The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake

Abstract

In this paper, a new version of the FLOw Redirection and Induction Dynamics (FLORIDyn) model is presented. The new model uses the three-dimensional parametric Gaussian FLORIS model and can provide dynamic wind farm simulations at a low computational cost under heterogeneous and changing wind conditions.
Both FLORIS and FLORIDyn are parametric models which can be used to simulate wind farms, evaluate controller performance and can serve as a control-oriented model. One central element in which they differ is in their representation of flow dynamics: FLORIS neglects these and provides a computationally very cheap approximation of the mean wind farm flow. FLORIDyn defines a framework which utilizes this low computational cost of FLORIS to simulate basic wake dynamics. This is achieved by creating so-called observation points (OPs) at each time step at the rotor plane which inherit the turbine state.
In this work, we develop the initial FLORIDyn framework further considering multiple aspects. The underlying FLORIS wake model is replaced by a Gaussian wake model. The distribution and characteristics of the OPs are adapted to account for the new parametric model but also to take complex flow conditions into account. To achieve this, a mathematical approach is developed to combine the parametric model and the changing, heterogeneous world conditions and link them with each OP. We also present a computationally lightweight wind field model to allow for a simulation environment in which heterogeneous flow conditions are possible.
FLORIDyn is compared to Simulator for Offshore Wind Farm Applications (SOWFA) simulations in three- and nine-turbine cases under static and changing environmental conditions. The results show a good agreement with the timing of the impact of upstream state changes on downstream turbines. They also show a good agreement in terms of how wakes are displaced by wind direction changes and when the resulting velocity deficit is experienced by downstream turbines. A good fit of the mean generated power is ensured by the underlying FLORIS model. In the three-turbine case, FLORIDyn simulates 4 s simulation time in 24.49 ms computational time. The resulting new FLORIDyn model proves to be a computationally attractive and capable tool for model-based dynamic wind farm control.

Full text

Wind Energ. Sci., 7, 2163–2179, 2022
https://doi.org/10.5194/wes-7-2163-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
The revised FLORIDyn model: implementation
of heterogeneous flow and the Gaussian wake
Marcus Becker1, Bastian Ritter2, Bart Doekemeijer3, Daan van der Hoek1, Ulrich Konigorski2,
Dries Allaerts4, and Jan-Willem van Wingerden1
1Delft Center for Systems and Control, Delft University of Technology,
Mekelweg 2, 2628 CD Delft, the Netherlands
2Control Systems and Mechatronics Lab, Technische Universität Darmstadt,
Landgraf Georg Str. 4, 64283 Darmstadt, Germany
3National Renewable Energy Laboratory, Golden, CO 80401, USA
4Faculty of Aerospace Engineering, Delft University of Technology,
Kluyverweg 1, 2629 HS Delft, the Netherlands
Correspondence: Marcus Becker (marcus.becker@tudelft.nl)
Received: 15 December 2021 – Discussion started: 4 January 2022
Revised: 20 June 2022 – Accepted: 17 October 2022 – Published: 1 November 2022
Abstract. In this paper, a new version of the FLOw Redirection and Induction Dynamics (FLORIDyn) model
is presented. The new model uses the three-dimensional parametric Gaussian FLORIS model and can provide
dynamic wind farm simulations at a low computational cost under heterogeneous and changing wind conditions.
Both FLORIS and FLORIDyn are parametric models which can be used to simulate wind farms, evaluate con-
troller performance and can serve as a control-oriented model. One central element in which they differ is in their
representation of flow dynamics: FLORIS neglects these and provides a computationally very cheap approxima-
tion of the mean wind farm flow. FLORIDyn defines a framework which utilizes this low computational cost of
FLORIS to simulate basic wake dynamics. This is achieved by creating so-called observation points (OPs) at
each time step at the rotor plane which inherit the turbine state.
In this work, we develop the initial FLORIDyn framework further considering multiple aspects. The underly-
ing FLORIS wake model is replaced by a Gaussian wake model. The distribution and characteristics of the OPs
are adapted to account for the new parametric model but also to take complex flow conditions into account. To
achieve this, a mathematical approach is developed to combine the parametric model and the changing, hetero-
geneous world conditions and link them with each OP. We also present a computationally lightweight wind field
model to allow for a simulation environment in which heterogeneous flow conditions are possible.
FLORIDyn is compared to Simulator for Offshore Wind Farm Applications (SOWFA) simulations in three-
and nine-turbine cases under static and changing environmental conditions. The results show a good agreement
with the timing of the impact of upstream state changes on downstream turbines. They also show a good agree-
ment in terms of how wakes are displaced by wind direction changes and when the resulting velocity deficit
is experienced by downstream turbines. A good fit of the mean generated power is ensured by the underlying
FLORIS model. In the three-turbine case, FLORIDyn simulates 4 s simulation time in 24.49 ms computational
time. The resulting new FLORIDyn model proves to be a computationally attractive and capable tool for model-
based dynamic wind farm control.
Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.

2164 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
1 Introduction
In recent years, the topic of wind farm control has gained
traction as renewable energies become more and more rele-
vant for the current and future energy mix. Maximizing the
power generated by a wind farm is not a trivial task as the
turbine-to-turbine interaction is characterized by the complex
flow, large delay times and an ever-changing environment.
In order to describe the wind field, parametric steady-state
approximations have been developed. These describe the
mean behavior of the flow with parametrized analytical ex-
pressions rather than differential equations. A first approach
was presented by Jensen (1983), which motivated years later
the development of more refined steady-state models, such
as the Zone FLORIS model (Gebraad et al., 2014). With
these low-computational-cost and easy-to-implement wake
descriptions, it is possible to develop a model-based control
algorithm. These control strategies have managed to improve
the power generated in high-fidelity simulations e.g., Ge-
braad et al. (2014) and in field experiments (Fleming et al.,
2017). The success of parametric steady-state models opens
up the question of whether it is possible to overcome one
of their great shortcomings: the lack of dynamics. A low-
computational-cost dynamic wake description can be used
to more accurately describe the wake behavior on smaller
timescales, during turbine state changes and during environ-
mental changes. This could lead to more sophisticated con-
trol approaches and wind farm analysis methods.
There have been efforts to implement parametric models
in a dynamic manner, some of which are described here. For
a more in-depth discussion of the current state of the art,
the interested reader is referred to the review by Kheirabadi
and Nagamune (2019) and more recently, Andersson et al.
