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DTU Wind Energy-Master-Series-M-0047(EN)
Long-term Corrections for Wind
Resource Assessment
Alfonso Perez-AndujarSupervised by:Alfredo Pena and Andrea N.
Hahmann
DTU Wind Energy, Ris Campus,Technical University of Denmark,
Roskilde, Denmark
December 2013
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Author: Alfonso Perez-AndujarSupervised by:Alfredo Pena and
Andrea N. HahmannTitle: Long-term Corrections for WindResource
AssessmentDepartment: DTU Wind Energy
Abstract (max. 2000 char)
This document is a MSc thesis developed for DTU WindEnergy at
Ris Campus. It is mainly a study of differentlong-term correction
methodologies, which estimate what theobserved wind climate might
look like, had measurements startedlong before. Long-term
corrections are commonly assumed torepresent the future long-term
wind climatology, so this assump-tion was also investigated.
Long-term corrections are derived from the relationshipbetween
the reference and the observed wind speed time series,in the time
window where both are concurrent. The time windowor concurrent
subset can be made to change in length andposition along the total
concurrent set, especially if observationsare long, as in this
thesis. Thus, for different concurrent subsetlengths and positions,
long-term corrected Weibull parameters
A and k, as well as the long-term corrected power density P
,were compared to those which had been actually observed atthe
site. This was done by means of bias ratios of long-termcorrected
to observed parameters. For each subset length, themean and
standard deviation of each bias ratio was calculated,over all
possible positions of that subset within the totalconcurrent set;
it was seen that 12 months is a long-enoughduration of the
concurrent period in order to observe a gen-eral stabilisation of
the three bias ratios. Furthermore, theWeibull method was the
absolute best of all non-regressionmethods at yielding bias ratios
closest to 1, while regardingthe regression methods, the Variance
Ratio method is the winner.
The pasts representativeness of the future long-term
windclimatology was explored as well: how representative the
concur-rent subset is of the full concurrent set clearly determined
howwell the LTC (derived from the concurrent subset) representsthe
future. Also, there is only a subtle difference between thecase
where the past is just long-term reference wind speed, andthe case
where it is long-term corrected wind speed.
DTU WindEnergy-Master-Series-00XX(EN)December 9, 2013
ISSN:ISBN:XXX
Contract no:XXX
Project no:XX
Sponsorship:XX
Cover:
Pages: 100Tables: 6Figures: 79References: 0
Technical Universityof DenmarkFrederiksborgvej 3994000
RoskildeDenmarkTel. [email protected]
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Contents
1 Introduction 7
2 Wind Power Meteorology 11
3 Theory: long-term correction methods 163.1 Regression methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2
Non-regression methods . . . . . . . . . . . . . . . . . . . . . .
. . . 18
4 Site description 21
5 WRF 23
6 General pre-processing and data treatment 266.1 General
pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . .
266.2 Data treatment . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 276.3 The effect of fixing invalid data on the
correlation . . . . . . . . . . . 28
7 Sensitivity analysis on the correlation 327.1 Sensitivity to
time-shifting the concurrent time series . . . . . . . . . . 327.2
Sensitivity to rotating the reference wind direction . . . . . . .
. . . . 327.3 Sensitivity to widening the averaging time-range
around minute 00 in
the observed 10-min average dataset . . . . . . . . . . . . . .
. . . . 33
8 The wind climate at Hvsre 358.1 The local wind speed and
direction . . . . . . . . . . . . . . . . . . . 358.2 The effect of
averaging on the WRF-observations correlation . . . . . . 378.3 A
description of each observed year . . . . . . . . . . . . . . . . .
. . 408.4 Similarity of concurrent WRF-derived and observed
parameters . . . . 44
9 Results I - Which is the best LTC method? 489.1 How many
months are enough to long-term correct? . . . . . . . . . . 499.2
The 12-month concurrent subset . . . . . . . . . . . . . . . . . .
. . 569.3 Optimisation . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 599.4 u and v: an alternative approach . . . . .
. . . . . . . . . . . . . . . 62
10 Results II - Can LTCs estimate the future? 6810.1 Description
of scheme . . . . . . . . . . . . . . . . . . . . . . . . . .
6810.2 Choice of concurrent year . . . . . . . . . . . . . . . . .
. . . . . . . 7010.3 LTCs representing the future for different
methods . . . . . . . . . . . 74
4 DTU Wind Energy-Master-Series-M-0047(EN)
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11 Discussion 77
12 Conclusions 82
13 References 84
A Appendix 86
DTU Wind Energy-Master-Series-M-0047(EN) 5
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I would like to thank my supervisors Alfredo and Andrea for the
time spent togetherduring the development of this thesis and
especially for their help during the final cor-rections. Thanks
also to Sonia Lileo, Knut Harstveit and Rickard Klinkert from
KjellerVindteknikk for their kind emails and constant help; Anthony
Rogers for his advice;Alan Mortimer for his time and help by phone;
Colin Ritter for his help and suggestions;Niels G. Mortensen for
the papers he printed for me; and finally to Wolfgang Schlezand the
guys from Garrad Hassan for the access they gave us to
WindFarmer.
Thanks to my friends Matteo and Philippe, to this beautiful
country where I metthe one and only Magic Mike; to Sandra and, of
course, to my family, who are alwaysa refuge.
6 DTU Wind Energy-Master-Series-M-0047(EN)
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1 Introduction
Projected wind farms are getting increasingly larger with time
in terms of turbine sizeand investment. A wind farm developer needs
to minimise the financial risk by calcu-lating the best possible
estimate of what the future long-term wind climatology at thesite
of interest will be like, i.e. estimating the future power
production. This, however,can only be done using measurements from
the past and implies assuming that this isa reasonable approach for
predicting the future climatology. Moreover, there may notbe more
than a year of wind speed and direction observations at the target
site, and toobtain a trustworthy all-time average that accounts for
the local interannual variations,around 810 years are needed. Such
long observations are of course very hard to findat target sites
because on-site measuring campaigns generally last not much
longerthan a year.
To circumvent the shortcoming of having only short-term on-site
observations, method-ologies known as long-term corrections (LTCs)
are commonly used in wind resourceassessment to give an estimation
of the long-term past wind climatology that couldhave been measured
at a target site. LTC methods work by exploring
relationshipsbetween the short-term observations at the site and
the short-term slice of a longerreference time series which is
concurrent to it. The long-term reference time seriescan be a
long-term observation from a nearby site, a dataset from analysis
or reanal-ysis data or results from numerical weather prediction
models. From the concurrentshort-term observed and reference
datasets, some correction factors are established, bymeans of which
the long-term time series can be transferred onto the target
site.
Figure 1: LTC general scheme for two imaginary concurrent time
series.
DTU Wind Energy-Master-Series-M-0047(EN) 7
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Figure 1 shows imaginary long-term reference and short-term
on-site observed timeseries. The longest possible concurrent
period, marked in green, therefore comprisesthe total observed set,
but comprises, on the other hand, just a slice of the long-term
dataset. The resulting LTC could go, in this case, as far back as
year 10 andthus constitute the wind climatology that could have
been measured at the targetsite, had measurements started earlier.
This is why LTCs are not in essence, as of-ten termed, predictions
of the future wind climatology, but rather
could-have-beenhypotheses regarding an already past time. The
energy yield of a long-term corrected(LTC) climatology is often
assumed to give a trustworthy idea of the future energyyield. This
is the same as assuming that the LTC climatology is representative
of thefuture, which is a reasonable assumption only if the
climatology of the area is knownto vary mildly with time.
Two main questions thus arise. First of all, how accurately do
the long-term ref-erence data describe the long-term wind
climatology at the site? Of course,since the whole purpose of using
long-term reference data is precisely to account for thelack of
long-term observations at the site, it may seem preposterous to try
to comparelong-term reference data with what is actually being
looked for. However, if there arelong-term data at both the
reference and the target sites for the same period (as inthis
thesis), it is interesting to see how similar reference long-term
data are to actuallong-term site observations. This consideration
has yet nothing to do with LTCs assuch, but of course, if the
reference wind climatology is not in the least representativeof the
sitess actual wind climatology, probably most LTC methods will give
biasedresults, since long-term reference data are the key
ingredient of a LTC.
Regarding the issue of similarity between reference data and
observations, Lileo etal. (2013) conducted an investigation on what
they termed the representativeness ofthe reference wind speed, i.e.
how well the reference wind speed represents the con-current site
wind speed. They investigated how well reference wind speeds
representobserved wind speeds, for 8 different reanalysis models
and 42 measurement sites interrain with low complexity. They
obtained the best results for those reanalysis refer-ence data
coming from the Weather Reanalysis Forecast (WRF) model. In this
respect,several different methods (introduced in section 3) will be
used in order to generateLTCs which can be later compared to actual
concurrent observations. These results willalso be compared to
those obtained by Lileo et al. (2013) and Rogers et al. (2005).
InLileo et al. (2013), the Knut & Harstveit (KH) method shows
the best agreement withobservations in terms of mean wind speed and
Weibull parameters A and k. Rogers etal. (2005) shows, on the other
hand, that the Variance Ratio (VAR) and the Mortimer(MOR) methods
are the closest. Note that in Lileo et al. (2013), long-term
referencewind speeds and directions come from reanalysis, whereas
Rogers et al. (2005) usedlong-term observations from a secondary
mast. Moreover, the methods investigated inone paper are not
investigated in the other.
