GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-3715-2015The Explicit Wake Parametrisation V1.0: a wind farm parametrisation in the mesoscale model WRFVolkerP. J. H.pvol@dtu.dkhttps://orcid.org/0000-0003-1944-4720BadgerJ.HahmannA. N.https://orcid.org/0000-0001-8785-3492OttS.https://orcid.org/0000-0003-2079-9422Wind Energy Department, Technical University of Denmark, Risø Campus, Roskilde, DenmarkP. J. H. Volker (pvol@dtu.dk)18November20158113715373115April201529April20153November20155November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/8/3715/2015/gmd-8-3715-2015.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/8/3715/2015/gmd-8-3715-2015.pdf
We describe the theoretical basis, implementation, and validation of
a new parametrisation that accounts for the effect of large offshore
wind farms on the atmosphere and can be used in mesoscale and
large-scale atmospheric models. This new parametrisation, referred
to as the Explicit Wake Parametrisation (EWP), uses classical wake
theory to describe the unresolved wake expansion. The EWP scheme is
validated for a neutral atmospheric boundary layer against
filtered in situ measurements from two meteorological masts situated
a few kilometres away from the Danish offshore wind farm Horns Rev I.
The simulated velocity deficit in
the wake of the wind farm compares well to that observed in the
measurements, and the velocity profile is qualitatively similar to
that simulated with large eddy simulation models and from wind
tunnel studies. At the same time, the validation process highlights
the challenges in verifying such models with real observations.
Introduction
Wind turbines capture the kinetic energy of the wind with their
turning blades, which transfer the energy to a transmission system
that drives an electric generator. In this process the flow in front
and behind a wind turbine is decelerated by the forces acting on the
rotating blades. In large wind farms, the interaction of the flow and
the wind turbines is further complicated by the interaction of the
wake of one wind turbine with neighbouring turbines. Besides the
changed velocity field around the turbines, there is also evidence
that wind turbines affect planetary boundary layer (PBL) processes due
to the changed turbulence .
Coastal regions are expected to become
major areas for wind energy production, since winds there are
generally strong, steady, and less turbulent. To obtain an optimal
yield, which is, among other things, a function of power production,
electrical cabling, and installation costs, it is often convenient to
group wind farms together. Examples are the Danish Rødsand 2 and
Nysted or the Belgian Belwind, Northwind, and Thornton wind farms, in
which the wind farm separation is only a few kilometres. In planning
new wind farms near existing ones it is important to estimate the
velocity perturbation from the nearby farm as accurately as possible,
because the power production is highly sensitive to the wind
speed. Velocity deficits behind wind farms can be considerable.
, for example, found for near-neutral atmospheric
stability, velocity deficits of 2 % up to 20 km downstream
of the Horns Rev I (25 km2) wind farm. The accurate
measurement of wakes from nearby farms becomes even more important in
light of the fact that as of 2015 large offshore wind farms cover
areas of up to 100 km2. Mesoscale models are suitable tools
to estimate wind energy resources in these sea areas
. However, the collective effect of the wind
turbines to the flow needs to be included in these models, and because
these occur at scales smaller than the model's grid size and remain
unresolved, the effects have to be parametrised.
The parametrisation of the effect of wind turbines largely depends on
the model and its mesh size. For instance large eddy simulation (LES)
models, while resolving the local flow around wind turbines, need to
parametrise the local drag forces on the turbine blade
. The grid spacing of mesoscale models, on the
other hand, is on the order of kilometres and tens of metres in the
horizontal and vertical direction, respectively. This means that the
profile of the turbine-induced velocity deficit can be captured to
some extent in the vertical direction. However, the downstream
development of the velocity remains completely unresolved for scales
smaller than the grid size. The major challenge in the parametrisation
is to account for the unresolved relevant processes in agreement with
the flow equations of the model. In recent years, steady progress has
been made in the parametrisation of the effect of wind farms in global
and mesoscale models: from the representation of wind farms by an
increased roughness length in and ,
up to the more advanced drag approaches in
, , , , , , and
. Apart from a local drag force, an additional
turbulence kinetic energy (TKE) source term is assumed in the schemes
proposed by , , , and .
In this article we develop a new approach, which is hereafter referred
to as the Explicit Wake Parametrisation (EWP). We define the TKE from
random fluctuations around the ensemble-averaged velocity, instead of
around the grid-cell-averaged velocity as done in the previous
parametrisations. Therefore, to be consistent with the flow equations
of the model, we apply a grid-cell-averaged drag force, and additional
TKE is only provided by the PBL scheme where there is an increased
vertical shear in horizontal velocity compared to the free-stream
velocity profile.
We implemented the EWP scheme in the open-source Weather Research and
Forecast (WRF) model . The WRF model already includes
a wind farm parametrisation option , denoted as WRF-WF,
which has been validated against wind farm measurements
. We validated the WRF-WF and EWP
parametrisations against long-term meteorological (met) mast
measurements in the wake of an offshore wind farm. To our knowledge,
measurements in the wake of a wind farm have not been used for the
validation of a wind farm parametrisation model in previous
literature.
In Sect. , we explain the theoretical basis of the EWP scheme
and its implementation in the WRF model. Section describes
the measurements, whereas Sect. introduces the WRF model
setup and the configuration of the EWP and WRF-WF scheme. In
Sect. both wind farm parametrisations are validated
against the met mast measurements in the wake of the wind
farm. A discussion of the results finalises the article in
Sect. , followed by the conclusions in
Sect. .
The Explicit Wake Parametrisation
We start by introducing the relevant mesoscale model
equations. Thereafter, the additional terms that represent the
effect of the wind turbines are derived and added to the model
equations. At the end of the section their implementation in the
mesoscale model is described.
A sketch of the downstream development of a turbine-induced velocity reduction. The x axis indicates the grid-cell
size. The grey line represents the instantaneous velocity and
the coloured lines the averaged values. The difference between
the average and instantaneous velocity defines the turbulence
fluctuation at each distance.
The mesoscale model framework
Wind turbines are well described by drag devices that slow down the
wind velocity from a free-stream value u. Downstream, due to mixing
of fluid particles from within and outside the wake, the velocity
deficit is gradually reduced to the point at which the background
conditions are restored.
We use a mesoscale model for the simulation of the wind farm wake and
its recovery. It uses the Reynolds-averaged Navier–Stokes (RANS)
equations,
∂u‾i∂t+u‾j∂u‾i∂xj+∂ui′uj′‾∂xj=-1ρ∂p‾∂xi-2εijkΩju‾k-δi3g+f‾di,
to describe the flow evolution. We use an overline to denote
ensemble-averaged quantities and a prime for a fluctuation around
the ensemble average. The exception is the average air density
ρ(x,t), where t denotes the time and x the
position. In Eq. (), u‾i(x,t) and
p‾(x,t)
represent the mean velocity components and the pressure, whereas Ωj
and g are the Earth's rotation vector and the gravitational acceleration
constant. Furthermore, the rightmost term
f‾di(x,t) is the ensemble-averaged horizontal
forcing due to the action of wind turbines (f‾d3=0).
Mesoscale models generally simulate the effects of turbulence in the
vertical direction only. The components of the Reynolds stress are
parametrised in a 1.5-order PBL scheme as
ui′u3′‾=-Km∂u‾i∂x3,
where the turbulence diffusion coefficient for momentum,
Km(x,t)=Smℓui′ui′‾12,
depends on the stability function Sm(x,t), the turbulence
length scale ℓ(x,t), and the TKE per unit mass,
ui′ui′‾/2. We write the most general form
of the TKE equation:
∂e‾∂t+T‾=p‾s+p‾b+p‾t-ϵ,
where on the left hand side (l.h.s.) e‾ denotes the TKE and
T‾ the transport, which includes the advection by the mean
flow, turbulence transport of TKE, and the divergence of the pressure
correlation. On the right hand side (r.h.s.), p‾s
represents the turbulence production from the vertical shear in the
horizontal velocity (shear production), p‾b the
turbulence production or destruction related to buoyancy forces,
p‾t the turbulence induced by the turbine rotor,
and ϵ the dissipation.