(2021). In the current literature, we have identified two major
trails of publications, which will be briefly discussed below.
The first research trail begins with the Aeolus SimWind-
Farm toolbox (Grunnet et al., 2010), which is publicly avail-
able. The toolbox uses the Jensen model (Jensen, 1983),
coupled with a dynamic description of the centerline and a
wind field grid. The centerline would imitate the wake me-
andering effect based on passive tracers, traveling with the
synthetically generated turbulent wind speed. A number of
limitations have been imposed for this toolbox: the mean
wind speed and direction are constant, the flow field is calcu-
lated in 2D, and the turbines operate with fixed yaw angles.
The toolbox has enabled the work of Poushpas and Leithead
(2014), who used the Frandsen multiple wake model (Frand-
sen et al., 2006) and added a description of turbine dynamics
to estimate fatigue loads. The model is then used to perform
induction control based on lookup tables of the thrust and
power coefficients with the goal to redistribute loads. This
work later inspired the dynamic wind farm simulator, intro-
duced in Bossanyi (2018). The model adds wake steering to
the fatigue load estimation and induction control capabilities.
To model the effect of yawing the turbine, the deflection for-
mulation of Jiménez et al. (2010) is used. Based on data from
the in-house code Bladed, the author formulates the effect of
yaw misalignment on the power coefficient by a polynomial
expression based on the blade pitch. The wind field is repre-
sented by low- and a high-frequency wind speed variations.
The low-frequency variations are correlated across the wind
farm and cause wake meandering and advection. The high-
frequency part is uncorrelated between the turbines and is
superimposed with the wake deficits. Lastly, the wake model
is switched to the Ainslie model (Ainslie, 1988).
A second trail of publications can be found starting with
Shapiro et al. (2017), where the authors use the previously
mentioned Jensen model and extend it to incorporate the im-
pact of time-varying extraction of kinetic energy of turbines
due to induction control. Assuming a constant wind direction
and wind speed, the authors derive a linear approximation of
the wake advection velocity based on the laws of momen-
tum conservation and mass conservation. The result is a one-
dimensional partial differential equation to describe the dy-
namic wake behavior. The model neglects possible changes
of the wake expansion due to a changing thrust coefficient
and also does not incorporate yaw angle changes. In Shapiro
et al. (2018), the authors extend their model to also take
the effects of yawing into account. Most recently, this ap-
proach inspired the development of the Floating Offshore
Wind Farm Simulator, published in Kheirabadi and Naga-
mune (2021). The authors extend the momentum conserva-
tion equations to incorporate time-varying free-stream wind
velocity effects. Additionally, they couple the model to a dy-
namic description of floating platforms, restricted by moor-
ing lines. The authors closely follow Bastankhah and Porté-
Agel (2016) to derive a parametric Gaussian velocity shape
for their model.
Alongside the two discussed trails of publications, the dy-
namic wake meandering (DWM) model was developed. The
DWM model, first presented by Larsen et al. (2008) and
later calibrated and refined by Madsen et al. (2010), pro-
poses an approach much closer to established CFD methods.
The model follows a pseudo-Lagrangian approach and cre-
ates turbulence boxes around the wake deficit which is cre-
ated by the turbine. These boxes are then subject to a syn-
thetic turbulent wind field, which allows the modeling of the
wake meandering effect. The DWM model puts a focus on
load estimation next to the power generated and simulates
the turbine by coupling a CFD actuator disc model with an
aeroelastic model. Compared to the other mentioned models,
the DWM model presents a synergy of CFD methods with
engineering approaches.
Another early attempt to derive a dynamic model from
a parametric steady-state model was published by Gebraad
and van Wingerden (2014), who utilized the just-published
FLORIS model (FLOw Redirection and Induction in Steady
state, Gebraad et al., 2014) and created the FLORIDyn model
(FLOw Redirection and Induction Dynamics). FLORIDyn
creates so-called observation points (OPs) at the rotor plane
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2165
which travel downstream at hub height with the effective
wind speed. Their path follows the zone boundaries de-
scribed by the FLORIS model. The wake deficit and shape
depend on the yaw angle and the induction factor. Changes
in these variables travel with the OPs and cause a delayed ef-
fect at downstream turbines. The authors derive a state-space
representation of the model behavior and validate it in a six-
turbine simulation against the high-fidelity large eddy sim-
ulation environment SOWFA (National Renewable Energy
Laboratory, 2020). The state-space representation is then
used to implement a Kalman filter for flow field estimation
(Gebraad et al., 2015). The model does have shortcomings:
due to the two-dimensional flow, shear and veer effects can
not be captured, the simulations only work in one wind direc-
tion and they do not capture turbulent effects. Furthermore,
due to the way the OPs travel, parts of the wake can overlap
and can create a faulty wake representation.
In this paper, we aim to overcome these issues and bring
the FLORIDyn approach into a form where it can incorporate
heterogeneous and changing flow conditions, wind shear, and
added turbulence levels. To achieve these changes, we re-
work the framework to use a Gaussian FLORIS model (Bas-
tankhah and Porté-Agel, 2016). This requires a new formu-
lation of the OP behavior. Due to these changes, the wakes
can also incorporate locally different and changing flow con-
ditions, such as wind speed, direction and ambient turbu-
lence intensity. To drive the model, a concept of a wind field
model is presented as well. The framework is then com-
pared to the simulation environment SOWFA in three- and
nine-turbine cases. Furthermore, in order to allow for collab-
oration and extension, the code is published in its entirety
(Becker, 2022a). The resulting Gaussian FLORIDyn model
is a capable, open-source alternative to the few other exist-
ing in-house parametric dynamic models, developed for wind
farm control purposes.
The remainder of this paper is organized as follows: Sect. 2
discusses the relevant characteristics of the former FLORI-
Dyn framework and how it is adapted. The simulation results
are presented in Sect. 3, which also discusses the computa-
tional performance. Section 4 concludes the paper and gives
recommendations for future work.
2 A new parametric dynamic wind farm model
In this section, the new Gaussian FLORIDyn model is in-
troduced. To prevent confusion, we will refer to the mod-
els of Gebraad et al. as the Zone FLORIS model (Ge-
braad et al., 2014) and the Zone FLORIDyn model (Gebraad
and van Wingerden, 2014). The Gaussian model by Bas-
tankhah and Porté-Agel (2016) will be referred to as Gaus-
sian FLORIS model.
