Introduction
Wind turbines are often clustered together in wind farms to save the cost of
land and cabling. However, aerodynamic interactions between the turbines in
the form of so-called wakes (low-speed regions) that form behind
wind turbines lead to power reductions in “waked” turbines of up to 50 %
compared to a lone-standing wind turbine in undisturbed flow
. These interactions are very important when
considering the topological placement of wind turbines in large wind farms.
In order to optimally design wind-farm layout, models are necessary that
accurately predict the aerodynamic turbine–wake interaction effects. Such
models need to be very fast, as wind-farm design optimisation needs to
consider the full spectrum of wind directions over a wind farm's operational
lifetime, thus requiring many thousands of model evaluations. Moreover,
wind-farm design is a multidisciplinary problem in which the aerodynamic
wake-interaction model is only one of the models, next to turbine load
models, lifetime analysis, economic investment models, etc. (see,
e.g., ). Today, the wake model that is
most used is the Jensen model . It is a simple and fast
model, but it is known to be inaccurate when looking at individual power
predictions of turbines in various waked conditions
(; ; ).
Layout optimisation of wind farms using fast wake models has been
investigated in numerous studies (;
; ; ;
; ;
; ;
). However, the accuracy of such optimisation results
has always remained a concern in view of the limited reliability of wake
models, and this has recently led to a renewed interest in the formulation of
accurate but fast wake models (;
).
In the last five years, the detailed simulation of
wind-farm–atmospheric-boundary-layer interaction and turbine wake
interactions based on high-fidelity simulation tools such as large-eddy
simulation (LES) have become very popular (see, e.g., Meyers and Meneveau,
2010; Calaf et al., 2010; Yang et al., 2012; Meyers and Meneveau, 2013; Wu
and Porté-Agel, 2013; Allaerts and Meyers, 2015), leading to many new
insights into the flow physics of wind farms. Given known and constant
meteorological conditions, these types of models provide a detailed
time-resolved prediction of the turbulent flow in a wind farm with resolution
of spatial flow structures in the order of 20 m and temporal fluctuations in
the order of 10 s. Although it is computationally infeasible in LES of wind
farms to resolve all the detailed flow physics, such as the turbine
blade-boundary layers (with length scale below a millimetre), these models do
lead to quite accurate predictions of wakes and wake merging when compared to
wind-tunnel and field experiments (; ;
). Unfortunately, LES of wind farms requires
supercomputing and simulation times that are several hours to days for one
single atmospheric condition. Hence, these models are not useful for layout
optimisation purposes.
In the current work, we investigate a hybrid approach in which the Jensen
model is used during optimisation, but we use LES to gradually adapt the
Jensen model and verify the optimisation results. To this end, the
wake-expansion coefficient in the Jensen model is iteratively fitted based on
LES. In itself, tuning of the wake-expansion coefficient (e.g. to
experiments) is quite common, but it is well known that the coefficient
depends on atmospheric conditions and farm layout, and it may also best depend on
streamwise distance into the farm . Therefore, a
coefficient that is tuned a priori will not fit all possible scenarios that
are encountered during layout optimisation of a wind farm over its relevant
range of atmospheric conditions. In a hybrid Jensen–LES approach, it is
possible to adapt the coefficient a posteriori during optimisation depending
on layout, wind direction, etc. The main focus of the current work is on the
formulation of an approach that is computationally feasible, given the very
high costs of performing LES (even in a hybrid Jensen–LES optimisation). We
demonstrate the proposed methodology on a moderately sized wind farm of
30 turbines in a 4 km by 3 km farm area.
This paper is organised as follows. In Sect. the
mathematical formulation for the optimisation problem is stated, and the
simulation models (both Jensen and LES) and the optimisation methodology are
introduced. In Sect. , results are presented. First, the
different steps in the algorithm are highlighted for a single wind-direction
optimisation case in Sect. –.
Subsequently, in Sect. , some results for optimisation with
multiple wind directions are discussed. Finally, conclusions are presented in
Sect. .
Problem description and methodology
In Sect. , the optimisation problem description is
introduced. Subsequently, the Jensen model is briefly reviewed in
Sect. . The LES simulation environment is discussed in
Sect. , and finally the hybrid Jensen–LES approach and the
optimisation method are presented in Sect. .
Problem description
Consider a set of Nt turbines that are to be placed in a fixed
domain Ω. Given constant atmospheric conditions and wind direction
(parameterised in a vector μ), the average power output of a
turbine at position xi in the wind farm is
P‾i(xi,μ)=1T∫0TPi(xi,t,μ)dt,
where Pi(xi,t,μ) corresponds to the instantaneous
power output of the turbine (given atmospheric conditions
μ), which is subject to turbulent wind fluctuations, and T
is a time averaging window that is sufficiently long to average out the
turbulence effects. Note that the Jensen model (see
Sect. ) directly predicts P‾i of turbines
in a wind farm, while, for example, experimental measurements as well as
results from LES (see Sect. ) yield
Pi(xi,t,μ) and thus explicitly require the above
time averaging.
The optimisation problem that we consider is formulated as
follows:
maximisexi∫∑i=1NtP‾i(xi,μ)fp(μ)dμsubjecttoxi∈Ω,∀i∈{1,⋯,Nt}‖xi-xj‖2≥dmin∀i,j∈{1,⋯,Nt},i≠j,
where Ω is the wind-farm domain in which turbines can be freely placed
and fp(μ) is the joint probability density function
of atmospheric conditions μ over which optimisation needs to be carried out (e.g. the yearly
wind-direction distribution, atmospheric stability class). Finally,
dmin is a constraint on the minimum distance between turbines. In
theory, the minimum distance between turbines is 1.0D (with D the rotor
diameter). In the current study, we will consider a dmin of 2.0D for
all optimisation cases.