The mesoscale model grid
For the mesoscale model, the previously defined variables in the
Eqs. () and () have to be redefined on the three-dimensional
model grid. For the parametrisation, we aim to obtain expressions for the
volume-averaged drag force, 〈f‾d〉, and the
volume-averaged turbulence induced by the turbine,
〈p‾t〉, that are consistent with the mesoscale model
flow equations. Here, the angle brackets denote the volume average.
A new volume-averaged velocity equation is derived by integrating
Eq. () over the grid-cell volume. This gives the expression
for 〈f‾d〉, which is derived in the
following section (Sect. ).
The derivation of the additional term 〈p‾t〉
depends on the definition of the velocity perturbation. Formally, a
velocity perturbation is the difference between the instantaneous and
ensemble-averaged velocity. For homogeneous flows, the spatially averaged
velocity can be used for the definition of a velocity fluctuation, since
the ensemble average can be approximated by the spatial average. However,
the flow around wind turbines is non-homogeneous and consequently the
spatial and ensemble average deviate. This has been illustrated in
Fig. . Double averaging (ensemble and spatial) allows for the total kinetic energy to be separated into three components.
The definition of the term 〈p‾t〉
in the TKE equation will then depend on which of the three components contribute
to the mean and which to the turbulence kinetic energy.
In Appendix A, we discuss in more detail the double averaging and the ways
mean and turbulence kinetic energy can be described.
In the EWP scheme, we follow and and define
a turbulence fluctuation around the ensemble mean (approach 1 in Appendix A). Therefore,
we include only the random motion in the TKE.
With this definition, the additional term becomes 〈p‾t〉=〈ui,h′fdi′‾〉,
where h is the hub height of the turbine. When we use fdi=-ρArctui,h2/2 for the drag force, where ct(u) is the thrust coefficient and
Ar the rotor area, we obtain:
〈p‾t〉=〈ui,hfdi‾〉-〈u‾i,hf‾di〉=-ρArct〈(u‾i,h+ui,h′)ui,h2‾〉/2+ρArct〈u‾i,hui,h2‾〉/2≈-ρArct〈u‾i,hui′,h2‾〉.
This term represents a sink of TKE due to the extraction of momentum. The
magnitude of this term is much smaller than the additional term in
the WRF-WF scheme (see Sect. ). Therefore, the additional
term 〈p‾t〉 is neglected in the EWP approach.
Additional turbulence is generated by shear production, which
we assume to be the dominant mechanism on the grid-cell average.
In the following sections, we derive the expression for the grid-cell-averaged 〈f‾d〉.
The sub-grid wake expansion
The velocity deficit expansion in the vertical direction within one grid cell is
not negligible, which leads to flow decelerations that extend beyond the rotor-swept
area. This part of the wake expansion is not accounted for in the
mesoscale model, hence we estimate it explicitly with a sub-grid-scale
(turbulence) diffusion equation. Then, a grid-cell-averaged force is
determined and added to the model RANS equations.
In this model, we assume the horizontal advection of velocity and the
turbulence diffusion to dominate in Eq. (). Considering first
only the flow behind the turbine rotor, we obtain the diffusion equation from Eqs. ()
and ():
u‾o∂∂xu‾o-u^=K∂2∂z2u‾o-u^+K∂2∂y2u‾o-u^,
which describes the expansion of the velocity deficit, u‾o-u^, behind the turbine. Here, u‾o=|u‾(h,t)|
denotes the advection velocity at hub height and u^(x) the
unresolved velocity in the stream-wise direction x in the wake of the turbine.
The turbulence diffusion, which causes the wake expansion, is described by a single
turbulence diffusion coefficient K(x,y,t)=Km(x,y,h,t) in Eq. () and
is given by the PBL scheme in WRF.
The velocity deficit profile
We define the vertical structure of the velocity deficit as
u‾d=u‾sξ, where
u‾s(x) is the maximum
velocity deficit at the centre of the wake and ξ(x,y,z)
a function that determines the wake expansion
. Equation () can be solved for the velocity
deficit profile:
u‾d=u‾sexp-12z-hσ2-12yσ2,
where the length scale, σ, that determines the wake expansion is
σ2=2Ku‾ox+σo2.
Equation () describes the ensemble-averaged profile of the
velocity deficit around hub height at a given point in the far
wake . Equation () represents the vertical
wake extension, resulting from turbulent diffusion of momentum, and it
is similar to the solution of Eq. (4.29) in for the
dispersion of plumes. Normally, the turbine wake is divided into a
near and far wake, where the far wake begins between 1 and 3 rotor
diameters . In the parametrisation, we
account for the near-wake expansion in the initial length scale
σo and describe the unresolved far wake expansion
with Eq. ().
We can find the velocity deficit profile for wind turbines by equating
the total thrust to the momentum removed by the action of the wind
turbine, i.e.
12ρctπro2u‾o2=∫-∞∞∫-∞∞ρu‾ou‾ddzdy=ρu‾ou‾s2πσ2,
where ro is the radius of the rotor. In Eq. (), the
l.h.s. represents the local forces at the rotor swept area, whereas the r.h.s.
describes the equivalent distributed
force for the expanded wake at any
x. From the Eqs. () and (), we find the
velocity deficit profile for a specific thrust force,
u‾d=ctro2u‾o4σ2exp-12z-hσ2-12yσ2.
When we insert the velocity deficit of Eq. () in the second
term of Eq. () and integrate in the cross-stream
direction y, we have the integrated thrust profile
f‾d=∫-∞∞∫-∞∞ρu‾ou‾ddzdy=∫-∞∞ρπ8ctro2u‾o2σexp-12z-hσ2dz.
The term on the r.h.s. within the
integral describes the equivalent distributed
thrust force in the
vertical direction at any distance x. Next, we derive from
Eq. () a single effective thrust force, which
represents the average wake expansion within a grid cell.
Turbine forcing in the mesoscale model
For the mesoscale model we derive an effective thrust force, which
describes the average wake expansion within a grid cell. Therefore, we
first determine the effective velocity deficit profile u‾e
by averaging the velocity deficit of Eq. () over the
cross-stream direction y and over a downstream distance L that the
wake travelled within the grid cell. It is convenient to approximate
this area-averaged velocity deficit profile
by a Gaussian-shaped profile:
u‾e=π8ctro2u‾oσeexp-12z-hσe2≅1L∫0L∫-∞∞u‾ddydx.
In Appendix B, we compare the area-averaged velocity deficit profile to the
approximated Gaussian-shaped profile. In Eq. (), σe is
the effective length scale that is related to the model grid size,
σe=1L∫0Lσdx=u‾o3KL2Ku‾oL+σo232-σo3.
From the definition of the effective length scale,
we can obtain the total effective thrust force f‾e by substituting
the length scale σ in Eq. () with the effective length scale
σe. This gives
f‾e=∫-∞∞ρπ8crro2u‾o2σeexp-12z-hσe2dz.
The grid-cell-averaged acceleration for the model RANS equations,
Eq. (), is now obtained when we divide Eq. () by the
mass and apply the grid-cell volume. For every vertical model layer
k, we then obtain from Eq. () the grid-cell-averaged
acceleration components,
〈f‾d1(k)〉=-π8ctro2u‾o2ΔxΔyσeexp-12z-hσe2cos[φ(k)]
and
〈f‾d2(k)〉=-π8ctro2u‾o2ΔxΔyσeexp-12z-hσe2sin[φ(k)],
in the x and y direction, respectively. In Eqs. () and
(), we use Δx and Δy for the horizontal
grid spacing in the x and y direction and φ(k) for the
wind direction. For the height z(k), we use the height at the centre
of layer k.
Implementation in the WRF model
We assume every turbine within a grid cell to be positioned at its
centre and use L=Δx/2 in Eq. ().
In the numerical model, Eqs. () and () are added
to the numerical approximation of Eq. (). Furthermore,
Eq. () is used to determine the effective length scale
σe of the vertical wake extension. At every time step the
total thrust force within a grid cell is obtained from
a superposition of the single turbine thrust forces.
