Introduction
France has one of the largest wind energy potentials in Europe
, but only the fourth largest installed capacity behind
Germany, Spain and the UK. Despite the governmental targets of 19 GW onshore
and 6 GW offshore installed capacity by 2020, the total installed capacity was
only 9.1 GW at the beginning of 2015. The windiest parts of France, where
most of the present wind farms are located, are the northwestern and
southeastern regions. The northwestern region, along the coastlines of the
English Channel, is located in the storm track so that there are often strong
winds coming from the Atlantic Ocean. The southeastern region is located
near the Mediterranean Sea and the valleys of this very mountainous region
channel the wind flows so that there are often strong and persistent
winds. Today, more wind farms are installed in other, less windy areas in
the northeastern and central parts of France. In these regions the mean
capacity factor was below 20 % in 2014 whereas it was on average
between 25 and 29 % in the windiest regions
.
As the wind industry developed, it was observed that wind farms
often produced less than expected, thus jeopardizing the projects'
profitability and undermining the whole industry. This led to concerns about
the overprediction of energy production. Aside from the errors coming
from turbine performance or availability, and from the natural
variability in wind, questions were raised about the methodology used to evaluate the
wind energy yield. One point of the methodology, questioned in this study, is
the common use of the Weibull distribution to model wind-speed statistics
instead of directly using the wind-speed series measured at the investigated
location. The Weibull distribution has become a widely used standard in wind
energy application due to its simplicity. It depends on two easily estimated
parameters and there are simple analytic expressions for the moments. It is a
reference used in wind energy softwares such as the Wind Atlas Analysis and
Application Program (WAsP), and it is included in regulations such as the IEC
61400-12 on wind turbine power performance testing. Many other aspects of the
resource assessment methodology can lead to bad estimates, such as the wind
measurement accuracy, the vertical extrapolation of wind measurements, their
temporal extrapolation with measure–correlate–predict techniques, the wind
flow and wake modeling or the use of inaccurate power curves. Each issue
is complex enough to be the subject of many dedicated research articles, but
this paper is aimed at evaluating the impact of the very step of wind-speed
statistic modeling that uses a Weibull distribution.
Very few studies compared the production of real wind farms to the estimated
production using either the series of wind measurements (called the
chronological method) or the wind statistics from a Weibull distribution fit
to that series (probabilistic method). Despite the widespread use of the Weibull
distribution, it has been shown to not accurately model all kinds of observed
distributions see, e.g., and many other
distributions have been proposed to model the wind-speed statistics,
especially mixed distributions. For a review, see, e.g.,
and references therein. However, in most references,
the influence of modeling the wind-speed statistics using a Weibull or another
distribution is assessed only for the wind energy content (i.e., the cubed
wind speed). Conversely, it has rarely been addressed in terms of power
output,
whereas the results are expected to be drastically different due to the
nonlinearity of the power curve. To the authors' knowledge, the
quantification of errors in the power output was addressed in only five
articles described hereafter
and always for a very limited number of locations (five, one, five and one,
respectively) except for the last one (178 buoys).
studied five wind farms in
northeastern Spain (in the Pyrenees Mountains). They compared the real and
estimated monthly energy production (MEP). They find slight underestimations
from the chronological approach and more underestimation from the
probabilistic approach. The errors due to the use of the Weibull distribution are
important at two stations where, from what appears in the wind histograms at
least, some of the distributions are bimodal. When the Weibull distribution does not
introduce too many errors, the authors underline that it does not come from a
good Weibull fit but from cancelations of under- and overestimation of the
production from the lower, intermediate and upper parts of the wind
distribution. A similar but less complete study is that of
over 48 months at one site in Taiwan. They find
underprediction using the chronological method, except in the low wind
months, and 5 % more underestimation from the probabilistic method
with the Weibull distribution. However, there are no indications of the shape of
the wind-speed statistics. There are also some theoretical studies, using
wind measurements and theoretical power curves, without any real production
data. studied five locations and found errors ranging
from -9 to +7 % in the monthly production using the Weibull
distribution. He notes that there are mostly underestimations at the two
locations with the lowest wind levels. studied
one site in Mexico, in a mountainous area where the wind distribution is
bimodal. They found that the Weibull distribution underpredicted the energy
production by 14 % and used instead a mixture of two Weibull
distributions to better represent the wind statistics.