The solution of the above optimisation problem requires a model for
P‾i(xi,μ). This is discussed next in
Sect. for the Jensen model and in Sect.
for the LES model. To solve the above optimisation problem, we use the
cross-entropy optimisation method in combination with a
hybrid Jensen–LES model as discussed in Sect. . Finally, note
that, for ease of notation, we drop μ as an argument in
P‾i. In fact, the conditions μ (e.g. wind
direction, turbulence intensity) are implicity contained on the set-up
and boundary conditions of the respective models below.
The Jensen wake model
We briefly review the Jensen wake model as originally developed by
and .
The model commences by assuming that each turbine generates a radially and
azimuthally uniform wake that linearly expands with downstream distance from
the turbine. Using simple mass conservation, this allows the
velocity deficit generated by turbine i to be described as
ΔUi(si)=U∞1-1-CT,i(1+kwsi/R)2,si>0,
where CT,i is the turbine thrust coefficient and si=(x-xi)⋅ef is the downstream axial
distance from the turbine, and ef the unit vector in the
mean-flow direction. Obviously, si>0. Upstream of a turbine, its own
generated wake has a velocity deficit ΔUi=0. Furthermore, kw is the linear wake-expansion coefficient, and R is the rotor radius.
Correlations exist that relate kw to the incoming atmospheric
boundary layer; for example
kw=u*U∞=κln(zh/z0)
is commonly used, with κ the von Kármán constant, zh
the turbine hub height, and z0 and u* the surface roughness and
friction velocity of the incoming atmospheric boundary layer. Note that, in
the current study, we will use LES to adapt kw in our optimisation
procedure as discussed in Sect. . Finally note that the wake
expansion is vertically restricted by the ground once the wake radius grows
larger than the turbine hub height. However, the ground is not directly
modelled, but instead mirror turbines are added below the ground, with wakes
that are included in the wake-merging model (see below).
In order to estimate the power output P‾i, the turbine's
incoming mean velocity is required. It is modelled as
Ui,in=U∞-ΔUi,in, with U∞ the
wind-farm inflow velocity at hub height and ΔUi,in the
upstream velocity deficit experienced by turbine i. The deficit ΔUi,in is heuristically modelled by quadratically adding upstream
wake deficits as follows:
ΔUi,in=∑j∈Si(ΔUj(sij))21/2.
Here Si is the set of all upstream turbines that have a wake
that geometrically intersects with turbine i and sij is the distance
along the wind direction between turbine i and j. In order to include the
effect of the ground on wake development, mirror turbines (below the ground)
are added to the set Si for each turbine whose wake is
restricted by the ground. It is furthermore possible that wakes only
partially overlap, in which case the rotor area of the inflow turbine is
split into regions with different overlaps. More details on the approach can
be found in .
Once the turbine inflow velocities Ui,in are determined, the
power per turbine is calculated as
P‾i(xi)=12CP,iρUi,in3,
where CP,i is the wind turbine's power coefficient. For an ideal
turbine, CP,i follows from axial momentum theory from, i.e.
CP,i=12CT,i[1-1-CT,i1/2].
For a real turbine, CP,i can be expressed as a function of
CT,i and wind speed, using either a mapping specific to the
turbine or blade-element momentum theory, and this can be straightforwardly
used in the Jensen model. In the current study, we will simply use above
ideal relationship, as our main focus is on the development and demonstration
of the hybrid Jensen–LES approach, and not so much on the specifics of the
selected turbine model.
Large-eddy simulation environment and simulation set-up
Simulations are performed using SP-Wind, developed at KU Leuven
. SP-Wind solves
the filtered incompressible Navier–Stokes equations, which are given by
∇⋅ũ=0∂ũ∂t+ũ⋅∇ũ=-1ρ∇p̃+∇⋅τM-f,
where ũ(x,t)=[ũ1,ũ2,ũ3] is the resolved velocity field, p̃ is the
pressure field, and τM is the sub-grid-scale (SGS)
model. We use a standard Smagorinsky model with Mason
and Thomson's wall damping to model the SGS stress.
Furthermore, -f represents the forces (per unit mass) introduced by
the turbines on the flow. In LES of wind-farm boundary layers, this
turbine-induced force is commonly modelled using an actuator-disc model
(ADM), as full meshing of the turbine blades and geometry leads to
computational grids that are too large for current-day computers. Expressed
for turbine i, this force corresponds to :
f(i)=12CT,i′V^i2i(x)e⊥;i=1⋯Nt,
where e⊥ represents the unit vector perpendicular to the
turbine disc, and
i(x) is a
geometrical smoothing function that distributes the uniform surface force of
the turbine over surrounding LES grid cells, with ∫Ω
i(x)dx′=A and A the turbine disc area. Moreover, V^i is the
disc-averaged turbine velocity, and CT,i′ is the disc-based
thrust coefficient. Unlike the conventional thrust coefficient CT
(used in the Jensen model), which is based on undisturbed velocity far
upstream of a turbine, CT,i′ is defined using the velocity at the
turbine disc. It results from integrating lift and drag coefficients over the
turbine blades, taking design geometry and flow angles into account (see
Appendix A in , for a detailed formulation). Based on axial
momentum theory, we have
CT=CT′(1+CT′/4)2,
which provides a direct relation between the thrust coefficient used in the
Jensen model and the disc-based thrust coefficient used in the LES model.
Finally, given the velocity field ũ(x,t) from a
LES, the average power output for turbine i is determined from
P‾i(xi)=1T∫0T∭f(i)⋅ũdxdt.
