Hindawi Publishing Corporation
ISRN Renewable Energy
Volume 2013, Article ID 657437, 9 pages
http://dx.doi.org/10.1155/2013/657437
Research Article
Wind Speed Estimation: Incorporating Seasonal Data Using
Markov Chain Models
Selin Karatepe1 and Kenneth W. Corscadden2
1
2
Department of Business Administration, Faculty of Economics and Administrative Sciences, Yalova University, 77100 Yalova, Turkey
Engineering Department, Faculty of Agriculture, Dalhousie University, Truro, NS, Canada B2N 5E3
Correspondence should be addressed to Selin Karatepe; [email protected]
Received 3 October 2013; Accepted 27 October 2013
Academic Editors: M. Benghanem, P. D. Lund, and S. Rehman
Copyright © 2013 S. Karatepe and K. W. Corscadden. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
This paper presents a novel approach for accurately modeling and ultimately predicting wind speed for selected sites when
incomplete data sets are available. The application of a seasonal simulation for the synthetic generation of wind speed data is
achieved using the Markov chain Monte Carlo technique with only one month of data from each season. This limited data model
was used to produce synthesized data that sufficiently captured the seasonal variations of wind characteristics. The model was
validated by comparing wind characteristics obtained from time series wind tower data from two countries with Markov chain
Monte Carlo simulations, demonstrating that one month of wind speed data from each season was sufficient to generate synthetic
wind speed data for the related season.
1. Introduction
One of the most challenging features of wind energy application is the uncertainty of the wind resource. Wind speed
and hence wind energy potential has a key influence on
the profitability of a wind farm and on the management of
transmission and distribution networks by utility systems
operators. Accurate and reliable data related to both longterm and short-term wind characteristics are essential for site
selection and technology specification [1]. Wind data acquisition is typically required for one to two years and is achieved
through the use of anemometers and 60 m meteorological
towers. Because of the stochastic nature of the wind resource,
long-term wind speed recordings are required to characterize
wind patterns, but in some cases wind data may be corrupted
or missing for short periods, making accurate and reliable
prediction of wind characteristics difficult. Operational wind
speed prediction is also a challenge with historic data playing
a major role in the prediction of future wind data, both of
which require long-term reliable and complete historic data.
The topic of wind speed prediction has been addressed by a
number of researchers with approaches typically based on the
analysis of historical time series wind data [2] or model based
using a range of meteorological and/or local topographical
input data. Lei et al. [3] provide a comprehensive overview of
wind speed forecasting and power prediction methods using
numerical and statistical approaches and conclude that different models have different strengths based on inputs and applications such as short-term or long-term prediction. Wind
speed data is typically analyzed to characterize the potential
annual resource, presented, for example, as a Weibull Density
Function [4] for the estimation of wind energy potential.
Foley et al. [5] describe wind speed forecasting as physical
based, utilizing statistical methods, learning based, using artificial intelligence, or hybrid, utilizing a combination of both
methods. Researchers have applied various combinations of
physical-based methods and further differentiated between
either temporal and/or spatial wind characteristics. Watts and
Jara [6] use statistical analysis for the prediction of wind
energy in Chile, concluding that wind is a potential contributor to the energy mix of this country. Cadenas and Rievra
[7] and Fyrippis et al. [8] have performed similar analyses to
evaluate wind potential in Mexico and Greece, respectively.
Costa et al. [9] present a review of alternative techniques
for short-term prediction of wind power. Ramirez-Rosado
et al. [10] compared two different methods of short-term
2
forecasting, producing daily wind predictions, and predicted
power loads and Landberg [11] uses a model-based approach
using meteorological data to provide short-term prediction,
of up to 36 hours, for wind power production. Researchers
have also used variants of physical-based systems, such as
Louka et al. [12] who use Kalman filtering in an attempt to
improve wind speed and wind power forecasting. Calif and
Schmitt [13] introduce a lognormal stochastic equation to
evaluate wind spectrum and identify a scaling equation for
evaluating the impact of wind turbulence on wind power
production. Several researchers have employed CALMET
[14, 15], a diagnostic wind model, to predict wind speed.
Learning-based techniques predominantly employ
advanced analysis using neural- or fuzzy-based approaches
or hybrid models. Guo et al. [16] present a hybrid wind speed
prediction system which combines historical daily wind
speed averages with artificial neural networks to forecast
daily average wind speeds for a one-month period, with
additional novel approaches presented by Liu et al. [17] and
Han et al. [18].