To a first approximation, we use the grid-cell-averaged velocity at hub height,
〈u‾o,h〉, as the upstream velocity
u‾o
for all turbines within the same grid cell (see Sect. ).
The turbulence diffusion coefficient for momentum, K in
Eq. (), comes from the selected PBL scheme. The
parametrisation, therefore, can be used with any PBL scheme. However,
to be consistent with the assumptions used in the derivation of the
wind farm parametrisation, a 1.5-order closure scheme with a turbulence
shear production term is recommended. A practical description of how to
use the EWP scheme in the WRF model is given in the section
“Code availability” at the end of the paper.
Location (a) and layout (b) of the offshore wind farm Horns Rev I
including two nearby masts (M6 and M7). The centre of the wind farm
and width of the sector (±15∘) used for the filtering
of the measurements are indicated by the dashed lines.
Wind farm and measurementsThe Horns Rev I wind farm and met masts
The parallelogram-shaped Horns Rev I wind farm is situated in the
North Sea about 15 km west of the Danish coast (Fig. ). It
is made up of 80 Vestas V80 (2 MW) pitch-controlled, variable-speed
turbines, resulting in a total rated wind farm capacity of 160 MW.
The turbines have a swept-area diameter of 80 m with the hub mounted
at 70 m a.m.s.l. The equally spaced turbines are placed in 10 columns
from west to east and 8 rows from north to south, labelled C1–C10 and
R1–R8 in Fig. b. The spacing between columns and rows is
560 m, which is equivalent to 7 turbine diameters.
describe in detail the various production data available at the wind
farm as well as their quality filtering and processing.
Turbine (C1, R7) is used as a reference turbine since it is not
affected by the wind farm wake under westerly flows; in Fig. b
it is marked with the solid bullet. The wind speed for each turbine is
estimated from the power production data, using the turbine-specific
power curve, and the wind direction is obtained from the adjusted yaw
angle .
For the validation, we use data from two met masts, M6 and M7, whose
positions are shown in Fig. b. The masts are located 2
and 6 km to the east of the eastern edge of the wind farm and are
thus directly in its wake for westerly winds. The 70 m tall masts
are identically equipped and their instrumentation includes high-quality
Risø cup anemometers for measuring wind speeds. The 10 min averaged
data from the two masts, which are independent of the wind farm data,
are used to validate the modelled wind speed reductions downstream from
the wind farm.
Data selection and averaging for the validation
For the validation of the results of the model simulations, and
especially because of the relatively large area-averaged fields in the
mesoscale model, it is very important to ensure that measurements and
model fields are comparable. This can be achieved by properly
selecting the wind direction and wind speed interval at the reference
turbine.
Regarding the wind direction selection, for narrow wind direction sectors, found a higher measured
turbine production at Horns Rev I compared to that estimated by wake models. Most
likely this is related to wind direction variations, which, for small
wind direction bins, exceed the bin size within the 10 min averaging
period. Since these variations are not accounted for in the model, we
follow the recommendation of averaging over a relatively large wind
direction interval. Furthermore, it is important that the flow
reaching the mast anemometers has passed through the wind farm, as
otherwise it does not characterise the wind farm wake. Therefore, we
select velocities whose directions are at the reference turbine
between 255 and 285∘. As demonstrated in Fig. b, the
flow from this sector is influenced by the wind farm wake.
In the period 2005 to 2009, we select the 10 min averaged wind speeds
from 6.5 to 11.5 ms-1 at the reference turbine and bin
them in 1 ms-1 intervals. In this range, we are guaranteed to be
above the cut-in wind speed of the turbines (4 ms-1) and
below the wind speed at which the control system starts to pitch the
blades (13 ms-1). To obtain as many data as possible,
we do not filter the measurements on stability. For strong westerly
winds, we expect the atmospheric stability near to the ground on
average to be neutral .
For every instance that the wind direction and wind speed at the
reference turbine was within the above-described range, the wind
speeds at M6 and M7 were accepted. Afterwards, the selected wind
speeds were averaged for each wind speed bin and normalised by the
average wind speed at the reference turbine
u‾ref (Table ).
Average wind speed at the reference turbine (u‾ref) and frequency of the measurements
at the met mast for all wind speed bins within the wind direction range 255–285∘.
Wind speed binu‾refNobs at M6–M7(ms-1)(ms-1)7±0.57.008878±0.57.959339±0.58.95109710±0.510.0599011±0.511.05729
The WRF model configuration used in the simulations.
Number of grid cells in the horizontal plane (nx, ny):80, 40Horizontal grid spacing (km):1.12Domain size in x, y, z (km):89.6, 33.6, 15Wind farm extension (nx×ny):5×5Boundary condition:OPENPBL scheme:Surface layer scheme:MYNN Monin–Obukhov similarity theoryTKE advection:YesPerturbation Coriolis:YesRoughness length (m):2×10-4Coriolis frequency (s-1)1.2×10-4
(a) Illustration of the model wind farm layout as
applied in the mesoscale model. The squares indicate the model
grid cells and the wind turbines are marked with
triangles. (b) Height (normalised by the turbine hub
height) of the model mass-levels for the three simulations: L40, L60,
and L80. The solid grey line and the dashed lines mark the
hub height and the upper and lower turbine blade tip, respectively.
WRF model configuration and averaging
In the simulations we used the WRF model V3.4. However, the WRF-WF
parametrisation has been updated to the version available in WRFV3.6.
The model domain is set up with 80×40 grid cells, with
a horizontal grid spacing Δx=Δy=1120m. This
horizontal grid spacing, which is twice the turbine separation,
guarantees a constant number of turbines per grid cell in the flow
direction. Equally to the Horns Rev I wind farm, the model wind farm
contains 80 V80 turbines and extends over five grid cells in the
west–east and north–south direction (Fig. a). The Vestas
V80 thrust and power curves were used for the turbine parameters.
The model was run in an idealised case mode with open lateral boundary
conditions p. 51, Coriolis forcing, and zero heat fluxes
from the lower boundary. At the surface, the no-slip condition holds.
The total domain is located over water, and we prescribe a constant
roughness length of z0=2×10-4m, which follows
the World Meteorological Organization (WMO) standards. The friction
velocity is obtained with the Charnock formula. The model simulations
were run with the MYNN 1.5-order level 2.5 PBL scheme ,
on which the WRF-WF scheme is dependent. These and other details of
the model configuration are summarised in Table .
We ran simulations for five wind speed intervals and for nine wind directions,
ranging from 255 to 285∘. Additionally, to investigate
the sensitivity to the vertical resolution, we set up three
experiments with 40 (L40), 60 (L60), and 80 (L80) layers in the
vertical direction. With these vertical resolutions there are 5, 7,
and 10 grid-cell volumes intersecting with the turbine rotor as shown
in Fig. b.
All 135 simulations (5 wind speeds, 9 wind directions, and 3 vertical
resolutions) were initialised with a constant geostrophic wind in
height in a dry and slightly stable atmosphere. After a 4-day
integration period, the wind converged in the whole domain to
a logarithmic neutral profile within a 650 m deep boundary layer.
The boundary layer was
capped by an inversion layer with a potential temperature gradient of
around 6 Kkm-1. Above the inversion layer the velocity remained
independent of height. The atmospheric state of each of these 135
simulations was used to drive a control simulation without wind farm
parametrisation, a simulation with the WRF-WF scheme, and a simulation
with the EWP scheme. We used the restart option in the WRF model to
initialise these simulations.
Each control or wind farm simulation lasted 1 day, resulting in
a total simulation length of 5 days. The wind speeds in the
control simulations were 7, 8, 9, 10, and 11 ms-1 at
70 m (hub height) after 5 days of simulation. We ended up with
the desired magnitude of the geostrophic wind by conducting several
experiments with different initialisations.
Details of the WRF simulations.