8 DTU Wind Energy-Master-Series-M-0047(EN)
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This takes us to the main motivation of this thesis, which is to
find out which LTCmethods give the best results. Indeed, even
though LTC methods are a common stepin a prediction process of the
future wind speed (as well as a relatively simple tool interms of
implementation, at least when compared to flow and wake modelling),
theyaccount for an average 2.5% of the total variability of the
entire predictive process.This is more than the flow and wake
variation put together, as seen in figure 2 of astudy carried out
in 2011.
Figure 2: Coefficient of variation [%] added by each of the
common steps in a prediction process of
the future wind speed. Taken with permission of Niels G.
Mortensen, Comparison of Resource and
Energy Yield Assessment Procedures, 2011.
Therefore, a consensus should be reached as to which LTC method
to use in a windresource assessment, and why.
Secondly, also an important motivation for doing this thesis:
can LTCs predict thefuture wind climatology? If so, it may seem
reasonable to hypothesise that the moredata gathered from the past,
the more accurate the description of the future will be.However,
does this hypothesis still hold reasonable, the longer the future
period to beestimated? The assumption of the past being
representative of the future has beenthe object of study in recent
years. Lileo et al. (2013) investigated, for an already pastperiod
of reanalysis data, how well different past windows (i.e. prior to
some date in-side the chosen period) of wind speed represent a
fixed future window of subsequent
DTU Wind Energy-Master-Series-M-0047(EN) 9
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years. They did this for each grid point over a certain focus
region, using wind speedsobtained from the Twentieth Century Global
Reanalysis Version II (20CRv2). In orderto get an idea of the pasts
representativeness of the future, they defined an error bytaking
the percentage difference in mean wind speed of the past and the
futureperiods. They concluded that the mean wind speed of the near
past is not necessarilythe best predictor of the future mean wind
speed, as well as that each grid point hasan optimum length of past
window, i.e. the number of past years needed to getthe best
prediction (i.e. the minimum percentage error) is specific to each
grid point.
In this thesis, a similar investigation is conducted. However,
LTCs from the pastare compared to future observations. Of course,
if the pasts representativeness ofthe future is to be studied, it
would always be safer to use past years of observationsto compare
them to future observations, rather than use past years of LTCs to
com-pare to future observations. However, as mentioned earlier,
there are usually no morethan 12 months of observations at a target
site, so comparing past LTCs to futureobservations can solve the
problem of lack of long-term observations, and tell us whichmethod
yields the best result.
The LTC methods used are explained in section 3. They are
classified as regression andnon-regression methods and easily found
in the literature (Riedel et al. (2001), Nielsenet al. (2001),
Woods and Watson (1997), Mortimer (1994), and also summarised
inLileo et al. (2013) and Rogers et al. (2005)).
Section 4 describes the site of interest, the area of Hvsre,
which is located in WesternDenmark; since WRF-derived wind speeds
are used as long-term references, the basicprinciple underlying the
model is explained in section 5. The filtering process applied
toinvalid values of wind speed and direction found in the observed
dataset is explained insection 6. Section 7 is a short
investigation on how the correlation for the concurrentwind speed
components varies under certain changing conditions. The climate at
thesite is described in section 8. Section 9 explores the ability
of the different LTC meth-ods to long-term correct different
parameters describing the wind climatology. Finally,section 10
tries to answer the question of whether we can predict the future
usinginformation from the past, at least for the specific case of
Hvsre and the choice ofinputs for this work.
10 DTU Wind Energy-Master-Series-M-0047(EN)
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2 Wind Power Meteorology
It is common practice to describe the frequency of 10-min,
30-min, or 1-hr averagewind speeds U at some site, over a
long-enough period (e.g. 1 year), by means of theWeibull
probability density function (p.d.f.),
f(U) = kUk1
Akexp
((U
A
)k), (1)
where A and k are the Weibull parameters. Equation 1 shows that
the frequency ofoccurrence of the wind, f(U), is driven just by A
and k. This section is a brief de-scription of the different
methods in which these two parameters can be calculatedfrom a wind
speed time series. Using A and k, together with wind direction, is
enoughinformation to characterise the site, at least for a study of
this kind.
Before investigating different methods of calculating A and k,
it is worth looking atcertain definitions, like for example the
mean wind speed , which is a particular caseof the non-central
moment when n = 1, and can be defined as
n =
0
Unf(U)dU. (2)
The variance 2 of the mean wind speed is another particular case
of the centralmoment, when n=2,
n =
0
(U 1)nf(U)dU. (3)A very useful relationship for this study shall
also be considered, involving non-centralmoments:
n = An(
1 +n
k
), (4)
where the gamma function is defined as:
(t) =
0
exxt1dx, (5)
and where t is a constant such that t > 1.
Square of the mean wind speedDividing the square of the first
non-central moment (the square of the mean) by thesecond
non-central moment (the mean of the square) gives an equation which
is afunction only of k. This can then be solved iteratively, since
it is a quotient of knownvalues,
DTU Wind Energy-Master-Series-M-0047(EN) 11
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212
=2(1 + 1
k
)(1 + 2
k
) . (6)This method will be referred to as 2NCM.
Cube of the mean wind speedDividing the cube of the first
non-central moment (the cube of the mean) by the thirdnon-central
moment (the mean of the cube) gives a result which is also just a
functionof k,
313
=3(1 + 1
k
)(1 + 3
k
) . (7)This method will be referred to as 3NCM.
Maximum Likelihood EstimatorThis method was developed by Harter
and Moore (1965). Let U1, U2, ..., UN be asample of N random and
independently distributed wind speeds drawn from a p.d.f.that
depends only on the wind speed U and on the parameter to be
estimated, . Thelikelihood function of the random sample Ui, i = 1,
..., N , is denoted L and is the jointdensity of all Ui from the
drawn sample,
L =Ni=1
f(Ui, ). (8)
The expression for L, when the p.d.f. is the Weibull probability
density function, is:
L(U,A, k) =Ni=1
kUk1
Akexp
((U
A
)). (9)
The two equations above are enough to solve iteratively A and
k,
lnL
A= 0 (10)
lnL
k= 0, (11)
and thus calculate which value of (A, k) maximises the
likelihood function. Thismethod will be referred to as MLE.
Least Square MethodThe Weibull cumulative distribution function
(c.d.f.), F (U), is obtained by integratingits p.d.f.,
12 DTU Wind Energy-Master-Series-M-0047(EN)
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F (U) =
U
f(U )dU = 1 exp((U
A
)k). (12)
Taking natural logarithms and rearranging equation 12 leads
to
ln( ln(1 F (U))) = k ln c+ lnU, (13)which can be minimised e.g.
via least squares. This method will be referred to as LSM.
The different Weibull parameters obtained from these four
techniques were appliedto a wind speed time series in order to
obtain four different distributions. These wereplotted alongside
the histogram of the dataset, in order to see the differences
betweenthem. Figure 3 shows the entire wind speed distribution,
whereas figure 4 shows anamplificaton for better visualisation,
since the f(U) curves from the different methodsare closely packed
together. The p.d.f representing the LSM method (blue curve)
givesnoticeably higher frequencies of occurrence for the speed
range 512 m/s.
0 5 10 15 20 25 300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Wind speed [m/s]
p.d.f.
Data histogram2NCM3NCMMLELSM
Figure 3: Histogram of the wind speed (bars), and Weibull
distribution p.d.f.s based on different
methods: 2NCM (square of the mean wind speed), 3NCM (cube of the
mean wind speed), MLE
(maximum likelihood estimator) and LSM (least square
method).
DTU Wind Energy-Master-Series-M-0047(EN) 13
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0 2 4 6 8 10 12 14 16 18 200.03
0.04
0.05
0.06
0.07
0.08
0.09
Wind speed [m/s]
p.d.f.
Data histogram
2NCM3NCMMLELSM
Figure 4: Amplification of the histogram of wind speed (bars),
and Weibull distribution p.d.f.s based
on different methods: 2NCM (square of the mean wind speed), 3NCM
(cube of the mean wind
speed), MLE (maximum likelihood estimator) and LSM (least square
method).
For a wind farm investor, besides A and k it is also very
important to estimate thefuture wind power density at the site of
interest, P . This third parameter is directlyderived from A and
k,
PA,k =1
2A3
(1 +
3
k
). (14)
However, the wind speed power density can also be calculated
directly from the timeseries speed values, by averaging over the
cubed values of the time series,
PU3 =1
2U3. (15)
Having two approaches is advantageous because it allows for a
direct comparisonbetween the single-valued PU3 (which is fixed for
any given time series), and PA,kcoming from each of the four
methods explained above. This comparison is shown intable 1 by
means of
P =PU3 PA,k
PU3. (16)
14 DTU Wind Energy-Master-Series-M-0047(EN)
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Methods
Parameters 2NCM 3NCM MLE LSM
A [m/s] 10.47 10.47 10.46 10.36
k 2.19 2.17 2.18 2.26
P [%] 0.67 0.00 0.41 6.02
Table 1: Percentage error between the power density calculated
as a function of the average cube
wind speed and the power density calculated as a function of A
and k obtained through different
methods. The expression used is P = (PU3 PA,k)/PU3 .
The p.d.f. curves derived from the four different methods
(figure 3 or 4) do not giveinformation on which of the four gives
the best description of the wind power density.However, table 1
does show how a small increase in terms of the k parameter
(from2.17 in 3NCM to 2.19 in 2NCM, i.e. 0.9%), keeping A constant,
means a differenceof 0.7% in power density (from 847 W/m2 to 853
W/m2). Furthermore, the LSMmethod is by far the worst in terms of P
, with just a 1.1% difference in A withrespect to the three other
methods. It can be concluded that, while 2NCM, 3NCM andMLE yield
very similar Weibull parameters, PA,k is so sensitive that only the
3NCMmethod is the best approach to estimating an accurate value of
the power density.