In Fig. a typical snapshot of a horizontal velocity field
ũ1(x,t) is shown, including an outline of the simulation
domain that is considered in the current study. The main domain size is Ly×Lx×Lz=8.0×6.0×1.0 km3, where x is
always the main flow direction and z is the vertical direction. The
wind farm is inserted in a subdomain Ω=4.0 km × 3.0 km
(also marked on the figure). At z=0 a classical high-Reynolds-number
wall-stress boundary condition is used , which
is parameterised by the ground surface roughness z0, for which we use
z0=0.1 m. At z=Lz a symmetry condition is used, and in the y
direction periodic boundary conditions are used. Finally, at x=0 an inflow
boundary condition is used.
The inflow is generated in a separate precursor simulation (also shown in
Fig. ), which employs shifted periodic boundary conditions
to avoid artificial spanwise locking of the typical low-speed streaks
observed in boundary layers (see , for details). For the
precursor simulation, a domain size of 8.0×6.0×1.0 km3 is
selected. The precursor simulation is driven by a constant pressure gradient,
which corresponds to ∇p∞/ρ=u*2/Lz, where
u*=(τw/ρ)1/2 is the friction velocity in the precursor domain.
In the current work, we are interested in region II operation of wind
turbines for which CT′ can be presumed to be constant. Given that
also z0 and zh are fixed, simulation results remain
dynamically equivalent for any selected value of u*, with velocity
scaling proportionally with u* and time scaling inversely
proportionally with u*. An output of our precursor simulation (given
z0=0.1 m) is the hub-height velocity uh≈17.5u*. Thus, to
obtain a realistic region II hub-height velocity of, for example, 8 m s-1, it
suffices to select u*=0.457 m s-1. However, since all later
results and comparisons with the Jensen model are normalised with inflow
velocity, or with first-row power output, the exact value of u* is
not further important (in our simulations, we just use u*=1).
Snapshot of an instantaneous velocity field in
the precursor domain and main simulation domain obtained from SP-Wind. Left panels: precursor, with side view (top) and plan view (bottom). Right panels: main, with side view (top) and plan view (bottom). Wind-farm area Ω is shown in the green dashed box, and the fringe region with the green
dash-dot line.
For the discretisation of the governing equations, SP-Wind uses a
pseudo-spectral method in the horizontal directions, applying the 3/2 rule
for dealiasing . In the vertical direction, a fourth-order
energy-conservative finite-difference discretisation scheme is used
. Non-periodic boundary conditions in the x direction
are implemented using a fringe-region technique, with a fringe region located
in the last 2 km of the domain (for details, see
). Mass is
conserved by using a Poisson equation for the pressure, which is solved using
a direct solver. Finally, time integration is performed using a classical
four-stage fourth-order Runge–Kutta scheme. For the simulations discussed in
this paper, a fixed time step of 0.4 s corresponding to a
Courant–Friedrichs–Lewy (CFL) number of approximately 0.4 is used. The
computational grid for the main domain corresponds to Ny×Nx×Nz =256×256×80; this is also the case for the
precursor domain. For nonlinear operations we use the 3/2 dealiasing rule,
so that all nonlinear operations in real space are performed on 384×384×80 grids for both domains. Simulation parameters are summarised
in Table .
In the current study, we consider a rectangular fixed wind-farm domain
Ω of 4.0 km by 3.0 km (see above), in which 30 turbines are to be
optimally placed. We take generic wind turbines with a diameter of
D=100 m and hub height of zh=100 m each. The selected
disc-based and standard thrust coefficients correspond to CT′=2.0 and CT=8/9 respectively. The choice of turbines,
simulation domain, and selected computational grids corresponds to the typical
case set-ups found in and , and we refer the
reader to these studies for detailed grid sensitivity analysis, for example.
Simulation parameters. Results remain dynamically equivalent for any
selected value of the friction velocity u*. The hub-height velocity
obtained in the precursor simulation corresponds to
uh ≈ 17.5 u*.
Total domain size (with
8 km × 6 km × 1 km
fringe region)
Total domain size
6 km × 6 km × 1 km
(without fringe region)
Optimisation domain size
4 km×3 km
Turbine diameter
100 m
Turbine height
100 m
Driving pressure
-u*2 / 1000 m s-2
gradient (precursor)
Surface roughness
0.1 m
Grid size
256 × 256 × 80
Cell size
31.25 m × 23.44 m × 12.5 m
Time step
0.4/u* s
Finally, simulations are initialised by first performing a spin-up of the
turbulence in the precursor simulation. Starting from a logarithmic mean
profile with random perturbations, the precursor simulation is advanced in
time for 15 000/u* s so that realistic turbulence can
develop. Subsequently, for every wind-farm layout, the precursor and main domain
are run simultaneously, and an additional spin-up period of
2000/u* s is simulated. This corresponds to at least 5u*
flow-through times of the main domain. At this point in time, time averaging
of LES results is started.
In Fig. 2, a detailed convergence analysis of the farm power and the power
output of a single turbine is shown for an aligned wind-farm layout
(corresponding to Case 4 in Table below). In
Fig. a, a power histogram is shown for the wind farm, as
well as for two individual turbines in the farm. Figure b
shows results of the relative error ϵP of the time average as a
function of the averaging time T (see Eq. 1), where
ϵP(T)=1T∫0TP(t)dt-P‾refP‾ref.
For reference P‾ref we use an average obtained over a
period of 40/u* h (with u*=0.457 m s-1 taken as a realistic
value, this corresponds to averaging over 88 h in physical time). It is seen
from the figure that, for limited averaging times, errors can be quite
significant, in particular when looking at the single-turbine average. In
fact, it is well known that the time average in turbulent flows converges as
T-1/2 see, e.g.,. This is also seen in
Fig. b:
errors decrease fast at low values
of T, but afterwards convergence stagnates. This is particularly
problematic when looking at the turbine average power, which requires roughly
15 to 20/u* h to converge within 1 % of the reference
average (requiring excessive computational costs – see below). It is further
seen that the error on the overall wind-farm power converges
significantly faster, i.e. an error of 1 % is reached after approximately
5/u* h. This is related to the fact that Nt partly
uncorrelated turbine power signals are accumulated. Therefore, in order to
limit computational effort related to LES in a hybrid Jensen–LES approach,
we will formulate our approach based on matching LES and Jensen farm power
levels. In order to avoid overfitting of the Jensen wake-expansion
coefficient, we use an ensemble of different wind-farm layouts that gradually
evolve during optimisation towards layouts that are more optimal in terms of
power extraction. This approach is further discussed in Sect. .