An emerging research area is the use of Markov chain
Monte Carlo (MCMC) simulation techniques to evaluate and
model wind speed characteristics. Several researchers used
first-order MC models. Jones and Lorenz [19] take 8 hour
averages of UK wind speed data recorded for one year with
an 11-states MC model. They concluded that the probability
density functions (PDFs) and the Autocorrelation Functions
(ACFs) were accurate but a second-order MC model could fit
better. Sahin and Sen [20] used hourly averages of wind speed
data recorded in 10 different stations in the northwestern
region of Turkey with an 8-states MC model. They concluded
that second and third autocorrelation coefficients are significant and that the MC model can sufficiently preserve most
of the statistical parameters of wind characteristics. Nfaoui
et al. [21] used hourly averages of wind speed data recorded
in Tangiers to form the model with 12 states. They also concluded that most of the statistical parameters of wind speed
time series are reproduced, and they propose that the lack of
similarity in ACFs is due to the nature of the Markov process.
Some researchers have included wind direction in an
attempt to improve the MC model. Ettoumi et al. [22] used
3 hour averages of wind speed and wind direction data
recorded in Algeria with a 9-states and 3-states MC model for
wind direction and wind speed, respectively, stating that both
models fit well. They suggest that a combination of Weibull
distribution with MC model would provide a better representation of wind data. Karatepe [23] used hourly averages
of wind speed data recorded in Turkey to directly generate
the economical value of wind electricity using a 16-states MC
model. The author states that these synthetic data could be
used as an input for long-term economical analysis of wind
farm projects as well as site assessments, when long-term
wind recordings are not available. This research indicates
that the MC model is a suitable method of reproducing the
general statistical characteristics of measured wind data, yet
the state size and order for optimal characterization still
require further investigation. Hocaoğlu et al. [24] used hourly
averages of wind speed data recorded in Turkey to show the
effect of state size on synthetic generation and concluded that
ISRN Renewable Energy
there is a direct correlation between the number of states and
model accuracy.
Kaminsky et al. [25] determined that real wind data
contain more low-frequency components than those which
are produced using first- and second-order MC models.
Further improvements are required to preserve the long
wave information even though a second-order MC model is
more representative of the real data in terms of ACFs and
PSDs. Shamshad et al. [26] expand to 12-states MC models
preserving the statistical characteristics but at the expense of
the ACFs, concluding that this is due to the intrinsic nature
of the MC models. The general shape of the spectral density
function is similar but both models failed to retain peculiar
peaks even though the second-order MC model performed
better. The authors suggest that removing periodic/seasonal
components before analysis may improve the results.
The following researchers used first-, second-, and thirdorder MC models for both wind speed and wind power
data. Papaefthymiou and Klöckl [27] used wind speed data
recorded for 2 years at several time steps. Applying models
based on 10 min, 30 min, and 60 min averages of wind speed
and wind power data resulted in a reduction of state size
without loss of information. Their research determined that
10 min averages and a third-order MC model provide a better
fit in terms of ACFs. Brokish and Kirtley [28] suggest that MC
models should not be used for time steps shorter than 15 to
40 minutes depending on the order of the MC and the state
size. They used different time steps of wind speed/power data
and different state sizes to determine when MC models are
appropriate for synthetic wind speed/power data generation.
Wu et al. [29] used the field-measured wind power data
recorded once every minute for 9 months to examine the
effect of different numbers of states and seasonal periodicity.
Seasonal periodicity was examined by using the data for a
specific month to form the transition matrix, which was then
utilized to simulate the synthetic data for the month. They
concluded that, when the number of states in the transition
matrix is properly selected, the MCMC method is an effective
technique to generate wind power time series, with relative
improvements in accuracy obtained by the incorporation of
seasonality in the simulation process.
Erto et al. [30] proposed a wind speed parameter estimation method to reduce the requirements of long-term on-site
anemometric monitoring by exploiting initial information
about parameters to be estimated via MCMC. The results of
the simulation showed that the proposed method, based on
one-month data and a priori information, could successfully
compete with the maximum likelihood estimates obtained
from one-year of measured data. Pang et al. [31] focused on
the Bayesian estimation of Weibull parameters using MCMC
techniques. The authors stated that the advantage of using
MCMC is that the posterior uncertainty about parameters
may be evaluated exactly using the MCMC sample without
any need for asymptotic approximations. They claim that
the MCMC method provides an alternative method for
parameter estimation of wind speed distributions.
