characteristics of wind data collected in several
regions of Mexico and most of them, only consider the Weibull distribution to fit the real wind speeds. Some of these works are summarized below. In Ref. [5] the characteristics of wind in Baja California
Sur, Mexico, are analyzed for a period of one year. This location is
considered representative of 15 wind farms that were installed. The
production of wind energy and the factor of wind turbines were estimated at 25%. The data of meteorological stations collected during two
years in the peninsula of Baja California are analyzed, [6]. Besides, the
power and energy production were determined and the useful hours of
each station as well. It is concluded that the north and south of the
peninsula have the highest intensity of wind and wind energy density,
and it is in this area where the highest concentration of urban population is found. Ref. [7] highlights the great potential of wind power in
the northern states of Mexico. The spatial and temporal information on
the wind resource in northern Mexico is studied, obtaining that the
daily pattern of wind speed behaves similarly in most of the cases
studied. The states of Chihuahua, Coahuila, Nuevo Leon and Tamaulipas exhibit a wind speed of more than 4.51 m/s in almost all territories. Also, along the coast of the state of Tamaulipas, an assessment of
wind energy potential was made, [8]. The results show that the wind
potential along the Tamaulipas is lower than that suggested by the official prevailing eolic-potential map in Mexico, and concludes that although the wind is a promising source of renewable energy along the
coast of Tamaulipas, for a better estimation of the wind potential is
necessary to have current high quality data measured in different locations and altitudes. In Ref. [9] the interpretation of the wind resource
is presented through a statistical analysis of wind speed distribution and
wind direction in the San Luis Potosí City, Mexico. In Ref. [10] a preliminary study is reported on forty meteorological stations located in
the state of Michoacán and islands of the Pacific Ocean with the objective of quantifying the wind resource in the state. In this study, it is
indicated that the state of Michoacan has moderate wind resources in
some places. On the other hand, Weibull bimodal probability distributions have been proposed to describe wind speeds within the
Mexican territory [11,12]. In recent years, the number of published
studies on a hybrid forecast models has increased due to their effectiveness in achieving more accurate wind power forecasts, in particular,
forecast models based on machine learning theories. In addition to the
works that have focused on the case of Mexico, it is important to
mention research that has pointed out other wind regimes in different
geographies, among them we list [13] in which the long-term wind
energy forecast is based on daily wind speed data using five machine
learning algorithms and their performance is shown by several case
studies. In Ref. [14], a novel model based on hybrid mode decomposition method and a robust and online sequential outlier robust extreme learning machine for short-term wind speed prediction was
proposed. The results of the experiment show that the hybrid mode
decomposition method is an effective form of wind speed decomposition, which can accurately capture the characteristics of the wind speed
time series, which improves the prediction performance, also, that
online sequential outlier robust extreme learning machine performs
better than offline models in practical forecasts. There are a variety of
probability density functions that have been reported in the scientific
literature to describe the frequency distributions of wind speed in
various regions of the world [15–17]. The Weibull PDF has been recognized as a suitable model [18,19] to estimate the wind power output
of a specific wind turbine. The influence of the degree of adjustment of
a PDF on the wind speed data to estimate the average power output of
the wind energy conversion system is analyzed in Ref. [20], for a set of
PDFs. In Ref. [21] the potential of wind energy has been estimated from
wind speed data recorded in two meteorological stations. Moreover, the
results were used to estimate the net energy production of seven
1.5 MW wind turbines, taking into account the correction of air density
and power losses in the wind farm. On the other hand, SDEs have been a
very active area of research for building new models for wind speed. An
Ornstein–Uhlenbeck Geometric Brownian Motion model in continuous
time along a partial differential equation is proposed in Ref. [22] to
model wind speeds, and the resulting wind power output statistics are
also illustrated to estimate the annual production of wind energy.
This research builds mainly on the approach published in Refs.
[22,23], in which a new technique for wind speed modelling based on
the Fokker–Planck equation is proposed, then following this technique,
we simulate trajectories of wind speeds for different PDFs, but unlike
the aforementioned research published we focus on showing the importance of selecting the wind speed PDF that best estimates the energy
production of a specific wind turbine at a particular location of Mexico.
For this purpose, the eight PDFs were assessed statistically and their
impact on the wind speed modelling and semiannual wind power
output have been reported. In addition, to show the effectiveness of the
numerical scheme and the computational burden involved, important
ingredients are reported, which are computational complexity and
stability analysis along the with sensitivity of wind speeds to the initial
condition.
This paper is organized as follows: Section 2 presents the analysis of
the real wind speed data; Section 3 describes the mathematical model
for the generation of wind speed trajectories; Section 4 describes the
numerical scheme, computational complexity, stability analysis and the
results of the simulations along the measures of accuracy; Section 5
presents the power curve model of the wind turbine under consideration, the results of the statistical analyses carried out and the semiannual energy production for each PDF; Section 6 provides the results
of applying the proposed methodology to a one-year wind speed dataset
generated by the MCP method; finally Section 7 gives the conclusions of
this work.
2. Data analysis
In this section, we analyze the wind speed data measured in
Juchitan de Zaragoza, Oaxaca. The data was provided by the independent system operator and consists of the average hourly wind
speeds recorded from March to August 2017, i.e., 4416 values. In order
to provide some details about the measurement campaign, the wind
turbine model installed in this area has wind resource measurement
systems and are installed at a height of 80 m [24,25]. In Fig. 1, the
location where the data was recorded is shown by a red circle. It is
worth mentioning that the meteorological phenomena that occur in this
region are the mountain wind, which causes wind speeds due to the
pressure gradient between mountains, and the effect of the sea breeze
blowing from the Gulf of Tehuantepec. In Table 1 we present the main
descriptive statistical properties that include: minimum speed (min),
maximum speed (max), average speed, standard deviation. To provide
more details on asymmetry, thickness or heaviness of the data distribution, additional characterizations such as skewness and kurtosis
are provided. Table 2 shows PDFs used in this work for wind speed
modelling, it is worth noting that almost all the selected density functions have been previously reported in the scientific literature as adequate to represent some wind regimes around the world. Table 3 provides the parameters of the best fitting PDF, while Table 4 shows the
results and ranking obtained by Kolmogorov–Smirnov, Anderson–Darling and Chi-square goodness-of-fit tests as reported by EasyFit Software. According to Table 4, the three-parameter Beta is the distribution
that best describes the behavior of the data in the three tests, while the
Weibull distribution, commonly selected as the conventional option,
appears as the fourth-best fit by Chi-square test and fifth-best fit by
Kolmogorov–Smirnov and Anderson–Darling tests. Moreover, for the
parameters displayed in Table 3, Fig. 2 shows the eight fitted PDFs from
Table 2. Additionally, a set of wind speed data generated by the MCP
2
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 1. Location where wind speeds were measured.