studied several statistical distributions with
data from 16 locations in the Canary Islands. There are no specific
results for the Weibull distribution but the interesting result is that the
relative errors in power production decreased as the goodness of fit
between the distributions and the observations improved.
studied 178 buoys located around North
America, where the wind speeds were measured at either 5 or 10 m above
sea level. They tested the two-parameter Weibull distribution as well as 13
other distributions. The error in power output was estimated using the theoretical power
curve of a Vestas V47/660 wind turbine. For the Weibull distribution, the
relative errors ranged from -10 to 7 %, while more complex
distributions (four-parameter Kappa, five-parameter Wakeby) gave much better
results. This is an interesting result even if the power curve is not really
suited: indeed, this turbine is supposed to have a hub height of at least 40 m, whereas the measurements are made at less than 10 m.
Therefore,
it probably puts too much weight on the high wind speeds.
The present study investigates the errors made in evaluating the annual
energy production when assuming that the wind-speed statistics follow a
Weibull distribution. It compares different ways of fitting the Weibull
distribution, which is essential because we show how the different
fitting methods lead to very different results. There are many articles
comparing different fitting methods for the Weibull distribution but none of
them quantify the arising errors in the power output. Moreover, we include the
WAsP fitting method, which is by far the most used in the industry but
almost never studied in articles. To our knowledge, this method is only
referred to in and for a completely different subject
of application. As a comparison to the Weibull, we also consider a more
complex mixed distribution, the Rayleigh–Rice distribution suggested in
. It has been chosen instead of a mixture of
Weibull distributions e.g.,, as
report a better description of the tails of the
distributions by the Rice-like distribution.
A limit of most of the studies evaluating the Weibull and other distributions
for wind energy applications is that they stop at the computation of
energy or use inappropriate power curves. Since the relation between the
available energy in the wind and the actual production of a wind turbine is
not at all linear, a good fit for the energy does not guarantee a good
estimation of production. To address this issue, this paper develops a
methodology to compute the production at any location using a realistic power
curve even when using surface measurements.
Another limit of most of the studies cited above is that they study very
small numbers of locations. To address this issue, the present study is based
on a large wind dataset of 89 weather stations, covering different
sub-climatic regions of France. This enables the discovery of some systematic
behaviors: systematic over- or underestimations depending on the wind
characteristics at the location and the fitting method. It enables
emphasis on the link between the goodness of fit of the distribution and the
production estimate errors
thus completing the work of.
Section presents the data and the methodology used to fit the
distributions and evaluate the errors in energy content and production
arising from the statistical modeling. Section presents the
resulting errors at all stations and Sect. discusses these
results. Finally, Sect. concludes the study.
Material and methods
Wind-speed data
In this article we use wind surface measurements (10 m a.g.l) from the
global NOAA ISD Lite database , already used in
and . This compilation of
observations from operational weather stations is the best publicly available
dataset over the considered region.
We use 4 years of measurements made between 1 January 2010 and
31 December 2013. The 10 min averaged wind speeds are recorded every hour.
We select the stations located in France that present a data availability
greater than 97 % over the 4 years, with a minimum of 85 % for
each month to ensure a good representation of the seasonal cycle. We also
keep only the stations with the best precision in the measurements. We
therefore keep 89 stations.
