In this work we relate uncertainty in background roughness length (

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

Currently (2016), microscale
models typically have computational resolutions finer than elevation maps;
commonly available elevation maps in most of the world today have typical
resolutions of

First we review the definition of roughness length, introducing and
demonstrating the statistical character of

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.

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.

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

From Eq. (

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.

Figure

In addition to the relatively wide distribution apparent for roughnesses
obtained from 30 min averages shown in Fig.

The IBL develops downwind from a roughness change with expansion slope
(

Because neutral conditions tend to be encountered most often

The increasing area of
surface affecting winds at increasing heights, and also associated averaging
issues, is beyond the scope of the current article

Avoiding substantial changes in surface characteristics and/or land use,
this can be useful towards the aim of gauging background

One can also calculate a more local roughness length
via Eq. (

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).

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

The geostrophic drag law can be simply expressed in scalar form
as

In general, uncertainty can be classified into two types

First,

The aleatoric (random) uncertainty inherent in roughness length can be said
to include that associated with the width of the observed distribution
of

When performing resource assessment, in practice wind engineers characterize
the surface via roughness length (as well as terrain elevation, which we do
not treat in this paper). Roughness characterization can occur via assignment
of

As shown in the section above, the
representative roughness length should be based on a

The epistemic components associated with the theory used to convert; wind
observations into observed

For the observation-based roughness lengths displayed in
Sect.

Means (geometric and arithmetic) and corresponding deviations in

However, the uncertainty in determining a representative roughness length –
via the appropriate (geometric) mean – is not the same as the width of the

For the Høvsøre homogeneous land sectors treated here and the
bootstrapped means, each calculated from 1 year's worth of resampled data,
the

One should be reminded that there are other methods to calculate

Even for an ideal homogeneous landscape, the wind industry, which is a
collection of wind engineers and companies, will as a group assign
different roughnesses to characterize the surface (whether actively or
inherited via acquired maps). This results in a distribution of

We provide a simple practical example of gauging such epistemic uncertainty
based on a systematic exercise: we asked separate groups of wind resource
assessment experts to individually evaluate the surface roughness length for
two commonly encountered land surface types. The groups of participants in
this exercise were polled at meetings of the Danish Wind Power Network
“Vindkraft-Net”

Image of the two areas (grassy and forested) used in roughness survey exercise.

The participants were shown a picture containing both a grassy area and a
forested area (the latter specified as having a mean tree height of 15 m)
and were asked to give

Note that Table

The variability in the user data differs between the polled groups and might
be affected by the limited sample size. Due to the limited distributions of
polled roughness lengths (not shown) gathered from each of the two expert
groups, an alternate estimate of collective user uncertainty (i.e.,
industry-wide) is provided by again applying a resampling method to the
distribution of surveyed

Bootstrapped statistics of mean roughnesses
from (resampled) user-provided

From Table

To summarize, in this subsection we saw that the equivalent (normalized
logarithmic) standard deviation from surveys of engineer or user-assigned
roughness is of the order of the expected roughness itself, as shown in
Table

The uncertainty in roughness length has an effect on a number of key
variables needed for wind resource assessment. Since the geostrophic wind
depends upon the surface friction velocity

By using the logarithmic wind profile (Eq.

From Eqs. (

Just as Eq. (

Error in predicted wind speed due to error in background roughness
at measurement site via Eq. (

The sensitivity of hub-height (predicted) wind speed to

Error in
predicted wind speed due to error in background roughness at prediction site
via Eq. (

Total error in predicted wind speed due to a bias
(

The estimated relative uncertainty in predicted wind speed (

Above we saw that wind speeds predicted via the GDL (Eq.

As one might expect, for measurement and observation sites with similar
background roughness, the change

In contrast to a possible bias in roughness assignment, one can imagine a
worst case scenario as having a negative error in

A more general situation is that of independent errors in roughness
assignment at different sites. In this limit, one foresees a distribution of

Distribution of
error in predicted wind speed, given distributions

In the figure, one can see the combined effect of different roughness
distributions and uncertainties, particularly for the case of grass to forest
(panel b in Fig.