method (see appendix A) is shown in Fig. 2 as well. The reference data
used to generate the MCP dataset was obtained from the ERA5 reanalysis data [26,27] (coordinates: N 16.5°, W 94.75°; height: 100 m).
To estimate the wind speed at a height of 80 m, we use the following
power law:
v1
h
= 1
v2
h2
3. The mathematical model
This section is dedicated to provide the construction of the mathematical model based on SDE to simulate wind speeds under uncertainty.
A general one dimension stochastic differential equation in the time
interval [0, T ] has the following form:
(2)
dXt = (Xt , t ) dt + (Xt , t ) dWt
(1)
where the real functions (Xt , t ) and (Xt , t ) are the drift and diffusion
(volatility) terms respectively. Wt is a standard Wiener process [29,30].
Moreover, since stochastic processes can be characterized by their PDF,
then, in this case, it is well known in the theory of stochastic processes
that the temporal evolution of the PDF of a stochastic process is described by the Fokker–Planck equation (or Kolmogorov Forward
equation), i.e., a stochastic process Xt modeled by the SDE (2) with PDF
f (x , t ) (also known as the transient PDF) that is observing the position x
at time t, its associated Fokker–Planck equation in one spatial dimension x has the following form:
where v1 and v2 denote the wind speeds at height h1 and h2 , respectively.
The power law exponent is assumed to be 1/6, a value in the range
reported for this zone [28]. The geographical location of this ERA5
dataset is 10 km from the place where the wind speed data was recorded. The dataset generated by the MCP method is introduced for
purposes of comparison with the real data as well as the assessment of
the long-term behavior of the wind resources. For this purpose, in
Section 6 we present an analysis using the approach proposed in this
paper for one-year dataset generated by the MCP method for this same
site.
Table 5 summarizes the closed expressions of mean and variance of
the distributions considered in the fitting process, while Table 6 shows
the closed expressions of skewness and kurtosis. It is worth noting that
all the expressions depend on the values of the fitting parameters.
Table 7 presents the theoretical values of the mean, variance,
asymmetry and kurtosis of the distributions considered in the fitting
process calculated using the expressions presented in Tables 5 and 6.
f (x , t )
=
t
x
[ (x , t ) f (x , t )] +
1 2
[
2 x2
2 (x ,
t ) f (x , t )]
(3)
From (3) we observe that the Fokker–Planck equation is a partial
differential equation that models phenomena that evolve on time described by their PDFs f (t , x ) , which in turn depend on the coefficients
of the SDE. In order to build a stationary process with a desired PDF and
an autocorrelation with exponential decay, we consider only the
Table 1
Descriptive statistics of the set of wind speeds.
Min [m/s]
Max [m/s]
Mean [m/s]
Standard Deviation [m/s]
Skewness
Kurtosis
0.07289
16.8052
6.455026
3.692978
0.400772
−0.656867
3
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Table 2
Probability distributions considered in the fitting process.
Distribution
2 (x )
PDF
Three-parameter
Beta (B)
fB (x ) =
1
3 B ( 1, 2 )
() ( )
1 1
x
x
3
3
2 1
if x > 0
3
0
Two-parameter
Gamma (G)
fG (x ) =
2
if x
x
2
1
x 1 1e
1 ( 1)
if x > 0
0
Three-parameter
generalized
Gamma (GG)
Log-Pearson 3
(LP3)
if x
1
3
2 ( 1)
fGG (x ) =
( )
x
e
2
fLP3 (x ) =
log (x )
2
3
(
log (x )
2
3
0
)
1 1
fN (x ) =
2 11
( 1) 21
x2 1
1 x2
2
1e
0
Two-parameter
Pert (P)
fP (x ) =
if x
41
4
x 2 ( 2 x)
4
1
5
+ 1, 5
2B
2
41
2
0
One-parameter
Rayleigh (R)
fR (x ) =
x2
2 2
x
2e
0
Two-parameter
Weibull (W)
fW (x ) =
1
2
if x
x
2 (x )
1 1
e
2
0
.
0
if x > 0 .
d
1
[ (x ) f (x )] +
[
dx
2 dx 2
2 (x ) f
f (x ) =
x
(x )] = 0.
0
d [Xt ]
=
dt
(4)
(9)
Xs
(10)
(11)
µ) e
(12)
t,
µ as t
as consequence [Xt ]
, which is the called mean reverting
property, i.e., tends to return to µ over time. The stationary autocovariance Cov (s, t ) is defined as:
(5)
Cov (s, t ) =
[(Xs
µ)(Xt
µ)].
