Introduction
Microscale flow models have been employed for decades in wind
energy assessment to estimate resources at one location based on wind
measurements at a different site . Furthermore, it has become
increasingly popular in the past decade to use mesoscale model output to
drive microscale models for the same purpose e.g.,.
Such flow modeling relies on characterization of the surface, including
terrain elevation and surface roughness. As input to atmospheric flow models,
both terrain elevation and roughness have uncertainties associated with their
assignment. In practice, terrain elevation uncertainty tends to be dominated
by the resolution of elevation maps
e.g.,. In contrast, there are a number of
significant uncertainties associated with roughness, which do not
(necessarily) depend on resolution; these include determination of roughness
length z0 from measurements and assignment of z0 in industrial
practice (based on land use, terrain type, and/or experience, for example). Overall,
uncertainty related to roughness tends to be dominant over elevation-related
uncertainty, particularly in wind-energy applications. In this work we
develop a practical treatment of the effect of roughness uncertainty upon
wind resource estimation, providing a formulation for estimation of
roughness-induced uncertainty in annual energy production.
First we review the definition of roughness length, introducing and
demonstrating the statistical character of z0, i.e., distributions
of z0 from measurements and the behavior of such; we statistically connect this
to a practical uncertainty metric. Then we present the
theoretical framework that is used for wind resource estimation
based on the geostrophic drag law as used in the European Wind
Atlas (EWA) methodology; and including its relation to roughness.
In Sect. we introduce uncertainty; this includes
basic characterization of the uncertainties inherent in (1) the roughness
definition and observed distributions of z0 (Sect. )
and (2) the variations in z0 prescribed in the wind energy
industry (Sect. ).
We continue by showing how uncertainty in the background roughness
can be translated into uncertainty in predicted wind distributions,
within the European Wind Atlas framework (Sect. ); here we
provide derivations of the sensitivity of predicted winds to input
roughnesses at observation and prediction sites.
Consequently, we examine the effect of user-assigned biases in roughness
assignment and more generally the combined effect of (independent)
roughness uncertainties on predicted wind speeds.
For practical use we also develop an analytical relation between
rated power, mean wind speed (Weibull-A parameter), and AEP; this is
accomplished via convolution of a generalized analytical power-curve form
and Weibull wind distribution. Thus, we translate z0 uncertainty into
uncertainty of annual energy production (AEP).
Though there are different methods possible for determining or calculating
roughness length, we concentrate here on the propagation of uncertainty in
background roughness to predicted wind speeds and annual energy production.
More details about and issues arising from alternate methods of roughness
length calculation are beyond the scope of this article and are the basis of
concurrent work to be included in a separate paper(s).
Lastly, we discuss approximate roughness uncertainty magnitudes expected
in practice and the consequences of them. This also includes, for
example, the result that sites with larger background roughness
tend to give larger relative uncertainty (i.e., %) in predicted
wind speeds and significant uncertainty in AEP.
We also discuss implications for the use of mesoscale simulation data
for driving microscale models, i.e., generalization of wind statistics.
Basis and framework
Physically, this work simply considers the use of wind measurements
(statistics) at some height above ground level at one location in order to
predict wind statistics at another location and height. Starting with ideal
(uniform flat) terrain, this prediction can be broken into components,
commonly labeled within the wind resource assessment community as vertical
and horizontal extrapolation. Subsequently, the theoretical
foundation of this work involves the two basic components related to the
physics modeled by such extrapolations: these are the wind profile for
vertical extrapolation and the geostrophic drag law (GDL) for relating
the wind statistics at different sites; they are covered in
Sect. and , respectively. The vertical wind profile
form (of which the simplest is the logarithmic law) requires a surface
roughness length, and the GDL also requires a characteristic (background)
roughness length. Because we wish to relate uncertainty in roughness to
uncertainty in wind energy estimates, i.e., finding the uncertainty in
accounting for the effect of the surface, we first begin by examining
roughness length, both in theory (i.e., definition) and in practice (e.g., its
statistical character).