Wind direction range (∘):258.75–281.25Wind direction interval (∘):3.75Wind speeds (ms-1):7.0, 8.0, 9.0, 10.0, and 11.0Number of vertical levels (nz):40 (L40), 60 (L60), 80 (L80)Wind farm parametrisation:None, WRF-WF, EWP
To summarise, we performed simulations for 9 wind directions, 5 wind
speeds, and 3 vertical resolutions, with and without wind farm
parametrisation, which gives a total of 405 (3 times 135) simulations
as outlined in Table . For the validation against
the mast measurements, we averaged the model wind speeds over the nine
wind directions with a uniform direction distribution. We used the
instantaneous wind speeds at the end of the simulation period for the
validation. Like for the observations, we normalise the
wind-direction-averaged wind speeds and hereafter simply refer to them as
“velocity”. The model wind speeds from the simulations without the
wind farm were used to normalise the wind farm flow.
Measured and simulated hub-height velocity within the wind
farm. The lines show the model-simulated velocities averaged over
wind direction with an initial length scale σo=1.7ro
for the three vertical resolutions (L40, L60, and L80). The diamonds
represent the measured turbine velocity averaged over each row and
the bars indicate their standard deviations. The crosses mark the
velocity at the grid-cell centre. The normalised velocity for
σo=1.5ro and
σo=1.9ro is shown with red
and blue dots.
Wind farm parametersThe EWP scheme
We use the wind speeds, derived from the power production data of the
turbines, to determine the initial length scale σo in
Eq. (). To a first approximation, the initial length scale
is defined to be independent of the upstream conditions, and it is
therefore the same for all turbines. The initial length scale is
assumed to scale with the rotor radius and accounts for the near-wake expansion.
For this calibration, we selected turbine production data at time
stamps when the derived upstream wind speed and wind direction at the
reference turbine ranged from 8.5 to 9.5 ms-1 and from
255 to 285∘. This wind speed bin was selected since it
contains the most turbine observations. It has a minimum of 850
observations for a turbine at the eastern edge of the wind farm and
a maximum of 1612 observations for a turbine in the first wind farm
column. The difference in the selected number of observations results
from the additional requirement that, for a given turbine, the local
upstream turbines have to be operational in order to guarantee wake
losses. For the power production data obtained in this way, we used
the power curve to derive the wind speed. The column-averaged velocity
was afterwards derived by averaging over the inner six rows
(R2–R7). The outer rows were excluded because they experience
different wake conditions under various wind directions.
Similarly to the model experiments in Sect. , we performed
simulations for nine wind directions between 255 and 285∘ for the
three vertical resolutions with a wind speed at hub height of
9 ms-1. To determine the best-fitting value, we varied for the EWP simulations
the initial length scale for all turbines from σo=ro
to σo=2ro, stepping with Δro=0.1.
For the comparison to the measurements,
the column-averaged wind speed was obtained by averaging over the 3
central wind farm grid cells in the cross-stream direction.
The lines in Fig. show the hub-height velocity from the simulations
with σo=1.7ro, which had the smallest overall bias
compared to the measurements. Additionally, the coloured dots indicate
the sensitivity to the initial length scale for σo=1.5
and 1.9ro.
The figure shows the same velocity reduction for all three
vertical resolutions. Consequently, the amount of energy extracted by
the turbines is independent of the vertical resolution. The
simulations with an initial length scale σo=1.7ro
follow the measured velocity reduction fairly well. We use this length
scale for all simulations in Sect. .
The WRF-WF scheme
In the validation, we also include results from the wind farm parametrisation
WRF-WF from WRF-V3.6. This parametrisation was introduced in WRFV3.3 by
. In this approach, the grid-cell-averaged
force, 〈f‾d〉, is approximated by the local forcing
term at the turbine rotor. The scheme applies a fraction of the total drag force to every model level
that intersects with the blade-sweep area of the wind turbine. Thus the
scheme simulates the local drag forces over the turbine rotor.
The additional TKE source term is parametrised as
〈p‾t,WRF-WF〉=ρAr(ct-cp)〈|u|‾〉3/2,
where cp is the power coefficient of the turbine and 〈|u|‾〉 the absolute value of the grid-cell
velocity. The application of one-dimensional theory
to Eq. () gives p‾t,WRF-WF=ρArcta〈|u|‾〉3/2 for the
additional TKE term. Here, a denotes the induction factor.
The same result,
〈p‾t,WRF-WF〉=ρArcta〈|u|‾〉3/2, is
obtained by defining a velocity fluctuation as the difference between
the grid-cell-averaged velocity and the instantaneous velocity
(approach 2 in Appendix A).
In Fig. , this velocity fluctuation has been denoted
by u′′.
An analytical derivation can be found in (their
Eq. 21, with their ξ=1). In this definition of TKE, the turbine-induced velocity reduction is counted as TKE. For the considered wind
speeds, the absolute value of
〈p‾t,WRF-WF〉 is around 30 times
larger than 〈p‾t〉 defined in Eq. (),
which comes from the different definition of TKE in the two schemes.
For example, for 10 ms-1 with ct=0.79, cp=0.43, and
〈ui′,h2‾〉=0.7m2s-2 at
hub height, the ratio between
〈p‾t,WRF-WF〉 and
〈p‾t〉 as has been defined in
Eq. () is 32.
In Sect. , we use the updated WRF-WF parametrisation
from WRFV3.6, which has no free parameters. The power and thrust
coefficients come from the Vestas V80 turbine.
Validation of the wind farm parametrisations
To obtain a complete picture of the modelled velocity field within and
downstream of the Horns Rev I wind farm, we compare
(1) the velocity decay in the wake of the wind farm from the two wind farm
parametrisations to the measurements for the 10 ms-1 wind
speed bin, and (2) the modelled velocities to the measurements at M6 (2 km
downstream) and M7 (6 km downstream) for all five wind speeds
(7, 8, 9, 10, and 11 ms-1). We recall that the initial length
scale used in the EWP scheme has been determined for 9 ms-1.
In a qualitative validation, we examine the velocity reduction behind
the wind farm and the profile of the velocity deficit. Furthermore,
we discuss the vertical structure of the modelled TKE field from the
discretised Eq. (), where the additional term
〈pt‾〉 has been parametrised in the WRF-WF scheme
and neglected in the EWP scheme. We use results from independent LES
simulations and wind tunnel experiments as a reference.
In the validation, we use the instantaneous model outputs from the
converged flow field after the 5-day integration period. Furthermore,
the velocities are normalised as described in the Sects.
and .
Hub-height velocity for the EWP (a) and WRF-WF (b) parametrisations for the L40, L60, and L80 simulations
and observations as a function of distance from the western edge of
the wind farm. The lines show the model simulated velocities, with
the crosses showing the velocity at the grid-cell centre. The
diamonds are used for the met mast measurements, and the bars are their
standard deviation. The vertical dashed lines show the wind farm
extension and the horizontal dotted line the velocity with the EWP
scheme at the easternmost grid cell.
Velocity recovery at turbine hub height
Figure shows the wind-direction-averaged hub-height
velocity (10 ms-1 bin) as a function of the downstream
distance for the EWP and WRF-WF schemes and all vertical resolutions.
For the EWP scheme (Fig. a) there is no identifiable
vertical resolution dependency on the velocity, while for the WRF-WF
(Fig. b) this is small. In the EWP scheme the velocity
within the wind farm decreases almost linearly with distance. On the other hand,
in the WRF-WF scheme it decreases more rapidly in the first turbine
columns and becomes nearly constant with distance at the end of the wind
farm. The behaviour in the EWP scheme suggests that no equilibrium has
been reached within the wind farm between the extracted momentum by
the wind turbines and the compensating flux of momentum from above. On
the other hand, from the nearly constant velocity at the end of the
wind farm for the WRF-WF scheme, it seems that the extraction of
momentum by the turbines is almost balanced by the flux of momentum
from aloft. At the end of the wind farm the velocity difference
between the two schemes is only 0.1 %. We have indicated the
velocity from the EWP scheme at the most easterly grid cell of the
wind farm with a dotted horizontal line in Fig. . This
agreement is noteworthy, given the two different methods used.