Calm winds correspond to wind speeds equal to zero in the dataset. They
represent 5.2 % of the entire dataset, with disparities among
the stations of course. The calm winds are removed before fitting the distributions
since they are not taken into account in the Weibull distribution. The wind
speeds are binned with intervals of about 0.514 m s-1 because the
data were previously recorded with bins of 1 knot. We add a small random
noise to the wind-speed data in order to remove the effect of this sampling.
The added noise is a continuous uniform distribution between -0.5 and
+0.5 knot (arising negative values are set to zero). We also tested a
Gaussian noise with a standard deviation of 1/3 knot, which led to the
exact same results.
Throughout the article the wind speed is noted as w and the series of hourly
observations is noted as (wi)i=1n, where n is the number of
observations, which would be 35 064 for a complete set over the years 2010
to 2013, but is fewer due to the missing values and removed calms. In
the following, the results are given for the 4 years of data but remain
similar when limiting the data to 1 year.
Wind-speed statistic models
We use two distributions for the wind speed: the commonly used Weibull
distribution and the Rayleigh–Rice distribution, defined in
. We note f their probability density
functions (PDF) and F their cumulative distribution function (CDF). We
explain here the different fitting methods for each distribution.
Weibull distribution
The Weibull distribution depends on the scale parameter A>0 and the shape
parameter k>0. Its PDF and CDF expressions are
fwbl(w;A,k)=kAwAk-1exp-wAkFwbl(w;A,k)=1-exp-wAk.
The Weibull distribution has simple expressions for its moments, such as the
average wind w‾ and the energy content w3‾:
w‾=AΓ1+1kw3‾=A3Γ1+3k,
where Γ is the gamma distribution defined by Γ(x)=∫0∞e-ttx-1dt.
There are many ways of fitting the Weibull distribution to a set of
observations; see for example or
for extensive comparison of some of the methods. In
this article we compare the three methods that are expected to be the most
used in the wind industry:
the maximum likelihood estimation (MLE), which maximizes the log likelihood function ;
the method of moments using the first and third moments (M1 & M3), which solves the set of Eqs. () and ();
the method used in WAsP using the third moment and the probability of
winds above the empirical mean wind speed . In WAsP, the data are divided into several direction sectors and one distribution is fit for
each sector. This is not the case here; there is no division according to the wind direction.
In each case, the first step is to iteratively solve a nonlinear equation
for k (the shape parameter) and, once k is known, to compute the value
for A (the shape parameter) from a simple relation.
Table gives those equations, with the following
notations for the observed mean, w^=1n∑i=1nwi ;
observed energy content (third moment), w3^=1n∑i=1nwi3 ; observed raw moment of order k, wk^=1n∑i=1nwik ; observed probability of winds above the
mean, p^=1n∑i=1n1{wi>w^}.
Sets of equations used to fit the Weibull distribution for each
method (see text for notations).
Method
Nonlinear equation to solve for k
Equation for A
MLE
∑i=1nwikln(wi)wk^-∑i=1nln(wi)-nk=0
A=wk^1/k
M1 & M3
w^3Γ1+3k-w3^Γ31+1k=0
A=w3^Γ1+3k1/3
WAsP
ln-lnp^-kln(w^)-13ln(w3^)+13Γ1+3k=0
Rayleigh–Rice distribution
The Rayleigh–Rice distribution is a mixture of a Rayleigh distribution
(parameter σ12) and a Rice distribution (parameters μ≥0 and
σ22) weighted by a parameter α (0≤α≤1). Its
PDF expression is
frr(w;α,σ12,μ,σ22)=αwσ22exp-w2+μ22σ22I0wμσ22+(1-α)wσ12exp-w22σ12,
where I0 is the modified Bessel function of the first kind and zero order.
There is no simple analytic expression for the CDF; therefore,
Frr is computed by numerical integration of frr
for a given set of parameters.
The Rayleigh–Rice distribution is fitted as in
by minimizing the right-tail Anderson–Darling statistics (Rn2)
(defined in ), calculated
by
Rn2=n2+2∑i=1nzi-1n∑i=1n(2i-1)ln(1-zn+1-i),
where we note zi=F(wi) for the series of observations
(wi)i=1n sorted so that w1≤…≤wn.