Convergence analysis of wind-farm and turbine power of an aligned
wind-farm case (Case 4 in Table ). (a) Probability density
function of
wind-farm power output and power output of a front-row and back-row turbine.
(b) Convergence error ϵP as a function of averaging
time T for the wind-farm power, and for the power of a front-row and
back-row turbine. Blue line: wind-farm power; green line: first-row
turbine; red line: last-row turbine.
In terms of computational cost, the spin-up of the precursor simulation is
the most expensive (but needs to be done only once), amounting to 32 h of
wall-clock time on the ThinKing cluster of the Flemish Supercomputer Centre,
using eight Ivy Bridge nodes consisting of two 10-core “Ivy Bridge” Xeon
E5-2680v2 CPUs (2.8 GHz, 25 MB level 3 cache) for a total of 160 cores.
Wind-farm spin-up takes around 14 h of wall-clock time on the same
processor layout. Subsequent averaging takes around 9 h of wall-clock time
per 3600/u* s of wind-farm time. In order to keep overall computational
costs under control, we limit time averaging in the current work to
3600/u* s (roughly corresponding to at least 9u* flow-through
times). This yields an expected error level on the power output of 2 % (see
discussion above and Fig. b). In practice, for
optimisation over a single atmospheric condition μ (e.g. a
single wind direction), it may be advisable to use at least 5/u* h for
the current case set-up. However, when considering optimisation over a range
of conditions, the impact of this variability will be further averaged out.
Hybrid Jensen–LES approach and cross-entropy optimisation
In the current manuscript, we propose a hybrid Jensen–LES approach for
wind-farm layout optimisation. To that end, the layout optimisation is based
on the Jensen model, but the wake-expansion coefficient kw is
iteratively used to fit the Jensen model to a set of LES data that is
gradually adapted to the layouts that are explored during optimisation. The
approach is summarised in Algorithm 2. Here, we describe the
approach considering a single atmospheric condition μ (see
Eq. ), e.g., a single wind direction. Generalisation
is straightforward, and optimisation over different wind directions will be
discussed in Sect. .
In a first step, a set of NL LES cases of regular and random
layouts are generated. This set is used to fit kw using
Algorithm 4 (see below). Subsequently, layout optimisation is performed using
the Jensen model and Algorithm 3 (see further below). The optimal layout is
then added to the set of LES cases, and a number of NR-1
(NR<NL) additional random layouts are added as
well. Moreover, the NR LES cases with lowest generated powers are
removed from the set. This new set is used to refit kw,
subsequently starting a new layout optimisation. By doing so, the LES data set used for fitting is
gradually taking more optimal layouts into account, while layouts that are
least optimal are removed from the set.
The procedure described above directly uses wind-farm power to fit the
wake-expansion coefficient and avoids using errors on individual turbine
power output. As discussed in Sect. , this reduces the need for
time averaging in the LES, and significantly lowers computational costs. Moreover, by including NL
different layouts, potential overfitting of kw is avoided, and the
influence of remaining LES convergence errors on the optimal fit is further
reduced.
For the layout optimisation in Algorithm 3 and the optimal
fit of kw in Algorithm 4, we employ the cross-entropy (CE) method. This method was originally developed to estimate the
probability of rare events. Later on, it was realised that it is also very
effective in solving difficult non-convex optimisation problems. The method
is explained in detail by and , among others. Here, we
briefly review the main features of the approach, as well as further detailing how we
use it in a hybrid Jensen–LES optimisation of wind-farm layout. In our
hybrid Jensen–LES optimisation approach of wind-farm layout, we use the CE
method both for Jensen-based layout optimisation, as well as for the adaptive
fitting of the Jensen wake-expansion coefficient against a range of LES
results (as further detailed below). However, it is important to emphasise
that any feasible optimisation method may be used for this. For instance,
recently, some work has focussed on the use of a gradient-based layout
optimisation approach in combination with engineering wake models
, while others have previously looked into the use of, for example, a
particle-swarm method and genetic algorithms
.
First of all, the optimisation problem Eq. () is
slightly reformulated in order to better cope with the second inequality
constraint (as further discussed below, the first constraint is more
straightforward to enforce directly). Therefore, we consider following a
non-smooth problem,
maxxi∑i=1NtP‾i(xi)+∑i=1Nt∑j=1i-1hij(xi,xj)subjecttoxi∈Ω,∀i∈{1,⋯,Nt},
where
hij(xi,xj)=-∞‖xi-xj‖2<dmin0otherwise.
This formulation is fully equivalent to Eq. ().
The CE method for solving the optimal placement problem now essentially
involves three steps. In a first step, a set of Ns uniformly distributed random samples of the optimisation
parameters xi are generated with a given mean value m(0)
and deviation d(0) (note that both m and d have
dimension 2×Nt). At startup (iteration 0), no prior
knowledge of the optimisation problem is available, so we chose the mean and
deviation such that the distribution spans the whole feasible parameter range
Ω. In the second step, samples are sorted according to their energy function value. The best Nb<Ns samples are chosen, and the mean
mb(k) and deviation db(k) of this set
(in iteration step k) is calculated. In a third step, a
next generation of samples (iteration step k+1) is then created using a
uniform distribution with a mean and deviation of
m(k+1)=m(k)+α(mb(k)-m(k)),d(k+1)=d(k)+α(db(k)-d(k)),
where the parameter α is selected in the [0,1] range, specifying how
conservative or exploratory the algorithm is. This procedure continues until
the end condition is met, which is usually set by specifying the maximum
number of iterations. We also transfer the optimum value in each generation
to the next generation, so that the energy function value of the optimum in
each generation increases monotonically.
