Introduction
Optimizing the control of wind turbines in a farm is challenging due
to the aerodynamic interactions among turbines. These interactions
come from the fact that downwind turbines are often operating in the
wakes of upwind ones . Two important wake
characteristics are (1) a flow velocity deficit and (2) an increase
in turbulence intensity. The former reduces power production of the
farm while the latter leads to a higher dynamic loading on downstream
turbines but also induces wake recovery. Individual turbine control
variables can influence the wake's flow velocity, turbulence intensity
and also location. Hence, by changing the control variables of
individual turbines, power production of and loading on these
controlled turbines and the downwind turbines can be manipulated. Wind
farm control aims to find control variables under changing atmospheric
conditions such that demanded power production and/or a minimization
of the loading can be guaranteed, improving the cost and quality of
wind energy. State-of-the-art closed-loop dynamic wind farm
controllers are based on computationally expensive wind farm models,
which make these methods suitable for analysis though unsuitable for
online control. The latter is important because it allows for model
adaptation to the time-varying atmospheric conditions using supervisory control and data acquisition
(SCADA)
measurements. As a consequence, more reliable control settings can be
evaluated. A survey on wind farm control can be found in Knudsen et al. (2015) and Boersma et al. (2017), for example. In the latter, a clear
distinction is made between model-based and model-free control
algorithms. This paper is focussed on the former in which it is
assumed that controllers are based on a mathematical model of the
system. Consequently, the controller performance depends highly on the
model quality. Modelling is therefore a crucial step towards
successful implementation of model-based wind farm control.
Overviews on wind farm models can be found
in Crespo et al. (1999), Vermeer et al. (2003), Sanderse (2009),
Sanderse et al. (2011), Annoni et al. (2014), Göçmen at al. (2016) and Boersma et al. (2017). The
spectrum of these models ranges from low fidelity to high
fidelity.
The latter tries to capture relatively precise wind farm
flow and turbine dynamics, while the former tries to capture only the
dominant characteristics (dynamic or static) in a wind farm. Examples
of high-fidelity wind farm models are Simulator fOr Wind Farm
Applications (SOWFA) , UTD Wind
Farm (UTDWF) , SP-Wind and
PArallelized LES Model (PALM) . These three dimensional
(3-D) high-fidelity wind farm models can easily have 106 or more
states. The resulting computation time can be of the order of days or
weeks using distributed computation for simulation times less than the
computation time. In other words, the computation time needed for large-eddy simulations
(LESs)
is in general more than the total time that is simulated. Clearly,
these types of models are not applicable for online model-based
control. Rather, these models serve as analysis or validation tools.
One way to reduce the high complexity of wake modelling is by using
two-dimensional (2-D) heuristic models that only capture specific wake
and turbine characteristics in a wind farm in the horizontal plane at
hub height. These types of models are found on the low-fidelity side of
the spectrum. Most of these wake models exclusively estimate a steady
state situation for given atmospheric conditions. Examples of static
models are the Frandsen model and the Jensen–Park
model .
One extension of the Jensen model
resulted in the parametric model called FLOw Redirection and Induction
in Steady-state (FLORIS) . Two examples of low-fidelity
dynamic models are SimWindFarm and the
model used in , in which relatively simple
approximations of the flow deficit are computed using heuristic
expressions.
Medium-fidelity models can be found in the middle of the spectrum as
they trade off the accuracy of high-fidelity models with the
computational complexity of low-fidelity models. These are in general
based on simplified versions of the Navier–Stokes equations. For
example, in the 2-D dynamic wake meandering (DWM) model ,
assumptions are made regarding the thin shear layer such that the
Navier–Stokes equations can be approximated using less computational
effort. The authors in present a model, which is
also based on the thin shear layer approximation, but according to the
authors it is applicable for non-axisymmetric wind turbine wakes. WakeFarm
(also referred to as FarmFlow) simulates the wind turbine wakes by
solving the steady parabolized Navier–Stokes equations in three
dimensions . Other wind farm models
based on the 3-D Reynolds-averaged Navier–Stokes (RANS) equations
are Avila et al. (2013) and van der Laan et al. (2015). In , time
averaging is applied to the Navier–Stokes equations, resulting in the
2-D RANS equations. The number of states is then reduced by employing
a state reduction technique.