Finally, Negra et al. [32] developed a synthetic wind speed
generator based on MC concepts. Instead of constructing a
transition probability matrix, they constructed a wind speed
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probability table from extrapolated wind speed statistical
data which was based on 7 years of wind measurements.
From this table, a set of synthetic time series was generated
and compared with observed time series in terms of ACFs,
averages, PDFs, and seasonal characteristics. The authors
claim that the models provide a good fit in PDFs and general
seasonal characteristics, but that second-order correlation
provided slightly undercorrelated results and third-order
correlation produced overcorrelated solutions.
This paper proposes the use of stochastic simulation techniques for synthetic wind data generation and the potential
development of a tool for the assessment of the profitability
of wind farm projects, accurate prediction for system operations, and wind energy markets. The research summarized
above suggests that MC models produce accurate results in
terms of PDFs and general statistical characteristics of data in
particular mean, median, standard deviation, variance, quartiles, minimum and maximum values, and Weibull distribution parameters. However, because of their short memory,
they are unable to reproduce the persistence and/or periodic
structure of wind data which manifests itself in ACFs and
PSDs. This paper addresses this issue and attempts to retain
the integrity of the data while reducing complexity. Instead
of using yearly data, the data sets are divided into subsets
(therefore keeping the individual seasonal information) and
using sufficiently less data to simulate the whole subset while
preserving the main statistical characteristics.
2. Experimental Design, Data, and Model
2.1. Data and Modeling Strategy. This research employs two
different data sets obtained from operational meteorological
towers located in the USA and Turkey. The first tower is
located at the National Wind Technology Center (NWTC),
Colorado, USA, and it provided hourly average wind speed
time series data at 10 m, 20 m, and 50 m for a period of 12
months (01.12.2005–01.12.2006). The second tower is located
in the Süpürgelik region of Yalova in Turkey, and it provided
hourly average wind speed time series at 30 m for a period of
9 months (01.12.2005–01.09.2006). The two countries are both
located in the northern hemisphere but have different seasonal characteristics and provide one method of determining
if the model can be applied to other data sets.
The objective of this research is to develop a model that
can accurately reflect seasonal variations using limited data
from an annual time series data set. To address this, both
data sets were arbitrarily divided into categories identified as
winter, spring, summer, and autumn. As both data sets have
been obtained from locations in the northern hemisphere,
seasons have been defined as winter: December, January, and
February; spring: March, April, and May; summer: June, July,
and August; and autumn: September, October, and November. The first month of each season was selected and used to
generate synthetic data for the related season with the aim of
using less data in the simulation process. First- and secondorder transition matrices were produced from the measured data for the selected months and used to generate
synthetic data. Figure 1 displays the histograms of the actual
3
data for each season. Figure 1(a) displays the frequency
distribution of each season for Turkey, identified as TR and
Figure 1(b) displays the frequency distribution of each season
for Colorado, identified as US.
2.2. Markov Chain Monte Carlo Simulation Technique. A
first-order MC was used to determine the next state of a
stochastic process depending only on its current state. In the
case of second-order MC, the next state depends on both the
present state and the most recent previous state. This could
be generalized for higher order MC where the probability of
a future state depends only on the given past history of the
process through the present state, not on any other past state.
The order of the chain represents the present or/and previous
time steps which have been taken into account to calculate the
transition probability of the next state. These probabilities are
included in the cells of a matrix called a transition probability
matrix. The size of the matrix depends on the states which
should be discretized by considering the nature of the random
variable as well as the modeling purpose. In order to calculate
transition probabilities, a further assumption is made which
depends on the definition of an MC.
A discrete-time stochastic process is an MC if, for all 𝑡 =
0, 1, 2, . . . and all states,
P (𝑋𝑡+1 = 𝑖𝑡+1 | 𝑋𝑡 = 𝑖𝑡 , 𝑋𝑡−1 = 𝑖𝑡−1 , . . . , 𝑋0 = 𝑖0 )
= P (𝑋𝑡+1 = 𝑖𝑡+1 | 𝑋𝑡 = 𝑖𝑡 ) .