Using the Weibull parameters is useful in that it describes the
local wind climatol-ogy through just two parameters. For example,
whenever sector-wise observed windspeeds are generalised, it is a
much better choice, in terms of computational cost, tohandle just
two parameters per sector instead of generalising value after speed
value.
DTU Wind Energy-Master-Series-M-0047(EN) 15
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3 Theory: long-term correction methods
Long-term correction methodologies need a short-term wind speed
or direction ob-served dataset at the site of interest, and a
long-term reference time series. It ismoreover necessary for both
time series to be concurrent during a certain period oftime. A wind
farm developer would usually use the entire short-term time series
andslice the corresponding piece of the reference long-term time
series which is concurrentin time with it. These two concurrent
time series of equal length can then be usedto calculate the LTC
factors, which, applied to the entire reference long-term
dataset,give the long-term correction and thus an estimation of the
sites long-term climatology.
Expressions such as reference concurrent and short-term
reference datasets areequivalent and will refer to the part of the
long-term reference time series that isconcurrent to the short-term
site-observed dataset, which will in turn be denotedshort-term
site, site concurrent or simply short-term observed time
series.
3.1 Regression methods
A plot of the concurrent site dataset vs. the concurrent
reference dataset is needed,from which to obtain a best fit that
will most accurately describe the relationship be-tween both
datasets. This can be done in an all-sector fashion, but it is
recommendedto correct sector-wise and ultimately recombine the
sector-wise corrections into anall-sector LTC. When sectorising
both concurrent short-term datasets, it is customaryto use the
direction of the short-term reference, i.e. to do as if the
short-term sitesdirection were the same as the concurrent reference
one. This is done for practicalreasons, since for most methods,
direction is not long-term corrected and thus the onlyavailable
long-term direction is the reference one.
Ordinary Least Square Method (OLS)It assumes that there is a
linear relationship between both concurrent time series.The aim is
to calculate the intercept and slope coefficients that will
minimise the
sum of the squared residuals,ni=1
2i , in the yaxis direction, where i = yi yi,i.e. the predicted
reference value minus the measured value. The regression linecan be
forced to go through the origin.
Total Least Square Method (TLS)This is equivalent to the
previous method, but the residuals are calculated as thedifference
between the reference predicted and the measured values in the
perpen-dicular direction with respect to the regression line,
instead of in the vertical one.The equations were taken from the
commercial software package WindFarmer Rs
16 DTU Wind Energy-Master-Series-M-0047(EN)
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manual, for its PCA Method (WindPRO 2.6 Manual, 2008). Again the
interceptcan be forced to be zero.
nth degree Polynomial Regression Method (PRn)In practice, an
-nth degree polynomial can be chosen to fit a data cloud. For
ascatter plot with a non-linear shape, it might be a reasonable
approach to fit ahigher degree polynomial and try to cover the data
cloud more accurately. In thiswork, the PRn method was applied
throughout by means of a third-order polyno-mial, henceforth
referred to as PR3.
Better results could be expected from this methodology, but, as
pointed out byRebbeck (1996), none of the non-linear models
investigated by him (higher or-der polynomials, cubic splines and
complex surface fitting) performed much betterthan a linear
regression.
Variance Ratio Method (VAR)This method was proposed by Rogers et
al. (2005) as a way to force the overallvariance of the LTC time
series to be equal to the overall variance of the observedtime
series, i.e. (y) = (y). This is done by forcing the slope parameter
to be(y)/(x); also, it avoids the problem of the variance of the
predicted wind speedabout the mean being smaller than the variance
of the observed wind speeds by afactor equal to the correlation
coefficient from the regression fit (Rogers et al.,2005).
DTU Wind Energy-Master-Series-M-0047(EN) 17
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0 10 20 30 400
10
20
30
40
Reference wind speed [m/s]
Sitewindspeed[m
/s]
w.s.OLSOLS f.t.oTLSPR3VAR
Figure 5: Different regression trend lines for concurrent
short-term site and short-term reference wind
speeds. Each trend line corresponds to a different fitting
method.
3.2 Non-regression methods
These methods firstly sectorise both the concurrent short-term
site and short-termreference time series. Parameters such as A, k
and wind power density P are thencalculated for each sector, so
that the correction factors can be applied sector-wise tothese
parameters. The resulting LTC is therefore not a time series, but a
collection of
sector-wise LTC parameters, from here on denoted A, k and P
.
Mortimer Method (MOR)This method was created by Alan A.
Mortimer, see Mortimer (1994). Both theconcurrent site and the
reference time series are firstly binned with respect to
thereference speed and direction: 1 m/s and 15, for example.
Secondly, a matrixrij is created, where each element ij contains
the mean of the quotient of con-current site and reference wind
speeds, i.e. the mean of vector vsst
vrlt. An analogous
matrix sij must also be built, to contain the standard deviation
of vectorvsstvrlt
.
(The subscripts stand for, respectively: site long-term (slt),
site short-term (sst),reference short-term (rst) and reference
long-term (rlt)).
sij is used to create a triangularly-distributed pseudorandom
number eij at eachspeed/direction bin ij, so that the final
governing equation can be applied:
yij = (rij + eij)xij, (17)
18 DTU Wind Energy-Master-Series-M-0047(EN)
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where y and x are the binned long-term corrected wind speed and
the binnedlong-term reference input, respectively.
Knut & Harstveit Method (KH)The KH method was developed by
Knut Harstveit and is used in the Norwegianwind assessment company
Kjeller Vindteknikk, see Klinkert (2012). A matrix ofshort-term
site observed wind speeds is constructed, Oij, where i and j are
direc-tion sectors in the concurrent reference and site datasets,
respectively. The elementij of the matrix contains all short-term
wind speed values at the site that fall intothe bin ij, i.e. those
wind speeds that belong to direction bin j but occur whenthe
concurrent short-term reference data value belongs to direction bin
i. Fromthis matrix, a population matrix Nij is derived; each
element is simply the numberof wind speeds found in each ij in
Oij.
A third matrix is also derived from Oij, containing the mean of
the observedshort-term site wind speeds contained at each ij. This
matrix is expressed as Oij.
A fourth matrix is computed as a probability matrix Pij derived
from Nij. Pijis obtained simply by dividing each value ij by the
sum of all the column j, i.e. itis the probability of directions
observed at the site occurring at the same time asreference
directions. Finally, a vector Qi is calculated, with as many
elements asdirection bins have been chosen. Each element contains
the quotient of long-termreference and short-term (concurrent)
reference wind speeds, each sectorised withits own direction. The
equation governing is expressed as
vjslt =12i=1
Oij Pij Qi, (18)
where vjslt is the LTC average wind speed calculated for bin
j.
Tallhaug and Nygaard Method (TN)This method is explained in
Tallgaud and Nygaard (1993). It follows the relation
vislt = visst + R
i islt
islt(virlt virst), which gives the site long-term mean wind
speed,
sectorised with respect to the reference wind direction. For
each sector, the Pearsoncoefficient R must be calculated, as well
as the standard deviation of both concur-rent, sector-wise
datasets. Finally, this predicted long-term mean wind speed mustbe
translated to the site wind direction by means of:
vjslt =ni=1
visltpji p
i
pj, (19)
where pji is a matrix containing the probability of site sector
j occuring at thesame time as reference sector i, while pj and pi
are the individual probabilities ofsectors i or j occurring at the
site and reference, respectively.
DTU Wind Energy-Master-Series-M-0047(EN) 19
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Woods & Watson (WW)This method is explained in Woods and
Watson (1997). Two matrices Wij andZij are created. The first one
contains the conditional probability of wind blowingin a certain
reference sector i, and in sector j of the site. The second
matrix
represents the inverse case. Both are built such thatnj=1
Wij = 1 andni=1
Zij = 1.
To calculate the long-term corrected wind speed at the site, the
authors proposedtwo options. In this thesis only the second option
is implemented, since, accordingto the authors, it is the choice
which yields the best results when the correlationbetween the
concurrent data sets is poor (and as will be seen, concurrency
ismoderate for the site):
vjslt = mj
(ni=1
Zij virlt
)+ cj (20)
Weibull Method (WBL)A very simple method found, among others, in
the WindPro R commercial softwarepackage (WindPRO 2.6 Manual 2008).
It needs both concurrent short-term siteand short-term reference
time series to be sectorised with respect to their own
direction values. The LTC site wind speed is defined as jslt
=isstjrst
jrlt. The su-
perscript j in jslt indicates that it is already sectorised for
the site direction j. represents any parameter calculated for a
specific bin, including frequency. This isthe only method to yield
a LTC frequency f , as implemented in this work.
Method Regression Non-regression Corrects direction
Developer
OLS X Yes if applied to u and v GL-GH, WindFarmerTLS X Yes if
applied to u and v GL-GH, WindFarmerPR3 X Yes if applied to u and
vVAR X Yes if applied to u and v Rogers, Rogers & ManwellMOR X
No Alan MortimerKH X No Knut HartsveitTN X No Tallhaug &
Nygaard
WW X No Woods & WatsonWBL X Yes EMD, Windpro
Table 2: Summary of the different LTC methods used in this
work.
20 DTU Wind Energy-Master-Series-M-0047(EN)
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4 Site description
The measuring station is located at DTU Wind Energys test center
for large windturbines at Hvsre, in Western Denmark.
Figure 6: Bing Maps R image of Hvsre test facility and its
surroundings.