The uncertainty in background roughness can also be translated into AEP
uncertainty by employing a relation between wind speed and AEP, i.e., via a
turbine (or perhaps wind farm) power curve. The propagation of

Sensitivity of
(error in) predicted normalized power due solely to error in background
roughness at measurement site vs. ratio of estimated to actual background

Figure

In order to give examples (and realistic numbers) useful to wind engineers,
in this section we translate the observation-based (Sect.

The relative uncertainties implied by roughness lengths calculated via
surface-layer wind speed measurements were outlined in
Sect.

For the uncertainties inherent in user-provided roughness lengths, we address
the two cases treated in Sect.

The magnitude of

For most general practical use, we ultimately consider roughness error
distributions and the consequent AEP error distributions, such as those shown
in Fig.

Uncertainty
in AEP vs. (relative) roughness uncertainty due to the combined effect of
observation and prediction site roughness-uncertainty; independent log-normal
roughness distributions assumed, with dimensionless width

Figure

We also note again that we have focused here on the AEP uncertainty caused by
uncertainty in background roughness rather than the

First we review the context of this work, i.e., the EWA method

The EWA method is implemented in WAsP and related software (e.g., windPRO, WindFarmer).

, which employs the geostrophic drag law (Eq.The GDL
applies to sites with approximately the same latitude and geostrophic-scale
forcing (roughly the distribution of geostrophic wind); the scale of spatial
variations in the geostrophic wind depends on the terrain complexity and can
vary from several tens of kilometers in simple terrain down to just a few
kilometers in very complex terrain or near coasts; see

Using the EWA method, uncertainty in

As mentioned in Sect.

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.

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

Additional uncertainties can also arise due to the use of a (mean) wind
profile expression, such as the simple log law (Eq.

The EWA roughness-averaging weighting function is
prescribed as

Vertical extrapolation has not been treated explicitly here, though it is
implicit in the vertical profile used to estimate

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

Another implication of this work applies to assessment in forested regions.
Some work on characterizing profile-amenable roughness over 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

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.

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

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

Uncertainty in

Uncertainty in EWA-predicted mean wind depends upon

For modest

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

EWA–GDL sensitivity expressions are applicable to treatment of WRF output for wind resources.

The specific “'filtered” data used in this paper
is going to be made available under

Here we elucidate the relations and approximations that allow translation of
the partial derivatives of hub-height wind speed with regard to roughness
(i.e., Eq.

First we approximate Eq. (

Because the roughness uncertainty (in

Equation (

Here

The error-scaling function can
also be written in terms of the exponential integral function

Total uncertainty vs. bias in background roughnesses

Just as above for the observation site background roughness, we can also
express the uncertainty in predicted wind speed due to uncertainty in the
roughness length for a prediction site. Following a similar procedure as
above, using Eq. (

Above it was written that predictions of wind speed (and thus AEP)
were relatively insensitive to observation and measurement height,
compared to the sensitivity to roughness.
The minor dependence upon

As one can see from the figure, the EWA method, i.e., via the
geostrophic drag law, predicted that

To propagate the uncertainty in mean wind speed into the annual energy
production (AEP), it is necessary to have a model for AEP in terms of mean
wind speed. Assuming a Weibull distribution for wind speeds, we are able to
relate the Weibull parameters to AEP for a given power curve. In this
appendix we produce a universal power-curve formulation, which allows us to
derive an expression for conversion of Weibull-

A canonical form for power curves including the smooth transition from
ideal to maximum power for mean winds approaching rated speed

We choose the order

For Rayleigh-distributed wind speeds
(Weibull, with

One can see from Fig.

Most succinctly, given a Weibull-

The effective wind-power exponent

The power-law exponent derived in Eq. (

For a given value of

The authors declare that they have no conflict of interest.

The authors would like to thank the reviewers for their time and effort
towards constructive criticism of the present article. Mark Kelly is also
grateful to Andrey Sogachev for discussion and for pointing toward the