At M6, 2 km downstream of the wind farm, the modelled velocity for both
schemes is within the uncertainty of the measurements. Nevertheless, the difference
between the schemes is increased from 0.1 % at the end of the wind farm to 4.7 % at M6.
This means that, downstream of the wind farm grid cells, where the wind farm parametrisations
are not active, the velocity diverges for the two schemes and its difference
becomes significant. The near-wake recovery is important, especially if the
velocity was used to estimate the power production on a neighbouring wind farm
that was located at this distance from the original wind farm. For example,
the Rødsand 2 and Nysted offshore wind farms in southern Denmark are separated
by a comparable distance.
Modelled (umod) minus measured wind speeds
(umeas) at M6 (circles) and M7 (triangles) as a function
of their measured wind speed, for the EWP (blue symbols) and the
WRF-WF scheme (red symbols) for five wind speed bins and three
vertical resolutions (shown with different symbol sizes). The coloured solid lines link
the M6 and M7 values for the same wind speed bin, whereas the dashed
lines are the values from M7 and the free-stream velocities at the end
of the wake. The crosses indicate the free-stream velocity.
Figure depicts wind-direction-averaged velocity recovery at
the two met masts for all five wind speed bins and all vertical
resolutions. It shows the differences in the WRF-modelled and measured
wind speed at M6 (circles) and M7 (triangles) compared to the measured
wind speed at the same masts. The WRF-modelled wind speed is obtained
from linear interpolation of the wind speeds in the nearest
grid cells. The modelled recovery rate can be deduced from the slopes
of the solid (between the values at M6 and M7) and dashed lines
(between M7 and the free-stream velocity). A negative slope is linked
to a slower modelled recovery compared to that measured, whereas
a positive slope shows a faster modelled recovery. There is no
vertical resolution dependency for a wind farm parametrisation when
the circles or triangles for a given wind speed bin lie on top of each
other.
The results show that the bias in velocity between the measurements
and the EWP simulations is small (<0.15ms-1) for all
wind speeds (except for the 7 ms-1 bin at M7). For the
EWP scheme, we find a positive velocity difference at M6 and a negative one at M7. Hence,
the modelled wake recovery oscillates between being slightly slower
from M6 to M7 and being slightly faster from the end of the wind farm
to M6, as well as from M7 to the end of the wake. The velocity of the
EWP scheme at M6 is nearly independent of the vertical resolution,
and at M7 the dependency is very weak.
The WRF-WF scheme shows a positive difference in velocity of up to
0.5 ms-1 at M6. This difference, between the WRF-WF scheme
and the measurements, becomes larger with increasing wind speed.
The higher modelled velocity at M6 is a consequence of the more rapid
recovery of the modelled wake compared to that of the measurements
from the end of the wind farm to M6. Between M6 and the point at
which the free-stream velocity is reached again, the modelled wake
recovery is slower than that measured. This overall positive
difference suggests that the modelled velocity with the WRF-WF scheme
is overestimated throughout the whole wake.
Horizontal view of the WRF-simulated velocity field at
hub height using the (a) EWP and (b) WRF-WF
schemes. The simulations are for 10 ms-1 in the
270∘ wind direction and L60. The dotted line indicates the
latitudinal centre and the solid rectangle the outer boundary of the
wind farm.
Figure shows the spatial structure of the modelled velocity
(10 ms-1 bin) within the wind farm and in the wake of the wind farm for
the L60 simulations in the 270∘ wind direction. A comparison
of the results from the two schemes confirms the faster initial wind
farm wake recovery in the WRF-WF scheme. The 10 % velocity deficit
contour, for example, extends for the EWP scheme to around x=15km (Fig. a), while for the WRF-WF scheme it extends to
x=8km
(Fig. b). The difference in the distance at which
a 7.5 % velocity deficit is reached is even larger: after x=21
and x=11km for the EWP and WRF-WF scheme, respectively. Further
downstream, after around 30 km, the velocity fields from the two
parametrisations become similar. Possible reasons for the difference
in the initial wake recovery are discussed in the next section.
Finally, Fig. shows a difference in the orientation of the
axis of the velocity deficit downstream from the wind farm. In both
simulations the steady-state wind direction of the free-stream flow
was 270∘. As the velocity decreases within the wind farm, the
velocity is expected to turn to the north, due to the changed Coriolis
force. In the wind farm wake, where the flow accelerates again, the
velocity should turn back again to the background direction. This
effect is seen for the EWP scheme (Fig. a). However, for the
WRF-WF scheme (Fig. b) the wake turns southward behind the
wind farm. A possible reason for this unexpected behaviour is the
turbulence transport of momentum from aloft (Ekman spiral) within the
wind farm. In the wind farm wake the flow keeps turning to the south
because of the flow acceleration from the momentum transport.
Vertical profile of TKE and velocity
To obtain a broader understanding of the mechanisms acting in the two
schemes, we compare the simulated TKE (per unit mass) and the velocity
deficit profiles for the 10 ms-1 wind speed bin in the
270∘ wind direction.
Figure shows the difference in TKE (wind farm minus control
simulation) for the L60 simulation. The cross sections in the x–z
plane are in the west–east direction and pass through the centre of
the wind farm. We used different colour scales in the two plots, due
to the relatively large differences in TKE production between the two
schemes. However, we have kept the outermost contour
(0.02 m2s-2) the same. The maximum TKE difference was
0.30 and 1.9 m2s-2 for the EWP and WRF-WF scheme,
respectively.
Vertical cross section of the TKE difference (〈e‾wf〉-〈e‾ref〉) (m2s-2) for the simulations for
10 m s-1 in the 270∘ wind direction and L60 for
the (a) EWP and (b) WRF-WF scheme. The region in
the model containing turbine blades is indicated by the rectangle.
Compared to the environmental TKE of the pure shear flow (not shown),
the TKE increases in the EWP scheme (Fig. a) at the end of
the wind farm by a factor of 2, whereas in the WRF-WF scheme
(Fig. b) it increases at hub height by a factor of 5.5. The
EWP scheme shows an increased and decreased TKE above and below
hub height compared to the reference simulation. The maximum increase
occurs behind the wind farm, where the velocity gradients are the
largest. In the WRF-WF scheme, the maximum TKE increase happens at
hub height within the wind farm. Recalling that in simulations with the WRF-WF scheme turbulence is generated by the source term 〈p‾t〉
and by the PBL scheme from turbulence shear production
(〈p‾s〉), we find that the intensity of
〈p‾t〉 dominates over that of the shear
production. The sum of the additional source term and the turbulence
shear production causes within the wind farms an increased turbulence
from the lowest model level upwards.
We use the results from the actuator-disc approach without rotation
from to obtain a qualitative impression of the
structure of the turbulence field from an LES model within a wind
farm. The actuator-disc approach from their LES model is most similar
to the drag approach in the mesoscale model. Their Fig. 5c shows that
higher and lower turbulence intensities dominate around the upper and
lower turbine blade tip. Also, a positive and negative shear stress
occur above and below hub height (their Fig. 7c). This indicates that
the shear in velocity dominates the turbulence production. Similar
features are present in the TKE field from the EWP scheme, where the
increased and decreased TKE above and below hub height are, in a similar manner, caused by an enhanced and reduced turbulence shear production
with respect to the neutral logarithmic velocity profile. Furthermore,
show an upper wake edge at around 4.5 turbine
hub heights for 10 aligned wind turbines after 60 turbine diameters
(their Fig. 12). They defined the wake edge at the point where the
velocity reduction for a given height was 1 %. Similarly, the edge
of the wake can be found from an increased TKE due to shear
production. For the outermost contour in Fig. , we obtain
a vertical wake extension of around 5 turbine diameters for the EWP
scheme at 4.8 km downstream (equivalent to 60 turbine diameters). At
the same distance, it is around 7 turbine hub heights for the WRF-WF
scheme.