Figure 6 shows the Hvsre site, marked in red. It is delimited to
the South by a U-shaped road and to the North by a creek. It is a
very flat area made of farmlands andgrasslands, and there are two
significant bodies of water: the North Sea to the Westand the
Bvling Fjord to the South. The farmland is cut mainly by the
limiting roadsaround. Along the coastline to the West and
protecting the 181-Road from the seawinds, there is a 5-m-high
embankment.
Figure 7 shows a closer view of Hvsre. The wind turbines lie in
a North-South array,each with its corresponding measuring mast
lying roughly 250 m to the West. The
DTU Wind Energy-Master-Series-M-0047(EN) 21
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meteorological mast (the station) is roughly 200 m South of the
southernmost turbine,from which Hvsres observations are
recorded.
Figure 7: Bing Maps R image of Hvsre test facility and its
surroundings.
The masts data feed can be followed in real time at DTU Wind
Energys website.These measurements are mainly wind speed and
direction at different heights, but alsotemperatures and
atmospheric pressure. In this thesis, however, only wind speed
anddirection measured by the meteorological mast (marked in blue in
figure 7) were used.The exact coordinates of the station are
562626893, and measurements wererecorded by a Ris P2546a cup
anemometer and vane placed 100 m above ground, forthe period
01012005 to 31122012. Both devices have a measuring frequency of10
Hz, but the data used in this thesis are 10-min average wind speed
and direction.The choice of 100 m height is suitable for large wind
turbines.
22 DTU Wind Energy-Master-Series-M-0047(EN)
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5 WRF
The Weather Research and Forecasting (WRF) model is a numerical
weather prediction(NWP) model widely used in research and industry
and that counts with up to 6000users (Skamarock et al (2008)). It
is a code-based tool, and it is accommodated inthe so-called WRF
Software Framework (WSF), which holds the different modulesthat
feed into the calculations. Thus, modules such as Physics Package
and WRF-Chem serve as input to the Dynamic Solvers (Advanced
Research WRF or ARW andNonhydrostatic Mesoscale Model or NMM) while
performing the calculations.
Figure 8: WRF software infrastructure, Skamarock et al
(2008).
WRF is highly user-configurable. As an example, it can be set to
use simplified physicsequations when calculating microphysics, or
be set to make use of its full capability(sophisticated mixed-phase
physics). It can either treat atmospheric radiation as a mixof long
and short waves or as a simple shortwave system. Surface physics
can be ac-counted for via a simple thermal model or via a more
complete model comprising allpossibilities (vegetation, moisture,
snow, ice, etc.). However it is this wide range ofpossibilities
what causes WRFs output to be highly dependent on the users
choicesand model tuning (Hahmann et al., 2013).
WRF output simulations were used in this work as reference data.
The simulationswere run at DTU Wind Energy Ris Campus by nesting
the model in a global atmo-spheric reanalysis, i.e. the
initialisation of WRFs mesoscale simulations, as well as theareas
boundary conditions, were taken from a global atmospheric
reanalysis. For this
DTU Wind Energy-Master-Series-M-0047(EN) 23
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work, version 3.2.1 of WRF was configured to use the ARW solver
on an outer domaingrid of size 15 km 15 km and on a nested domain
grid of size 5 km 5 km.
Figure 9 shows the real boundaries of a part of northwestern
Jutland (including Hvsvre),overlapped by WRFs nested grid land
mask. This is the configuration used in order toobtain the
simulated wind speed and direction, which were used as reference
data inthe thesis.
Figure 9: Representation of the land mass and the ocean as seen
by WRFs nested grid. The pink x
marks the location of the meteorological mast. The two red dots
located East and West of the mast
mark the two closest v-component grid output points. The green
points North and South mark the
u-component grid output points (it is a staggered grid).
The four points (represented as red, green and white dots in
figure 9) belong to thea horizontal slice of the 3D grid, thus
representing only the pressure level roughlyequivalent to 100 m in
height. The values of the zonal and the meridional wind speed
24 DTU Wind Energy-Master-Series-M-0047(EN)
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at the four dots were then horizontally interpolated so as to
obtain a single value ofWRF-derived horizontal wind speeds at the
middle point (white dot in the figure). Thisfinal output point is,
as mentioned, at a height of roughly 100 m.
As for the 5 km 5 km horizontal grid resolution, figure 9 shows
that this causesa large difference between the modeled and the real
horizontal boundaries. Indeed,WRFs land mask mismatch implies that
winds modeled as northeasterly winds atHvsre blow over water when
reaching the mast, when in reality, northeasterly windsblow over
land. This change in roughness length between the real (observed)
and themodelled, WRF-derived winds is one of the reasons behind
deviations between betweenboth at coastal sites such as Hvsre.
On the other hand, the coarseness of WRFs grid does not present
a problem atHvsre in terms of unseen obstacles, since, as seen in
the previous section, the siteis mainly flat terrain. Also, no new
significant buildings were erected that could havenot been included
in WRFs topography input. All this makes Hvsre a unique site
interms of observations and reference data.
Choosing the reference time series to be WRF-derived should be
validated by repeat-ing the experiment with wind speeds derived
from another NWP model, or even fromlong-term observations from a
nearby mast (e.g. from the two neighbouring wind farmsseen in
figure 6).
DTU Wind Energy-Master-Series-M-0047(EN) 25
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6 General pre-processing and data treat-ment
The time series coming from Hvsres meteorological station
comprises 10-min aver-age wind speed and direction observations at
100 m. As mentioned in section 4, theperiod used in this thesis
goes from 01012005 to 31122012.
The reference time series used comes from the WRF mesoscale
model (section 5),which outputs instantaneous hourly values of the
horizontal wind velocity componentsu and v. These were transformed
to speed and direction for the period 01011999 to31122012.
The maximum possible concurrent period for both observations and
reanalysis is there-fore 01012005 to 31122012 (the duration of the
observations).
6.1 General pre-processing
Both original or raw observed speed and direction time series
had to be pre-processedbefore they could be put to use. This was
done in 5 steps, of which only the last appliesto the WRF
dataset:
1. Remove extra time stamps.In the observed time series, extra
values were found sometimes in between twooutput time stamps, e.g.
an extra output value at 05 between 00 and 10 min.Therefore, in
this case, if valid values of wind speed were found at both 00
and10 time stamps for that hour, the value at 05 was removed.
Otherwise, the extratime stamp was shifted in place of the missing
one (see next point).
2. Shift time stamps.Values corresponding to time stamps which
were not 00, 10, 20, 30, 40 or 50 minwere shifted, if needed. As an
example, a value at minute 09 was shifted to minute10 if the wind
speed value at minute 10 was missing, or time stamp 04 was
shiftedto 00 if 00 did not previously exist (in both cases, the
time series would showvalues only at 00 and 10).
3. Reduce the length of the observed time seriesSince the WRF
time series comprises instantaneous, hourly wind speed and
di-rection values, the observed time series contains 6 times more
values for anyconcurrent period. However, in order to see how they
correlate to each other,both time series must have the same number
of data points. This means that the6 observed speed and direction
values in each hour must be substituted by just one.
26 DTU Wind Energy-Master-Series-M-0047(EN)
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This only value was chosen as the 10-min average value
corresponding to thetime frame 010 min. Indeed, averaging over the
6 values in each hour, in orderto obtain a single value per hour,
would have meant a greater loss of information.
4. Choose a fixing scheme to treat invalid data.Invalid
recordings were not seen to come necessarily in pairs, since
flagged timestamps were found that either (i) contained a flawed
record only of speed (ii)contained a flawed record only of
direction, or (iii) contained flawed speed anddirection. Two
different paths can be taken when any of the two previous cases
isencountered:
(a) Time stamps containing invalid data are removed.
(b) Time stamps containing invalid data are filled with some
estimated value.
These two possibilities are investigated in subsection 6.2
below.
Note: as well as non numeric wind speed and direction outputs,
invalid wind speedsare also (i) super high readings (usually taken
as wind speeds above three timesthe overall standard deviation),
and (ii) time windows with a constant wind speedor direction.
However, none of these two cases were seen to occur in the
observedtime series.
5. As for the WRF dataset, the only pre-processing it required
was the interpolationcalculated at a height of 100 m from the three
different isobaric surface levelsat which the model outputs its
computations: roughly 14, 70 and 125 m. Thisinterpolation was
carried out at each time stamp (at each hour) in order to
obtainhourly u and v simulated velocity components at 100 m.
6.2 Data treatment
After shifting and reducing the observed time series, all
remaining invalid data had tobe treated. In the case of Hvsres
hourly observed time series, invalid data accountfor a 2% of all
the values. As mentioned in point 4. above, two paths were
followedwhen a flagged wind speed or direction value was
encountered: the time stamp itselfwas either removed or filled with
some numeric data:
1. Time stamps containing invalid data are removed.
Time stamps containing either an invalid speed or direction
value were removed.The resulting wind speed time series will
hereafter be denoted chopped time se-ries.
DTU Wind Energy-Master-Series-M-0047(EN) 27
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2. Time stamps containing invalid data are filled with some
estimated value.
2.1. Months were treated separately when following this scheme,
as in Salmonand Taylor (2013), i.e. missing wind speeds were
substituted by the monthly aver-age. However, since missing wind
speeds at Hvsre are usually grouped in chunksof 100 or more
consecutive invalid values, the resulting time series, after
suchsubstitution, showed unphysical behaviour. This could be seen
in a simple WRF-derived vs. observed wind speed scatter plot as odd
horizontal alignments of points.
The resulting wind speed time series will hereafter be denoted
monthly-averagetime series.
2.2. A Matlab R function named inpaint nans.m was chosen
instead. This func-tion interpolates between the values at the
beginning and the end of a missingchunk of data in any time series.