The treatment of the constraint xi∈Ω is straightforward.
Whenever a turbine location in a sample falls outside Ω, the location
is simply orthogonally projected on the boundary of Ω. Note that
turbines in samples in the initial generation always fall in Ω, but in
later generations, this is not always the case. Though the projection on
Ω will slightly change the distribution, as relatively more sample
points can end up on the boundary, we did not find this to hamper the
convergence of our algorithm. Finally, the treatment of the distance
constraint is implicitly handled by the energy function formulation and does
not, in principle, require any further attention.
Given the Jensen model, and an input for the wake-expansion coefficient
kw, the cross-entropy layout optimisation is summarised in
Algorithm 2, and specific choices are documented. We run
the cross-entropy optimisation scheme for Niter=2000 iterations;
however, we find it beneficial for convergence and computational efficiency
to omit hij in the energy function during the first M iterations, and to
only enforce the hij constraint for k>M. We take M=200 in our
implementation.
The standard deviation of samples in the cross entropy scheme eventually
converge to zero. Once the standard deviation has become small, and if the
algorithm is locked in a local optimum, it will no longer break away from
it. To reduce the chance of this happening, we reset the calculated value of
d after 1000 iterations. For turbines with x coordinate less than
0.5 km or bigger than 3.5 km, we reset their corresponding deviation to
[0.5,0.5], and for the rest we reset the deviation to [Lx/2,Ly/2]. This
can be interpreted as running the cross entropy in two stages. Both run for
1000 iterations: the first runs starting with a uniform distribution in
Ω, and the second starts with the optimum layout of the first stage as
the mean value for its initial population. In the interest of simplicity,
this detail is not included in the outline of Algorithm 2.
A second algorithm that is used in Algorithm 1 is the fitting
of the wake-expansion coefficient kw to the LES data. Fitting
kw is also a non-convex optimisation problem, and therefore we
simply use the CE method again, but now for a scalar field. This is
summarised in Algorithm 3. For this fitting, we found a number
of iterations, Niters, of 50 sufficient for good convergence.
We remark here that Algorithm 3 can in principle be used to fit
more complicated relations for kw. For instance, introducing the
heuristic dependence kw=a+bx (or similar expressions), and fitting
a and b instead of the mean value of kw, may be an interesting
approach to represent the downstream development of kw in the
wind farm related to increased turbulence levels. In the current work, we did
not further explore this type of parameterisations of kw, as a
simple fit of the mean value already leads to very satisfactory results (see
next section).
Finally, we remark that the CE method is a global optimisation method.
However, its convergence to the global optimum in a finite number of
iterations can only be formally proven for some specific conditions; in
practice, convergence depends on a number heuristic choices and is difficult
to formally prove see, e.g.,for details. In fact,
this is a disadvantage that all global optimisation methods share. However,
the main advantage of using a global method is the fact that the algorithm
does not get trapped in local optimums that easily. Moreover, the
disadvantage of the high number of function evaluations required for such
global methods to work well is not really an issue, as Jensen-model
evaluations are extremely cheap. In fact, as further discussed below, the
main cost in our overall hybrid method remains associated with performing the
LES.
Results
In the current section, optimisation results are discussed. First of all, in
Sect. , the initial LES database for calibration of the Jensen
model is constructed. Next, optimisation results of the Jensen only model are
discussed in Sect. . Subsequently, hybrid Jensen–LES
optimisation results are presented in Sect. . Finally,
optimisation for multiple wind directions is discussed in
Sect. .
Set-up of LES database for initial calibration
Large-eddy simulation results for different wind-farm layouts. Power
output normalised with respect to total power of a wind-farm consisting of
“first-row” turbines. Average LES power is 69.97 %.
Case no.
Description
Relative wind-farm power
1
Aligned with 5D × 5D spacing
51.81 %
2
Aligned with 6D × 5D spacing
56.76 %
3
Aligned with 7D × 5D spacing
60.80 %
4
Aligned with 8D × 5D spacing
64.36 %
5
Staggered with 8D × 5D spacing
83.60 %
6
Gradually staggered with 8D spacing
87.40 %
7
Randomly generated with dmin = 2D
79.28 %
8
Randomly generated with dmin = 3D
76.16 %
9
Randomly generated with dmin = 4D
78.66 %
10
Randomly generated with dmin = 5D
80.49 %
Layout and relative turbine power output for four of the cases
listed in Table . Relative power results are obtained from
large-eddy simulations. Turbine locations are marked with coloured disk: size
and colour scale by relative power. Plot boundary (red line)
corresponds to boundaries of domain Ω
(see Fig. ).
A first step in Algorithm 2 is the generation of a LES
database that is a starting point for the calibration of the Jensen model.
Here we choose a mix of staggered, aligned, and randomly generated
layouts. An overview of the different cases and their generated power is
provided in Table . We normalise all results with the power
output of a “wakeless” wind farm, i.e. a wind farm consisting of turbines
that all have undisturbed inflow. In order to normalise all LES results in
the same way, we use the averaged power output of turbines located in the
first row of the aligned and staggered layouts and multiply it by Nt (= 30) to find the “wakeless” wind-farm output. We then state every
wind-farm power output as a percentage of this “wakeless” wind-farm output.
When looking at the results of Table it is apparent that the
aligned cases perform quite poorly in terms of relative power output and
considerably worse than the staggered cases, but they also perform worse than any of the
random layouts that we investigated.