The influence of the TKE field to the velocity profile is analysed in
Fig. . The figure shows the velocity deficit profile Δ〈u‾x〉/〈u‾o,h〉 for
both schemes and all vertical resolutions. Here
〈u‾o,h〉 denotes the free-stream velocity
at hub height. The velocity deficit is defined as
Δ〈u‾x(z)〉=〈u‾(z)〉-〈u‾free(z)〉, where 〈u‾free(z)〉 is the free-stream velocity profile
from the reference simulation and 〈u‾(z)〉
the velocity profile from the wind farm simulation. The free-stream
velocities from the EWP and WRF-WF simulations were visually
indistinguishable. We choose the second and third grid cell within the wind
farm (C2 and C3) and the second point behind the wind farm (C7) that
corresponds approximately to the location of M6.
Comparison of the vertical profiles of simulated velocity
deficit for the second (C2), third (C3), and seventh (C7) grid cell
containing wind turbines from the first westernmost turbine:
(a) L60 simulations, (b) L40 and L80 simulation
for the EWP, and (c) L40 and L80 simulation for the WRF-WF
scheme. The turbine hub height is indicated by the horizontal solid
line and the turbine blade bottom and top by the dashed lines. The
free-stream wind speed was 10 ms-1 in the 270∘
wind direction.
Figure a shows the velocity deficit profiles from the EWP
and WRF-WF scheme from the L60 simulation. The profiles indicate
a stronger diffusion in the WRF-WF scheme compared to that in the EWP
scheme. This can be recognised by the vertical extension of the
vertical deficit profile at C7, behind the wind farm.
For the EWP scheme (Fig. b), we find a maximum velocity
deficit at hub height and symmetric features around the maximum for
the L40 and L80 simulations. Also, results from LES simulations, wind
tunnel experiments, and measurements
have a maximum velocity deficit at hub height in the far turbine wake
for a neutral boundary layer. The profiles in (their
Fig. 36) and in (their Fig. 4) additionally show
symmetric features around the maximum with a shape similar to the
velocity profile of the EWP scheme. Figure b demonstrates
the vertical resolution independence of the EWP scheme within the wind
farm: the velocity deficits of the L40 and L80 simulations are almost
identical. A small difference is found below hub height in the wind
farm wake.
With the WRF-WF scheme (Fig. c) the maximum velocity
deficit is displaced vertically above hub height, which reaches the
upper wind turbine blade tip at mast M6 (C7). The dependency of the WRF-WF
scheme on the chosen vertical resolutions is weak; differences are
found within the wind farm (C2, C3) above hub height. Also, the
profiles from the WRF-WF scheme show increased (area-averaged)
velocities inside the wind farm at the lowest model level. The wind
farm simulations from (their Fig. 13) do not support
this behaviour.
Discussion
We use wind farm parametrisations implemented in a mesoscale model to
simulate the effect of wind farm wakes in areas on the order of
hundreds of kilometres. However, the models have a limited horizontal
resolution and hence the local processes within a wind farm remain
unresolved.
In the proposed parametrisation, we use the classical wake theory
to describe the sub-grid-scale wake expansion.
Compared to empirical fits from, for example, and
, it offers the advantage that the wake expansion is
described as a function of atmospheric stability. In this study, we have validated the
approach for different wind speeds in a neutral atmospheric boundary layer.
Its performance as a function of atmospheric stability, which requires
information of the profiles, will be
investigated in future. In their LES model results, found
a sensitivity of the velocity reduction to the wind farm layout. In
current implementations of wind farm parametrisations, all turbines
within a grid cell experience the same upstream velocity. Although
these parametrisations are not meant to estimate the local velocity
field within the wind farm, differences in velocity reduction within
the wind farm could affect the velocity in the wake of it.
Future implementations may account for the local
flow within the wind farm by using data from high-resolution models,
which can be input to the mesoscale model via a look-up table as
suggested by and . However, we currently have
no measurement data in the wind farm wake to validate
these approaches for different wind farm layouts. At the Horns Rev I
wind farm, we were restricted to the geometry of met masts positions,
which did not allow for study of the sensitivity of the velocity field in
the wind farm wake for additional wind direction sectors.
A fair comparison between the mesoscale model and long-term
measurements can be realised in several ways. One method is to match
the simulation period to that observed. By selecting corresponding
time frames, one can then compare the fields of interest. The main
disadvantage of this method is that errors in the background flow
simulated by the mesoscale model also exist, which complicates the
analysis. The second method is to sample the data and the model
simulations in rather strict idealised conditions of wind speed and
direction. We have chosen this second method using the WRF model in
idealised case mode and compare its results to properly averaged
measurements. Here the equilibrium solution without the wind farm
effects is purposely made to match the free-stream velocity, and thus
background errors in the simulations are absent. Besides a more
detailed analysis, this method also allows for the study of the vertical
dependency of the velocity reductions, since the atmospheric
background conditions are almost identical for the simulations with
different vertical resolutions.
Before we validated the results of the schemes in the wake of the wind
farm, the a priori unknown initial length scale of the EWP scheme had
to be determined. We did this using the turbine power measurements
from the most frequently observed wind speed bin. This limitation
could not be avoided, since, to our knowledge, no other long-term
measurements from large offshore wind farms are available. For
the most frequently observed wind speed bin, we found an initial length scale of
σ0=1.7ro that fitted the turbine measurements the best.
We recommend this constant for similar wind turbines. Future wind turbine
measurements are needed to determine this value for
other turbine types, such as low-induction turbines.
This constant was then used for the validation of all wind speed bins.
On the other hand, for the WRF-WF scheme we have used the turbine-specific
thrust and power curves, which are its only input parameters. The
difference of 0.1 % between the velocity deficit simulated by the
WRF-WF and EWP scheme at the end of the wind farm for the
10 ms-1 wind speed bin facilitated the comparison
between the schemes in the wind farm wake, where the parametrisations
are not active anymore.
One major difference between the EWP and WRF-WF approach is in the
treatment of the grid-cell-averaged TKE budget equation. The TKE
production regulates the vertical profiles of momentum, temperature,
and moisture within the PBL. Differences in TKE production would thus
affect the local weather (e.g. temperature, humidity, and possibly
clouds) response to the presence of large wind farms.
Both wind farm schemes use a PBL scheme that parametrises the TKE equation
in terms of grid-cell-averaged variables. Therefore, in the wake of the wind
turbines TKE is generated by the increased vertical shear in horizontal
velocity. Then, the different definition of the unresolved velocity
fluctuation in the WRF-WF and EWP scheme leads to a different term
〈p‾t〉 that is the direct consequence of the
presence of a drag force. In the EWP scheme, a velocity fluctuation is
defined around the ensemble-averaged velocity. With this definition
〈p‾t〉 is small and can be neglected. Instead,
in the WRF-WF scheme, velocity fluctuations are defined around the grid-cell-averaged velocity and a parametrisation of 〈p‾t〉
is added to the model TKE equation. The simulations have shown that, in the
WRF-WF scheme, 〈p‾t〉 dominates over the shear
production and that its total TKE is more than 3 times larger than that in the
EWP scheme. However, it is unclear how well the actual grid-cell-averaged
shear production is approximated by the shear production calculated with the
PBL scheme in WRF, on which the EWP scheme relies. Therefore, future
measurement campaigns of the actual structure and intensity of the TKE field
within and around wind farms under suitable atmospheric conditions can help
to settle this issue.
Conclusions
We introduce a wind farm parametrisation for use in mesoscale
models. The EWP approach is based on classical wake theory and
parametrises the unresolved expansion of the turbine-induced wake
explicitly in the grid cell that contains turbines, where the largest
velocity gradients occur. The associated turbulence shear production
is then determined by the PBL scheme in the mesoscale model. The
approach has been implemented in the WRF mesoscale model and can be
used with any PBL scheme. However, we recommend PBL schemes that model
the TKE equation.