It also takes into account the general patternbefore and after the
missing values, in order to best simulate the pattern of
thegenerated data.
In the case of Hvsres hourly observed dataset, this function was
applied component-wise, i.e. separately to the u and v datasets.
The reason for doing so is that thefunction did not work well when
interpolating direction values (especially around0), so it was
chosen to convert speed and direction into components before
usingthe function. This conversion into components, however,
requires both speed anddirection values to be valid. Therefore, it
was enough that a time stamp containedeither invalid speed or
invalid direction, to mark it as flagged. The flagged timestamp was
then filled by applying the inpaint nans.m function to the u and
vdatasets. The fixed time series were ultimately combined back into
speed and di-rection.
The resulting wind speed time series will hereafter be denoted
painted time series.
6.3 The effect of fixing invalid data on the correlation
This subsection investigates the effect that fixing invalid data
in Hvsres observeddataset has on how it correlates to the
concurrent reanalysis dataset. In order to do so,the observed time
series was subjected to an increasing number of artificially
injectedinvalid data. Since Hvsres hourly observed time series
already contained invalid data,the starting dataset, which had to
be free of invalid data, was in reality the paintedtime series.
This dataset was then iteratively corrupted.
At each iteration, each of the fixing schemes described above
was applied to the
28 DTU Wind Energy-Master-Series-M-0047(EN)
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corrupted dataset, after which both the fixed and the concurrent
reference datasetwere correlated, and the correlation coefficient
r2 calculated.
In the case where invalid data were substituted by either a
monthly average or aninterpolation, the reference dataset remained
untouched and both kept as many datapoints. On the other hand, in
the case of time stamp removal, the infected dataset andthe
reference dataset lost the same (concurrent) values in order to
make correlationpossible.
The artificial injection of invalid data into Hvsres hourly
observed time series wasdone in two ways:
1. Invalid values (100 individual, randomly scattered) were
added to the time seriesat each iteration. See figure 10.
2. Invalid chunks (each comprising 100 consecutive values) were
randomly added tothe time series at each iteration. See figure
11.
As mentioned, for both cases, after injecting invalid data at
each iteration, the differentfixing schemes were applied.
100 200 300 400 500 600 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Invalid data (x100 individual values)
r2
Chopped time seriesPainted time seriesMonthly average time
series
Figure 10: Correlation coefficient of Hvsres observed time
series after fixing its invalid wind speed
values, and the concurrent reference time series, as a function
of the number of invalid data values.
The invalid values were randomly injected, 100 values each
time.
DTU Wind Energy-Master-Series-M-0047(EN) 29
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100 200 300 400 500 600 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Invalid data (packs of 100 consecutive values)
r2
Chopped time seriesPainted time seriesMonthly average time
series
Figure 11: Correlation coefficient of Hvsres observed time
series after fixing its invalid wind speed
values, and the concurrent reference time series, as a function
of the number of invalid data values.
The invalid values were randomly injected, blocks of 100
consecutive values each time.
Note that the chopped time series loses, at each iteration, as
many data points asinvalid values were added (and at the same exact
positions). Thus, the fact that thisis at the same time the least
representative dataset of Hvsre and the most stablein terms of
correlation, as seen from figures 10 and 11, means that r2 is not a
reliableparameter for quantifying the amount of information lost to
invalid values. Other ap-proaches such as comparing parameters (A,
k and P ) calculated from the corrupted(and subsequently fixed)
observed dataset and parameters from the reference datasetmay be
more accurate.
Another distinctive feature of figures 10 and 11 is the huge
difference for the paintedtime series between the case where 100
individual invalid values are randomly added ateach time (figure
10), compared to when randomly-scattered packs of 100
consecutiveinvalid values are added (figure 11). The former case
allows the interpolating functionto keep both time series similar,
whereas in the latter case, the wider gaps make itmore difficult
for the fixing scheme to be successful. This is backed up by taking
alook at figure 11: from value 520 onwards (along the x-axis),
there is no more spaceto assign whole packs of 100 invalid values,
and so these are, for increasing numberof invalid data, injected
individually (as in figure 10): indeed, from this point on,
thedecay of r2 is much less acute.
The correlation coefficient r2 is therefore seen to be
ineffective at determining howmuch representativeness has been lost
to invalid data. In the case where the number ofdata points
decreases in the two concurrent time series (red curve, figures 10
and 11),both time series are still actually representative of each
other, so r2 does not decrease.It also does not decrease when
invalid data are replaced by similar (interpolated) data,
30 DTU Wind Energy-Master-Series-M-0047(EN)
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as seen from the blue curve in figure 10. r2 decreases however
considerably when thegaps or holes are replaced by surrogate data
which is very different from the localpattern around the gap (blue
curve, figures 10 and 11).
For remaining calculations in this thesis, the observed time
series used will be the oneresulting from fixing Hvsres real
invalid data with the Matlab R function; this schemekeeps the right
number of datapoints and does not show the unphysical patterns
seenin the monthly-average time series.
DTU Wind Energy-Master-Series-M-0047(EN) 31
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7 Sensitivity analysis on the correlation
It is interesting to investigate the correlation between
reference and observations, as afunction of three different
situations: (i) time-shifting the two concurrent time series,(ii)
rotating the reference wind direction and (iii) widening the
averaging time-rangearound minute 00 in the observed 10-min average
dataset. For this section, the corre-lation coefficient r2 was
calculated separately for uref and uobs (in blue in the
figuresbelow), and separately for vref and vobs (in red). This was
done for the period 01012005 to 31122012 using WRF simulations as
reference data.
7.1 Sensitivity to time-shifting the concurrent time series
The maximum correlation is obtained at a 1 hour shift between
WRF and measure-ments, as seen in figure 12. This was expected,
since the reference data time stampswere not initially time-shifted
(to account for the 1-hour difference between referenceand
observations time-zones).
20 15 10 5 0 5 10 15 20
0.2
0.3
0.4
0.5
0.6
0.7
0.8
r2
Time shift [hours]
u-componentv-component
Figure 12: Effect of time a time shift between concurrent the
observed and the reference wind speed
time series on the correlation between them.
7.2 Sensitivity to rotating the reference wind direction
A rotation of the reference wind direction was carried out, in
order to detect any possiblemisalignment between reference and
observed wind speed. The procedure was to addor subtract some
degrees to the reference direction time series, and then calculate
new
32 DTU Wind Energy-Master-Series-M-0047(EN)
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misaligned uref and vref velocity components with which to
correlate to uobs and vobs,
which stayed the same. The result is shown in figure 13.
20 15 10 5 0 5 10 15 20
0.65
0.7
0.75
0.8
Angle rotation of reference direction []
r2
u-componentv-component
Figure 13: Effect of rotating the reference wind direction on
the correlation between the observed and
the reference wind speeds.
There is an offset in direction, but this shows only in the
correlation coefficient betweenthe varying uwrf and the fixed uobs.
Indeed, while the correlation is symmetric for thev and has a
maximum value at 0, the u components maximum is displaced 5.It was
assumed that the wind vane was correctly calibrated throughout the
measuringperiod, so the offset can be associated exclusively to a
systematic error in WRF.
7.3 Sensitivity to widening the averaging time-range
aroundminute 00 in the observed 10-min average dataset
It was explained in section 6 that the number of time stamps in
Hvsres measuredtime series had to be reduced from 6 per hour to 1
per hour, in order to correlate it tothe concurrent reference
dataset. As seen, the procedure consisted of picking out onlythe
10-min average value corresponding to the 00 min time stamp. It is
interesting,however, to see what happens if a broader range (always
around 00 min) is used toaverage and obtain a single hourly value
of wind speed, i.e. 10, 20, 30 min,and so on, instead of just the
raw value at 00 min. The results of such procedure areshown in
figure 14.
DTU Wind Energy-Master-Series-M-0047(EN) 33
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00 min +/10 min +/20 min +/30 min +/40 min +/50 min
0.76
0.77
0.78
0.79
0.8
0.81
0.82
Averaging range around minute 00
r2
u-component
v-component
Figure 14: Effect on the correlation of a changing breadth of
the averaging range around 00 m in the
observed time series.
As seen in figure 14, the correlation increases (although very
slightly) with increasingwidth of the averaging range. Indeed, it
is easier for two concurrent averages (overcertain time window) to
correlate well than for just a single point from uobs or vobs(i.e.
at 00 minutes) to correlate well to the concurrent hourly uwrf or
vwrf . Averag-ing smooths both WRF and observed time series, as
will be seen in the next section,causing the correlation
coefficient to increase. In this case, the difference is so
smallbecause the difference in width of the averaging range is very
small as well.
After these sensitivity analyses, the behaviour of the
correlation with respect to atime shift, a rotation and an
averaging is known. The version of the WRF-derived windspeed that
is used henceforth is the one to which a 1-hr time shift has been
applied,to which no rotation has been applied, and to which no
extra averaging is applied (i.e.the instantaneous value of the
10-min average wind speed directly outputted by themodel at minute
00).
34 DTU Wind Energy-Master-Series-M-0047(EN)
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8 The wind climate at Hvsre
From on-site hourly observations spanning the period 20052012,
there is a clear pat-tern at Hvsre: the wind comes mainly from the
North Sea as a northwesterly wind,with a mean speed of 9.3 m/s at a
height of 100 m. Figure 15 shows the wind speeddistribution for
observations and the concurrent WRF output at Hvsre for the
8-yearperiod.
In this section, similarities between observed and WRF time
series will be investigated,for different averaging periods.