Roughness length: theory and practice
The concept of roughness length began with characterization of the velocity
profile in ideal engineering flows (e.g., pipes), where roughness has a direct
physical interpretation ; it was further adopted
to describe the wind profile in the atmospheric surface layer (ASL), whereby
it has an implicit (and not directly physical) definition
. The basic role of roughness length and its
definition, can be seen through the ideal expression for the mean wind
profile U(z) over a homogeneous flat surface in neutral conditions (without
thermal stability effects):
U(z)=u*κlnzz0.
In Eq. () z0 is the roughness length and z the height
above (distance normal to) the surface, expressed in the same units; κ
is the von Kármán constant, generally accepted to be 0.4
. The friction velocity u* is defined by u*2≡-〈u′w′〉, i.e., as mean momentum transport towards the
surface through turbulent stream-wise (u′) and vertical or surface-normal
(w′) velocity fluctuations. The roughness z0 can also be seen as an
integration constant since Eq. () results from integrating
dU/dz=u*/(κz); the latter is typically derived via
dimensional analysis, through the Buckingham Pi theorem
e.g.,. The logarithmic wind profile
(Eq. ) depends upon a number of assumptions: u* is effectively
constant from the surface up to height z i.e., du*2/dz≪σu2/ℓABL, the surface is
flat and uniform, there is horizontal homogeneity (no variations parallel to
the surface), there is no height dependence in the forcing of the flow, and
there are (no effects due to) temperature variations, i.e., the only variables
determining dU/dz are u* and z.
Calculation of roughness length from wind measurements
From Eq. () one can see that for U measured at two heights
{z1,z2}, the roughness can be calculated by
ln(z0)=U(z2)lnz1-U(z1)lnz2U(z2)-U(z1).
While one can also obtain the roughness via the shear exponent
e.g., that is often used in wind energy,
Eq. () does not involve approximations and directly follows
from the definition of roughness. One can also use friction velocity measured
in the surface layer and wind speed from one (or more) height(s) to derive
roughness (e.g., z0=zexp[-κU(z)/u*] from Eq. ), but
doing so requires sonic anemometers, which are not yet commonly used in the
wind energy industry. Thus, we use Eq. () for the observed
roughness data analyzed and shown in this paper, and leave alternate
z0 estimation methods for concurrent work and dissemination that focuses
solely upon roughness. This choice is further supported by the focus of the
present article – we are concerned here with the impact of roughness length
on wind energy estimates – and because we develop and use an
uncertainty-estimation framework that is generally applicable to z0,
regardless of whether z0 is derived from Eq. () or via
U/u*.
Roughness as a statistic
Even in seemingly ideal conditions – such as measuring wind profiles in the
surface layer at a site where the terrain is flat and appears uniform, with
non-neutral cases excluded – in practice one still observes a broad range of
roughnesses. This is demonstrated in Fig. , which shows the
roughness length calculated from 10 and 40 m measurements at the Danish
National Wind Turbine Test Station at Høvsøre for upwind directions
corresponding to flat and homogeneous surfaces (east of the meteorological
measurement mast). Here we have filtered out non-neutral conditions by
keeping only cases unaffected by thermodynamic stability by using
z/|L|<0.001, i.e., for Obukhov lengths L much greater than the heights of
measurement.
(a) Distribution of z0 for homogeneous land sectors
(30∘ wide) east of Høvsøre. (b) Joint distribution of
z0 and wind direction ϕ; darker represents most common values, and white
is no occurrence. Calculation follows Eq. (), with z1= 10
and z2= 40 m, and it is limited to neutral conditions (|L|-1<0.001 m-1). (c) Visual map east of site (red pointer;
southern border of homogeneous zone at ∼ 130∘ denoted by
yellow line).