As the new Gaussian FLORIDyn model is building upon
previous work, Sect. 2.1 and 2.2 briefly introduce the termi-
nology and properties of the underlying Gaussian FLORIS
Figure 1. Sketched shape of the wake with the different sections,
the deflection and areas of equal relative reduction by the Gaussian
shape.
model and the Zone FLORIDyn framework. The novel Gaus-
sian FLORIDyn model makes changes to the Zone FLORI-
Dyn framework. These are discussed in Sect. 2.3. Section 2.4
describes how heterogeneous environmental conditions are
taken into account. To get the power coefficient (CP) and
the thrust coefficient (CT) values closer to the validation
platform SOWFA, a lookup table was generated (Sect. 2.5).
Lastly, a basic wind field model is given in Sect. 2.6. It is built
to provide the heterogeneous field conditions to evaluate the
FLORIDyn model.
In the wake coordinate system, K1,x1describes the down-
wind direction, y1the horizontal crosswind direction and
z1the vertical crosswind direction (Fig. 1). In this coordinate
frame, the rotor center is always located at [0,0,0]>. This
coordinate system is not to be confused with the longitudi-
nal (x0), latitudinal (y0) and vertical (z0) world coordinate
system K0.
2.1 The Gaussian FLORIS model
The core of the used Gaussian FLORIS model is based
on the work of Bastankhah and Porté-Agel (2016). This
work describes a parametric, three-dimensional wake with a
Gaussian-shaped wind speed recovery. As it has been applied
and described in previous publications (e.g., Farrell et al.,
2021), only the basic terminology is introduced here as well
as the wake shape. In the present work of this paper, the
model has been extended with the calculation of added tur-
bulence as proposed by Crespo and Hernández (1996). The
power calculation has been extended by the cos(γ)ppadapta-
tion to the yaw angle (Medici, 2005) and an efficiency term η
for tuning (Gebraad et al., 2014). Figure 1 depicts an illustra-
tion of the wake with its three areas: the potential core, the
near-wake area and the far-wake area. For all areas a reduc-
tion factor r=1u/ufree can be calculated, where ufree is the
free wind speed and 1u is the wind speed deficit. The poten-
tial core is a region from jets in a coflow (Lee and Chu, 2003).
Here, it is used to approximate the immediate region behind
the rotor plane. Within the potential core, ris constant. In the
near- and far-field area rreduces to 0, following a Gaussian
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2166 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
shape with the extremum at the centerline or border of the
potential core. The recovery rate is based on σyand σzin the
respective crosswind directions. The potential core width is
described by wy,pc and wz,pc , which continuously decrease
for the length of the potential core xc. Lastly, the deflection δ
returns the position of the centerline.
The mentioned variables are dependent on turbine states,
such as the thrust coefficient CTand the yaw angle γ, the
ambient turbulence intensity I0, and a set of 10 parameters.
The parameters adjust wake properties such as the recovery
rate, the expansion rate, the sensitivity to added turbulence
levels and the influence of the yaw angle. The values of the
parameters are listed in Table 1 in Sect. 3.
2.2 The Zone FLORIDyn model
An initial FLORIDyn model was published in Gebraad and
van Wingerden (2014). The model is based on the previ-
ously published Zone FLORIS model, which approximates
the wake shape with three zones: near field, far field and
mixing zone (Gebraad et al., 2014). Every zone has a for-
mulation of the velocity recovery in downstream direction.
To introduce dynamics, observation points (OPs) are created
at the rotor plane at each time step.
The OPs serve the purpose to describe the local FLORIS
wake characteristics at their location. To do that, they in-
herit the turbine states at the time of their creation which are
necessary to calculate the FLORIS wake. With time, each
OP travels downstream, representing a mass of air traveling
in the wind. Their travel path is determined by the borders of
the FLORIS wake zones. The speed they travel with is equal
to the effective wind speed they represent. Figure 2 shows the
basic concept. Initial OPs are colored black to stress that they
inherited the same state. The OPs created after the yaw step
are colored white, showing that their inherited state differs.
With this framework, the steady-state wake represents the
known FLORIS wake, but other than in FLORIS, changes
propagate through the wake instead of instantly affecting tur-
bines downstream. If, for instance, the yaw angle of the tur-
bine changes, the new generation of OPs will inherit the new
angle while old OPs still travel according to the previous an-
gle.
In the case of overlapping wakes, an OP travels into the
wake of another turbine. The OP locates the closest up- and
downstream OPs from the foreign wake and interpolates their
reduction factor at its location. In this model, the resulting
reduction of the free wind speed is calculated as follows:
ueff,OP ufree,OP, rown, rf,OP=ufree,OP (1−rown)
nT
Y
i=1
(1−ri)
| {z }
rf,OP
,(1)
where ufree,OP is the free wind speed at the OP’s location.
This wake interaction model could also be exchanged for an-
other formulation. The wind speed reduction rown is based
Figure 2. Creation and propagation of the OPs: in panel (a) a set
of OPs is created, inherits the turbine state and travels downstream,
following the FLORIS wake shape, shown in panel (b). In panel (c)
the turbine state changed and the new OPs inherit a different state
(now colored white) and follow the new, dark-indicated wake shape
in panel (d).
on the OP’s own wake, and riis the interpolated reduction of
one of the nTupwind turbines.
To calculate the effective wind speed at the rotor plane,
the model calculates an effective velocity reduction factor rT
for every turbine at every time step. The algorithm com-
bines the reduction of each upstream turbine by a root sum
square. Within one wake, the reduction factors of the zones
are summed, weighted by the overlapping area with the rotor
plane.
2.3 Changes to the FLORIDyn approach
Due to the changed underlying FLORIS model, the FLORI-
Dyn approach needs to be adapted. Specifically, the move
to a three-dimensional flow field requires a fitting distribu-
tion of the OPs, which is discussed in Sect. 2.3.1. This opens
up the possibility to reformulate the calculation of the ef-
fective wind speed at the rotor plane, which is presented
in Sect. 2.3.2. The travel speed of the OPs is addressed in
Sect. 2.3.3. In this section, we use the wake coordinate sys-
tem K1, indicated by the lower index 1 (e.g., y1,OP). The re-
lation between world and wake coordinate system will be ex-
plained in Sect. 2.4.