8.1 The local wind speed and direction
0 5 10 15 20 25 30 35 400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Wind speed [m/s]
p.d.f
A = 10.5 m/s
k = 2.2
A = 10.47 m/s
k = 2.17
WRFObservations
Figure 15: All-sector histogram and Weibull distribution
function at Hvsre, 20052012. Observations
in blue and WRF-derived wind speeds in red.
Figure 15 shows that the observed and WRF-derived all-sector
wind speed distributionsare in good agreement. Both p.d.f. curves
overlap for all wind speeds. For sector-wiseand yearly
representations, see figures 53 and 54 in Appendix A, which also
show accor-dance between WRF-derived and observed wind speed
distributions (except for sector1 in figure 53, which shows the
wake effect of the test center facility).
DTU Wind Energy-Master-Series-M-0047(EN) 35
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Figure 16 shows the observed and WRF-derived wind rose for the
entire period 20052012, again for the hourly time series. There is
also a good agreement for site and WRFdirections, with three
exceptions. Firstly, northerly winds are smaller in magnitude inthe
observed wind rose than in the reference wind rose, most probably
due to the wakeof the wind turbines North of the mast (WRF does not
take the effect of the turbinetest center into account).
Secondly, there is a slight mismatch in the northerwesterly
winds, probably due tothe fact that Hvsre is a coastal site and, as
mentioned in section 5, small directionmisalignments between WRF
and the observations (in direction sectors with a sea-landboundary)
may cause large and abrupt changes in the roughness which is fed to
themodel, thus affecting the modelled wind speed.
Lastly, observations have slightly higher maxima in wind speed
values than those pre-dicted by the model (this is also shown in
figure 15 but it is not as clear). This is dueto the fact that the
horizontal resolution in the mesoscale model is not small enoughto
correctly predict extreme events, e.g. storms, which contribute to
these wind speedmaxima.
1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
100 m REF 20052012
Wind speed [m s1]
(a)
1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
100 m OBS 20052012
Wind speed [m s1]
(b)
Figure 16: All-year (20052012) reference and observed wind rose
at Hvsre site.
36 DTU Wind Energy-Master-Series-M-0047(EN)
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8.2 The effect of averaging on the WRF-observations
correla-tion
For an overall impression of the observed wind speed it is also
interesting to take a lookat the time series itself for the entire
period 20052012, as expressed through differentaveraging periods,
i.e. hourly, daily, monthly and yearly average wind speed. (Note
thatthe hourly average version of the observed time series comes
from merely picking the00 values; the reference dataset already
comes, on the other hand, as hourly values,as explained in section
6).
1 2 3 4 5 6 7x 104
5
10
15
20
25
30
35
Time [s]
Windspeed[m
/s]
Hourly Daily Monthly Yearly Mean
Figure 17: Observed wind speed at Hvsre, 20052012, as expressed
by different averaging periods.
As seen from figure 17, a time series is smoothed down to
different levels by succes-sively averaging over longer periods of
time; added to this, the longer the averagingperiod, the fewer the
values comprising the time series.
More importantly, averaging both the observed and reference time
series (as in figure17) has a direct impact on the mutual
correlation. Indeed, hourly-averaged WRF andobservations (figure
18) correlate poorly in comparison to yearly-averaged versions
ofthe same time series (figure 21).
DTU Wind Energy-Master-Series-M-0047(EN) 37
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50 100 150 200 250 300 350
5
10
15
20
25
30
35
Time [hours]
Windspeed[m
/s]
WRFObservations
Figure 18: Hourly-averaged observed and WRF-derived wind speeds
at Hvsre (first 120 hours of
January 2005 depicted).
2 4 6 8 10 12 14
10
15
20
25
Time [days]
Windspeed[m
/s]
WRFObservations
Figure 19: Daily-averaged observed and WRF-derived wind speeds
at Hvsre (first 15 days of January
2005 depicted).
Figures 18 and 19 show the same time window in hours and in
days, respectively. Theobserved 40 m/s wind speed storm spike
occurring at hour 192 or day 8 (January 2005)stands out in both
plots and the effect of averaging is most noticeable in figure
19.
38 DTU Wind Energy-Master-Series-M-0047(EN)
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Figure 20 below shows the same spike for month 1 (at the very
left of the x-axis) andit does barely reach 15 m/s.
10 20 30 40 50 60 70 80 906
8
10
12
14
Time [months]
Windspeed[m
/s]
WRF Observations
Figure 20: Monthly-averaged observed and WRF-derived wind speeds
at Hvsre, 20052012.
2005 2006 2007 2008 2009 2010 2011 20128.6
8.8
9
9.2
9.4
9.6
9.8
Time [years]
Windspeed[m
/s]
WRFObservations
Figure 21: Yearly-averaged observed and WRF-derived wind speeds
at Hvsre, 20052012.
Table 3 summarises the effect that averaging the two time series
has on the numberof data points. The table also quantifies how well
observed wind speeds are matchedby WRF simulations, by calculating
the mean of the absolute value of the percentage
DTU Wind Energy-Master-Series-M-0047(EN) 39
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difference between all observed and simulated values (in a
yearly, monthly, daily andhourly basis). This mean percentage
difference or relative error between reference andobservations is
highest on an hourly basis (see figure 18), whereas yearly
averaging,on the other hand (figure 21), shows an apparent biggest
similarity between both timeseries.
Averagingperiod
Mean absolute percentage differenceObservationsWRF [%]
Correlation coefficient r2
ObservationsWRFNumber of pointsin both time series
Yearly 1.07 0.93 8
Monthly 4.47 0.92 96
Daily 18.18 0.78 2922
Hourly 35.17 0.64 70128
Table 3: Absolute value mean percentage difference calculated as
the mean of
100| ((Uobs/Uref ) 1) |. r2 coefficients between reference and
observed time series and num-ber of data points are also displayed,
as a function of different averaging periods. Data taken from
datasets spanning 20052012.
This is however misleading, since it is really the
yearly-averaged values of WRF thatare closest to the
yearly-averaged values of observations: it is therefore important
toexplicitly state which averaging period is being used in an
investigation of this kind,moreover when dealing with correlation
coefficients between observations and WRFsimulations.
Moreover, to describe a long-term wind climatology through its
wind speed distri-bution, it it is not necessary to capture an
hour-to-hour wind speed behaviour. Ofcourse anyone would want the
reference time series to be identical to the observed onefor the
concurrent period, which would imply r2=1, but a reference time
series with alower correlation need not necessarily be worse at
estimating average parameters suchas A, k or P . As seen in table
3, if the concurrent WRF time series is yearly-averaged,the
correlation is high, but the LTC time series comprises just 8
points and thus suffersfrom the biggest loss of information. This
was seen already in section 6.
8.3 A description of each observed year
In section 9, where LTC methods will finally be applied, it will
be important to knowhow similar single years of observations are to
the entire observed period at Hvsre.
A simple investigation on similarity of years is conducted in
this section, and thisis important because a bad LTC whose
correction factors were calculated from someyear in particular
could be attributed to that observed years dissimilarity to the
entireperiod 20052012. Therefore, single estimators were firstly
calculated for each year of
40 DTU Wind Energy-Master-Series-M-0047(EN)
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the hourly observed time series: A, k and wind power density P ,
as seen in figure 22.
05 06 07 08 09 10 11 12
10
10.5
11
A[m/s]
WRF OBS
05 06 07 08 09 10 11 122.1
2.2
2.3
2.4
k[
]
05 06 07 08 09 10 11 12700800900
1000
P[W/m
2]
Year
Figure 22: All-sector yearly reference (WRF-derived) and
observed A, k and P parameters, 20052012.
Figure 22 depicts the three all-sector observed parameters. The
variability around themean is shown in table 4, and it was
calculated as the relative error of the all-year(20052012)
parameter with respect to the mean value of the parameter each
year,
e.g. for year i, the error in the A parameter is Ai = 100(
AiAtot 1)
.
Percentage difference [%]
Year A k P
2005 1.02 0.73 2.682006 5.31 1.38 16.552007 5.09 5.40 17.722008
1.46 5.12 7.672009 4.43 2.44 17.442010 7.09 4.80 29.192011 4.72
0.29 12.862012 2.99 4.61 4.12
Table 4: All-sector percentage difference of yearly observed
parameters with respect to the all-year
(20052012) parameters.
Year 2010 is clearly the outlier in the case of the three
estimators. P shows the
DTU Wind Energy-Master-Series-M-0047(EN) 41
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biggest difference because it was calculated as in equation 14,
i.e. A is cubed.
Years 2006, 2007 and 2009 also present large deviations in mean
power density. Notethat although 2006 and 2007s respective errors
in A are roughly equal in magnitudebut opposite in sign, the fact
that they are consecutive creates a steep 2-year changein P . This
can also be seen, even more acutely, in the case of years 2009 and
2010.
The number of yearly counts above certain wind speeds helps to
explain the differ-ence in year-to-year P . It can be seen, for
example, why year 2010 has such a low P .See table 5.
Observed counts above wind speed:
Year 10 m/s 15 m/s 20 m/s 25 m/s 30 m/s
2005 3528 991 170 24 5
2006 3118 765 110 6 0
2007 3792 1285 273 41 3
2008 3537 1156 234 18 0
2009 3197 677 80 5 2
2010 2978 595 65 0 0
2011 3678 1157 248 32 4
2012 3825 985 146 11 1
Table 5: All-sector observed wind speed counts above certain
values for each year.