We analysed the results of simulations from the scheme in the wake of
a wind farm. For the validation, we used the averaged wind speeds
within a 30∘ wind direction sector at two met masts in the
wake of the Horns Rev I wind farm. The model was run for several wind
direction bins that cover those sampled in the observations. For each
wind speed bin, we compared a wind-direction-averaged wind speed to
the similarly averaged measurements. We found that, for all five velocity
bins, the velocity modelled with the EWP scheme agreed well with the
met mast measurements 2 and 6 km downstream from the edge of the
wind farm. The EWP scheme reproduces the wind farm wake within the
standard deviation of the measurements.
To our knowledge, no long-term data sets are available to validate the
details of the vertical structure of the velocity deficit and
turbulence in the wake of the wind farm. In a qualitative comparison,
we found the vertical structure of the modelled TKE field to agree
with that of actuator-disc simulations by LES models ,
with an increased and decreased TKE around the upper and bottom rotor
tip, respectively. The TKE field in the EWP scheme led to a symmetric
velocity deficit profile around hub height, similar to velocity
deficit profiles in and . Also, the vertical wind
farm wake expansion in the EWP approach was similar to that described
in the aforementioned studies. While validation of the WRF-WF
parametrisation has been carried out before with measurements within
a wind farm , this is the first time that the
validation has been done in the wake of a wind farm, at the scales the
mesoscale model was designed to simulate.
We use the notation and symbols of , with the exception
that the ensemble average is used instead of the time average. The instantaneous velocity,
ui, can be decomposed in a spatial average with a fluctuation around it,
ui=〈ui〉+ui′′, and an ensemble average with a fluctuation,
ui=u‾i+ui′. Figure 1 illustrates the instantaneous velocity (grey line),
as well as the spatial (red line) and ensemble-averaged (blue line) velocity
in the vicinity of a wake.
We can decompose the total kinetic energy:
12〈ui2‾〉=12〈u‾i 2〉+12〈ui′2‾〉,=12〈u‾i〉2+12〈u‾i′′2〉+12〈ui′2‾〉.
In Eq. (), we applied an ensemble and spatial averaging to the kinetic energy
and we have decomposed the ensemble-averaged kinetic energy in an average and fluctuating
part. Here, 12〈u‾i2〉 is the spatial average of
the ensemble-averaged kinetic energy and 12〈ui′2‾〉
the spatial average of the kinetic energy from random velocity fluctuations.
By decomposing the first term on the r.h.s. of Eq. (), we obtain
Eq. (). In Eq. (), we now have three contributions to the
total spatial and ensemble-averaged kinetic energy. The first term,
12〈u‾i〉2, is the kinetic energy of the spatial and
ensemble-averaged velocity. The second term,
12〈u‾i′′2〉, is the spatially averaged
kinetic energy of the heterogeneous part of the mean flow, which is the difference between the
ensemble and spatially averaged kinetic energy. This term arises only in non-homogeneous flow
conditions and is also called “dispersive kinetic energy” by .
For each contribution on the r.h.s. of Eq. () to the total kinetic energy,
a budget equation can be derived. The complete set of equations can be found
in, for example, . We can combine the three components in Eq. ()
with kinetic energy of the mean flow (MKE) and turbulence kinetic energy (TKE) in two ways, which we refer to as approach 1 and
2. The MKE is not directly resolved by the model. However, the definition of TKE determines how the
effect of wind turbines to the TKE is parametrised.
Approach 1 follows Raupach and Shaw (1982) and Finnigan and Shaw (2008) and defines MKE = 12〈u‾i2〉=12〈u‾i〉2+12〈u‾i′′2〉 and TKE = 12〈ui′2‾〉. Here, the MKE is equal to the spatial average of the ensemble-averaged
kinetic energy, and it contains all kinetic energy of the organised motion. With this
definition only random motion contributes to the TKE. The presence of the drag force gives rise to
the term 〈p‾t〉=〈ui′fdi′‾〉, where
f′ is the fluctuation of the drag force around the ensemble-averaged force. This approach is
used in the EWP scheme, and in Sect. 2.2 the additional term is derived.
Approach 2 is the second way the three components in Eq. (A2) can be assigned to
the MKE and TKE, namely MKE = 12〈u‾i〉2 and TKE = 12〈u‾i′′2〉+12〈ui′2‾〉. In this case, the MKE contains only kinetic energy from the
spatially averaged velocity. However, the TKE now also contains energy from the heterogeneous part of
the mean flow additional to the energy from random motion. In this approach, a fluctuation can be
decomposed in u′′=u‾i′′+ui′. Therefore, the source term becomes
〈p‾t〉=〈ui′′fdi′′‾〉, where
f′′ is the fluctuation of the drag force around the spatially averaged force.
In the WRF-WF, this approach is used (see Sect. ).
In the EWP scheme, we approximate the velocity deficit profiles on the
r.h.s. of Eq. () by a Gaussian-shaped velocity profile on the l.h.s. of
Eq. ().
To show that these profiles are to a good approximation similar, we
compare the difference between the average of 5000 single profiles at
distances 0<x<500m to the approximated Gaussian velocity
deficit. We normalise both profiles by the depth, Δy, of the
wake in the cross-stream direction. For the comparison we used
u‾o=8ms-1, ro=40 m,
σo=60m, cT=0.8, K=6m2s-1,
and Δy=1120m. The wake centre is defined at z=0m.
Panel (a) shows a comparison between the distance average
of the velocity deficit profiles (blue line) and the Gaussian
profile with the average spread 〈σ〉 (red
line). Panel (b) shows the difference between the spatially averaged
Gaussian profiles and the Gaussian profile with the average spread.
The result in Fig. shows that the differences between the
spatially averaged Gaussian profiles and the Gaussian profile with the
spatially averaged spread is far less than 0.001ms-1 in the entire velocity deficit region.
Code availability
In this section a short guideline of the usage of the EWP scheme in
the WRF model is given. The EWP approach can be run in serial,
shared-memory, or distributed memory options. Currently, it is not
possible to run the approach with the mixed shared and distributed
memory option. The scheme can be used for idealised and
real case simulations. For the real case simulations,
the wind farm parametrisation can be activated in any nest of the
simulations. The additional namelist.input option bl_turbine
should be used to select the wind farm parametrisation.
The EWP scheme needs additional input files in ASCII format. In the
first file the positions and types of all wind turbines should be
listed. The file name has to be specified as a string in the
additional namelist.input option windturbines_spec. For every
turbine used, the turbine characteristics, i.e. the hub height and
diameter and the thrust and power coefficients as a function of wind
speed, need to be listed in a file. The power coefficient is used
optionally to estimate the turbine power production. This file name has
to start with the turbine type used in the first file, followed by the
extension .turbine.
Please contact pvol@dtu.dk to obtain the code of the EWP wind farm
parametrisation.
Acknowledgements
Funding for research was provided through the European Union's Seventh
Programme, under grant agreement no. FP7-PEOPLE-ITN-2008/no238576 and no. FP7-ENERGY-2011-1/no282797. The
authors gratefully acknowledge the suggestions and helpful
discussions with Jerry H.-Y. Huang and Scott Capps, both from the
Department of Atmospheric and Oceanic Sciences, UCLA. We would like
to acknowledge Vattenfall AB and DONG Energy A/S for providing us
with data from the Horns Rev I offshore wind farm and Kurt S. Hansen
for the data processing and filtering. Finally, the authors want to thank the reviewers for their helpfull comments.