2.3.1 Distribution of the observation points
By changing the underlying FLORIS model, the travel path
of the OPs and their distribution has to be rethought. The
Gaussian FLORIS model does not have defined borders, and
it is three-dimensional. To cover the crosswind wake area
regularly for any number of OPs, an algorithm based on the
sunflower distribution was used (Vogel, 1979). The algorithm
returns a relative crosswind coordinate (νy,νz)∈ [−0.5,0.5]
for a given number of OPs. We used 50 OPs per time step. To
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2167
cover the majority of the Gaussian wake influence, the wake
width was chosen to be ±3σyand ±3σzfrom the centerline
and the potential core. The following equation is used to cal-
culate the position of an OP in the wake coordinate system:
y1,OP νy,OP,σy, wy,pc,δ=νy ,OP 6σy+wy,pc+δ, (2)
z1,OP νz,OP, σz, wz,pc=νz,OP 6σz+wz,pc.(3)
Note that this model only assumes a horizontal deflection.
To add a vertical deflection, due to rotor tilt for instance,
Eq. (3) needs to be adapted accordingly. For simplicity’s
sake σyis used, which represents σy,nw for 0 < x1≤xcand
σy,fw for x1> xc. Respectively, σzis defined the same way.
The variable δdescribes the deflection of the centerline. If
OPs travel below z1=0 they are ignored. Since νyand νzare
not changed during the simulation, they can be calculated a
priori. They are then used in every time step for the new gen-
eration of OPs. OPs with the same relative coordinate follow
each other and form what is called a chain. The number of
chains is equal to the number of OPs created at each time
step.
2.3.2 Wind speed at the rotor plane
Since OPs are created at the rotor plane and they interact with
foreign wakes, they can be used to estimate the effective wind
speed for the power generation. To do that, they have to be
distributed across the rotor plane rather than the wake area:
y1,OP νy,OP, γ |x1=0=νy,OPDcos γ, (4)
z1,OP νy,OP|x1=0=νz,OPD. (5)
The next step is to determine the area represented by ev-
ery OP. This is done offline by generating a Voronoi pattern
(Voronoi, 1908a, b) with the OPs’ relative location as seeds
and a circular boundary with radius 0.5. The area of the re-
sulting polygons is normalized by the rotor area and used as
weight. All weights are stored in the vector w.
During the simulation, the OPs calculate the reduction
of foreign wakes rf,OP on themselves as shown in Eq. (1).
Stored in a vector rf= [rf,1,···, rf,nOP]>the effective wind
speed at the rotor plane is calculated as follows:
ueff =w>(rf◦u),(6)
where ◦stands for the element-wise multiplication and urep-
resents a vector of the free wind speeds at the locations of the
OPs. An OP considers itself influenced by a foreign wake if
the closest foreign OP is less than 1
4Daway. This is an arbi-
trary chosen threshold to reduce the number of OPs for the
interaction interpolation. As the outer wake OPs represent
the most recovered sections of the wake, this still results in a
smooth influence transition.
2.3.3 Travel speed
In the former version of the FLORIDyn model, the OPs travel
with the effective wind speed they represent. Regions in the
center of the wake with lower effective wind speeds therefore
propagate the changes slower than the outer areas. While this
seems an intuitive choice, it leads to problems. Initial simula-
tion results showed that, in comparison to the SOWFA simu-
lation, the effects of a state change arrive noticeably slower in
FLORIDyn at downstream turbines. Also, due to the differ-
ence in OP travel speed, the outer regions adapt their shape
earlier in a downstream location, which leads to overlapping
areas with the slow regions, which have not adapted yet. This
makes the wake representation not injective anymore: multi-
ple OPs occupy and describe the same space at the same time
with varying properties.
In this article, the OPs are assumed to propagate with the
speed of the free-stream wind rather than the effective wind
speed in accordance with Taylor’s frozen turbulence hypoth-
esis (Taylor, 1938). The decision is supported by experimen-
tal results from Schlipf et al. (2010) and has also been used
by other similar codes, e.g., Grunnet et al. (2010). This also
solves the issue of the overlapping wake areas since neigh-
boring OPs travel at the same speed and follow the same state
changes. Another implication of this adaptation is that OPs
no longer need to calculate the influence of foreign wakes
at every time step. This would be used to determine their
effective wind speed and thus how far they travel down-
stream in one time step. The only OPs which need to cal-
culate the foreign influence are the ones at the rotor plane
in order to determine the effective wind speed according to
Eq. (6). These model assumptions also significantly decrease
the computational load during the simulation. The downside
of the change is that the effects of state changes now arrive
too fast and abrupt at downstream turbines, which will be
seen and discussed with the simulation results in Sect. 3. In
future work, the wake propagation speed could be a tuning
parameter which is set depending on atmospheric conditions
such as the turbulence intensity for instance (Andersen et al.,
2017).
2.4 Including directional dependency and observation
point propagation
In this section, we address how the OPs, and therefore the
wakes, react to a wind direction change. We assume that a
wind direction change only affects the wake orientation and
that the wake structure and downstream evolution (as defined
by the underlying FLORIS model) can be seen independent
from the free-stream behavior. It is therefore possible to split
the two aspects into two coordinate systems: the world coor-
dinate system K0and the wake coordinate system K1. The
free-flow conditions are described in K0, whereas the wake
properties are described in K1. An OP links these two coor-
dinate systems.
The underlying FLORIS model is described in K1, where
the origin x1=y1=z1=0 is located in the center of the ro-
tor plane. The downwind distance is denoted as x1,y1de-
scribes the horizontal crosswind distance and z1describes
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2168 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
Figure 3. This figure visualizes the working of Eq. (9), which is applied for each OP individually. In panels (a) and (b), the position update
of an OP in a time step with a constant wind direction is depicted. Panels (c) and (d) show the position update when the wind direction
changes. In this case, the wake coordinate system is rotated around the OP’s location to match the new downstream direction. This causes
the apparent origin of the wake in the world coordinate system to change, which is visualized by the grey turbine.