(1)
Equation (1) implies that the probability of the state at
time 𝑡 + 1 depends only on the state at time 𝑡 (𝑖𝑡 ). Consider
the following:
P (𝑋𝑡+1 = 𝑗 | 𝑋𝑡 = 𝑖) ,
(2)
for all states 𝑖 and 𝑗 and all 𝑡.
If one assumes that the conditional probability stated in
(2) is independent of 𝑡, then,
P (𝑋𝑡+1 = 𝑗 | 𝑋𝑡 = 𝑖) = 𝑝𝑖𝑗 ,
(3)
where 𝑝𝑖𝑗 is the probability that given the system is in state
𝑖 at time 𝑡, it will be in a state 𝑗 at time 𝑡 + 1. The 𝑝𝑖𝑗 ’s are
often referred to as the transition probabilities for the MC.
Moreover, (3) is often called the stationary assumption which
implies that the probability of moving from state 𝑖 to state 𝑗
during one period does not change over time [33].
The transition probability matrix (TPM) for a first-order
MC with 𝑠 states can be written as
𝑋𝑡+1 󳨀→
𝑝11 𝑝12
[𝑝21 𝑝22
[
P = 𝑋𝑡 ↓ [ ..
..
[ .
.
[ 𝑝𝑠1 𝑝𝑠2
⋅ ⋅ ⋅ 𝑝1𝑠
⋅ ⋅ ⋅ 𝑝2𝑠 ]
]
.. ] .
⋅⋅⋅ . ]
(4)
⋅ ⋅ ⋅ 𝑝𝑠𝑠 ]
The elements of P are nonnegative and, given that the state at
time 𝑡 is 𝑖, the process must be somewhere at time 𝑡 + 1 which
means that the elements in each row must sum to one.
4
ISRN Renewable Energy
160
500
140
400
100
Frequency
Frequency
120
80
60
40
300
200
100
20
0
0.0
3.5
7.0
10.5
14.0
Wind speed (m/s)
17.5
21.0
0
−4
0
4
8
12
16
20
24
Wind speed (m/s)
Variable
Winter 30 m TR
Spring 30 m TR
Summer 30 m TR
Variable
Winter 20 m US
Spring 20 m US
Summer 20 m US
Autumn 20 m US
(b)
(a)
Figure 1: Histograms of seasonal wind speed time series: (a) TR at 30 m and (b) US at 20 m.
For all 𝑖 and 𝑗,
If the transition probability in the 𝑖th row at the 𝑘th state is
𝑝𝑖𝑗𝑙 , the cumulative probability can be calculated [26] by using
the following:
(i) 𝑝𝑖𝑗 ≥ 0,
(ii) ∑𝑠𝑗=1 𝑝𝑖𝑗 = 1, 𝑖 = 1, 2, . . . 𝑠.
𝑘
The formula in (5) is used to calculate the transition probabilities as follows:
𝑛𝑖𝑗
𝑝𝑖𝑗 = 𝑠
,
(5)
∑𝑗=1 𝑛𝑖𝑗
where 𝑛𝑖𝑗 represents the number of transitions from state 𝑖 to
state 𝑗 during one period.
If the transition probability in the 𝑖th row at the 𝑘th state
is 𝑝𝑖𝑘 , the cumulative probability can be calculated [20] using
(6) as
𝑘
𝑝𝑖𝑗 = ∑𝑝𝑖𝑗 .
(6)
𝑗=1
A second-order transition probability matrix can be written
as
𝑋𝑡+2 󳨀→
P = 𝑋𝑡 , 𝑋𝑡+1
𝑝111
[𝑝121
[
[ ..
[ .
[
↓[
[ 𝑝1𝑠1
[𝑝211
[
[ ..
[ .
[ 𝑝𝑠𝑠1
𝑝112 ⋅ ⋅ ⋅
𝑝122 ⋅ ⋅ ⋅
..
..
.
.
𝑝1𝑠2 ⋅ ⋅ ⋅
𝑝212 ⋅ ⋅ ⋅
..
..
.
.
𝑝𝑠𝑠2
𝑝11𝑠
𝑝12𝑠 ]
]
.. ]
. ]
]
𝑝1𝑠𝑠 ]
].
𝑝21𝑠 ]
]
.. ]
. ]
⋅ ⋅ ⋅ 𝑝𝑠𝑠𝑠 ]
𝑝𝑖𝑗𝑘 = ∑𝑝𝑖𝑗𝑙 .