Figure starkly demonstrates that even at a homogeneous,
well-studied, and presumably simple site, roughness length has a distribution
of significant width. Note that we plot the distribution of roughness length
in logarithmic space; this is done because it is ln(z0) that directly
affects the wind profile, as in Eq. (). This also highlights the
breadth of the distribution (several orders of magnitude) and that we must
subsequently approach roughness uncertainty in a multiplicative
(dimensionless) way and not in an additive way. We also remind that the
roughness lengths generally used in wind flow modeling and resource
assessment actually correspond to some geometric mean, which
should be based on the z0 distribution alternately one can
express wind profiles in terms of the distribution P(lnz0) and
corresponding arithmetic mean; cf.,; unfortunately z0
is not (yet) defined explicitly as such in typical wind engineering
practice. Thus, in this paper we focus on roughness uncertainty within the
“implied mean-roughness” framework implicit in standard wind engineering.
In addition to the relatively wide distribution apparent for roughnesses
obtained from 30 min averages shown in Fig. (and slightly
wider for 10 min averages, not shown), one can also see some local – and
nonideal – details. One sees the minor effects of a barn and a small
building located roughly 800 m upwind at ∼ 80 and
∼ 110∘,
respectively, as well as the larger effect of the seasonally varying
marsh–fjord coastline 800–900 m to the southeast
(∼ 130–135∘). Such roughness changes tend to violate the
assumptions behind the logarithmic profile over a range of observation
heights falling within the nonequilibrium internal boundary layer (IBL)
transition region . The more drastic semi-coastal roughness change
contaminates the shear measured between 10 and 40 m enough to give the
larger apparent z0 shown in Fig. b as ϕ→135∘ and the subsequently wider distribution P(z0) shown in
Fig. a for the 120∘ sector.
Because neutral conditions tend to be encountered most often stability
distributions have their peak around L-1=0; see ,
the distribution of shear exponent P(α) can also be related in terms
of an effective roughness length without filtering stability to exclude
non-neutral conditions . Thus, the wind profile can
indeed give information about the surface, though the shear at higher z
includes the effect of increasingly more terrain further upwind (potentially
including hills as well as roughnesses).
Avoiding substantial changes in surface characteristics and/or land use,
this can be useful towards the aim of gauging background z0.
One can also calculate a more local roughness length
via Eq. () using measurements of U and u* within the surface
layer (filtering out non-neutral conditions via measured heat fluxes), but
doing so requires sonic anemometers, which are not (yet) commonly used in the
wind industry. For example, using U and u* measured at z= 10 m for
the case above gives P(z0,ϕ) that is insensitive to the inhomogeneities
described above, i.e., it does not jump as ϕ increases above
∼ 130∘. Although the resultant z0(U/u*) tends to better
conform to the assumptions behind surface-layer theory and
Eq. (), it is consequently limited to ASL heights – which in
stable conditions (e.g., nighttime, winter) only extend to ∼ 10–20 m.
Furthermore, the z0 derived from U/u* in the ASL is local, only pertaining
to the nearest several hundred meters, perhaps less in stable conditions.
However, the widths of P(lnz0) derived from U/u* (not shown) are on
par with those obtained from U at two heights and displayed in
Fig. .
Thus, in the present article concerned about uncertainty, we do not address
the implications of surface-layer theory nor its conditional violation, but
rather focus on the effect of roughness uncertainty – as it would be
measured (or assigned) in industrial practice – upon resource assessment,
particularly through horizontal extrapolation from an observation mast to
a separate turbine location(s).