Cov (s, t ) , then Cov ( ) saand if be the lag time t s and Cov ( )
tisfies the following evolution equation similar to (11):
dCov ( )
=
d
Cov ( ), with Cov (0) =
2,
(13)
where
is the variance of the stationary stochastic process Xt . Solving
(13) leads to a closed expression for the autocovariance:
2
Cov ( ) =
d (x )
1 (x ) df (x )
+
,
dx
2 f (x ) dx
(Xs ) dWs
0
[Xt ] + µ ,
[Xt ] = µ + ( [X 0]
2 (y )
dy
2 (y )
t
µ ds +
which is an ordinary differential whose solution is given by:
where the constant C is obtained from the density function property of
f (x ) . The representation (5) is known as the Wright’s equation [31],
which is a peculiar formula that combines the main ingredients of the
stochastic process Xt : drift, volatility and the corresponding PDF. Now
assuming that the PDF f (x ) is known and that (x ) = 0 if f (x ) = 0 , then
straightforward calculation based in the integration of Eq. (4) leads to
the following relationship between the drift and volatility, which can be
solved for (x ) and (x ) as follows:
(x ) = (x )
0
where X0 denotes the initial condition. Since the Itô stochastic integral
has expected value equal to zero, then taking the expectation operator
and derivating both sides of (10), we get the evolution equation:
Solving (4), we get a closed expression of f (x ) :
C
exp
2 (x )
µ f (y ) dy if f (x )
if f (x ) = 0.
t
Xt = X0
0
stationary case of Eq. (3), i.e., , and f do not depend on time t. Then,
the stationary PDF f (x ) is the solution of the stationary Fokker–Planck
equation:
d2
y
It is worth mentioning that the drift representation given by (8) is a
sufficient condition for obtaining stochastic processes with exponential
decay autocorrelation. A motivation of this fact can be found in the
theory of stochastic processes, since the regression theorem states that
for Markov processes in which the mean values obey equations of linear
evolution [32]. First we observe that Eq. (8) the SDE (2) belongs to the
class of mean reverting processes, which can be written under the Itô
interpretation as follows:
1
if x
x
0
0
x
2
(8)
µ),
2
f (x )
=
.
if x > 0 .
if x
( )
0
0
if x > 0
41
2
(7)
if f (x ) = 0.
(x
if x > 0 .
if x
if x > 0
0
where µ is the mean of the desired probability distribution f (x ) , and
is the autocorrelation coefficient. Inserting Eq. (8) in (7) yields this
closed expression for 2 (x ) :
if x > 0 .
0
Nakagami (N)
0
(x ) =
3
if x
1
e
x | 2 | ( 1)
.
0
.
0
(y ) f (y ) dy if f (x )
Therefore, for an arbitrary PDF f (x ) , and if one of the functions (x )
or (x ) is known, the other function can be obtained by solving (6) or
(7), respectively. On the other hand, the approach adopted in this work
considers only mean reversion Ornstein–Uhlenbeck process with nonlinear volatility, i.e., the drift term is defined throughout this work by:
0
x
2
1 3 1
x
2
f (x )
=
2e
(14)
.
As a consequence of the above, the autocorrelation Cor ( ) becomes:
(6)
Cor ( ) = e
(15)
.
Table 3
Parameters of fit.
Parameter
1
2
3
B
G
GG
LP3
N
P
R
W
1.5292
2.5983
17.324
3.0552
2.1128
0.36888
11.279
3.483
2.9953
−0.44167
2.9628
1.0048
55.302
4.4566
20.57
5.1504
1.6686
7.2792
4
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Table 4
Tests statistics.
Distribution
Kolmogorov–Smirnov
Rank
Anderson–Darling
Rank
Chi-squared
Rank
B
G
GG
LP3
N
P
R
W
0.01119
0.05442
0.01778
0.01818
0.06198
0.02338
0.05501
0.03165
1
6
2
3
8
4
7
5
1.1043
51.177
3.7832
2.4717
40.949
6.5733
39.178
8.1585
1
8
3
2
7
4
6
5
14.935
299.05
37.9015
21.6995
240.665
53.7315
246.815
51.4295
1
8
3
2
6
5
7
4
Remark 1. From Eqs. (8), (14) and (15), it is worth noting that the
stochastic process build by the technique described above has the
property of exponential decay for both the autocovariance and
autocorrelation, which is an important property when it comes to
wind speed modelling on an hour scale in certain geographic areas. For
more details about the derivation of expressions (8) and 14, we refer to
Refs. [23,32].
Table 5
Means and variances of the distributions considered in the fitting process.
Distribution
Mean
B
µB =
1 3
1+ 2
G
µG =
1 2
GG
In Table 8 we summarize the closed form expressions of 2 (x ) under
different PDFs, it is important to mention that these expressions were
validated by the Mathematica software package and for the sake of
simplicity some 2 are represented in terms of the so-called incomplete
gamma function (u , v ) and generalized incomplete gamma function
(u , v , w ) = ( u , w )
(u, v ) . Concerning the algorithm developed in
this work, Fig. 3 depicts a flowchart that describes the whole process
from data processing, wind speed simulation, wind power output to
error analysis. This process is described in two main stages: The first
consists in the simulation of wind speed trajectories, the analysis of
statistical characteristics and the error involved during the simulation;
the second estimates the power output of a particular wind turbine
which is determined from the simulated wind speed trajectories and a
wind turbine power curve build from the manufacturer real data, finally
the analysis of statistical characteristics and error is carried out.
Variance
2
µGG =
LP3
2) 1
2
1
1+
N
µN =
µP =
R
µR =
W
µW =
3
e 3
(1
1
2
4 1+ 2
6
2
2
G
=
2
GG
1 2 32
( 1 + 2 )2 ( 1 + 2 + 1)
2
1 2
2
(1 + )
1
1
=
2
N
=
2
P
=
2
R
=
2
W
2
2
=
2
LP3
( 1)
1
=
1
( 1)
µLP3 =
P
1+
2
B
( 1)
(1
2
1
1+
1
2 2) 1
2
2
2
1+
1
(1
2
2) 1
1
2
( 1)
2
1+
1
1
(4 1 + 2 )(5 2
252
)
2 (4
1
2
=
2
3
2( )
1
1
3
e2 3
4 1)
(1 + )
2
1
2
(1 + )
1
1
we implement an implicit integration scheme that belongs to the family
of implicit Milstein schemes [33], which is used to approximate the SDE
(2). The first step consists in discretizing the time interval [0,T] as
T
< tN with N = is the number of simulated
follows: 0 = t 0 < t1 <
steps, tn + 1 = tn + for 0 n N and is the step size. Since different
trajectories are simulated to estimate the wind speed statistical
4. Wind speed simulations
To generate wind speed trajectories we solve numerically Eq. (2),
Fig. 2. Wind speed frequency distribution and probability density functions obtained in the fitting process.