Also considered as medium-fidelity models are the ones presented
in Boersma et al. (2016b) and Soleimanzadeh et al. (2014). These wind farm models are
based on the discretized 2-D Navier–Stokes equations. However, these
models do not contain a turbulence model that allows for wake
recovery. In addition, these 2-D models do not take any neglected 3-D
effects into account and no yaw actuation of the individual turbines
is included.
In this paper, a model will be presented that can be considered as
a building block for the closed-loop control framework as illustrated
in Fig. .
General dynamical closed-loop control framework with measurements zk
and its estimation z^k and state estimation q^k.
The signals rk and wk are reference and control signals, respectively.
In this paper we present a dynamic model that is compatible with this framework.
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In current practice, signals such as power can be measured from a wind
farm, but current research is also focussing on estimating wake
characteristics using a lidar device . These and
other wind farm measurements are called SCADA data and can be used by
an estimator that is able to adapt the model parameters to current
atmospheric conditions and/or estimate the full state space, e.g.
all the flow velocities at hub height in the farm. The work presented
in illustrates the latter and employs the
dynamic wind farm model presented in this paper. Subsequently, the
estimation can then be used to compute optimal control variables using
a model predictive controller. The work presented in
illustrates the application of such a model predictive wind farm
controller using the dynamic model presented in this work.
The online closed-loop control paradigm as depicted in
Fig. demands a control-oriented dynamic
wind farm model that will be presented in this paper. Characteristics
of such control-oriented models are
low computational cost such that online model update, state estimation and control signal evaluation is
possible;
their dynamical nature such that they
can deal with varying atmospheric conditions within relatively small
timescales.
The dynamic control-oriented wind farm model presented in this paper,
referred to as WindFarmSimulator (WFSim), is applicable in the
framework discussed above and satisfies the two points above. It is
based on corrected 2-D Navier–Stokes equations and contains
a heuristic turbulence model. The Navier–Stokes equations are
modified in order to partially correct for the neglected vertical
dimension. Each turbine is modelled using the actuator disk
model (ADM) and features yaw and axial induction actuation. An
important model feature is the exploitation of the sparse system
matrices, leading to computational efficiency. WFSim will be compared
to high-fidelity flow data and used in a practical control
application.
The remainder of this paper is organized as follows. In
Sect. , the mathematical background of the medium-fidelity wind farm model will be explained including a discussion on
the rotor and turbulence model. This section ends with an analysis
regarding the wind farm model's computation time. In
Sect. , WFSim will be validated in two cases
using flow velocities in the longitudinal and lateral directions at
hub height and turbine power signals computed with two different LES-based wind farm models. The first case considered is a two-turbine
wind farm with turbines modelled using the ADM. The second case is
a nine-turbine wind farm with turbines modelled employing Fatigue,
Aerodynamics, Structures and Turbulence
(FAST) . This paper is concluded in
Sect. .
Formulation of a dynamic control-oriented wind farm model
In the current section, a simplified wind farm model is formulated
that is sufficiently fast for online control but retains some of the
elemental features of three-dimensional turbulent flows. In order for
the model to be fast, we envisage a 2-D-like model, but adapted to
account for three-dimensional flow relaxation. We will dub the
resulting model WFSim (WindFarmSimulator).
As a starting point we use the standard incompressible three-dimensional filtered Navier–Stokes equations,
as used in LES,
i.e.
∂ṽ∂t+(ṽ⋅∇)ṽ+∇⋅τM+1ρ∇p̃-f=0,momentum equation,∇⋅ṽ=0,continuity equation.