Geostrophic drag law: European Wind Atlas method
The geostrophic drag law (GDL) allows wind statistics observed at one site to
be applied at potential wind farm sites nearby that may have different
surface characteristics (i.e., roughness and terrain elevation); it is the
basis of the EWA method used widely for
wind resource estimation. The GDL arises from matching the dimensionless
surface-layer profile of mean wind in neutral conditions (i.e., the log law
divided by u*) to dimensionless solutions of the mean horizontal equations
of motion away from the surface, as affected by the Coriolis force
. The mean atmospheric boundary layer
(ABL) flow is driven by a large-scale mean pressure gradient ∇P,
also expressible as the geostrophic wind G≡-k^×∇P/(fρ)={-∂P/∂y,∂P/∂x}/(fρ), where k^ is the vertical unit
vector and f is the latitude-dependent Coriolis parameter; the pressure
gradient force is balanced (vectorially) by the Coriolis force and momentum
transfer to the surface. Thus, the GDL essentially relates the large-scale forcing
(expressible as the geostrophic wind above the ABL) to the surface-layer
momentum flux (friction velocity), depending on the surface roughness.
The geostrophic drag law can be simply expressed in scalar form
as
G=u*κlnu*/fz0-A02+B02,
where A0 and B0 are empirical constants (taken by the EWA to be
1.8 and 4.5). Thus, for two sites that can be assumed to have
the same large-scale forcing (distribution of G), then the wind
statistics at one site can be translated to wind statistics at the other.
From the wind profile relation (Eq. ) one can obtain u* from
measured U over one roughness z0,1, and subsequently G from
Eq. (); then at the prediction site one can solve Eq. ()
to get u* at a potential turbine site and subsequently find U there
over a roughness z0,2. Below, we will show the impact of roughness
uncertainty upon wind speed and AEP estimates via Eqs. ()
and ().
Conclusions
First we review the context of this work, i.e., the EWA method , which employs the
geostrophic drag law (Eq. ) to perform horizontal extrapolation:
mean wind speed measured at a site with some background roughness(es) can be
used to predict the mean wind at another location with potentially
different surface characteristics, assuming the sites are forced by the same
pressure gradient (geostrophic wind). For separate measurement and/or prediction
sites where the EWA method is valid, resource assessments that
account for background roughness length (z0) tend to be better than
assessments that ignore z0 such as those based only on observed
shear exponent; see. This is especially true for sites
in terrain with different background roughness; consequently, the EWA method
has been used in wind energy for decades. The need for and justification of
this method is also implied by Fig. , which displays the
sensitivity of EWA-predicted winds to turbine-site roughness (z0,2); it
can thus also be used to show how much the predicted mean wind changes due to
z0,2 differing from the measurement-site roughness z0,1. For
z0,2/z0,1 deviating significantly from 1 (taking the x axis of
Fig. a as this ratio), a significant ΔUpred can result, and the EWA method is needed to account for
such. One can see that if z0,2 differs from z0,1 by a factor of 5,
the predicted mean wind may be affected by ∼ 5–25 %; subsequently,
the AEP could change by a factor of up to ∼ 2.5 times this, i.e., as
much as ∼ 60 %.
Using the EWA method, uncertainty in z0 leads to uncertainty in resource
predictions that can be significant, as shown in Sect. . Both
user-implicit (Sect. ) and definition-related
(Sect. ) uncertainties in roughness length are found to
effectively be (treatable as) roughly of the same order of magnitude, and
they lead to an uncertainty in prediction of mean wind speed and AEP. The
uncertainty in prediction is slightly more sensitive to measurement-site
roughness z0,1 than prediction-site roughness z0,2, as seen in
Eqs. ()–() and displayed in
Figs. –. However, there is also a minor
dependence on measurement and prediction heights via the vertical wind
profile used within the EWA method (log law implicit in Eqs. ,
); shown by Fig. in
Appendix .
As mentioned in Sect. , even in ideal (steady, neutral)
conditions, the mean roughness 〈z0〉 obtained from
observations and Eq. () via different calculation methods in the
surface layer, such as using wind speeds at multiple heights or alternately
wind speed with friction velocity, differs by an amount that appears to
greatly exceed the uncertainty derived for any given method. For example,
bootstrapped distributions of 〈z0〉 for the homogeneous flat
grassland sectors at Høvsøre had relative widths (approximate
uncertainty) well under 10 % when using Eq. () and
Uobs in the surface layer, whether calculated with or without
u*; however, the ratio of the means (or peaks of P(〈z0〉)) from
the different calculation methods was roughly 3. In contrast, the uncertainty
of z0 estimated from polls of two groups of wind resource assessment
experts (for grassland and forest) in Sect. was on the
order of z0 itself, i.e., w/〈z0〉∼ 1 when estimated from
single values of z0 as in Table ; such uncertainty
shrinks, however, if assuming that wind engineers gauge roughness from a
collection of accepted sources, as in the example of
Table .