As for wind direction, one way to see which of the observed
years is anomalous is byvisual inspection of Hvsres yearly observed
wind roses. From figures 23 and 24 it isclear that, regarding
direction, year 2010 is also anomalous: its wind speed does notcome
mainly from the North-West, but is evenly distributed between
North-West andNorth-East directions. Hvsres 8-year observed wind
rose is shown in figure 16.
42 DTU Wind Energy-Master-Series-M-0047(EN)
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1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2005
Wind speed [m s1]
(a)
1%2%
3%4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2006
Wind speed [m s1]
(b)
1%2%3%4%5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2007
Wind speed [m s1]
(c)
1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2008
Wind speed [m s1]
(d)
Figure 23: Yearly observed wind roses at Hvsre, for height 100
m, and hourly direction time series.
Years 20052008 displayed.
DTU Wind Energy-Master-Series-M-0047(EN) 43
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1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2009
Wind speed [m s1]
(a)
1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2010
Wind speed [m s1]
(b)
1%
2%
3%
4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2011
Wind speed [m s1]
(c)
1%2%
3%4%
5%
W E
S
N
0 - 55 - 1010 - 1515 - 2020 - 2525 - 3030 - 3535 - 4040 - 45
OBS 2012
Wind speed [m s1]
(d)
Figure 24: Yearly observed wind roses at Hvsre, for height 100
m, and hourly direction time series.
Years 20092012 displayed.
8.4 Similarity of concurrent WRF-derived and observed
param-eters
Regarding the LTCs that will be calculated in section 9, it is
also important to deter-mine how similar reference and
observational estimators are, on a yearly basis. Thissubsection is
therefore a check of WRFs ability to describe the observed wind
climateon a year-to-year basis. This is important because, if a LTC
is biased, it might happenthat the concurrent period its correction
factors arose from shows a low similarity be-tween the observed and
reference datasets.
Firstly from a correlation point of view, figures 25 and 26 show
the value-to-valuerelationship of reference and observed speed and
direction, separately for each year.
-
0 10 20 300
10
20
302005
r 2 =0.66
Uobs[m
/s]
0 10 20 300
10
20
302006
r 2 =0.6
0 10 20 300
10
20
302007
r 2 =0.66
0 10 20 300
10
20
302008
r 2 =0.7
0 10 20 300
10
20
302009
r 2 =0.57
Uobs[m
/s]
Uref [m/s]0 10 20 300
10
20
302010
r 2 =0.54
Uref [m/s]0 10 20 300
10
20
302011
r 2 =0.69
Uref [m/s]0 10 20 300
10
20
302012
r 2 =0.62
Uref [m/s]
Figure 25: All-sector hourly observed vs. reference wind speed,
on a yearly basis.
0 2000
100
200
300
r 2 =0.84
2005
dobs[]
0 2000
100
200
300
r 2 =0.81
2006
0 2000
100
200
300
r 2 =0.87
2007
0 2000
100
200
300
r 2 =0.85
2008
0 2000
100
200
300
r 2 =0.85
2009
dobs[]
dref []
0 2000
100
200
300
r 2 =0.85
2010
dref []
0 2000
100
200
300
r 2 =0.82
2011
dref []
0 2000
100
200
300
r 2 =0.82
2012
dref []
Figure 26: All-sector hourly observed vs. reference wind
direction, on a yearly basis.
DTU Wind Energy-Master-Series-M-0047(EN) 45
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Note that r2, in the case of direction (figure 26), has been
calculated taking only theblue values into account, which are those
which represent a difference dobs dref 6200. This filters out, on
average, 4% of all values each year, and has been applied inorder
to avoid the two corner clouds, which would otherwise unfairly bias
the correla-tion. As seen in both figures, the yearly direction
correlation is, when calculated thisway, higher than that of the
wind speed (shown in figure 25).
As for wind speed, it is easy to visually verify from figure 25
that WRF-derived windspeeds matches observations with moderate-high
success on a year-to-year basis. Doesthis mean, however, that the
reference data are representative of the observed data?Table 6
below shows each years hourly r2 (between WRF simulations and
observedwind speeds), but also the percentage difference between
yearly WRF and observedparameters (A, k and P ). Numerically, the
biggest difference between values ofr2 occurs between years 2008
and 2010, with a difference of 20%. At first glance thiscould
explain 2008s P , which is double that of 2010. However, two other
years whichhave equal r2, such as 2005 and 2007, have the second
biggest percentage differencebetween years, namely 570%, showing
that r2s effect on yearly parameter similarity isnot so clear.
Furthermore, the year with the largest correlation coefficient
(2008 withr2 = 0.70) has at the same time the second largest
difference between yearly simulatedand observed power density: P =
2.54%.
The hourly correlation between observed and reference time
series is therefore seento have no connection to the yearly
differences between the two datasets parameters.However, this does
not mean that the difference between yearly WRF and
observedparameters should be trusted over r2, in terms of
representativeness.
All in all, when analysing LTCs in sections 9 and 10, a certain
years odd result will beattributed to either:
1. That years large yearly difference in observed P , with
respect to the averageobserved P in 20052012
2. That years big difference between WRF and observations.
3. That years low r2.
Points 2. and 3. both measure WRFs ability to match observations
but are, as seen,unrelated. Which one of the two has a more
determinant effect on the LTCs will beseen in sections 9 and 10. As
seen at the beginning of this section and in section 6, r2
may be more a measure of the mutual synchronisation between two
concurrent timeseries than a measure of representativeness (Lileo
et al. (2013)).
46 DTU Wind Energy-Master-Series-M-0047(EN)
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Percentage difference [%]
Year A k PHourly correlation coefficient r2
ObservationsWRF [%]2005 0.48 2.67 0.82 0.66
2006 0.11 0.31 0.62 0.60
2007 1.45 1.34 5.49 0.66
2008 0.48 1.09 2.54 0.70
2009 2.36 7.53 1.48 0.57
2010 1.26 3.05 1.30 0.54
2011 1.23 2.32 1.71 0.69
2012 0.97 1.43 1.81 0.62
Table 6: All-sector percentage difference of yearly observed
parameters with respect to yearly WRF-
derived parameters.
The relative difference between yearly observed and reference
parameters was also rep-resented graphically in figure 22, and for
further detail, it is worthwhile looking at itssector-wise version
in Appendix A (figures 55 through 57).
As for the yearly difference between WRF and observed wind
directions, a visual in-spection is carried out in Appendix A
(figures 58 and 59), where the wind roses ofboth are displayed for
each year. Such a study is most important when wind directionis
long-term corrected, and this is done in subsection 9.4.
DTU Wind Energy-Master-Series-M-0047(EN) 47
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9 Results I - Which is the best LTC method?
There is a wide variety of empirical methods with which to
estimate the past long-termwind climatology of a target site. As
already mentioned, LTCs describe the climatologythat could have
been recorded at the target site, had a mast started recording at
thesite much earlier.
The connection between the long-term reference time series and
the short-term obser-vations at the site is the time period where
both datasets are concurrent. The concur-rent period is therefore
as long as the shortest of both time series, i.e. the
observations.The concurrent period used to calculate the correction
factors, however, can be cho-sen in such a way that it comprises
the entire observed time series, or just a subset of it.
Part of the uniqueness of Hvsres data resides in the fact that
the measured setspans a long time: 20052012. Therefore, in this
case, the short-term site observationsare in reality a long-term
set (8 years), which allows for the creation of a wide varietyof
subsets of different lengths and positions within the total
20052012 set.
Thus, following the approach explained in Rogers et al. (2005),
different subset lengthswere defined, starting from just 3 up to 27
months, in steps of 3 months, i.e. 9 dif-ferent subset lengths. For
each subset length, the subset was placed in
successivenon-overlapping positions along the whole period
20052012; for each position, cor-rection factors were computed,
with which to calculate a LTC spanning the period20052012. Finally,
each LTC was validated against what had been actually observedat
Hvsre for 20052012. The optimum position for each subset length was
also de-termined in subsection 9.3, i.e. the position which yielded
the LTC closest to actualobservations.
To better explain the above scheme, it is helpful to take as an
example the concurrentsubset of length 6 months: in this case, as
depicted in figure 27, there are 16 differentpossible positions in
the period 20052012 (16 possible 6-month-long
non-overlappingsubsets); for each of these positions, each LTC
method was applied sector-wise; fromeach LTC obtained for each
subset position, sector-wise and ultimately all-sector pa-
rameters A, k and P were calculated.
48 DTU Wind Energy-Master-Series-M-0047(EN)
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Figure 27: Possible positions of the concurrent reference and
observed subsets, for the case of 6-
month-long non-overlapping subsets, within the total (20052012)
concurrent set.
A set of bias ratios was defined, as in Rogers et al.
(2005):
bA =A
A, (21)
bk =k
k, (22)
and
bP =P
P, (23)
which express the quotient of LTC vs. observed parameters A, k
and P for the period20052012. The next step was to calculate a mean
value and a standard deviation foreach subset length (over all
positions), i.e. bA and (bA) in the case of the A parameter.The
mean and the standard deviation simplified the presentation of the
data and provedenough to determine how many months of concurrent
time are enough for each LTCmethod to produce a successful LTC at
Hvsre and for the period of observations.
9.1 How many months are enough to long-term correct?
In this work in particular, the available observations span the
period 20052012, thusproviding, along with the reference data, 8
years to choose the concurrent time from.Real life projects, on the
other hand, usually have no more than a couple of years
ofobservations; therefore, unless the the observations at hand are
long (3 years or more),chopping them into subsets and evaluating
the effect of changing the subset positionon the LTC may be an
overkill. However, when observations are long as in Hvsre, this
DTU Wind Energy-Master-Series-M-0047(EN) 49
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methodology does reveal how different LTC methods work under
different conditions.