the vertical one. K0does not have a special orientation apart
from z0=0 being the ground level and the z0axis pointing
upwards. In this work, x0describes the west–east axis, and
y0describes the south–north axis. To transform a location
vector r1, described in K1of a turbine with the rotor-center
location t0, into r0the rotational matrix R01 is used:
r0=

x0
y0
z0

=t0+R01(ϕ)r1=

x0,T
y0,T
z0,T


+

cosϕ−sin ϕ0
sinϕcos ϕ0
0 0 1 


x1
y1
z1

.(7)
This equation assumes a uniform wind direction ϕat every
location. This will not be the case for the formulation used
for the OP propagation later on in Eq. (9). Each OP has two
location vectors, r0,OP and r1,OP, one for each coordinate
system. The OP’s position update and its reduction factor is
calculated in K1.K0is used to calculate the wake interaction
and to determine the wind speed, the wind direction and the
ambient turbulence intensity. At the OP’s creation, r1,OP is
determined by the Eqs. (4) and (5) for the crosswind coor-
dinates; the downwind coordinate is set to 0. Its world lo-
cation, r0,OP, is then determined by Eq. (7) with the wind
direction ϕ0,Tat the turbine location. To iterate the location
of an OP from time step kto time step k+1, the downwind
step is calculated first in K1:
x1,OP(k+1) =x1,OP(k)+uOP1t , (8)
where 1t is the time step duration and uOP is the magni-
tude of the wind vector u0,OP at the OP’s location r0,OP. The
direction will be applied in Eq. (9). For the scope of this
work, u0,OP can only have non-zero components in the x0
and y0direction. With x1,OP(k+1) the new crosswind loca-
tions y1,OP(k+1) and z1,OP(k+1) can be calculated with the
Eqs. (2) and (3), respectively. This completes the transition
r1,OP(k)→r1,OP(k+1). Note that only x1,OP(k) is needed
to determine the OP’s location in K1. At the cost of calcu-
lating y1,OP(k) and z1,OP(k) again at each time step, they do
not have to be stored as states. To update r0,OP(k), the step
which the OP took in K1has to be translated into K0:
r0,OP(k+1) =r0,OP (k)+R01 ϕ0,OPr1,OP(k+1) −r1,OP (k),(9)
where ϕ0,OP is the wind direction at r0,OP(k). Note that
ϕ0,OP refers to one OP’s individual wind direction; other OPs
may have different values. This means that each OP prop-
agates on its own and non-uniform wind directions can be
simulated. Figure 3 shows the OP step in the wake and world
coordinate system. In Fig. 3a and b the wind direction is con-
stant, indicated by the arrow left to the y0axis. The OP cal-
culates its step in the wake coordinate system (dotted arrow)
and updates its location vectors. These are here simplified
to r0and r1. In Fig. 3c the wind direction changes, and the
former FLORIS wake description is invalid and greyed out.
With the new wind direction R01(ϕ0,OP) is calculated differ-
ently. The OP can calculate its step in the wake coordinate
system as before, but its translation K1→K0changed. Note
that neither r0nor r1is influenced by the changed wind di-
rection. Their magnitude and orientation remain the same in
their respective coordinate systems; however, their orienta-
tion towards each other changes.
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2169
Figure 4. Greedy control settings of the un-yawed 10 MW DTU
reference turbine based on the effective rotor wind speed.
2.5 Calculation of CTand CP
The thrust coefficient CTis often approximated following the
actuator disc theory: CT(a)=4a(1 −a), where ais the ax-
ial induction factor. To circumvent this approximation, simu-
lations or experiments can be used to create lookup tables.
Since most equations of the Gaussian FLORIS model are
dependent directly on CTrather than a, we used lookup ta-
bles generated in SOWFA to align FLORIDyn’s thrust coef-
ficient with what the turbines in the validation environment
experience. For completeness, we also use lookup tables for
the power coefficient CP. The tables are generated for the
DTU 10 MW reference turbine (Bak et al., 2013). It has to
be added that these tables are generated from a grid of high-
fidelity simulations, where the coefficients were read after the
simulation converged to a steady state. The tables can there-
fore only approximate the effect a changing turbine state and
changing wind field conditions onto CTand CP. Control ap-
proaches for axial-induction-based controllers, such as the
one presented by Annoni et al. (2016), successfully use sim-
ilar lookup tables, which is why we assume these to be suffi-
cient. Nevertheless, an extension for dynamic circumstances
would be a valuable addition for future work but is also con-
nected to a significant computational effort.
In the tables, the coefficients are described dependent on
the blade pitch angle βand the tip speed ratio λ(ω, ueff),
where ωis the angular velocity of the rotor. However, nei-
ther FLORIS nor FLORIDyn can provide λand β. What
they can provide is ueff. Combined with the assumption that
each turbine follows a greedy control strategy and maxi-
mizes CP(λ, β ) for the given wind, we can formulate the co-
efficients dependent only on ueff: first maximize CPwithin
the physical limitations of the wind turbine for all wind
speeds, and then use the λP,max and βP,max to calculate the
respective CT. The resulting curves can be seen in Fig. 4.
Unfortunately, the resulting CT(ueff) values can get very
high, especially for low wind speeds. This conflicts with
some FLORIS equations which comprise the term √1−CT
and become complex for CTvalues above 1. To avoid these
issues, CT(ueff) is limited to its value at the Betz limit:
CT|a=1/3=0.8 (Bianchi et al., 2007). Another complication
is the calculation of the added turbulence levels as it is the
only equation which requires the axial induction factor. In
this case, the calculation of CT(a) was inverted to deter-
mine a(CT), based on the actuator disc theory, as follows:
a=1
21−p1−CT.(10)
Yaw effects on CTand aare neglected here. In future work
this expression could be substituted, for instance by the
polynomial approximation of Madsen et al. (2020). It ex-
tends a(CT) to CTvalues above 1. However, as CTis limited
in this work, this extension is not necessary. The power coef-
ficient is the remaining aspect which was used unaltered from
the lookup tables. For the tested wind speeds below 11m s−1,
the power coefficient is constant at CP=0.4929. The ef-
fect of γis approximated by multiplying CPwith cos(γ)pp.
For simplicity’s sake we assume ppto be a constant value.
This could be extended by the work presented by Liew et al.
(2020) which takes the presence of other wakes into account.
Similarly, Howland et al. (2020) presents an adaptation for
locally varying wind profiles.