In Fig. the layout and relative power output of
individual turbines are shown for an aligned and a staggered layout as well as for two of the
random layouts. Wind direction is always from left to right. First of all, we
remark that there is still considerable variability at turbine level that is
due to the limited averaging period of 3600/u* s. As shown in
Fig. b, variability in the turbine power average is in the
order of ±5 % or more, and this is in line with the variability
observed in the first row of Fig. . We verified
that first-row turbine averages all converge to a relative power of 100 %
when averages of up to 15/u* h are used. Finally, we note
that the accumulated farm power is much better converged.
Comparison of Jensen model and LES results
Without access to reference results that can serve to tune kw in
the Jensen model, it is possible to resort to Eq. () to
determine kw. Using this equation for our simulation set-up leads
to
kw=0.41ln(100/ 0.1)=0.060.
Here we briefly compare the Jensen model using this value with LES results.
To do so, we use the 10 layouts presented in Table .
A comparison of flow fields as generated by the Jensen model and LES is
shown in Fig. . It is seen that the
averaged flow data of LES are much smoother as a result of turbulent mixing.
In contrast, in the Jensen model, wakes have a sharp boundary, also leading to
sharply marked overlap regions. Note that mirror wakes also occur more
downstream in the farm. Some features are not represented at all by the
Jensen model. For instance, in the random layout, it is seen that
side-by-side wakes can influence each other. Such behaviour is not
parameterised in the Jensen model.
Comparison of Jensen model and LES for an aligned and random
layout.
Comparing outputs of LES and the Jensen wake model with
kw = 0.060.
Case
Relative
Relative
Error
power
power
(LES)
(Jensen
model)
Aligned 5D × 6D
51.21 %
52.30 %
-1.09 %
Aligned 6D × 6D
55.93 %
57.98 %
-2.05 %
Aligned 7D × 6D
60.13 %
62.88 %
-2.75 %
Aligned 8D × 6D
63.34 %
66.83 %
-3.50 %
Staggered 8D
82.33 %
86.81 %
-4.48 %
Gradually staggered 8D
85.77 %
89.18 %
-3.41 %
Random1
78.29 %
85.20 %
-6.91 %
Random2
74.77 %
82.30 %
-7.53 %
Random3
77.96 %
84.95 %
-6.99 %
Random4
79.17 %
84.04 %
-4.87 %
However, the most relevant property from a power optimisation point of view
is the total error in the predicted power. In Table ,
the average power output from LES and the Jensen model is compared. It is
seen that the Jensen model using kw=0.060 is very accurate
for some cases, but not so for others. In particular, the cases that have a
higher relative power extraction are generally predicted worse by the Jensen
model, than the cases with a lower relative power (the most prominent
exception is Case 6). Another trend is that the regular cases are better
predicted than the irregular cases. However, in the context of optimisation, it is
not important for the Jensen model to be accurate over a wide range
of different layouts. Far away from the optimal layout, the required accuracy
can be allowed to be considerably lower than close to the optimum. In this
sense, Algorithm 2 gradually adapts the Jensen model through
its wake-expansion coefficient to better fit more performing layouts.
Finally, when looking at turbine level in Fig. for
one of the random layouts (i.e. Case 10), it is seen that errors at the turbine
level are much larger than the error on the accumulated power reported in
Table . Again, from an optimisation point of view,
this is less of an issue as long as a coupled approach in combination with LES
is used to adapt the model and verify the overall results close to the
optimum. We further notice here that the statistical errors on the averaged
turbine power output from LES are still significant due to the limited time
of averaging (in the order of 5 % – see discussion in
Sect. ).
Comparing the wind-turbine power generation obtained from LES data
(black numbers) and Jensen model (red numbers). Turbine locations are marked with
coloured disk: size and colour scale by relative power. Plot boundary
(red box) corresponds to boundaries of domain Ω
(see Fig. ).
Hybrid Jensen–LES optimisation
Using Algorithm 2, we now optimise the wind-farm layout with
a single constant wind direction given the set-up in
Fig. and wind coming from the left. Strictly speaking,
this corresponds to the situation where fp(μ) in
Eq. () corresponds to a Dirac delta function centred
around an eastern wind direction, so that the integral over atmospheric
conditions in Eq. () drops out. Optimisation over
different wind directions is briefly discussed in Sect. .
For the single wind-direction case considered here, only three outer
iterations are required in the algorithm to converge to an optimal layout and
optimally tuned Jensen model. Intermediate results of these iterations are
discussed below.
In iteration 1, we start Algorithm 2 with the initial
cases shown in Table . Using these 10 cases, we use
Algorithm 4 to optimise the value of kw, finding a
value of kw=0.055. Subsequently, this value is used to optimise
the layout using Algorithm 3. The resulting optimal layout
is shown in Fig. . Table
summarises the relative LES and Jensen power, as well as errors for the 10 initial
training cases and for the newly obtained optimal layout. The
relative power generated by the newly found optimum corresponds to 90.5 %
(evaluated using the LES), but the error with the Jensen model is still
noticeable, i.e. -2.62 %.
Optimal layout and relative power for a single wind direction
obtained after iteration 1. Relative power results are obtained from
large-eddy simulations. Turbine locations are marked with coloured disk: size
and colour scale by relative power. Plot boundary (red line)
corresponds to boundaries of domain Ω
(see Fig. ).
Iteration 1: comparing outputs of LES and Jensen wake model with
kw = 0.055.