The scheme explained in the introduction to this section was
carried out sector-wise,the sector distribution chosen to have 12
sectors, 30 each, with sector 1 facing North.This configuration is
a common choice in wind power meteorology.
Figure 28: Sector distribution chosen for sector-wise
calculations.
For each subset length and position, each observed subset was
sectorised with respectto its concurrent WRF subset; sector-wise
correction factors were thus obtained, whichwere applied to each
corresponding sector of the entire 20052012 reference wind
speed(long-term reference data had been previously sectorised with
respect to the reference
direction). To convert sector-wise values of A, k and P to
single all-sector values,the procedure explained in Troen and
Petersen (1989) was followed. This all-sectorprocedure is as
follows:
1. Each sectors mean wind speed is calculated, and multiplied by
its sector frequency(i.e. the number of data points). This is
repeated for all sectors and the result isadded. The result is
divided by the total frequency (the sum of all
sector-wisefrequencies) in order to find the all-sector mean wind
speed . (This is a weightedaverage).
2. The sector-wise quadratic mean wind speed, u2 = A2(1 + 2
k
), is calculated, and
weighted over all sectors, as was done with the mean wind
speed.
3. The all-sector parameter 2/u2 is calculated, with which to
solve the equation2/u2 = 2 (1 + 1/k) / (1 + 2/k) and obtain the
value of the all-sector k pa-rameter.
50 DTU Wind Energy-Master-Series-M-0047(EN)
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4. The all-sector A parameter is computed as A = /(1 + 1
k
).
5. The all-sector mean wind power density P is calculated with
equation 14.
Note: the frequency mentioned in points 1. and 2. refers to the
LTC frequency. Theonly LTC method (as implemented in this thesis)
which yields a LTC frequency is theWBL method. No other
non-regression method in this work yields such a result, buta
specific procedure will be explained in subsection 9.4 by which to
use regressionmethods to obtain a LTC direction d. Until then,
however, f is assumed to be equalto the long-term reference
frequency.
Figures 29 through 31 show all-sector mean bias ratios bA, bk
and bP for both re-gression (solid lines) and non-regression
methods (dashed lines).
3 6 9 12 15 18 21 24 27
0.98
0.99
1
1.01
1.02
1.03
1.04
Subset length [months]
b A[]
OLSTLSPR3VARMORKHTNWWWBL
Figure 29: Mean bias ratio bA as a function of the concurrent
subset length (months). For each
subset length, the mean value was obtained by averaging over the
bias ratios found at all possible
non-overlapping positions of the concurrent subset within the
total (20052012) concurrent set.
Figure 29 shows all bA curves within just 1% of the exact match
with observationsfrom 9 months subset length onwards. Only the
non-regression methods MOR and WWshow a larger difference of
average +3% and 3%, respectively, for all subset
lengths.Significant initial drops and jumps are seen for three of
the four regression methods(TLS, PR3 and VAR), as well as for the
non-regression KH method. This suggeststhat LTCs coming from
non-regression methods (except MOR and WW) may depictconcurrent
observations of A more accurately than regression methods for
short-lengthsubsets (36 months). Moreover, two methods clearly
stand out from among the rest:OLS and WBL. These are the simplest
linear and the simplest sector-wise transforma-tion, respectively,
yet show the best corrections of the A parameter.
DTU Wind Energy-Master-Series-M-0047(EN) 51
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It is worthwhile looking at figures 60 through 65 in Appendix A,
which are a sector-wise representation of bA (for regression and
non-regression methods, respectively).Sector 2 in figure 60 shows
the only anomalous value of bA for the regression
methods,specifically for PR3 at subset length 6 months. The
remaining regression methods donot show this behaviour for the same
concurrent subset length, so the cause for thisspike is most
probably the inability of the cubic polynomial to correctly
describe therelationship between WRF and observed wind speeds. The
cubic fit was seen to haveeither explosive or curly shapes for high
wind speeds, but these odd fits are less fre-quent as the subset
length grows, and indeed no anomalous spike can be seen for thePR3
method (for none of the three bias ratios) for subsets longer than
6 months.
In their study regarding the length of reference period to be
taken, Lileo et al. (2013)obtained a very similar shape for their
curve of mean absolute prediction error of themean wind speed, even
though they took the reference period in years.
Regarding the correction of the k parameter, figure 30 also
depicts all the methodsbehaviour as a function of concurrent subset
length.
3 6 9 12 15 18 21 24 270.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Subset length [months]
b k[]
OLSTLSPR3VARMORKHTNWWWBL
Figure 30: Mean bias ratio bk as a function of the concurrent
subset length (months). For each
subset length, the mean value was obtained by averaging over the
bias ratios found at all possible
non-overlapping positions of the concurrent subset within the
total (20052012) concurrent set.
The mean bias ratio bk has on the other hand a wider spread, its
value being con-strained roughly as |bk| < 7% for subset lengths
larger than 9 months, for all methodsexcept MOR, OLS and PR3. Note
that for this thesis, when applying the non-regressionmethods KH,
TN and WW, sector-wise k was assumed to be equal to the
concurrent
52 DTU Wind Energy-Master-Series-M-0047(EN)
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observed subsets k. One would therefore expect to see the red,
green and pink dashedcurves in figure 30 overlapping each other;
this however happens only in the sector-wisedepiction of the bias
ratios, and figure 30 shows all-sector values.
While in Rogers et al. (2005), the VAR method showed that |bk|
< 5% from 6 monthssubset length onwards, figure 30 shows a
smaller difference of roughly 1% from 12months onwards. Figure 30
also shows a perfect match of WBLs the VARs bk, anda steady bias of
roughly +1% for the KH method, from 12 months on. Rogers et
al.(2005) also showed very unbiased results for bk for the MOR
method, which contrastswith the constant bk = 0.8 seen in the
figure. However, the results depicted here forthe regression TLS
method are better than those seen in Rogers et al. (2005) for
theirlinear regression method (which showed a difference of around
constant +40%).
As for the OLS and TN methods, which worked very well in the
correction of A, theypresent on the other hand large deviations for
the k parameter. PR3, which workedwell for A (in the all-sector
case) yields, together with OLS, the worst result witharound +20%
bias for all subset lengths. It also shows the same bias when
representedsector-wise in figure 61.
What matters, however, from a power production point of view, is
how well P iscorrected. This is indeed the crucial parameter in
wind farm assessment. As seen fromlooking globally at all three
figures 29, 30 and 31, and as could be suspected fromequation 14,
for a good correction of P , good corrections of both A and k are
needed.Indeed, all ratios which are biased in the estimation of
either A or k are also biased inP ; but only those which show small
bias ratios in both A and k are truly unbiased inthe correction of
P . i.e. the TN, VAR, KH and WBL methods.
All in all, looking at the all-sector figures above, it can be
seen that bA and bP sta-bilise to a constant value for all sectors
after 912 months. The initial jumps or dropscan be associated to
small subset lengths. However, after this all methods look
quiteinsensitive to increasing length of the concurrent subset,
since the three bias ratios donot vary wildly along the way up to
the maximum length of 27 months (neither insector-wise nor in
all-sector representations).
Also, certain sectors seem to have systematically worse results,
as seen in figures 60through 65; in sectors 1 and 2, the WW method
yields especially biased results forbA and bk, while PR3 fails for
bP and bk. Both these sectors happen to comprise thefewest number
of data points for each subset length: figure 32 shows the
sector-wisefrequency of occurrence, plotted as the mean frequency
of each subset length, over allthe possible positions. Sectors 1
and 2 have the smallest values of mean frequency ffor all subset
lengths. Moreover, this may explain the low sector-wise mean
correlationcoefficient r2 for these two sectors, depicted in figure
33. In the case of sector 1, there
DTU Wind Energy-Master-Series-M-0047(EN) 53
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3 6 9 12 15 18 21 24 27
0.9
1
1.1
1.2
1.3
1.4
Subset length [months]
b P[]
OLSTLSPR3VARMORKHTNWWWBL
Figure 31: Mean bias ratio bP as a function of the concurrent
subset length (months). For each
subset length, the mean value was obtained by averaging over the
bias ratios found at all possible
non-overlapping positions of the concurrent subset within the
total (20052012) concurrent set.
is also a wake disrupting the observations, which could in turn
explain the invertedpattern of r2.
3 6 9 121518212427
200
400
600
Sector 1
f[
]
3 6 9 121518212427
200
400
600
Sector 2
3 6 9 121518212427200400600800
10001200
Sector 3
3 6 9 121518212427
500
1000
1500Sector 4
3 6 9 121518212427
500
1000
1500Sector 5
f[
]
3 6 9 121518212427200400600800
10001200
Sector 6
3 6 9 121518212427200400600800
10001200
Sector 7
3 6 9 121518212427500
1000
1500
2000Sector 8
3 6 9 121518212427500
100015002000
Sector 9
Subset length [months]
f[
]
3 6 9 121518212427500
1000150020002500
Sector 10
Subset length [months]3 6 9 121518212427
5001000150020002500
Sector 11
Subset length [months]3 6 9 121518212427
200400600800
10001200
Sector 12
Subset length [months]
Figure 32: Mean correlation coefficient f [-] as a function of
the concurrent subset length (months).
For each subset length, the mean value was obtained by averaging
over the f found at all possible
non-overlapping positions of the concurrent subset within the
total (20052012) concurrent set.
54 DTU Wind Energy-Master-Series-M-0047(EN)
-
3 6 9 121518212427
0.21
0.22
0.23Sector 1
r2[
]
3 6 9 1215182124270.260.28
0.30.320.34
Sector 2