The minimization of Rn2 is solved by a Nelder–Mead algorithm to find the
best parameters. It is a little difficult to converge because there are four
parameters, among which the α parameter has a nonlinear effect. To
overcome this, we first fit the distribution for only three parameters and a
fixed value of α, repeat this for a series of different α
values and choose the best of all fits. This best fit is then used as a
first estimate to fit with four parameters and it converges rapidly.
Energy estimation
We study the available energy content E, i.e., the cube of the wind speed,
and the production P, i.e., the energy yield from a wind turbine. E and
P are computed for each distribution using their density function
(probabilistic method) and compared to the reference value based on the
series of observations without any statistical modeling (chronological
method).
Power curve
In order to compute the power production, we use a power curve derived from a
Vestas V90/2000 wind turbine, which has a 90 m diameter rotor and 2 MW
nominal power. This model is one of the most common turbines in France as
well as the rest of the world. The problem is that the hub height of such a wind
turbine is typically 100 m whereas we use wind measurements at 10 m.
Moreover, using a single power curve for all the stations is not possible
because the stations have very different average winds; this would lead to an
unrealistically large production at some stations and almost no production
at others.
Therefore, we use a flexible power curve Pa(w) depending on a
parameter a to adapt to the wind characteristics at each station. The
initial power curve is transformed linearly so that it is equivalent to
multiplying the wind speeds by the value of a. This can be seen as a
vertical extrapolation of the surface wind. For example, a value a=1.35
corresponds to the coefficient that would be used in an extrapolation of the
10 m wind speeds to the altitude of 85 m using the one-seventh power law. As we
normalize the power curve, it could also be seen as using a smaller wind
turbine adapted to lower winds.
The initial V90/2000 power curve is drawn in Fig. (in
red) as well as a modified one with parameter a=1.35 (in dashed blue). At
each station, the a parameter is adjusted so that the capacity factor of
the turbine reaches 30 %. The real production would be a little lower
since we removed the calm winds, and because we use an ideal power curve and
do not account for any losses in the production process.
Example of the power curves used to compute the power production
P. In red (P1) the power curve of a Vestas V90/2000 wind turbine
normalized by its rated power (2 MW) is shown. In dashed blue
(P1.35) the
power curve adapted from the previous one by a linear transformation of
factor a=1.35 is shown.
Maps of ΔE, i.e., the relative errors in wind energy content
due to the statistical modeling, at the 89 stations for either the Weibull
distribution fit by maximum likelihood (a) or the Rayleigh–Rice
distribution (b).
Energy
The reference energy computed from the series of observations (wi)i=1n
is
Eref=1n∑i=0nwi3.
The energy computed from f, the PDF of a distribution fitted to the
observations, is
E=∫0∞f(w)w3dw.
The error in energy of the probabilistic method (Eq. ) relative
to the chronological method (Eq. ) is
ΔE=E-E refE ref.
Maps of ΔP, i.e., the relative errors in wind power production
due to the statistical modeling, at the 89 stations for either the Weibull
distribution fit by the maximum likelihood method (a), the first
and third moments method (b), and the WAsP method (c),
or the Rayleigh–Rice distribution (d).
Production
Based on the power curve Pa, the reference production is
Pref=1n∑i=0nPa(wi).
This is the mean power output in watts but since we normalized Pa
by its nominal power, Eq. () actually gives the mean capacity
factor. The a parameter of the power curve is adjusted so that the capacity
factor reaches 30 %. This value of a is then used to compute P from
the four distributions.
The average power computed from the probabilistic method from the PDF f is
P=∫0∞f(w)Pa(w)dw.
The error in production of the probabilistic method relative to the
chronological method is
ΔP=P-P refPref.