5
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Table 6
Skewness and kurtosis of the distributions considered in the fitting process.
Distribution
Skewness (S ) & Kurtosis (K )
B
SB =
:
KB =
G
:
2( 2
1) 1 + 2 + 1
( 1 + 2 + 2) 1 2
.
6( 13 + 12 (1 2 2) 2 1 2 ( 2 + 2) + 22 ( 2 + 1))
1 2 ( 1 + 2 + 2)( 1 + 2 + 3)
SG =
2
KG =
6
1
.
1
GG
1
2 3 1+
3
SGG =
3 ( 1)
( 1)
2
:
1+
1
+ 2 ( 1)
( 1)
1+
KGG =
2
LP3
SLP3 =
e3 3 (2(1
:
3(1
KLP3 =
N
1
:
KN =
P
Sp =
:
R
(5 2
KP =
14
5 1
SR =
2
:
KR =
1+
(
6 2
(4
1+
SW =
1
KW =
1+
1+
3
(1
3
1
3
2
2
1+
3
3
2)
1 + (1
1
4 2)
3
.
4 1 ) ( 1 ) (2 1 )
1
1
1
2
2 ( + 1)
1
2
1+
)+
6
5
.
1) 2 (2 1 )
.
.
1+
1+
:
.
4
3
( 1)
3
5
4 1+ 2
41
24 + 16
)2
3
3
1
6 4 1+
+ 23 4 1 (4 1
2
2
1
2
2
1+
1 ( 1)
2
1
5 2
3)
)3/2
(4
W
1
7 ( 2 2 1)
4 1 )(4 1 + 2 )
(
3
3/2
1
2
3 2 1+
1
( 1) 2 1 +
2
2 1 ) 4 ( 1)
1 (1
1+
3
1+
1
6 2 1+
3
3
3
+ 2 ( 1)
3
3 1 3(1 2 ) 1 (1
2)
2
2) 1 + (1 3 2) 1 )
2 1 ))3/2
(e 2 3 ((1 2 2) 1 (1
2)
4 1 + 6(1 2 ) 1 (1
2 1 4(1 3 ) 1 (1
2)
2
2)
2
2 1 )2
((1 2 2) 1 (1
2)
3( )
1
1
2
4
1
2 3 1+
+4 1
2
SN =
2
3
1+
1
1+
3
2
1+
12 ( 1 )
3
2
1+
2
2
1
1
2 1+ 1
1
1
1
6 4 1+
+ 12
1
1+
1
1+
2
1
1
+2 3 1+
1
3/2
2 1+ 1
1
4
1+
2 1+ 1
1
Mean [m/s]
Standard Deviation [m/s]
Skewness
Kurtosis
B
G
GG
LP3
N
P
R
W
6.4184
6.455
6.4254
6.4696
6.594
6.3994
6.4551
6.5035
3.6947
3.693
3.6589
3.7912
3.4381
3.5993
3.3742
4.0051
0.39641
1.1442
0.38014
0.53539
0.62914
0.53978
0.63111
0.89421
−0.63917
1.9639
−0.46919
−0.39658
0.2428
−0.27819
0.24509
0.85086
1+
1
1+
2
2
1
1
2
3 2 1+
+
1
1+
4
.
1
1
Wni = W i (tn) W i (tn 1) be the sequence of independent increments of
the Wiener process such that Wni ~N (0, ) . The discretized version of
the SDE (2) produce the following recursive:
Table 7
Theoretical statistical values.
Distribution
3
Xni = Xni
+
X0i
= µ,
1
1
2
+ 2 ( (Xni )) + (Xni 1)) + (Xni 1) Wni
(Xni 1) (Xni 1)(( Wni )
2
)
(16)
where the initial condition X0i depends on the mean value of the PDF
p (x ) as shown in Table 5. The coefficients (Xni ) and (Xni ) are explicitly
shown in Tables 5 and 8 respectively for each probability distribution.
The following pseudo-code describes the main 8 steps of the algorithm
from the parameters setting until the statistical measures of the generated trajectories:
properties, then an upper subscript i is added to the discretized process
Xtin +1 , that is, the position of the stochastic process at time n of the ith
trajectory, where 1 i N and N is the total number of simulated
trajectories. For the ease of notation let Xtin +1 : =Xni + 1, and let
6
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Algorithm 1. Simulation of trajectories
>
4.1. Wind speeds trajectories and statistical measures
• Computational complexity:
The computational complexity is very important in computation
theory because it allows the determination of the time needed to run a
certain mathematical model. Since nowadays real life problems have a
huge amount of variables and require an excessive time of data processing to generate a set of solutions, the estimation of running time has
become increasingly important in engineering processes or those that
require decisions to be taken online within minutes or even seconds. In
this regard, the attempt to reduce the burden of the number of simulations on the computational complexity was achieved by developing a
Fortran-90 code to run the simulation of N = 10 4 trajectories for each
PDF. From Table 11 we could appreciate how the running time strongly
depends on the PDF, moreover all the times still within reasonable
limits. It is worth mentioning that the simulation running time was
estimated using a system with the following characteristics: Intel Core
i7-6700HQ CPU @ 2.60 GHz, 16.0 GB RAM, x64 based processor.
• Stability:
This section is dedicated to addressing the issue of the long term
behavior of the implicit Milstein scheme (16). In this respect, the stability theory is considered as the framework developed to give more
insights about the accuracy of the numerical scheme as t
and to
maintain the propagated errors bounded. More explicitly, the stability
theory is concerned about the choices of the step size that reproduces
the stability property of a given numerical scheme. It is important to
mention that many definitions of numerical stability have been introduced, but only for particular test equations [33–35]. Following the
same approach stated in [36], the numerical scheme (16) meets the
conditions of stability distribution in mean square if the following
properties are satisfied:
In this section, a comparative statistical analysis is made between
the simulations and the real wind speed data presented in Section 2.