(8)
𝑙=1
For synthetic data generation using random numbers, the
transition probability matrix should be transformed into a
cumulative probability transition matrix (CPM) by taking
successive summations of its elements in each row. There
is a significant tradeoff between model complexity and the
model accuracy. By increasing the dimension of the state
space, more accurate modeling results could be obtained
[24]. However, higher discretization increases the model
complexity by introducing a large number of parameters that
are difficult to assess from data [26].
The MCMC simulation procedure for synthetic generation of wind speed time series is therefore achieved using the
following steps.
Step 1. Define states of the associated MC and construct
TPM.
Step 2. Construct CPM.
(7)
Step 3. Generate uniformly distributed random numbers
between 0 and 1.
Step 4. Select an initial state randomly, say, 𝑖.
Step 5. Compare the value of the random number with the
elements of 𝑖th row of the CPM. The next state is found
ISRN Renewable Energy
5
200
140
120
100
Frequency
Frequency
150
100
80
60
40
50
20
0
−5
0
5
10
15
20
0
25
−4
0
4
Wind speed (m/s)
Variable
Actual
Simulation (MCI)
Mean
6.359
6.239
StDev
4.597
4.632
N
2159
2159
8
12
16
Wind speed (m/s)
Variable
Actual
Simulation (MCI)
140
160
120
140
100
120
80
60
StDev
3.226
3.219
N
2208
2208
15
18
100
80
60
40
40
20
0
24
(b)
Frequency
Frequency
(a)
Mean
4.276
3.772
20
20
0.00
1.75
3.50
5.25
7.00
Wind speed (m/s)
8.75
Mean
3.578
3.810
3.743
StDev
1.739
1.846
1.770
Variable
Actual
Simulation (MC1)
Simulation (MC2)
10.50
N
2208
2208
2208
(c)
0
−3
0
3
6
9
12
Wind speed (m/s)
Mean
4.113
3.861
4.034
Variable
Actual
Simulation (MC1)
Simulation (MC2)
StDev
3.012
2.755
2.965
N
2184
2184
2184
(d)
Figure 2: Histograms of observed and generated wind speed time series (Colorado, US, 20 m): (a) winter, (b) spring, (c) summer, and (d)
autumn.
where the value of this random number is greater than the
cumulative probability of the previous state but less than or
equal to the cumulative probability of the following state, say
𝑗. In this case, the next random number should be compared
with the elements of the 𝑗th row of the CPM.
Step 6. A transition from state 𝑖 to state 𝑗 in CPM can be
converted into wind speed data by using the following:
𝑉 = 𝑉𝑗−1 + 𝑍𝑖 (𝑉𝑗 − 𝑉𝑗−1 ) ,
(9)
6
ISRN Renewable Energy
120
160
140
100
120
Frequency
Frequency
80
60
100
80
60
40
40
20
0
20
0.0
3.5
7.0
10.5 14.0
Wind speed (m/s)
Variable
Actual
Simulation (MCI)
Mean
6.678
6.231
17.5
StDev
4.010
3.852
0
21.0
N
2159
2159
0.0
2.5
5.0
7.5
10.0 12.5
Wind speed (m/s)
Variable
Actual
Simulation (MCI)
(a)
Mean
5.559
5.501
StDev
2.967
3.006
15.0
17.5
N
2208
2208
(b)
160
140
Frequency
120
100
80
60
40
20
0
0
3
6
9
12
Wind speed (m/s)
Variable
Actual
Simulation (MC1)
Simulation (MC2)
Mean
5.910
5.317
5.570
15
StDev
3.254
2.813
2.995
18
N
2208
2208
2208
(c)
Figure 3: Histograms of observed and generated wind speed time series (Yalova, TR, 30 m): (a) winter, (b) spring, and (c) summer.
where 𝑉𝑗−1 and 𝑉𝑗 are the lower and upper boundaries of the
state and 𝑍𝑖 is the uniform random number.
By repeating Steps 5 and 6 for each uniform random
number, any desired length of synthetic wind speed data can
be generated.
3. Results
In order to test the validity of the model, a combination of
visual evaluation, comparison of general statistical parameters (descriptive statistics and Weibull distribution parameters), and goodness of fit tests were used. The transition
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7
Table 1: General statistical parameters in m/s for observed wind
speed time series (Colorado, US, 20 m).