We note that more exact quantification of measured roughness uncertainty
involves consideration of numerous other factors, from ABL physics and fluid
dynamics to inhomogeneous boundary conditions and turbulent transport.
Likewise, more accurate characterization of epistemic user-based
(industry-wide) uncertainty would likely require a much wider survey for a
greater number of roughnesses. Here we have made a basic evaluation of the
main roughness uncertainty components and their approximate magnitudes,
focusing first on what resultant uncertainty can be expected in a wind
resource prediction, given some level of roughness uncertainty. The latter
focus leads to analysis culminating in Fig. , which
visualizes a primary result of this work: the uncertainty in AEP (or scaled
mean wind) predicted via the EWA method for a given uncertainty in
background roughness length and pair of surface types (roughnesses) at
separate prediction and measurement sites. From Fig. we see
that the basic trend for uncertainty in mean wind speed or AEP behaves as
approximately (w/〈z0〉)6/7 in the dimensionless roughness
uncertainty regime w/〈z0〉<∼ 200 %, i.e., just within
the range we have estimated.
There are other sources of uncertainty implicit in the use of the EWA method, in
addition to the roughness lengths. Additional uncertainties include the
applicability of the GDL (see footnote 6), the constants
(A,B) within Eq. (), and the actual form and/or use of
Eq. () with arguments averaged in an ensemble (or spatial) sense.
These are beyond the scope of the current paper. However, as for
applicability of the GDL, regarding the distance between measurement and
prediction sites, we remind the reader that (fine-resolution) mesoscale
models give an indication of the spatial extent (and direction) of variations
in the geostrophic wind, and we refer the reader to and
, for example. As to the distance over which one may horizontally
extrapolate in more complex terrain, this depends upon the observation and
prediction heights, along with the terrain complexity (as ruggedness
index RIX, , or local elevation variability
, ); we point the reader to
and for uncertainty in complex
terrain. The minor uncertainties due to GDL constants (A,B) are the subject
of ongoing work e.g.,, and the GDL averaging issue
is currently seen to be secondary due to the well-behaved nature of
Eqs. () and () and the magnitude of z0
variations expected.
Additional uncertainties can also arise due to the use of a (mean) wind
profile expression, such as the simple log law (Eq. ) invoked
here. One uncertainty is due to the applicability of a given profile model.
Following and due to the statistical dominance of neutral
conditions , we have used the (surface-layer) form
(Eq. ) applicable in neutral conditions; furthermore, we limit our
observational analysis to neutral steady conditions and observations to be
within the surface-layer, where the logarithmic profile is valid and the
roughness length is simply defined. However, deviations from logarithmic may
occur above the surface layer, such as for the prediction height considered
in the figures (100 m), in the case of very shallow ABL depths i.e.,
depths less than ∼ 2zpred,or 200 m in this
case that occasionally occur . This
ABL-depth effect is negligible for zpred close to
zobs (and z0,1/z0,2 near 1) and is minor for the heights
considered. However, an additional uncertainty dependent upon the ABL depth
could be modeled following and
, or alternately a better profile form
e.g., could be invoked along with the GDL,
particularly to reduce uncertainties for predictions well above 100 m or in
areas where lower-level jets are expected. Another uncertainty arising
implicitly from the profile model, as analyzed here, is due to considering
the same z0 for use in both the profile model and the GDL. That is, the
wind profile reacts to a more local roughness, whereas the GDL
reacts to a geostrophic-scale z0. In the latter is obtained by
taking a weighted geometric spatial average of z0, where lnz0 is
integrated upwind from a given location with a weighting function that decays
with distance; thus, the local and geostrophic z0 can differ slightly.