Fig. 4 shows the autocorrelation of the wind speed dataset for time lags
of up to 84 h and although a periodicity due to the daily cycle of wind
speed can be observed, an exponential fit is obtained according to expression (15). Moreover, in this case, the exponential fit parameter
which is an important ingredient of the wind speed modelling as stated
by Eq. (15) is given by = 0.0257 , and it can be concluded that the
proposed model can be adopted to describe the wind speed in this
particular place of Mexico.
Fig. 5 shows the normalized histogram and autocorrelation of the
real wind speed data as well as the normalized histograms and the
average of the autocorrelations, while Table 9 presents the average of
the statistical characteristics of the 10 4 trajectories of wind speed simulations generated for each probability distribution considered in the
fitting process. To facilitate the presentation we summarize the results
of Tables 1 and 9 in a single figure to illustrate the differences between
the statistical characteristics of the real data and the wind speed simulations as shown in Fig. 6. The average statistical characteristics with
the smallest difference between the simulations and the real data are
given for four of the six characteristics considered (maximum, standard
deviation, skewness and kurtosis) by the three-parameter Beta distribution, while for the three-parameter generalized Gamma and oneparameter Rayleigh distributions the smallest difference was obtained
for the minimum and mean speed respectively. It is expected that the
data set generated by the MCP method will better reproduce the
minimum value, the mean and the standard deviation, due to the formulation of the MCP algorithm (see Appendix A).
In order to get more insight and for ease of reading, we display
MAE, RMSE and R2 values between the real data and the simulations as
shown in Table 10 and Fig. 7. The three-parameter Beta distribution
shows the best performance, while the Weibull distribution is ranked as
the fifth-best fit, according to the results of these evaluation criteria. It
should be noted that the data set generated by the MCP method reaches
the mid-point of the ranking, according to the three evaluation criteria.
i. supn 0 Xnx 2 <
for all n 0 and x
.
Xnx Xny 2 = 0 uniformly for (x , y )
ii. limn
compact set.
K
2
where K is a
Since from an analytical point of view it is not possible to formulate
a stability condition for the numerical scheme (16), because (x ) depends on each PDF as shown in Table 8. Thus, the properties i and ii are
corroborated numerically for each PDF, it is worthy of mention that this
stability analysis is carried out only for the initials conditions x and y
which are respectively the theoretical mean and the real data mean
shown in Tables 5 and 1. From Fig. 8 we could appreciate that the
sensitivity of the second moment is very small, which is an important
property when it comes to choose adequately the initial condition, that
4.2. Computational complexity and stability
This section is dedicated to provide some details on the running
time and the stability of the numerical scheme for each PDF under the
corresponding fitting parameters.
7
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Table 8
The Volatility coefficients.
Dist.
Volatility
B
2 (x )
=
G
2 (x )
=2
GG
LP3
2 ( 3 x)x
1+ 2
2x
2
3
2 (x )
=2 x
2 (x )
= 2 | 2| (1
N
P
( )
2
1
2 (x )
=
2 (x )
=
2 (x )
=
2 (x )
= 2
R
1
3
x
2
1
x
1
1 3
x
2
2
1
1
e
2x +
2
2x
1
2
( )
2
x
1
1
(
x)
41
+1
2
1
e 2 1 erfc
(
2
x
x2
2
1e
+
(
3 1+ 1
2x
2
1e 3
1 x2
2 x1 2 1
41
2 ( 2
W
e
2)
( )
3
x
2
x
2
1
1
3
( )
,
log(x )
x
3
2
)
1
)
1
(
1,
1+
1
2
(1
2)(log(x )
2
3)
,
log(x )
2
3
)
1 2
1, x
2
( 1)
41
2
( )
x
2 1
1
1
1+
3
1 2
x
2
1,
3
( 1)
2
1
2
)
1
2
+ ,
1
1+
3
x
1
1
,
( )
x
2
1
( )
1
1
Fig. 4. Autocorrelation of the hourly mean wind speed data and its exponential
fit.
5. Wind power output
In this section, the estimation of the power output of a wind turbine
is based on the simulated wind speeds obtained in the previous section
and the power curve of a wind turbine installed at the same site where
the wind speed data was recorded is estimated by the spline method
using a set of empirical measurements provided by the manufacturer as
described in the following section.
5.1. Wind power curve model
Fig. 3. Diagram of the proposed methodology.
The Acciona Wind Power AW70/1500 class I with a unit power of
1.5 MW is among the wind turbine installed in this location of Mexico.
The manufacturer provides the power curve at a density of 1.180 kg /m3
for the wind turbine model mentioned above. This turbine has a threeblade rotor, active nacelle windward yaw control, with blade pitch
change and variable rotor rotation speed Table 12. Besides the physical
characteristics of the wind turbine, another important ingredient is the
is, the simulated trajectories are very proximate for x and y as initial
conditions in each case. In Fig. 9 it becomes clear that the second
moment is bounded, moreover, it approaches µ2 + 2 as time evolves.
8
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 5. Normalized histogram and autocorrelation of real wind speed data and normalized histograms and average autocorrelations of the simulations.
Table 9
Average descriptive statistics of wind speed simulations.
Distribution
Min [m/s]
Max [m/s]
Mean [m/s]
Standard Deviation [m/s]
Skewness
Kurtosis
MCP
B
G
GG
LP3
N
P
R
W
0.07289
0.110303
0.358240
0.087157
0.145483
0.204258
0.148720
0.197459
0.126489
17.085386
16.525916
23.814224
18.030927
17.936184
19.631555
17.936608
19.256636
23.232353
6.455148
6.411695
6.449004
6.419172
6.462705
6.588238
6.392991
6.449378
6.496780
3.692758
3.651965
3.634885
3.615556
3.745309
3.392254
3.554577
3.329197
3.948806
0.459377
0.392986
1.066654
0.368643
0.528972
0.602731
0.529956
0.604640
0.852387
−0.4646418
−0.620744
1.521374
−0.477380
−0.387931
0.157515
−0.283162
0.159506
0.674446
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Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 6. Differences between the characteristics of the real wind speed data and the average characteristics of the simulated wind speed trajectories.