Case
Relative
Relative
Error
power
power
(LES)
(Jensen
model)
Aligned 5D × 6D
51.21 %
49.59 %
1.62 %
Aligned 6D × 6D
55.93 %
55.40 %
0.53 %
Aligned 7D × 6D
60.13 %
60.17 %
-0.04 %
Aligned 8D × 6D
63.34 %
64.22 %
-0.88 %
Staggered 8D
82.33 %
85.78 %
-3.46 %
Gradually staggered 8D
85.77 %
92.27 %
-6.50 %
Random1
78.29 %
84.52 %
-6.23 %
Random2
74.77 %
81.23 %
-6.46 %
Random3
77.96 %
84.51 %
-6.55 %
Random4
79.17 %
83.27 %
-4.10 %
Optimum iter. 1
90.51 %
93.13 %
-2.62 %
Optimal layout and relative power for a single wind direction
obtained after iteration 2. Relative power results are obtained from
large-eddy simulations. Turbine locations are marked with coloured disk: size
and colour scale by relative power. Plot boundary (red line)
corresponds to boundaries of domain Ω
(see Fig. ).
In iteration 2, we add optimal layout 1 and four additional random
layouts to the LES database and remove the 5 layouts with lowest relative
power. Using Algorithm 4, we find a new value kw=0.036 that best fits the Jensen model to the LES data. Subsequently, using
Algorithm 3, a new optimal layout is found, which is shown
in Fig. . Furthermore, an overview of relative
powers from LES and Jensen is shown in Table . It is
seen that the new optimal layout leads to a relative power of 92.8%
(evaluated using LES), but in contrast to the first iteration, the error with
the Jensen model remains now limited to 0.17 %.
As can be seen, the two optimum layouts, although obtained using different
values of kw, have the same general structure.
Iteration 2: comparing outputs of LES and Jensen wake model with
kw = 0.036.
Case
Relative
Relative
Error
power
power
(LES)
(Jensen
model)
Staggered 8D
82.33 %
79.01 %
3.32 %
Gradually staggered 8D
85.77 %
93.06 %
-7.29 %
Random1
78.29 %
81.81 %
-3.52 %
Random3
77.96 %
82.27 %
-4.31 %
Random4
79.17 %
77.80 %
1.37 %
Random5
79.54 %
82.06 %
-2.52 %
Random6
76.01 %
78.67 %
-2.65 %
Random7
80.96 %
83.50 %
-2.54 %
Random8
76.25 %
78.04 %
-1.79 %
Optimum iter. 1
90.51 %
90.17 %
0.34 %
Optimum iter. 2
92.04 %
91.88 %
0.17 %
In iteration 3, we repeat the procedure a third time and find (almost)
the same value for kw. Only the fourth digit differs, and the
resulting new optimal layout remains the same. In fact, we observed that up
to changes in the second digit, the value of kw does not
significantly influence the optimal layout. Finally, the error between the
Jensen model and the LES is below 1 %, which corresponds roughly to the
statistical averaging accuracy of the LES. We conclude that the algorithm is
converged.
Optimum values of kw obtained in different iterations of
Algorithm 2.
Iteration
optimum
Relative LES
no.
kw
power of
corresponding
optimum
layout
1
0.055 %
90.51 %
2
0.036 %
92.04 %
3
0.036 %
NA
NA: not available.
After initial set-up of the LES database, each main optimisation step
requires 2.5×106 Jensen evaluations per iteration and five LES
evaluations. Wall time for the Jensen evaluations (per iteration) corresponds
roughly to 1.25 h on one Ivy Bridge node of the ThinKing cluster of the
Flemish Supercomputer Centre. Total wall time for LES (per iteration, and
excluding the precursor spin-up time – see Sect. ) amounts to
approximately 70 h on eight nodes of the Flemish Supercomputer, equivalent
to 560 node hours. Even though the Jensen model is 500 000 times more
evaluated per iteration than LES, the total LES cost is roughly 500 times
more expensive and the LES wall time is roughly 50 times longer.
Given this single wind direction, the optimal layout leads to a relative
wind-farm performance of around 93 % of a wakeless wind farm, which is
considerably higher than a typical aligned or staggered layout. Moreover,
when looking at the layout that was found in Fig. , it
is observed that turbines are grouped into two main clusters – one at the
front of the farm and one at the back of the farm, leaving a large
stream-wise distance in between for wake recovery. Obviously, this result is
particular for a single wind direction. In the next section, we study the
cases with multiple wind directions.
Finally, we remark that it is difficult to prove formal convergence of the CE
method that we use for our optimisation (see discussion at the end of
Sect. ), and optimisation is terminated based on a maximum number of
iterations in Algorithms 2 and 3. Therefore,
we checked the dependence of our results versus the initial starting point of
the optimisation. We found that a change in initial distribution leads to
slight shifts in the turbine locations, but this does not significantly
influence the value of the power extraction. Moreover, we also experimented
with the use of gradient-based optimisation using the CE optimum as a
starting point of the gradient-based method. To this end, we employed
Matlab's fmincon routine. Unfortunately, including all nonlinear distance
constraints did not work (given 30 turbines, there are 435 distance
constraints). Omitting these in the gradient-based method, we found that
turbine locations again slightly shift, but that power increases by 0.4 %
only, indicating the obtained CE optimum is well converged. Overall, we find
that the energy function is relatively flat near the region of optimal power
production – i.e. small shifts in the turbine locations do not lead to
significant changes in power output.
Optimisation for multiple wind directions
We now consider optimisation over a wind-direction distribution. Two cases
are considered. The first corresponds to a uniform wind distribution over an
angle of ±7.5∘, representing a case with a dominant wind
direction. The second corresponds to a uniform wind distribution over an
angle of ±180∘, representing a case without a dominant wind
direction.
In order to properly represent power output over the wind distribution using
the Jensen model, we sample the uniform distributions in 1.5∘
increments, and the integral in Eq. () is discretised
using a Riemann sum based on these intervals each with constant probability.
For LES evaluations, we use a much coarser sampling: for the dominant
wind-direction case we use only three directions and for the uniform 360∘ case eight directions.