The velocity field
ṽ=(ṽ1,ṽ2,ṽ3)T
and pressure field p̃ represent filtered variables;
∇=(∂/∂x,∂/∂y,∂/∂z)T; the air density ρ, which is assumed to be constant; and
τM represents the subgrid-scale model, which will be
defined in Sect. . As is common in LESs of
high-Reynolds-number atmospheric simulations with grid resolutions in
the metre
range, direct effects of viscous stresses on the filtered
fields are negligible, so that these terms are left out. Finally, the
term f represents the effect of turbines on the flow, as
further detailed in Sect. .
Although LES filters are usually implicitly tied to the LES grid and
filter length scale in the subgrid-scale model, we presume here that
ṽ corresponds to a top-hat filtered velocity
field, with filter width D, where D is the turbine diameter. Thus,
ṽ(x,y,z)=1D3∫z-D/2z+D/2∫y-D/2y+D/2∫x-D/2x+D/2v(x′,y′,z′)dx′dy′dz′.
From a wind farm simulation perspective, we are mainly interested in
the flow velocity field at hub height zh, i.e.
ṽ(x,y,zh). Moreover, to evaluate turbine forces
and power, it suffices to know the velocity at turbine locations
tn=(xn,yn)T (with n=1⋯ℵ and ℵ the
number of turbines in the farm) since
ṽ(xn,yn,zh) is a reasonable representation of
the turbine disk-averaged velocity.
Therefore, we focus on formulating a 2-D-like set of equations for
ṽ(x,y,zh). To this end, we define
u=ṽ1(x,y,zh)ṽ2(x,y,zh)T,=uvT,
and w=ṽ3(x,y,zh) and p=p̃(x,y,zh)/ρ. Moreover, we assume that
w≈0, so that the LES equations given in Eq. () can
be reformulated in terms of u as
∂u∂t+(u⋅∇H)u+∇H⋅τH+∇Hp-f=-∂(uw+τM,13)∂ze1-∂(vw+τM,23)∂ze2,∇H⋅u=-∂w∂z,
with ∇H=(∂/∂x,∂/∂y)T, τH a 2-D tensor containing the
horizontal components of the subgrid stresses
τM, and e1 and e2 the unit
vectors in the x and y directions, respectively.
Finally, we further simplify the equations above using two additional
assumptions. First of all, we presume ∂w/∂z≈∂v/∂y. When centred at the turbine axis, this is one
of the conditions required for axial symmetry, though axial symmetry
also requires further conditions on ∂v/∂z and
∂w/∂y, which are not imposed. In general, ∂w/∂z≈∂v/∂y implies equal
divergence–convergence of streamlines in y and
z directions. Although this is a very simple condition, we presume it
to be good enough to resolve the lack of relaxation of purely 2-D
models. If necessary, a more general form (w≈0 and ∂w/∂z≈c⋅∂v/∂y), with c
a tuning parameter (e.g. obtained through state estimation) could
be considered, but results in the current work indicate that this may
not be necessary. Secondly, we simply neglect the right-hand side of
Eq. (). Though this is a rather crude
assumption, the rationale is that the modelling term
τH will suffice in the context of
a control model, where model coefficients can be updated online based
on feedback (see also the discussion in
Sect. ). Hence our final 2-D-like model
corresponds to
∂u∂t+(u⋅∇H)u+∇H⋅τH+∇Hp-f=0,∇H⋅u=-∂v∂y.
We emphasize here that the model above is not a classical 2-D model due to
the difference in formulation of the continuity equation. In contrast
to a standard 2-D model, this allows for flow relaxation in the third
direction when, for example, encountering slow down by a wind turbine. This
can be seen in Fig. , in which simulation results obtained with the model above, a standard 2-D dynamic wind farm
model and LES are shown. The simulation case itself will be discussed in more
detail in Sect. . Here we depict the normalized flow
deficit in the wake at 5D downstream of the turbine along the
cross-stream axis. The figure illustrates that the standard 2-D
Navier–Stokes equations lead to a significant speed up at the wake
edges. This is a result from conservation of mass in two dimensions
and the flow deceleration in front of the turbine, pushing part of the
air around the turbine. In the WFSim model, this speed up is smaller,
as mass can also flow around the turbine in the third dimension. In
Fig. it can be seen that LES data are better
estimated when imposing flow relaxation in the third
dimension. Finally, note that partially modelling the missing vertical
dimension as proposed above is novel with respect to the work
presented in .