Table 10
MAE, RMSE and R2 of the normalized histograms of the real data and the simulations.
Distribution
MAE
RMSE
R2
MCP
B
G
GG
LP3
N
P
R
W
0.002403135
0.001477
0.004915427
0.0019063
0.001798118
0.004096345
0.002126495
0.004196217
0.002629609
0.004831736
0.003268266
0.010055312
0.004419069
0.003681935
0.00812216
0.004589661
0.008275252
0.004858317
0.934799966
0.970168418
0.717620957
0.94546153
0.962138859
0.815759756
0.941169488
0.808748948
0.934080635
Table 11
Running time.
Tiempo [min]
B
G
GG
LP3
N
P
R
W
0.783
1.333
7.516
4.55
7.216
1.833
2.45
4.3
power curve, which is responsible for the estimation of power output.
Although several mathematical models have been proposed to represent the power curve of a wind turbine [37–41]. A regression of
polynomial splines based on the cubic spline is the one used in this
work, moreover, the cubic spline interpolation method which consists
in fitting different cubic polynomial between each pair of data points
has shown to be more appropriate to estimate power output from the
Fig. 7. Comparison between simulations with different probability distributions in terms of the MAE, RMSE and R2 .
10
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 8. Sensitivity of the second moment.
the power curve of the wind turbine is expressed as:
M
(ai v 3 + bi v 2 + ci v + di )
P (v ) =
[vi 1, vi] (v )
(17)
i=1
where M is the number of the data points and
function defined as:
A (x )
=
1 if x
0 if x
A
is the indicator
A
A
Fig. 10 shows a set of black dotted corresponding to the wind turbine power curve along with its fitting curve given by a blue line.
The power output trajectory of the wind turbine is obtained considering the real wind speed data. The statistical characteristics of this
are presented in Table 13.
For each wind speed trajectory, the power output trajectory of the
wind turbine is obtained, then the mean, standard deviation, skewness
and kurtosis of each power trajectory is calculated. Finally, the average
vector of mean, standard deviations, skewness and kurtosis is calculated
Fig. 9. Second moment behavior.
manufacturer power curve [42,43]. Therefore the fitting equations for
Table 12
Technical characteristics of studied wind turbine.
Description
Manufacture
Rated output [kW]
Diameter [m]
Aerodynamic regulation
Acciona AW 70/1500 Cl I
Acciona
1500
70
Pitch control
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Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 10. Power curve AW70/1500 wind turbines.
for the 10000 power trajectories. The results of the power output obtained from the simulations using the wind turbine power curve model
are shown in Table 14. The differences between the characteristics of
the power output considering the real data and the average of the
output power characteristics considering the simulated wind speeds for
each probability distribution are shown in Fig. 11.
From Tables 13 and 14, we observe that the average statistical
characteristics power output with the smallest difference are given for
three of the four characteristics considered (mean, skewness and kurtosis) by the three-parameter Beta distribution, while with the LogPearson 3 distribution the smallest difference was obtained for the
standard deviation. On the other hand, Commonly an important parameter for estimating the power of a wind farm is the mean power. In
this case, the entire wind farm from which the data were obtained has a
total of 167 wind turbines, so an incorrect selection of the probability
distribution for wind modelling could give a greater underestimate of
the mean power of the wind farm. Normalizing the output power results
obtained through simulations and the real wind speed data, Table 15
shows the MAE, RMSE and R2 results between the wind turbine power
output obtained through the simulated data set and the real data set. It
is noted that the best result for these evaluation criteria is given by the
three-parameter Beta distribution. Once again, the data set generated
by the MCP method is at the mid-point of the ranking, according to the
three evaluation criteria. Besides, an overview of the results obtained
from the comparison and evaluation procedures, it becomes evident the
superiority of the three-parameter Beta distribution for wind speed
modelling, which is achieved by the approach proposed for the case
study presented in this work. Further, the poor performance shown by
the Weibull distribution could be due to the fact that the measurement
period is too short as reported in different investigations [40,44] (see
Fig. 12).
Table 14
Average descriptive statistics of power output simulations.
Standard Deviation [kW]
Skewness
Kurtosis
423.1422611
510.1765921
1.026974274
−0.409119974
Mean [kW]
Standard
Deviation [kW]
Skewness
Kurtosis
MCP
B
G
GG
LP3
N
P
R
W
415.207694
418.5595497
373.7887806
415.5896845
418.2905908
403.7261188
399.3916595
385.1272548
405.670835
501.518108
500.4996098
474.3920511
488.0661657
506.7991143
471.4237966
486.1584772
460.4637413
503.4067387
1.06029097
1.046083454
1.300519395
1.050358594
1.065034937
1.146151348
1.141410545
1.21867966
1.138636547
−0.28580117
−0.255209262
0.464880221
−0.188737864
−0.235557291
0.106679865
0.017072759
0.328607261
−0.055894063
5.2. Semiannual energy
This section provides an evaluation of the performance of each
probability distribution on the semiannual energy production by comparing the energy produced considering the real wind speed data set
and the average energy produced considering the wind speed simulations with the different probability distributions. It is clear that the
three-parameter Beta distribution shows superiority when it comes the
semiannual energy production, because its percentage error with respect to the real data semiannual energy production is the smallest
among the eight probability distribution considered is this evaluation.
On the other hand, it could be appreciated from Table 16 that all the
probability distributions as well as the MCP method lead to an underestimation of the semiannual energy production, which is very small in
almost all the cases. This is generally due to differences in the distribution of the real data set and the probability densities, i.e., some
speeds in the real data set (approximately between 9 and 14 m/s, see
Fig. 5) shows a higher frequency than those obtained by the simulation
process, and although the simulations reach higher speeds than the real
data set as a result of the fits, the energy calculated for the simulations
does not exceed the energy calculated for the real wind speed data set.