Edited by: S. Unterstrasser
ReferencesAbkar, M. and
Porté-Agel, F.: A new wind-farm parameterization for large-scale
atmospheric models, J. Renewable Sustainable
Energy, 7, 013121, 10.1063/1.4907600,
2015.Adams, A. S. and
Keith. D. W.: A wind farm parametrization for WRF, 8th WRF Users
Workshop, 11–15 June 2007, Boulder, abstract 5.5, available at:
http://www2.mmm.ucar.edu/wrf/users/workshops/WS2007/abstracts/5-5_Adams.pdf, 2007.Badger, J.,
Volker, P. J. H., Prospathospoulos, J., Sieros, G., Ott, S.,
Rethore, P.-E., Hahmann, A. N., and Hasager, C. B.: Wake modelling
combining mesoscale and microscale modelsm, in: Proceedings of
ICOWES, Technical University of Denmark, 17–19 June 2013, Lyngby, p. 182–193, available at:
http://indico.conferences.dtu.dk/getFile.py/access?resId=0&materialId=paper&confId=126,
2013. Baidya Roy, S.: Simulating impacts
of wind farms on local hydrometeorology, J. Wind Eng. Ind. Aerod.,
99, 491–498, 2011. Baidya Roy, S. and
Traiteur, J. J.: Impact of wind farms on surface air temperature,
P. Natl. Acad. Sci. USA, 107, 17899–17904, 2010. Baidya Roy, S.,
Pacala, S. W., and Walko, R. L.: Can large wind farms affect local
meteorology?, J. Geophys. Res., 109, 2156–2202, 2004.Barrie, D. B. and Kirk-Davidoff, D. B.: Weather response to a large wind
turbine array, Atmos. Chem. Phys., 10, 769–775, 10.5194/acp-10-769-2010, 2010.Blahak, U., Goretzki, B.,
and Meis, J.: A simple parametrisation of drag forces induced by
large wind farms for numerical weather prediction models, EWEC
Conference, 20–23 April 2010, Warsaw, p. 186–189, available at:
http://proceedings.ewea.org/ewec2010/allfiles2/757_EWEC2010presentation.pdf, 2010.Calaf, M., Meneveau, C.,
Meyers, J.: Large eddy simulation study of fully developed
wind-turbine array boundary layers, Phys. Fluids, 22, 015110,
10.1063/1.3291077, 2010.
Christiansen, M. B. and Hasager, C. B.: Wake effects of large offshore
wind farms identified from satellite SAR, Remote Sens. Environ., 98,
251–268, 2005. Crespo, A. and
Hernández, J.: Turbulence characteristics in wind-turbine
wakes, J. Wind. Eng. Ind. Aerod., 61, 71–85, 1996.Finnigan, J. J.
and Shaw, R. H.: Double-averaging methodology and its application
to turbulent flow in and above vegetation canopies, Acta Geophys.,
56, 534–561, 10.2478/s11600-008-0034-x, 2008. Fitch, A. C., Olson, J. B.,
Lundquist, J. K., Dudhia, J., Gupta, A., Michalakes, J., and
Barstad, I.: Local and mesoscale impacts of wind farms as
parameterized in a mesoscale NWP model, Mon. Weather Rev., 140,
3017–3038, 2012. Fitch, A. C., Lundquist, J. K.,
and Olson, J. B.: Mesoscale influences of wind farms throughout a
diurnal cycle, Mon. Weather Rev., 141, 2173–2198, 2013a.Fitch, A. C., Olson, J. B.,
and Lundquist, J. K.: Parameterization of Wind Farms in Climate Models,
J. Climate, 26, 6439–6458,
10.1175/JCLI-D-12-00376.1, 2013b. Frandsen, S. T.,
Jørgensen, H. E., Barthelmie, R., Badger, J., Hansen, K. S.,
Ott, S., Rethore, P.-E., Larsen, S. E., and Jensen, L. E.: The
making of a second-generation wind farm efficiency model complex,
Wind Energy, 12, 445–458, 2009. Gaumond, M.,
Réthoré, P.-E., Ott, S., Peña A., Bechmann, A., and
Hansen, K. S.: Evaluation of the wind direction uncertainty and its
impact on wake modeling at the Horns Rev offshore wind farm, Wind
Energy, 7, 1169–1178, 2014.Hahmann, A. N.,
Vincent, C. L., Peña, A., Lange, J., and Hasager, C. B.: Wind
climate estimation using WRF model output: method and model
sensitivities over the sea, Int. J. Climatol., 35, 3422–3439,
doi:10.1002/joc.4217, 2014. Hansen, K. S.,
Barthelmie, R. J., Jensen, L. E., and Sommer, A.: The impact of
turbine intensity and atmospheric stability on power deficits due to
wind turbine wakes at Horns Rev wind farm, Wind Energy, 15,
183–196, 2012. Hansen, M. O. L.: Aerodynamics of
Wind Turbines, James & James, London, UK, 2003. Hasager, C. B.,
Rasmussen, L., Peña, A., Jensen, L. E., and Réthoré, P.-E.:
Wind farm wake: the Horns Rev Photo Case, Energies, 6, 696–716,
2013. Iungo, G. V., Wu, Y.-T., and
Porté-Agel, F.: Field measurements of wind turbine wakes with
lidars, J. Atmos. Ocean. Tech., 30, 274–287, 2013. Jacobson, M. Z. and
Archer, C. L.: Saturation wind power potential and its implications
for wind energy, P. Natl. Acad. Sci. USA, 109, 15679–15684,
2012. Jiménez, P. A.,
Navarro, J., Palomares, A. M., and Dudhia, J.: Mesoscale modeling of
offshore wind turbine wakes at the wind farm resolving scale:
a composite-based analysis with the Weather Research and Forecasting
model over Horns Rev, Wind Energy, 18, 559–566, 2014. Keith, D. W.,
DeCarolis, J. F., Denkenberger, D. C., Lenschow, D. H.,
Malyshev, S. L., Pacala, S., and Rasch, P. J.: The influence of
large-scale wind power on global climate, P. Natl. Acad. Sci. USA,
101, 16115–16120, 2004.Lu, H. and
Porté-Agel, F.: Large-eddy simulation of a very large wind farm in
a stable atmospheric boundary layer, Phys. Fluids, 23, 065101,
10.1063/1.3589857, 2011. Nakanishi, M. and
Niino, H.: Development of an improved turbulence closure model for
the atmospheric boundary layer, J. Meteorol. Soc. Jpn., 87,
895–912, 2009. Porté-Agel, F.,
Wu, Y.-T., Lu, H., and Conzemius, R. J.: Large-eddy simulation of
atmospheric boundary layer flow through wind turbines and wind
farms, J. Wind Eng. Ind. Aerod., 99, 154–168, 2011.Raupach, M. R. and
Shaw, R. H.: Averaging procedures for flow within vegetation
canopies, Bound.-Lay. Meteorol., 22, 79–90, 10.1007/BF00128057,
1982.
Sathe, A., Gryning, S.-E., and
Peña, A.: Comparison of the atmospheric stability and wind
profiles at two wind farm sites over a long marine fetch in the
North Sea, Wind Energy, 14, 767–780, 2011. Skamarock, W., Klemp, J.,
Dudhia, J., Gill, D., Barker, D., Duda, M., Huang, X., Wang, W., and
Powers, J.: A Description of the Advanced Research WRF Version 3,
NCAR Technical note, Massachusetts, USA, 2008.Tennekes, H. and
Lumley, J. L.: A First Course in Turbulence, The MIT Pess, Boulder,
USA, available at: http://www2.mmm.ucar.edu/wrf/users/docs/arw_v3.pdf, 1972. Vermeer, L. J.,
Søensen, J. N., and Crespo, A.: Wind turbine wake aerodynamics,
Prog. Aerosp. Sci., 39, 467–510, 2003.Wang, C. and Prinn, R. G.: Potential climatic impacts and reliability of very
large-scale wind farms, Atmos. Chem. Phys., 10, 2053–2061, 10.5194/acp-10-2053-2010, 2010. Wu, Y.-T. and
Porté-Agel, F.: Simulation of turbulent flow inside and above wind
farms: model validation and layout effects, Bound.-Lay. Meteorol.,
146, 181–205, 2013. Wyngaard, J. C.: Turbulence in
the Atmosphere, Cambridge Press, Cambridge, UK, 2010.Xie, S. and Archer, C.: Self-similarity and
turbulence characteristics of wind turbine wakes via large-eddy simulation,
Wind Energy, 18, 1815–1838, 10.1002/we.1792, 2015.Zhang, W., Markfort, C. D.,
and Porté-Agel, F.: Wind-Turbine Wakes in a Convective Boundary Layer:
A Wind-Tunnel Study, Bound.-Lay. Meteorol. 146, 161–179,
10.1007/s10546-012-9751-4, 2013.