Results of two-turbine simulations. Normalized time-averaged wake deficit at hub height
5D downwind from the downwind turbine using standard 2-D Navier–Stokes equations (red crossed),
our model with the adapted continuity equation (blue), and LES data (black dashed).
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Schematic illustration of the mixing length.
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This section is further organized as follows. First, in
Sect. , the subgrid-scale model will be
introduced. Then, in Sect. , the turbine model will
be explained. The discretization of the equations is presented in
Sect. , and boundary and initial conditions
are discussed in Sect. .
Turbulence model
In the literature, many subgrid-scale models are documented, and to
date, model accuracy remains a challenge in LES research (see e.g.
). However, in the current paper, an important
factor in the selection of a model is simplicity and computational
efficiency, rather than accuracy. In fact, in contrast to conventional
modelling, in a control-oriented model completeness of the turbulence
model is not a major issue since unknown model coefficients can be
calibrated online using measurements and
feedback , thus also controlling the overall
error. Therefore, in this work we fall back to one of the simplest and
first known turbulence models, Prandtl's mixing length model.
We formulate the stress tensor τH using
an eddy-viscosity assumption, i.e.
τH=-νtS,
with S=12(∇Hu+(∇Hu)T) the 2-D rate-of-strain
tensor, and νt the eddy viscosity. The latter is further modelled
as in :
νt=lu(x,y)2∂u∂y,
where lu(x,y) is the mixing length. It could be interesting to
define the mixing length for each position in the wind farm
separately, but this will lead to too many tuning variables. Moreover,
in , the authors illustrate that in a turbine's near
wake the mixing length is roughly invariant for different downstream
locations, but in the far wake, the mixing length increases linearly
with downstream distance. We use this to formulate a simple heuristic
parametrization for the mixing length model so that the number of
decision variables will be reduced drastically. From now on we assume
that the wind is coming from the east, but can have a direction
defined by φ. Then, the wind farm will be divided in segments
as illustrated in Fig. .
Each segments has its own (xn′,yn′) coordinate system located in the global (x,y) coordinate system. Now we propose the following mixing length parametrization:
lu(x,y)=G(xn′,yn′)*lun(xn′,yn′),if x∈X and y∈Y.0,otherwise,
with G(x,y) a (smoothing) pillbox filter with radius 3, * the
2-D spatial convolution operator, X={x:xn′≤x≤xn′+cos(φ)d}, Y={y:yn′-D2+sin(φ)xn′≤y≤yn′+D2+sin(φ)xn′} and φ defined as the
mean wind direction (see Fig. ), which we
bound by |φ|≤45∘ in this work. In addition we constraint d by
cos(φ)d≤|xq-xn|, with xn a turbine x coordinate
and xq its downwind turbines x coordinate. We can see
lun(xn′,yn′) as the local mixing length that belongs to turbine
n and denote it as
lun(xn′,yn′)=(xn′-d′)ls,if xn′∈Xn′ and yn′∈Yn′.0,otherwise.
with Xn′={xn′:d′≤xn′≤d} and
Yn′={yn′:|yn′|≤D} and tuning parameter
ls that defines the slope of the (linearly increasing) local mixing
length parameter. In fact, this parameter could be related to
turbulence intensity, i.e., the amount of wake recovery. In this
work we will not investigate this relation further. With the
formulation above, the number of tuning variables that belong to the
turbulence model (ls,d,d′) is reduced to 3ℵ. Additionally,
we assume that ls, d and d′ are equal for each turbine in the
farm, which reduces the number of tuning variables that belong to the
turbulence model to three, a quantity that could be dealt with by an
online estimator. However, in order to have only three tuning variables,
the included turbulence model is defined as a simplified mixing length
model found heuristically using and adapting information
from .
Turbine model
Turbines are modelled using a classical non-rotating actuator disk model (ADM).