2.6 Wind field model
In order to drive the FLORIDyn model, the wind field needs
to be able to simulate heterogeneous, changing environmen-
tal conditions. The implemented solution is inspired by the
work of Farrell et al. (2021). The basic assumption is that
measurements of the wind field variables are available at cer-
tain locations. This could be due to satellite data, lidar mea-
surements, met masts or other sensors. The value of a mea-
surement for the location of an OP is then interpolated be-
tween the measurements available. To reduce the computa-
tional effort of an interpolation at every time step, a nearest-
neighbor interpolation (NNI) is desirable. To get a sufficient
resolution of the measurements to justify a NNI, the sparse
measurements mhave to be mapped to dense measurement
grid points mg:
mg=Mm,(11)
where the matrix Mdescribes the mapping and can be calcu-
lated offline. The ith row in Mdescribes the percental com-
position of mg,i from m. As a result, the sum of every row in
Mis equal to 1. This way, a more complex interpolation can
be reduced to a matrix multiplication and a NNI at runtime.
In this work, a linear interpolation is used to map the mea-
surements to the grid points, which are spaced in a 20×20 m
grid. OPs outside of the grid defined by mguse the closest
grid point. This method is also independent from the quan-
tity measured. In this work, the wind speed, the wind direc-
tion and the ambient turbulence intensity were interpolated
with the presented method.
However, the presented method is only meant for values
changing in the x0and y0direction. The wind speed is the
only field measurement which is also changed in the z0di-
rection; wind direction and ambient turbulence intensity are
assumed to be constant in the vertical direction. Following
Farrell et al. (2021) the power law is applied:
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2170 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
Table 1. Parameters used in the simulation with the values they influence.
FLORIS FLORIDyn Wind
Wake expansion Added turbulence Potential core Power chains, OPs shear
kakbkf,a kf,b kf,c kf,d α∗β∗η ppncnOP αs
0.38371 0.003678 0.73 0.8325 0.0325 −0.32 2.32 0.154 0.8572 2.2 50 200 0.08
u(z0)=z0
z0,mαs
uz0,m,(12)
where z0,mis the height of the measurement and αsis
the shear coefficient. The shear coefficient approximates the
combined effect of atmospheric stability and surface rough-
ness. A small value describes unstable flow conditions. Ex-
amples for characteristic αsvalues due to surface rough-
ness are 0.11 over water, 0.16 over grass, 0.20 over shrubs,
0.28 over forests and 0.40 over cities (Emeis, 2018). In this
work z0,mis equal to the hub height zhof the turbine.
3 Simulation results
In this section, the Gaussian FLORIDyn model is com-
pared to SOWFA with the focus on turbine interaction. Two
wind farm layouts are considered for comparison: three con-
secutive turbines and a nine-turbine cluster arranged in a
3×3 configuration. The DTU 10 MW reference turbine is
used for all simulations. Table 1 summarizes the FLORIS
and FLORIDyn parameters used in the simulations. The
FLORIS parameters kaand kbare from Niayifar and Porté-
Agel (2015), kf,a to kf,d are set based on FLORISSE_M
(Doekemeijer et al., 2021), and α∗and β∗follow the find-
ings of Bastankhah and Porté-Agel (2016). The efficiency η
was tuned based on turbine T0 in the three-turbine base-
line case; ppwas tuned based on the three-turbine yaw case
(Sect. 3.1.1 and 3.1.2 respectively). For FLORIDyn, ncre-
lates to the number of OP chains per turbine and nOP to the
number of OPs per chain. The value of nOP was set to cover
the entire relevant downstream domain of a turbine; ncwas
set to maintain a sufficient density of OPs at the location of
other turbines. In FLORIDyn, one time step is 4.0 s long.
Table 1 also includes the wind shear coefficient, αs, which
was approximated based on the free flow in SOWFA. The in-
flow boundary conditions for SOWFA are provided by a pre-
cursor simulation which simulates a horizontally homoge-
nous, conventionally neutral atmospheric boundary layer in-
cluding Coriolis effects. The SOWFA settings differ for the
three-turbine case and the nine-turbine case and will be ex-
plained in the respective sections.
Figure 5. Scaled layout of the three-turbine case with the wind di-
rection indicated by an arrow on the left. The 0, 10 and 20◦yaw ori-
entations from T0 are indicated as turbine symbols with the accord-
ing orientation. The colored background areas indicate the zones of
cell refinement.
3.1 Three-turbine case
The three turbines are placed 5D=892 m apart from each
other in downwind direction. Turbine T0 is located at
(608, 500 m), and T1 and T2 are at (1500, 500 m) and
(2392, 500 m) respectively. The mean wind speed at hub
height is approximately 8.2 m s−1with an ambient turbu-
lence intensity of roughly 6 %. The mean wind direction
is constant along the xaxis. The full SOWFA flow field
domain spans 3 ×1×1 km and was simulated with a time
step of 1t =0.04 s. The base cells of the flow field are
10 ×10 ×10 m. The refinement areas are centered in the do-
main and have no offset from the ground. The first refinement
is 2.4×0.8×0.5 km with 5 ×5×5 m cells, the second one
is 2.2×0.6×0.35 km with 2.5×2.5×2.5 m cells. Figure 5
shows the to-scale layout including the areas of cell refine-
ment. In SOWFA, the turbines are modeled with the built-in
actuator line method (ALM) implementation (Sorensen and
Shen, 2002).
To give a better idea of the low-frequency, less-turbulent
dynamics, the power generated in SOWFA is also presented
filtered by a zero-phase second-order low-pass filter. This
non-causal filter is added to aid the visual interpretation
of the simulation results. The filter has a damping ratio of
d=0.7 and a natural frequency of ω=0.03 s−1. This allows
for a more equal comparison as FLORIDyn is sampled at a
lower frequency and turbulence is only included as a flow
field metric.
A regular second-order low-pass filter with the same dand
ωis used for the FLORIDyn data. This causal filter visualizes
how low-pass filtering would affect the predicted power gen-
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2171
Figure 6. Wind farm start-up and steady-state comparison of the power generated in SOWFA and in FLORIDyn. The unfiltered data are
plotted in panel (a), filtered data are plotted in panel (b). The SOWFA data are filtered with a zero-phase (noncausal) low-pass filter and
FLORIDyn with a causal low-pass filter.
erated. This could have advantages due to the changes made
to the OP travel speed in Sect. 2.3.3, which can lead to a very
abrupt wake interaction, as will be discussed in Sect. 3.1.1.
However, the filter also naturally adds a phase shift to the
data, an effect which might not be desired.