It should be mentioned, that since in this paper we consider only a
single wind turbine the wake effect was neglected in the stochastic
modelling, moreover, the wake effect is of utmost importance when
modelling wind turbine array or an entire wind farm [45].
Table 13
Descriptive statistics of power output considering real wind speed data.
Mean [kW]
Distribution
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Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 11. Differences between the characteristics of the power output considering the real wind speed data and the simulated wind speed trajectories.
Table 15
MAE, RMSE and R2 of the normalized histograms of power output considering
real wind speed data and simulations.
Table 16
Comparison of semiannual energy produced considering real data and simulations.
Distribution
MAE
RMSE
R2
Distribution
Semiannual Energy [kWh]
%
MCP
B
G
GG
LP3
N
P
R
W
0.00437802
0.003404031
0.006827566
0.004198397
0.00378997
0.007922569
0.003980151
0.007746742
0.00489699
0.005099887
0.004293557
0.008822355
0.00570716
0.005192474
0.010939231
0.005194509
0.010513847
0.006930335
0.99726136
0.998058898
0.991804355
0.996570316
0.997161017
0.987399502
0.997158791
0.988360417
0.994942655
Real
MCP
B
G
GG
LP 3
N
P
R
W
1868595.525
1833557.176
1848358.971
1650651.255
1835244.047
1847171.249
1782854.541
1763713.569
1700721.957
1791442.407
−1.875117
−1.082982
−11.663534
−1.784842
−1.146544
−4.588526
−5.612876
−8.983944
−4.128936
Fig. 12. MAE, RMSE and R2 of power output considering real wind speed data and simulations.
13
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Fig. 13. Comparison between simulations with different probability distributions in terms of the MAE, RMSE and R2 .
Fig. 14. MAE, RMSE and R2 of power output considering wind speed MCP data and simulations.
i.e., the Weibull distribution, which does not exceed the fourth-best fit
in the goodness-of-fit test ranking, for the one, three and five years data
sets generated by the MCP method (more details on the ranking of the
probability distributions considered for these data sets are reported in
Appendix B). The reason why the Weibull distribution is not performing
well for the dataset generated by the MCP algorithm could be attributed
to the high frequency of low wind speeds.
From the above, we can conclude that it is not sufficient to evaluate
the distribution functions with goodness-of-fit tests, although these tests
are convenient to identify the appropriate distributions before performing more detailed analyses. Table 17 provides an evaluation of the
performance of each probability distribution on the annual energy
production, comparing the energy produced considering the data set
generated through the MCP method and the average energy produced
considering the wind speed simulations with different probability distributions. The Log-Pearson 3, three-parameter generalized Gamma,
and three-parameter Beta distributions present the smallest error in the
annual energy estimate. Finally, it should be noted that the Log-Pearson
3 distribution presents a slight overestimation of energy, while the
three-parameter generalized Gamma and three-parameter Beta distributions present an underestimation.
Table 17
Comparison of annual energy produced considering MCP data and simulations.
Distribution
Annual Energy [kWh]
%
MCP
B
G
GG
LP3
N
P
R
W
5428219.13
5258805.23
4764267.86
5283208.12
5439246.52
5232341.56
5135470.59
5035693.47
5044172.75
−3.12%
−12.23%
−2.67%
0.20%
−3.61%
−5.39%
−7.23%
−7.08%
6. Long-term evaluation of the wind resource
In order to carry out an annual assessment of the wind resource at
this particular site of Mexico and due to the lack of a broader data set,
the MCP method is proposed to complete the historical series of wind
speeds of one year. The reference data for this purpose were those
described in Section 2 recorded in the period from September 2016 to
August 2017, moreover, in the appendix B we present the ranking of the
probability distributions according to the goodness-of-fit tests. From the
application of the proposed methodology to this new data set generated
by the MCP algorithm, the results of MAE, RMSE and R2 evaluation
criteria are displayed in Fig. 13 for both the data set generated by the
MCP method and the simulations, while Fig. 14 displays the results of
the power output evaluation criteria. The results of the evaluation
criteria show that, the best performance in wind speed modelling is
obtained by the three-parameter generalized Gamma distribution,
while wind speed described by the Log-Pearson 3 distribution produces
the best performance in power output modelling. The results of the
goodness-of-fit tests and the evaluation criteria considered contrast
with the common choice for wind resource assessment and modelling,
7. Conclusions
In this work, the power output of a wind turbine is estimated considering a set of probability distributions obtained from a goodness-offit to a set of wind speeds collected every hour during six months in a
location of Mexico. A series of wind speed trajectories are simulated
through a model based on stochastic differential equations and the
stationary representation of the Fokker–Planck equation, which allows
the construction of a stationary stochastic process with a desired
probability distribution and exponential decay of the autocorrelation,
characteristics that have been observed in the real wind speed data.
Employing the cubic spline method, the power output of a wind turbine
14
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
installed in the same place where the data were collected is obtained,
first, we consider the set of real wind speed data and then the simulated
wind speeds. From the simulations, some statistical tests were carried
out to observe how the power output of a wind turbine is affected by
modeling the wind speed with different probability distributions. The
methodology proposed in this paper was also applied to an annual set of
wind speed data from the site of interest generated through the application of the MCP method. For both case studies, the results of the MAE,
RMSE and R2 criteria, show a superiority of the three-parameter Beta,
three-parameter generalized Gamma and Log-Pearson 3 distributions
over the Weibull distribution, commonly selected as the conventional
option.
curation. Mohamed
Supervision.
CRediT authorship contribution statement
The authors would like to thank the National Center of Energy
Control (CENACE) for having provided the data for wind speeds used in
this article.
Badaoui:
Conceptualization,
Methodology,
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
J. Pablo Arenas-López: Methodology, Software, Investigation, Data
Appendix A. Preliminary mathematical tools
This section is aimed to introduce some basic standard mathematical tools such as:
• The parameter estimation method which consists in finding the values of the parameters that maximize the probability (likelihood function) of
obtaining the actual observations.
• The goodness of fit test that measures the compatibility of the observed probability distribution with the theoretical probability distribution.