Results
The errors in the energy assessment are computed for four distributions: the
Weibull distribution fitted by the three different methods and the
Rayleigh–Rice distribution.
Energy
In terms of energy content E, both the method of moments and the WAsP
method for fitting the Weibull distribution make no error since the energy
content is the third moment w3‾ and it is fixed to the observed
energy content when solving Eq. () to find the A and k
parameters. For the other distributions, the results are shown in
Fig. . With the Weibull distribution fitted by maximum
likelihood (Fig. a), the errors are low in the northeastern
region but much larger in the southern region. The absolute errors are above
5 % at 32 stations and above 10 % at 10 stations (the maximum
being 29 %). The energy is almost always underestimated, except in
some places in the valleys of the southeastern region. With the
Rayleigh–Rice distribution (Fig. b), the errors are in
general very low, with some exceptions. The absolute errors are below
3 % at 80 % of the stations; only 10 stations are above 5 %
(including 2 above 10 %).
Representation of the energy and production calculations at Melun
station (2010–2013). (a) Histogram of the observed wind-speed
series and probability density function f(w) of the fitted distributions
(Weibull fit by MLE, moments, or WAsP methods and Rayleigh–Rice fit).
(b) Wind energy content as a function of wind speed w (i.e., f(w)w3) for each distribution and associated histogram for the observations.
(c) Power curve Pa adapted to the station in order to
have a capacity factor of 30 %. (d) Wind power output as a
function of wind speed (i.e., f(w)Pa(w)) for each distribution
and associated histogram for the observations.
Production
When it comes to the power output P, errors arise for all four cases, even
when there was no error in the energy content. Indeed, the fact that ΔE=0 does not mean that the observed distribution is well fitted by the
Weibull distribution. There may be positive and negative errors balancing one
another when integrating over the whole distribution. Since the power curve
is not a linear function of w3, these errors do not balance anymore in the
power calculations and this may lead to large values of ΔP.
For the Weibull distribution fitted by maximum likelihood
(Fig. a), ΔP is of the same order of magnitude but of
opposite sign as ΔE (Fig. a). The mean absolute error
(MAE) is 5.2 %, two-thirds of the stations have an absolute error above
3 % and the maximum error is 32 %. With the method of moments
(Fig. b), the errors are similar to those for MLE; the spatial
pattern is the same, the values are just slightly lower (MAE of 4.3 %,
maximum error of 17 %). Conversely, with the WAsP method the
errors are small, the MAE is 1.7 %, only 13 stations have an absolute
error above 3 % and the maximum error is 9 %. Finally, with the
Rayleigh–Rice distribution, we find small errors everywhere, with a slight
bias towards overestimation: the errors range from 0.2 to
3.1 %, with an average of 1.4 %.
Sensitivity of the results
The figures above are given for the whole dataset (years 2010 to 2013). The
results are similar when limiting the wind series to only 1 year, with only
small differences due to the interannual wind variability. For the
computation of the production, the power curve has an important role. We
tested other shapes, among which were simpler power curves with a ramp between
cut-in and rated wind speed as a linear function of either w or w3. They
all led to similar results. Choosing a different capacity factor, such as
25 or 35 %, to adjust the a parameter of the power curve has
also a small impact. The errors tend to be smaller when using a larger
capacity factor, except in the case of the MLE method where the errors tend
to increase. When the capacity factors are much lower, the errors of the
Weibull MLE switch to negative values. This could represent what happens in
seasons with low wind. This is consistent with the findings of
, where underprediction of the production mainly
appears in the least windy locations and months.
We also tested the sensitivity of the results to the sampling of the wind
data. We added a small random noise to the wind data because the speeds were
binned with a 1-knot interval. When fitting the distributions to the raw
sampled data instead of the smoothed data, the results are very similar
except for the WAsP method. In that case, fitting the Weibull to the raw data
leads to large negative or positive errors (MAE around 5 %), without
any spatial coherence. It is unclear why.