In this method, each wind turbine is represented by a uniformly distributed force
acting on the grid points where the rotor disk is located. Figure
depicts a schematic top-view representation of a turbine with yaw angle γ.
Using such an approach, the force exerted by the turbines can be expressed as
f=∑n=1ℵfn,withfn=cf2CTn′[Uncos(γn)]2cos(γn+φ)sin(γn+φ)×HD2-||s-tn||2δ(s-tn)⋅e⟂,n,
with s=(x,y)T, H[⋅] the Heaviside function and
δ[⋅] the Dirac delta function, e⟂,n the unit
vector perpendicular to the nth rotor disk with position
tn. Furthermore, we have CTn′ the disk-based thrust
coefficient following , which can be expressed in
terms of the classical thrust coefficient CTn using the
following relation: CTn′=CTn/(1-an)2, with an the
axial induction factor of the nth turbine. Interestingly, the
coefficient CTn′ can be directly related to the turbine
set point in terms of blade pitch angle and rotational speed (see,
e.g. Appendix A in ). In the WFSim model,
CTn′ and yaw angle γn are considered as the control
variables and can thus be used to regulate the wakes and hence wind
farm performance. Furthermore, the scalar cf in
Eq. () can be regarded as a tuning variable and will in
this work be set equal for all turbines in the farm.
Schematic representation of a turbine with yaw angle γn and flow
velocity U=([u(xn,yn)]2+[v(xn,yn)]2)1/2 at the rotor. Figure adapted from .
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One cell for the x momentum equation (grey, ui,J), one
for the y momentum equation (yellow, vI,j) and one for the
continuity equation (pink, pI,J). All three cells have equal dimensions and overlap.
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Power
From the resolved flow velocity components, the power generated by the
farm is computed as
P=∑n=1ℵ12ρACPn[Uncos(γn)]3.
It is stated in (Appendix A) that when there is no
drag and swirl is added to the wake, CTn′=CPn. Since this is
an idealized situation, a loss factor will be introduced such that
CPn=cpCTn′. The scalar cp can be seen as a tuning
variable and will be set equal for all turbines in the farm. In the power expression
above, we have the factor cos(γn)3 with
exponent 3. In literature such as
and p. 37, for example, numerical values for the exponent
were given according to LES and wind tunnel data,
respectively. However, to date, the exact value for it is still under
research and since this is outside the scope of this study, the value
of the exponent will be 3.
This concludes the formulation of the WFSim model. In order to resolve
for flow velocity components and wind farm power, the governing
equations given in Eqs. () and () need to be
discretized, a topic that will be discussed in the following section.
Discretization
The set of equations are spatially discretized over a staggered grid
following . This is carried out by employing the
finite volume method and the hybrid differencing scheme. Temporal
discretization is performed using the implicit method that is
unconditionally stable . This comes down to
deriving the integrals:
∫Δt∫ΔV∂u∂t+(u⋅∇H)u+∇H⋅τH+∇Hp-fdVdt=0,∫Δt∫ΔV∇H⋅u+∂v∂ydVdt=0,
with ΔV the volume of one cell (see Fig. ) and
Δt the sample period. One obtains, for each cell, the
following fully discretized Navier–Stokes equations (for detailed
derivation we refer to Appendix ):
x momentum equation for the (i,J)th cell (black in Fig. ):
ai,Jpxui,J=ai,Jnxai,Jsxai,Jwxai,Jex×ui,J+1ui,J-1ui-1,Jui+1,JT-δyj,j+1pI,J-pI-1,J+fi,Jx+……+ai,Jnwxai,Jswxai,Jnexai,Jsex×vI-1,j+1vI-1,jvI,j+1vI,JT
y momentum equation for the (I,j)th cell (yellow in Fig. ):
aI,jpyvI,j=aI,jnyaI,jsyaI,jwyaI,jey×vI,j+1vI,j-1vI-1,jvI+1,jT-δxi,i+1pI,J-pI,J-1+fI,jy+……+ai,Jnwyai,Jswyai,Jneyai,Jsey×ui,Jui,J-1ui+1,Jui+1,J-1T
continuity equation for the (I,J)th cell (pink in Fig. ):
0=δyj,j+1ui+1,J-ui,J+2δxi,i+1vI,j+1-vI,j.