Note that the two filters have different purposes: the non-
causal SOWFA filter aims to help to interpret the simula-
tion results, while the causal FLORIDyn filter explores if and
when the use of a low-pass filter would be advantageous or
if it would decrease the quality of the results.
3.1.1 Comparison of the wind farm start-up and steady
state
In Fig. 6 the power generated by the turbines in FLORIDyn
is compared to the SOWFA simulation. The dynamics in this
simulation are the turbulent wind field and the settling of the
wake. In the unfiltered data, the interaction in FLORIDyn
sets in earlier and more abrupt than in SOWFA. This is due
to the OPs traveling at the free wind speed, as explained in
Sect. 2.3.3. The slight curvature of the drop at t≈100 s can
be explained by the wind shear: OPs at a lower altitude travel
at a slower free wind speed than OPs at a higher altitude and
therefore arrive later at the downstream turbine and therefore
affect the turbine later. There are two major aspects to ad-
dress in order to close the gap between the SOWFA and the
FLORIDyn start-up: on the one hand, the way state changes
propagate through the wake; on the other hand, how a down-
stream turbine reacts to the new wind field. To give an idea
of how a change of the latter aspect would influence the plot,
Fig. 6 also shows low-pass-filtered FLORIDyn data in com-
parison to zero-phase-filtered SOWFA data. The FLORIDyn
data align much more with the SOWFA data but still show
discrepancies in terms of dynamic response and steady-state
quality of the solution. It should be emphasized that the in-
FLORIDyn-applied filter does not affect the wake, it only
adds an artificial dynamic response to the power calculation.
This is important when heterogeneous and changing wind di-
rections are taken into account.
The power generated after the wind farm start-up remains
steady in FLORIDyn. This is because there are no turbulent
wind speed changes in FLORIDyn. In this state, FLORIDyn
is equal to the underlying FLORIS model; therefore, errors in
this state need to be solved by adapting the FLORIS model.
This could be done by parameter tuning, which has only
been done partially in this work (ηand pp; see introduction
Sect. 3).
To incorporate the turbulent wind speed changes, at least
to a certain degree, an estimation of the wind speed at the tur-
bine location would be necessary. This could be done by in-
cluding wind speed sensor data or estimating the wind based
on the power generated (Gebraad et al., 2015) or based on a
torque balance equation (Ortega et al., 2013).
A notable aspect of this simulation is the influence of the
added turbulence. Because T0 adds turbulence to the wind
field, the wake of turbine T1 recovers faster. Turbine T2 thus
experiences higher wind speeds and generates more power
than it would without the additional turbulence. This effect
can also be observed in the SOWFA data. The old Zone
FLORIDyn model is not able to capture this effect due to the
underlying Zone FLORIS model. It shows how the FLORI-
Dyn framework is inherently dependent on the capabilities of
the employed FLORIS model.
3.1.2 Comparison during a yaw angle change
In this simulation, the yaw angle γof turbine T0 is changing
from 0 to 20◦in steps of 10◦, starting at t=200 and 800 s.
The change rate of γis set to 0.3◦s−1. Figure 7 shows the un-
filtered SOWFA data in comparison to the unfiltered FLORI-
Dyn data on the left, as well as the filtered data on the right.
Filtering was performed as described in the introduction of
Sect. 3.1.
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2172 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
Figure 7. Comparison of the power generated in SOWFA and in FLORIDyn with changing yaw angles. The transparent bars indicate the
time window in which turbine T0 increases its yaw angle by 10◦. Panel (a) shows the unfiltered data, and panel (b) shows the filtered data.
The SOWFA data are filtered with a zero-phase (noncausal) low-pass filter and FLORIDyn with a causal low-pass filter.
Figure 8. Comparison between the zero-phase-filtered (zp.f.) SOWFA data in the baseline case (bl.) and in the yaw case. (a) In absolute
values; (b) with the difference between the yaw case and the baseline case. The transparent bars indicate the time window in which turbine T0
increases its yaw angle, first from 0 to 10◦and then to 20◦.
In FLORIDyn, turbine T1 shows a slight reaction to the
yaw changes of turbine T0 at roughly t≈320 s and more
significantly at t≈920 s. The influence of the state change
then travels further and impacts T2 at t≈430 s and, as well
more significantly, at t≈1030s. In SOWFA, the reaction is
obscured by turbulent influences. However, an increase in av-
erage power can be seen for T1 and T2 throughout the entire
simulation. Figure 8 shows the baseline simulation in com-
parison to the SOWFA simulation, in absolute values and
the difference. The data of both simulations can be com-
pared since they use the same wind field. It allows for a
more accurate determination of the reaction time to the up-
stream change. Turbine T1 starts to react to the first yaw an-
gle change at t≈320±8 s, T2 at t≈434±8 s. Given the dis-
tance to T0, this translates to a travel speed of the first influ-
ence between [6.98, 7.97] ms−1to T1 and [7.38, 7.90] m s−1
to T2. This indicates that first effects of the yaw angle change
do travel at almost free-stream velocity and the times align
with the FLORIDyn prediction. However, FLORIDyn does
lack the dynamic nature of the interaction, which means the
response of the wake to the state change of the upstream tur-
bine and the response of the downstream turbine to changes
in the wake. Given that all OPs travel at their free-stream
velocity, turbine state changes are directly picked up by the
OPs and transported, and the FLORIDyn turbine reacts im-
mediately when the OP arrives. The low-pass-filtered results
provide an idea of how a dynamic response could change the
results. The unfiltered difference between the SOWFA sim-
ulations is given in the Appendix A1. Figure 7 also shows
that FLORIDyn underestimates the gain in generated energy
in the steady-state region. The error likely lies with the un-
derlying FLORIS model as it is a steady-state error. Better
parameter tuning would likely decrease or even eliminate the
error.
3.2 Nine-turbine case
In order to test the model in a changing environment, a sim-
ulation with nine turbines was performed. The turbines are
arranged in a three-by-three grid with 900 m distance to the
closest turbines and 600 m to the edge. The setup is presented
in Fig. 9 as well as the numbering of the turbines. The wind
field performs a 60◦uniform wind direction change from
15 to 75◦, as indicated in Fig. 9. The change starts at t=
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2173
Figure 9. Complete nine-turbine FLORIDyn flow field in comparison to SOWFA at t=600 s. The wind direction change is indicated in the
lower-left corner of the SOWFA plot.