Discussion
Examples at two stations
To better understand these results, we first focus on two example stations:
Melun, which is located in northern central France, near Paris
(48.6∘ N, 2.7∘ E), and Orange, which is located in the
Rhône River valley between the Alps and the Massif Central Mountains, close to
the Mediterranean Sea (44.1∘ N, 4.8∘ E). The wind-speed
histograms at these two stations are shown in Figs. a
and a, respectively, as well as the PDF of the Weibull
distributions (fitted by the three different methods) and of the
Rayleigh–Rice distribution. At Melun the histogram is more peaked than can be
modeled by a Weibull distribution. This is a common behavior at most
stations, which is very pronounced at some locations in southern France. At Orange,
the distribution is bimodal; there are two peaks and this cannot be modeled
accurately by the unimodal Weibull distribution, but can be modeled by the
more flexible Rayleigh–Rice distribution. This type of bimodal distribution
is found at several locations in the southern valleys of France.
Representation of the energy and production calculations at Orange,
representative of a bimodal case. Same description as Fig. .
The computation of energy is very sensitive to the adjustment of the right
tail of the distribution since the very high winds, once cubed, have an
important weight despite their low frequency. The Weibull distribution,
especially the maximum likelihood fit, tends to underestimate the frequency
of these very high winds and therefore underestimate E. This phenomenon is
visible in Fig. b for Melun in the range 9–15 m s-1.
Most stations, especially in the southern part of France, present such an
underestimation of the very high winds by the Weibull distribution, with
different magnitudes.
At some other locations, such as at Orange, the wind-speed distribution has
two peaks. In that case, the Weibull distribution, which cannot model two
peaks, passes through both: it underestimates the wind frequency at the two
peaks but overestimates the winds in between the two peaks and the very
high winds beyond the second peak. At Orange, we can see that the Weibull
distribution, whatever the fitting method, overestimates the probability of
winds above 15 m s-1 (Fig. a) and therefore their
contribution to the energy (Fig. b). With the moments and WAsP
methods, this overestimation is smaller and balanced by the underestimation
in the range 8–15 m s-1 (corresponding to the second peak). In the
case of the MLE method, it is not completely balanced and it leads to an
overestimation of the energy by more than 10 %.
When it comes to the estimation of the energy yield from a wind turbine, the
very high winds are not so important since the power output is constant
between the rated wind speed and the cutout wind speed of the wind turbine.
The power curves used at Melun and Orange are drawn in Figs. c
and c, respectively, and the power output is shown in Figs. d
and d. Nevertheless, an underestimation of the very high winds
is associated with an overestimation of winds around the rated wind speed,
which have the largest contribution to the production. This is why we get
opposite errors in E and P with the Weibull fit by MLE. With the WAsP
method, the Weibull fit is better adjusted to the high winds: we can see
clearly in Fig. b that there is less underestimation for the
winds above 8 m s-1 and less overestimation for the winds below
8 m s-1 than with the two other methods. Therefore, the errors in P
are much reduced in most cases.
In the bimodal cases, the overestimation of the very high winds is
associated with an underestimation of middle-high winds and therefore an
underestimation of the energy yield. This is particularly critical at
Orange, where the winds with the largest weight in the production are
exactly around the second peak, largely underestimated by the Weibull
distribution. Finding an underestimation of the production in a bimodal case
is consistent with the literature, such as the Mexican case of
, and at least some of the largest
underestimations found in (the wind
histograms are only shown for some cases).
Importance of the goodness of fit between the observed and theoretical distributions
The fact that modeling the wind-speed statistics using the Weibull distribution
introduces errors in the energy estimate can be related to the poor fit of
this distribution to the tail of the observations, i.e., the high wind speeds,
which contribute a lot to the energy. It was shown in
that the Weibull distribution does not fit the tail well. Conversely, the Rayleigh–Rice distribution was shown to
have good agreements on the tail. Indeed, here the few locations where we find
large errors ΔE with the Rayleigh–Rice distribution (see
Fig. b) are the stations where the very high winds are not
well fitted by the Rayleigh–Rice.