The states
u•,•,v•,•,p•,• are
defined for the time k+1 while the coefficients
a•,•• and the forcing terms
f•,•• depend on the state at time
k. Detailed definitions of these coefficients are given in
Appendix , Table . Note the appearance of the previously explained
factor 2 (see Eq. ) in
Eq. ().
Next, the state vectors uk,vk, and pk and control variable
vectors νk and γk at time step k will be defined:
uk=u3,2⋮u3,Ny-1u4,2⋮u4,Ny-1⋮uNx-1,2⋮uNx-1,Ny-1,vk=v2,3⋮v2,Ny-1v3,3⋮v3,Ny-1⋮vNx-1,3⋮vNx-1,Ny-1,pk=p2,2⋮p2,Ny-1p3,2⋮p3,Ny-1⋮pNx-1,3⋮pNx-1,Ny-2,νk=CT1′CT2′⋮CTℵ′,γk=γ1γ2⋮γℵ,
with Nx and Ny the number of cells in the x and y directions,
respectively, and ℵ the number of turbines in the wind
farm. Each component in uk, vk and pk represents a flow
velocity and pressure at a point in the field defined by
the subscript. For clarity reasons, an example of a staggered grid is
depicted in Fig. .
Example of a staggered grid with cells each having volume ΔV. In WFSim, the grid is quadrilateral.
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Mean computation time per simulation time step Δtcpu
vs. number of states nq. The red dashed line is WFSim as presented in
Eq. () and blue is WFSim as presented in Eq. ().
Note that the number of cells is approximately nq/3 with nq the number of states.
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Mean computation time per simulation time step Δtcpu vs. number of states nq for
the WFSim representation as given in Eq. (). Computations are performed on a regular notebook with one core.
nq
Δtcpu (s)
nq
Δtcpu (s)
nq
Δtcpu (s)
nq
Δtcpu (s)
3×103
0.02
27×103
0.22
115×103
1.19
239×103
3.1
6×103
0.04
43×103
0.37
147×103
1.66
258×103
3.5
9×103
0.06
64×103
0.60
182×103
2.12
268×103
3.7
14×103
0.10
88×103
0.88
221×103
2.50
276×103
3.8
Boundary and initial conditions
All the components that are not contained in the vector, uk,vk and
pk, but do appear in the staggered grid need to be defined. For the
flow velocity components, first-order conditions on the west side of
the grid are prescribed assuming the wind is coming from the
east. These Dirichlet inflow boundary conditions are related to the
ambient inflow defined as ub and vb and can vary over time. Zero
stress (also referred to as Neumann) boundary conditions are
prescribed on the other boundaries. Therefore, for the flow velocity
components on the boundaries we define
u2,J=ubfor J=1,2,…,Ny,ui,Ny=ui,Ny-1for i=3,4,…,Nx,ui,1=ui,2for i=3,4,…,Nx,uNx,J=uNx-1,Jfor J=2,3,…,Nx-1,v1,j=vbfor j=2,3,…,Ny,vI,Ny=vI,Ny-1for I=2,3,…,Nx,vI,2=vI,3for I=2,3,…,Nx,vNx,j=vNx-1,jfor j=3,4,…,Ny-1.