Winter
Mean
St. deviation
Variance
Coef. of variation
Minimum
First quartile
Median
Third quartile
Maximum
Weibull shape
Parameter (ML)
Weibull shape
Parameter (WAsP)
Weibull scale
Parameter (ML)
Weibull scale
Parameter (WAsP)
Actual data (m/s)
Spring
Summer
Table 3: General statistical parameters in m/s for observed wind
speed time series (Yalova, TR 30 m).
Autumn
6.36
4.60
21.13
72.29
0.31
2.84
5.25
8.79
27.15
4.28
3.23
10.41
75.44
0.31
2.25
3.33
5.21
23.86
3.58
1.74
3.02
48.60
0.56
2.35
3.17
4.46
11.49
4.11
3.01
9.07
73.23
0.48
2.09
3.09
5.11
20.12
1.437
1.493
2.188
1.507
1.371
1.074
1.769
1.114
7.021
4.785
4.053
4.600
6.900
3.936
3.790
3.806
Table 2: General statistical parameters in m/s for generated wind
speed time series (Colorado, US, 20 m).
Winter
Mean
St. deviation
Variance
Coef. of variation
Minimum
First quartile
Median
Third quartile
Maximum
Weibull shape
Parameter (ML)
Weibull shape
Parameter (WAsP)
Weibull scale
Parameter (ML)
Weibull scale
Parameter (WAsP)
6.24
4.63
21.44
74.21
0.25
2.69
5.07
8.67
27.47
3.77
3.81 (3.74)
3.22
1.85 (1.77)
10.36 3.41 (3.13)
85.34 48.46 (47.30)
0.25
0.75 (0.51)
1.81
2.43 (2.53)
2.90
3.34 (3.25)
4.57
4.81 (4.64)
23.95 10.97 (10.99)
3.86 (4.03)
2.75 (2.96)
7.59 (8.79)
71.35 (73.49)
0.50 (0.50)
2.06 (2.11)
3.02 (3.20)
4.89 (4.90)
19.76 (19.70)
1.418
1.336 2.203 (2.245) 1.532 (1.499)
1.336
0.972 1.743 (1.699) 1.180 (1.077)
6.888
4.142 4.318 (4.240) 4.319 (4.504)
6.757
3.359 3.996 (3.863) 3.731 (3.629)
6.68
4.01
16.09
60.04
0
3.78
6.19
9.17
23.23
5.56
2.97
8.80
53.37
0
3.37
5.17
7.24
17.83
5.91
3.25
10.58
55.06
0.01
3.32
5.50
8.19
18.34
1.596
1.945
1.868
1.785
1.917
1.976
7.378
6.263
6.645
7.561
6.241
6.736
Table 4: General statistical parameters in m/s for generated wind
speed time series (Yalova, TR, 30 m).
Synthetic data (m/s)
Winter Spring Summer
Autumn
Mean
St. deviation
Variance
Coef. of variation
Minimum
First quartile
Median
Third quartile
Maximum
Weibull shape
Parameter (ML)
Weibull shape
Parameter (WAsP)
Weibull scale
Parameter (ML)
Weibull scale
Parameter (WAsP)
Actual data (m/s)
Spring
Summer
Winter
Mean
St. deviation
Variance
Coef. of variation
Minimum
First quartile
Median
Third quartile
Maximum
Weibull shape
Parameter (ML)
Weibull shape
Parameter (WAsP)
Weibull scale
Parameter (ML)
Weibull scale
Parameter (WAsP)
Synthetic data (m/s)
Spring
Summer
6.23
3.85
14.83
61.80
0
3.51
6.01
8.50
23.37
5.50
3.00
9.03
54.65
0
3.26
5.27
7.26
17.99
5.32 (5.57)
2.81 (2.99)
7.91 (8.97)
52.91 (53.76)
0 (0.004)
3.09 (3.17)
5.16 (5.21)
7.38 (7.83)
16.54 (15.89)
1.474
1.784
1.919 (1.923)
1.874
1.955
2.176 (1.939)
6.790
6.117
5.965 (6.271)
7.316
6.233
6.148 (6.249)
Note: values in parenthesis are the simulation results for 2nd order MC.
Note: values in parenthesis are the simulation results for 2nd order MC.
probability matrices of both first- and second-order MC
models have the probability mass concentrated on and
around the diagonal elements. This result implies that the
next wind speed will be most likely in the same state as
the current wind speed and the probability of a transition
between far states is infrequent. General statistical parameters
of observed and generated wind speeds for both locations
are presented in Tables 1, 2, 3, and 4. Comparison of the
descriptive statistics and the Weibull distribution parameters
of simulations indicate that the results are very close to the
realizations.