This is not likely to have a major effect on the analysis here since the
Høvsøre sectors considered were ideal and without significant inhomogeneity,
such that the upwind-averaged roughness is within 10 % of the local
z0. However, it is worth noting that for large roughness changes (e.g.,
coastlines) within ∼ 10 km upwind of a site, the geostrophic z0
will differ from the site's z0;
Eqs. ()–() can be recast for such. The
effect on roughness uncertainty incurred through such spatial averaging is
expected to be (much) smaller than the crude factor w/〈z0〉∼ 3 (200 %) found and presented above, though systematic
evaluation of this effect is still a subject of ongoing research.
Analogously, the height-dependent effect of inhomogeneities upon roughness
(i.e., above the ASL) – in particular its uncertainty – is also under study,
but is expected to be minor for simple terrain.
Vertical extrapolation has not been treated explicitly here, though it is
implicit in the vertical profile used to estimate u* from observed wind
for use in the GDL. Such treatment, in conjunction with taking the profile
roughness and geostrophic-scale roughness to be the same, is a choice that
we have made to facilitate systematic modeling of roughness-induced
uncertainty; thus, we have been able to estimate the effect of roughness,
which occurs through both the wind profile (vertical extrapolation) and
through invocation of the GDL (horizontal extrapolation). A separate
model for the uncertainty in vertical extrapolation using a logarithmic-based
profile (as in the EWA and popular wind software, e.g., WAsP), but without
considering roughness uncertainty, is given in and
. Treating the z0-related uncertainties
separately, per the geostrophic drag law and wind profile, is the subject of
continuing work beyond the scope of the current article.
Applications and implications
In increasingly complex terrain, the actual surface roughness becomes less
significant compared to terrain slope with regards to affecting the flow.
However, for horizontal extrapolation, the aggregate effect of the (complex)
terrain-induced drag leads to an increase in the effective geostrophic-scale
roughness . Thus, the geostrophic-drag and
roughness uncertainty analysis given in this work can also be applied towards
improved use of microscale models in complex terrain when horizontal
extrapolation is involved. In particular, computational fluid dynamics
solvers (e.g., RANS and LES), when employed using different simulation domains
for measurement and wind farm sites, are typically used to calculate
terrain-induced flow perturbations (speed-up factors) at the respective
sites. However, for domains with different degrees of complexity (or potentially
different resolutions) – and thus different large-scale drag – then the use
of the geostrophic drag law (or any analogous empirical algorithm or method)
demands that measured wind statistics must additionally be transformed
properly, accounting for differences in the effective domain-scale mean
roughness in the two domains (per wind direction). Thus, uncertainty in
characterizing the effective roughness due to terrain drag can be translated
into a corresponding uncertainty in mean wind (or AEP) via the framework
presented here. Alternately, for a given pair of (observation, prediction)
sites, the uncertainty in mean wind prediction due to neglect of terrain drag
can be estimated: a bias is introduced, whereby the effective geostrophic
roughness is underestimated. From Fig. one can see, for
example, that for sites with the same effective roughness (complexity)
of z0,eff∼ 1 m and with an underestimation of 1 order of
magnitude (abias≃0.1), a positive error ΔUpred∼ 2 % is incurred.
Another implication of this work applies to assessment in forested regions.
Some work on characterizing profile-amenable roughness over forest
e.g., implies that z0 over
forest is larger than what has been typically assigned in wind resource
assessment (i.e., z0> 1, not z0 ≲ 1), despite such
underestimates being used for decades in the wind industry
. We now see an explanation for
this looking at Fig. : systematic underestimation leads to
smaller errors in wind speeds predicted via the EWA method compared to a
positive bias on z0, particularly for typical application where both
measurement and turbine sites are in high-roughness areas (dash–dot line in
Fig. ) such as forest.