The errors in the production are also related to the goodness of fit between
the distributions and the observations, which can be measured for example by
the right-tail Anderson–Darling statistics (Rn2, Eq. ).
Figure shows ΔP (in absolute value) as a function
of Rn2. With the Weibull distribution fitted by maximum likelihood
(squares) and by the method of moments (circles), |ΔP| values are
highly correlated with Rn2 (Pearson correlation coefficient of 0.9).
This can be linked with , who find that the error
in the production decreases when the quality of fit of the distributions
increases.
With the WAsP method (diamonds), the relation is weaker (correlation of
0.36) because this method does not necessarily give a very good fit to the
whole distribution (thus Rn2 values may be higher than for the MLE method) but
favors a better fit to the range of winds that are important for the energy
production (thus ΔP values are lowered).
Relative error in the wind power production, ΔP (in absolute
value) as a function of the right-tail Anderson–Darling score, Rn2, used
as a goodness-of-fit estimate for the Weibull distribution. The distribution
is fit by the maximum likelihood (squares), moments (circles) and WAsP (diamonds)
methods at each station. Logarithmic scales.
Conclusions
In this article we investigated the errors in the
wind resource assessment that could result from the use of a statistical
model, especially with the commonly used Weibull distribution. We showed
the importance of evaluating the errors in the production instead of the
energy (i.e., the cubed wind speed), as it is mostly done in the literature.
Indeed, a perfect fit to the energy does not guarantee that the
distribution really fits the observed data. It may come from the cancellation
of opposite errors and may lead to large errors in the production due to the
nonlinear effect of the power curve. Furthermore, the energy content is not a good
indicator of the goodness of fit of a distribution because it puts too much
weight on the tail of the wind-speed distribution, which actually contributes
very little or not at all to the energy production due to the shape of the power
curves.
We found large errors in the production (MAE around 4 or 5 %) when
modeling the wind-speed statistics using a Weibull distribution fit with either
maximum likelihood estimation or the first and third moments method. We
found lower errors with the WAsP method, which is reassuring for the wind
industry since WAsP is among the most commonly used software programs in wind
resource assessment. Still, even this method may lead to important errors at
some locations so we advise against the use of the Weibull distribution.
Apart from the WAsP method, the Weibull fits lead to an overestimation of
the production at most locations in France. This bias could have contributed
to the observed overestimation of the production. We also found more errors
in the areas closer to high topography, where the Weibull distribution is
less adapted, and also a tendency towards more errors when the capacity
factors are lower. As these two conditions correspond to the new areas
targeted by the wind industry in France, these are more reasons not to use
the Weibull distribution in the future.
The Rayleigh–Rice distribution shows very good skill at predicting the energy production
at all locations. The fit is always very close to the observations over the
whole distribution. Therefore, the errors are always very small for whichever part
of the wind distribution is used for the production. There is a slight
overestimation, but it is not problematic since this bias is systematic and
could be anticipated. However, even with such a good distribution, the very use of
a statistical model is questionable. The Weibull distribution was very
useful in the early age of the wind industry because it
simplified the computations considerably. Today computers can handle very long wind
series without any problem, limiting the need for any modeling when there
are actual measurements. Indeed, the measurements are much more precise and
contain much more information than any two- or four-parameter model.
This study benefits from the use of a large dataset, covering all regions of
France. The drawback of this dataset is that they are surface measurements,
often located in areas not actually adapted to wind project development. The
results apply to all the studies using surface measurements, and could be
followed by more precise studies at precise locations, using real wind
project measurements at higher levels above ground. A question is whether or
not the wind distribution varies a lot with the altitude and whether the
shapes are closer to the Weibull distribution higher up.