For the initial conditions, we define all longitudinal and latitudinal flow velocity components in
the field as ub and vb, respectively, which are the boundary velocity
values. The initial pressure field is set to zero. Note that by
defining the boundary conditions as given above, the assumption is
that the wind is coming from the east in Fig. , which
coincides with the definition of the mixing length (see
Sect. ). Finally, the equations given in
Eqs. () and () can be transformed to the
difference algebraic equation:
Ax(uk,vk)Axy(uk)B1Ayx(uk)Ay(uk,vk)B2B1T2B2T0︸E(qk)uk+1vk+1pk+1︸qk+1=A11000A220000︸Aukvkpk︸qk+b1(uk,vk,νk,γk)b2(uk,vk,νk,γk)b3︸b(qk,wk),
with nq=nu+nv+np and uk∈Rnu,vk∈Rnv,pk∈Rnp containing all flow
velocities in the longitudinal and lateral direction and the pressure
vector at time k and control variable
wk=νkTγkTT∈R2ℵ. The non-singular square
descriptor matrix E(qk) contains the coefficients
a•,••, appearing in Eqs. ()
and (), that depend on the state at time k. The square constant
matrix A solely depends on grid spacing and sample period Δt. Note
that the state vector contains three states for every cell; hence an increase
in grid resolution results in an increase in matrix dimensions. However, the
system matrices that occur in Eq. () are sparse and efficient numerical solvers are available for these
types of problems. This will be demonstrated in Sect. . The vector b(qk,wk) contains the forcing terms (turbines) and boundary conditions.
By defining Nx,Ny,ΔxI,I+1,ΔyJ,J+1, and the
turbine positions, a wind farm topology is determined. Next, ambient
flow conditions ub and vb, tuning
parameters cf,cp,d,d′,ls and the control variable wk need to
be specified. The system given in Eq. () is then
fully defined and can be solved.
Summary of the WFSim simulation set-up.
Domain size Lx×Ly
2×0.63(km2)
Turbine rotor diameter D
126.4 (m)
Grid size Nx×Ny
50×25
Turbine arrangement
2×1
Cell size Δx×Δy
40×23 (m2)
Turbine spacing
5D
Times
Δt=1,Δtcpu=0.02 (s)
Atmospheric conditions
ub=8, vb=0 (m s-1), ρ=1.2 (kgm-3)
Force and power factor
cf=1.7, cp=0.95
Turbulence model
d=530, d′=122 (m) ls=0.06
Excitation signals for the two-turbine simulation case. The yaw angles are set to zero.
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Mean flow centreline at four time instances through the farm. The vertical red dashed lines indicate the positions of the turbines.
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Wind farm power from PALM (blue dashed) and WFSim (black).
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Computation time
When discretizing partial differential equations, a trade-off has to
be made between the computation time and grid resolution. Typically,
a higher resolution results in more precise computation of the
variables but also increasing computation time. In WFSim,
computational cost is reduced by exploiting sparsity and by applying
the reverse Cuthill–McKee algorithm . The latter is applicable due to the fact that the matrix
structure is fixed. The interested reader is referred
to for more information on the Cuthill–McKee
algorithm in WFSim.
In this section, the mean computation time needed for one time step
Δtcpu will be analysed. The presented results are
obtained on a regular notebook with an Intel Core i7-4600U
2.7 GHz processor employing one core and MATLAB. Since the
objective is to do online control, i.e., it is desired to reduce
computational complexity, this section introduces a second WFSim
representation. The first representation was given in
Eq. () while the second is defined as
Ax(uk,vk)0B10Ay(uk,vk)B2B1T2B2T0︸E(qk)uk+1vk+1pk+1︸qk+1=A11000A220000︸Aukvkpk︸qk+b1(uk,vk,νk,γk)b2(uk,vk,νk,γk)b3︸b(qk,wk).
The difference can be found in the descriptor matrix. In the
representation above, the elements Axy(uk),Ayx(uk) that occur in
Eq. () are set to zero. This can be justified by
the fact that their contribution is negligible since these matrices
contain elements that, for our case studies, are of the order of
O(1) while the elements in Ax(uk,vk) and
Ay(uk,vk) are of the order of O(3). Consequently, no
significant change in the computed flow field has been observed,
but a decrease in Δtcpu has (see
Fig. ). Therefore, the remainder of this paper will continue
with the WFSim representation given in
Eq. (). Table depicts more
numerical values of Δtcpu for this WFSim
representation.
From Table we can conclude that Δtcpu increases between quadratic and linear with respect
to the number of states nq for nq<221×103. It depends
on the computer properties how much you can increase the number of
states until the CPU is out of memory.