The frequency distributions of observed and generated
wind speed time series for both locations are presented in
Figures 2 and 3. Both the visual evaluation of histograms
and quantitative comparison of the Weibull distribution
parameters of observed and generated data indicate a good
8
ISRN Renewable Energy
Table 5: Ansari-Bradley test results.
Actual data
Winter 20 m US
Spring 20 m US
Summer 20 m US
Autumn 20 m US
Winter 30 m TR
Spring 30 m TR
Summer 30 m TR
Synthetic data
Winter 20 m (1st order MC)
Spring 20 m (1st order MC)
Summer 20 m (1st order MC)
Summer 20 m (2nd order MC)
Autumn 20 m (1st order MC)
Autumn 20 m (2st order MC)
Winter 30 m (1st order MC)
Spring 30 m (1st order MC)
Summer 20 m (1st order MC)
Summer 20 m (2nd order MC)
Equal medians
Yes
No
No
Yes
Yes
Yes
Yes
Yes
No
No
accordance among the series for both locations. Even though
30% of the simulations failed in the goodness of fit tests, the fit
of the PDFs is satisfactory. To test the accordance among the
probability distributions of simulated series and actual series,
the Ansari-Bradley test was used (Table 5).
The Ansari-Bradley test is a nonparametric test which
requires that random variables are mutually independent and
coming from a continuous population with equal medians
but does not require the assumption of normal distribution.
The null hypothesis is that 𝑋 and 𝑌 variables have the same
probability distribution which is not specified. The alternative
hypothesis is that 𝑋 and 𝑌 variables come from distributions
that have the same median and shape but different dispersions
[34].
Considering that the variability of the simulation output
is affected by the stochastic nature of the input wind speed
data as well as by the random nature of the MC model itself,
the MCMC simulation technique is sufficient to preserve
most of the statistical characteristics and stochastic behavior
of wind speed time series. Also, it is possible to improve the
accuracy by using a second-order MC model.
The simulation outputs for 10 m and 50 m heights for the
Colorado data were also examined. While the 10 m output fits
better than the 20 m, the 50 m output is a poorer fit than the
20 m. However, by increasing the state size and considering
the shape of the PDF of the observed wind speed to determine
the state intervals, it is possible to improve the accordance.
These results prove that 1 month of wind speed time series is
sufficient to generate synthetic wind speed time series for the
related season, accurately.
4. Conclusion
The collection of long-term wind speed measurements is
essential for the economic evaluation and hence the potential
ability to obtain financial support for any wind energy project.
In some cases, prolonged and continual data for resource
assessment may not be available and there is a need for tools
that provide accurate and reliable simulations of wind speed
data based on limited data sets. This creates an opportunity
for the production of synthesized data sets based on statistical
parameters of a wind regime. In this paper, a limited data
set was used to produce synthesized data. By using observed
Ansari-Bradley test (P values)
0.2493
—
—
0.0143
0.7394
0.0408
0.1361
0.099
—
—
Decision (𝛼 = 0.01)
H1 rejected
—
—
H1 rejected
H1 rejected
H1 rejected
H1 rejected
H1 rejected
—
—
wind speed data from selected months, synthetic wind speed
data was generated for related seasons. The comparisons
between the actual data and the simulations showed that
the statistical characteristics were satisfactorily reproduced.
Therefore, the most important result is that only one month
of wind speed data was sufficient to reproduce most of the
general statistical characteristics and the stochastic behavior
of wind speed time series for the related season. This result
implies that Markov chain models could be used to complete
missing data. The study also showed that the models used
in this approach are impacted by the characteristics of the
data set which prevalently manifests itself by probability
distributions. Examining and considering these probabilistic
characteristics in discretization of the states of a Markov chain
model would provide a better representation of the actual
wind pattern. A further study is needed to determine the sensitivity of the simulation outputs with regard to the different
probability distributions. Also, it is expected that an application of a continuous-time Markov process may improve the
accuracy especially in terms of reproduction of missing data.
Acknowledgments
The authors would like to thank Dr. Ahmet Duran Şahin
for providing the wind speed data for Turkey. The authors
gratefully acknowledge the financial support for this research
which was provided by Agri-Futures Nova Scotia.
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