The roughness sensitivity–uncertainty analysis developed here also has
application to – and implications on – the treatment of mesoscale model
output for use in microscale wind flow models. In so-called
meso-to-microscale downscaling or wind climate generalization
, mesoscale wind output (or statistics of such)
is treated in order to avoid “double-counting” of local surface-induced
effects by the microscale model that have already been included in the
mesoscale modeling. Additionally, the meso–micro downscaling procedure
facilitates driving of the microscale flow simulation with mean winds that
are appropriate as per the roughness input to both the microscale and
mesoscale models, i.e., an effective geostrophic wind via the EWA method.
Since any given planetary boundary layer (PBL) scheme in a mesoscale model
can react differently for a given model resolution, it may be necessary to
scale input roughnesses used in the generalization procedure
. For (homogeneous ideal) output wind profiles from a
particular PBL scheme and resolution, the ratio of profile-implied z0 to
input z0 can be used with the analytic sensitivity relations developed
herein to systematically adjust the input roughness map and/or to scale the
wind inputs to microscale models.
An additional application following from the roughness analysis herein – and
consequently ongoing research – involves a limitation inherent in using a
single characteristic (mean) roughness length. Due to the statistical nature
of roughness and the significant width of measured roughness distributions
(e.g., Fig. ), an improvement would be to use P(z0) instead of
mean z0 in wind assessment and atmospheric flow modeling, following the
suggestion of . This becomes yet more significant (and
complicated) considering that the width of P(z0) tends to depend on
direction and vary from site to site, and it also involves correlations with
other variables e.g., stability;. Given the
limited applicability of the EWA method to time series (the GDL was not
explicitly derived in a statistical mean sense), refined wind resource
estimates – which are essentially statistical atmospheric fluid mechanics –
using (joint) distributions of roughness and stability offer potential
improvement over current mean methods and are a subject of continued study.
One final application follows from the analytical form introduced here to
approximate common production power curves, in a general or universal way under
the assumption of Weibull-distributed wind speeds. From this, the exponent in
the power-law expression relating annual energy production and mean wind
speed was derived, allowing us to relate uncertainty in roughness length to
uncertainty in AEP. More flexible power-curve forms can also be made from
logistic functions e.g., generalizing those of as
well. Regardless of the exact form, such analytical treatment also
facilitates quick computation of power for a given set of Weibull parameters,
which is applicable to large data sets such as the Global Wind Atlas .
Lastly we re-iterate that issues in the definition of roughness length, and
specific limits of its validity, are beyond the scope of this article.
However, current ongoing work includes closer examination of the (turbulent)
mechanisms involved in the observation of roughness length from wind
measurements and heterogeneity; subsequent links to refined uncertainty
characterization may follow such investigation.
Summary of conclusions and implications
The EWA method (e.g., WAsP) exploits surface roughness information to improve
resource predictions at one site based on measurements at another, but there
is uncertainty wz0 in the roughness length.
Uncertainty in z0 leads to uncertainty in predicting resources using the EWA
method.
Uncertainty in EWA-predicted mean wind depends upon
wz0, and to a lesser extent also upon {z1,z2}.
wz0 (half-width of P(z0)) is of the same order as
the mean, i.e., wz0∼〈z0〉 for both user- and
observation-derived z0.
For modest z0 uncertainties wz0≲2〈z0〉, the uncertainties {ΔU,ΔAEP}∝wz0/〈z0〉6/7.
In complex terrain and/or forest, ignoring the effect of form drag
causes a positive bias in predictions.
Underestimation of aggregate forest roughness leads to
smaller error than overestimation.
Analytical form for power curve PC(U/Vrat)
gives AEP(〈U〉) and thus uncertainty in AEP, i.e., ΔAEP(Δ〈U〉).
EWA–GDL sensitivity expressions are applicable to
treatment of WRF output for wind resources.