A.1. Maximum likelihood estimation
Suppose that X1 , …, Xn are independent and identically distributed random variables with PDF f (x; ) , where
The likelihood function is defined as
is a single unknown parameter.
n
L
;x1, …, x n =
f xi;
.
(A.1)
i=1
The maximum likelihood estimates are obtained by maximizing the likelihood for , i.e., the value of
solving the following equation
L
that maximizes L ( ;X ) or equivalently by
;X = 0.
(A.2)
If the necessary condition for the maximum is fulfilled, then the solution
of (A.2) is the maximum likelihood estimator.
A.2. Goodness of fit test
The goodness of fit test is the procedure by which we measure the compatibility of the empirical distribution function F (x ) of the data with the
hypothetical cumulative distribution function F (x ) (CDF). Indeed, the goodness of fit is performed in order to test the following hypothesis:
H0 : F (x ) = F (x )
x
.
.
H1: F (x )
F (x ) foratleastonevalueof x
• Kolmogorov–Smirnov
Let Fn be the CDF based on a sample size n. The Kolmogorov–Smirnov statistic denoted by Dn is defined as follows [46]:
Dn = sup x F (x )
Fn (x )
(A.3)
which calculates the maximum difference in absolute value between the aforementioned cumulative distributions functions.
• Anderson–Darling
The Anderson–Darling test is used to verify if a sample of data came from a population with a specific distribution. It is a modification of the
Kolmogorov–Smirnov test and gives more attention to the tails. The Anderson Darling test makes use of the specific distribution in the calculation
of critical values. The advantage is that this sharpens the test, the disadvantage is that the critical values must be calculated for each hypothetical
distribution. The Anderson–Darling statistic is defined as [47]:
A2 =
n
1
2
n
2i
1 (logF (x i ) + log(1
F (x n
i + 1))).
(A.4)
i=1
• Chi-Squared
In the Chi-square test, the range of the random variable is divided in k intervals, and the data consists of the number of observations found within
each interval. The test statistic is given by [46]:
15
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
k
2
npi ) 2
npi
(oi
=
i=1
(A.5)
where oi denotes the number of data points that fall within the i-th interval, npi denotes the expected number of data points that should fall within
the i-th interval, k is the number of intervals, n is the number of data points, and pi is the probability of interval i occurring and can be calculated
as follows:
pi = F^ (x i )
F^ (x i 1)
where x i and x i
1
(A.6)
are the lower limit value and the upper limit value in the interval i, respectively.
On the other hand, for the tests introduced above, choosing a significance level of 0.05 is typically used for most applications for which the
associated critical value is established according to the significance level and the sample size for each goodness of fit test. More explicitly, if the test
statistic is greater than or equal to the critical value, the null hypothesis H0 (the data follow the specified distribution) is rejected, otherwise, if the
test statistic is less than the critical value, we conclude that there is not enough statistical variation within the data set to reject the null hypothesis.
A.3. Evaluation criteria
To assess the performance of the stochastic model under different probability distributions, three common statistical metrics are used: the mean
absolute error (MAE), the root mean square error (RMSE) and R-squared (R2 ). Their mathematical definitions are shown below:
• Mean absolute error
MAE =
1
N
N
| y (i )
z (i )|,
(A.7)
i=1
• Root mean square error
RMSE =
• R-squared
R2 = 1
1
N
N
z (i)) 2 ,
(y (i )
(A.8)
i=1
N
(y (i )
i=1
N
(y (i)
i=1
z (i)) 2
y¯)2
,
(A.9)
where y (i) is the calculated value of the PDF obtained through the real data in the i-th interval, z (i) is the calculated value of the PDF obtained
through the simulation in the interval i-th and N is the number of intervals. Although, in general, smaller values indicate the smallest difference
between simulations and real data, the MAE is a better reflection of accuracy when the importance of outliers in the evaluation is limited, while the
RMSE summarizes the errors at square, and is significantly affected by large error values or outliers [48]; however, smaller measures still indicate
better performance. Unlike MAE and RMSE, the R-squared measure has values in the interval [0, 1] where higher values represent better performance.
A.4. The Measure-Correlate-Predict (MCP) method
The MCP method is used to predict the wind resource at a site of interest by taking a set of long-term measurement data from nearby locations
(weather stations, a neighboring airport, reanalysis data, etc.). This is done by establishing the statistical relationship of the concurrent data period
between the short-term wind data and the long-term wind reference data, and then applying the relationship to the long-term data period. A variety
of MCP methods have been proposed in the literature [49], which differ in terms of general approach, model definition, use of leadership sectors, and
so on. In this paper, the variance method was used to determine a linear relationship between hourly wind speed averages for the site of interest and
the reference site. The linear model for which the predicted values are expected to have the same mean and overall variance as the observed values is
[50]:
y =
y
x
x + µy
y
x
µx
(A.10)
where y is the estimated value of the site of interest, x is the wind speed at the reference site and µ y , µx and y, x are the mean and standard
deviations of the two concurrent data sets. In this case, the values of estimated negative wind speeds were considered as the minimum value of the
data of the site of interest.
Appendix B. Ranking of goodness-of-fit tests for one, three and five years of data generated by the MCP method
See Table B.1.
16
Energy Conversion and Management 222 (2020) 113152
J.P. Arenas-López and M. Badaoui
Table B.1
Ranking of goodness-of-fit tests for one, three and five years of data generated by the MCP method, according to the results of the EasyFit Software.
Years
1
Sept. 16-Aug. 17
3
Mar. 16-Feb. 19
5
Jan. 15-Dec. 19
Tests
K-S
A-D
C-s
K-S
A-D
C-s
K-S
A-D
C-s
B
G
GG
LP3
N
P
R
W
3
8
2
1
7
4
6
5
2
7
1
8
6
3
5
4
2
7
1
N/A
6
3
4
5
3
8
2
1
7
4
6
5
2
7
1
8
6
3
5
4
2
7
1
N/A
6
3
5
4
3
7
1
2
8
4
6
5
2
8
1
7
6
3
5
4
2
7
1
N/A
6
3
5
4
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