The error between Jensen model and LES is only defined relying on
these distinct directions. In this way, the overall computational costs
related to LES remains limited compared to the additional Jensen model
evaluations that are performed.
We first focus on the dominant wind-direction case and perform an
optimisation using the Jensen model and kw=0.036 obtained in the
previous section. An overview of the errors between Jensen model and LES for
the optimal layout is given in Table . It is seen
that errors are already below 2 % for all directions, and therefore we do
not further perform iterations using Algorithm 2 here. The
overall optimal power output corresponds to 93.67 %, and the related
layout is shown in Fig. . It is seen
that the optimal layout for the dominant wind direction very much resembles
the layout for the single wind-direction case. Turbines are again clustered
in two large groups, one in front and one at the back of the wind farm.
Optimal layout and relative power for dominant wind-direction case
(angle of ±7.5∘). Relative power results are obtained from
large-eddy simulations. Turbine locations are marked with coloured disk: size
and colour scale by relative power. Plot boundary (red box)
corresponds to boundaries of domain Ω
(see Fig. ).
Finally, we look at optimisation for the uniform wind distribution. Again we
perform optimisation using the Jensen model and kw=0.036. An
overview of the errors for the optimal layout is provided in
Table . Also, errors are now overall relatively low, and so, for the sake of saving computational resources, we do not perform further
iterations using Algorithm 2. We further find that, overall,
the average power output of the optimised layout corresponds to 93.45 %.
This compares to 71.73 and 75.25 % for the aligned 8D×6D
and for staggered layout respectively.
The optimal layout itself is shown in Fig. .
In contrast to the layout found for the dominant wind direction, turbines
are now spread out much more evenly throughout the domain. Moreover, a number
of turbines, i.e. seven, are located on the domain boundary. We remark here
that, for similar optimisation cases in the literature, turbines sometimes end up
at the domain corners (see, e.g., or
), but this is not the case for all studies
e.g.. Currently, we are not sure whether this is
possibly related to domain shape, size, and number of turbines, or whether
this is related to the existence of local optimums or convergence of the
optimisation method. Using a hybrid method that combines a global method with
a gradient-based approach, as proposed by , and exploring a
large number of optimisation starting points, may be required for studying
this in more detail. This is an interesting topic for further research.
Dominant wind-direction case – evaluation of optimal layout.
Relative power for three different wind directions comparing outputs of LES and
Jensen wake model with kw = 0.036.
Wind direction
Relative
Relative
Error
(degrees)
power
power
(LES)
(Jensen
model)
0
93.08 %
91.91 %
1.17 %
7.5
94.27 %
93.21 %
1.06 %
-7.5
93.44 %
92.43 %
1.02 %
Average (0, 7.5, -7.5)
93.67 %
92.51 %
1.08 %
Average (-7.5, 1.5, 7.5)
–
92.53 %
–
Optimal layout and relative power for the uniform wind-direction
case (angle of ±180∘). Relative power results are obtained from
large-eddy simulations. Turbine locations are marked with coloured disk: size
and colour scale by relative power. Plot boundary (red box)
corresponds to boundaries of domain Ω
(see Fig. ).
Uniform 360∘ case – evaluation of optimal layout. Relative
power for eight different wind directions, comparing outputs of LES and Jensen
wake model with kw = 0.036.
Wind direction
Relative
Relative
Error
(degrees)
power
power
(LES)
(Jensen
model)
0
87.11 %
89.16 %
-2.05 %
45
96.08 %
96.06 %
0.03 %
90
94.75 %
94.37 %
0.39 %
135
96.27 %
95.27 %
1.00 %
180
87.97 %
89.41 %
-1.45 %
-135
97.36 %
96.35 %
1.01 %
-90
94.59 %
94.37 %
0.23 %
-45
95.46 %
94.91 %
0.55 %
Average
93.45 %
93.55 %
-0.10 %
Conclusions
In the current work, we proposed a hybrid Jensen–LES approach for layout
optimisation of wind farms. The Jensen model is a wake model that is
sufficiently fast to allow, in principle, wind-farm optimisation over different
wind directions and using global optimisation approaches that take into
account the non-convex nature of the optimisation problem. Large-eddy
simulations are much more accurate than the Jensen model, but they are by orders
of magnitude too slow to be used for wind-farm layout optimisation.
Therefore, we introduce a nested optimisation approach in which the Jensen
model is used as a surrogate model. In the inner loop, the Jensen model is
used to perform the layout optimisation, while in an outer loop, the wake-expansion coefficient in the Jensen model is adapted to better fit LES
results of the gradually evolving optimal layouts.
In the current study, layout optimisation of a wind farm of 30 turbines on a
4 km × 3 km area is considered. For this set-up, we found
that an iterative fitting of the average wake-expansion coefficient in the
Jensen model during optimisation to be sufficient, leading to errors below
1 % for the optimal layout. For larger wind-farm layouts, wind-farm areas
that are more complex, or including different atmospheric stratification
regimes, it may be necessary to consider a more complex parameterisation of
the wake-expansion coefficient. This may include dependence of the wake-expansion coefficient on wind direction, Obukhov scale, or downstream
location in the wind farm. These are topics for further research.
Finally, the layouts found for the current set-up differed greatly depending
on the wind-direction scenario. In the case of a dominant wind direction, turbines were clustered together
at the front and back of the wind-farm area, allowing for maximum wake
recovery in between. For a 360∘ uniformly distributed wind rose,
turbines are evenly spread out over the domain. However, this is a result of
optimisation of energy yield only, given a number of turbines and wind-farm
area, and the effect of wake–wake and wake–boundary-layer interaction. In
practice, wind-farm layout optimisation is a multidisciplinary problem that
includes effects and costs of turbine loading, costs of installation,
maintenance, cabling, etc. The full inclusion of a hybrid Jensen–LES model
in such an optimisation framework is also an important topic of further
research.