600 s with 0.2◦s−1and ends at t=900 s. The change in wind
direction is achieved by using SOWFA’s built-in utility to
specify the wind speed and wind direction at a certain height
and time. For the remainder of the simulation, the wind field
conditions remain steady. To keep the computational load of
the SOWFA simulation low, the DTU 10 MW turbines were
simulated with the actuator disc method (ADM). This also
allows a coarser grid and time resolution: the domain is dis-
cretized in 10 ×10 ×10 m cells, and the SOWFA time step
length is set to 0.5 s. The flow field spans 3 ×3×1 km. The
average wind speed during the simulation is 8.2 m s−1, and
the ambient turbulence intensity is approximately 6 %. Dur-
ing the simulation, the turbines maintain a yaw angle of 0◦
and turn with the wind. For simplicity we assume ideal wind
direction tracking capabilities and apply a prescribed mo-
tion. For more information see the dataset which contains
the SOWFA files for the case and the precursor simulation
(Becker, 2022b).
Figure 10 shows the flow field during the wind direction
change, starting at the time instances t=700, t=800 and
900 s. The SOWFA slices are taken at hub height. To show
the center of the FLORIDyn flow field, only OPs between
0.5zh< zOP <1.5zhare plotted. As the wake expands, more
chains leave these bounds, which leads to a sparser descrip-
tion of the wake in the plot. Due to the changes to FLORI-
Dyn described in Sect. 2.3.3, the OPs do not influence each
other in the field and OPs with a higher velocity can appear
among OPs with a lower velocity. The net effect of multiple
OPs is only calculated at the rotor plane. The grey arrows
in the FLORIDyn plots indicate the current wind direction.
The plots of both simulations visualize how the wakes slowly
transition to the new wind direction, forming a bow shape in
the process. FLORIDyn seems to describe the general path
of the SOWFA wakes quite well. It also capture some effects
like shorter, wider wakes of T4 and T5 at t=900 s.
To more accurately judge the timing of FLORIDyn,
Fig. 11 shows the generated power of all nine turbines. The
plots are arranged in the same way as the turbines in the
flow field plots. All plots show the filtered and unfiltered
Figure 10. Nine-turbine flow field at hub height during the wind
direction change at t=700 s (a, b),t=800 s (c, d) and t=
900 s (e, f). The FLORIDyn flow field is on the left and includes
grey arrows as an indicator of the current wind direction. On the
right is the corresponding snapshot from the SOWFA simulation.
data of FLORIDyn and SOWFA. The filtering is identical
to the filtering in Sect. 3.1. The grey area marks the time
window of the wind direction change. Looking at the mag-
nitude of the generated power, FLORIDyn predicts the aver-
age of the free-stream turbines T0, T3, T6, T7 and T8 quite
well, but the remaining turbines show a noticeable offset in
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2174 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
Figure 11. Power generated in the nine-turbine case. The grey area marks the time window in which the wind direction linearly changes by
60◦. The plots are arranged to fit the layout of the wind farm in Fig. 9. The data show the zero-phase-filtered (zp. f.) and the unfiltered (unf.)
SOWFA data, as well as the filtered (f.) and unfiltered FLORIDyn data.
Table 2. Points in time at which the power generated in the nine-turbine case is minimal due to wake interaction.
Turbine 1 Turbine 2 Turbine 4 Turbine 5
Min. 1 Min. 2 Min. 1 Min. 2 Min. 1 Min. 1
SOWFA (s) 833.5 996.5 822 972 826.5 809
FLORIDyn (s) 832 992 836 992 832 836
Error (s) −1.5−4.5+14 +20 +5.5+27
generated power. This could be due to speed-up effects and
is briefly discussed in Appendix A2. The interesting aspect
is the timing of the wake interaction from upstream turbines
with downstream turbines. The generated power by T4 shows
the passing of the wake of T6 during the wind direction
change. Noticeable is the accuracy with which the unfiltered
FLORIDyn data align the unfiltered SOWFA data. Table 2
lists the points in time at which the power generated is mini-
mal in SOWFA and in FLORIDyn, as well as the difference.
This shows that FLORIDyn predicts the maximal wake in-
fluence 5.5 s later than in SOWFA.
The filtering of the FLORIDyn data significantly wors-
ens the quality of the result, in contrast to the simulations
of the three-turbine case. The filtering was applied on the
data from the rotor plane. Thus, only modifying the way a
turbine perceives the incoming, foreign wake in FLORIDyn
will not improve the simulation for changing environments.
As a result, future research has to improve the way a tur-
bine dynamically influences its own wake. Turbine T1 shows
similar behavior to T4: the generated power shows the wake
influence of T3 first, but it also shows, after the wind direc-
tion stopped changing, the influence of the outskirts of the
wake of T6. While the timing of this interaction shows good
agreement, the magnitude of the interaction is considerably
lower in FLORIDyn than in SOWFA. This could be due to
a too fast recovering FLORIS wake, an inadequate wake su-
perposition method or due to local turbulence levels, which
FLORIDyn can not capture. T5 shows the overlapping influ-
ence of the wakes of T6 and T7. The two overlapping Gaus-
sian influences do form a longer period of reduced gener-
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2175
Table 3. Computational performance.
Number of turbines 2 3 4 9
Total number of OPs 2 ×1043×1044×1049×104
tcomp.time per step (s) 2.44 ×10−25.87 ×10−21.09 ×10−16.13 ×10−1
ated power. This can be seen in both simulations. Table 2
shows the largest timing error between SOWFA and FLORI-
Dyn for this wake influence. This could stem from an inac-
curate wake interaction model and the way added turbulence
is treated. T2 shows the most overlapping influences by the
wakes of T3, T4, T6 and T7 in this order. While the first two
overlapping interactions show good agreement, FLORIDyn
shows poorer agreement with SOWFA for the influence of T6
and T7. Again, the SOWFA simulation suggests a larger de-
crease in generated power. A reason could be that the way
wakes combine is not described accurately enough. The tim-
ing of the wake of T7 seems to be a bit too late as well. How-
ever, the SOWFA simulation recovers to its steady state at
about the same time as FLORIDyn. Additionally, all turbines
with the exception of T6 experience a small influence of the
upstream turbines in the steady-state configuration. This ef-
fect is not noticeable in the SOWFA simulations. Conclud-
ing, FLORIDyn describes the timing of passing wake influ-
ences quite well. However, there are discrepancies in terms
of magnitude and possibly in the way wakes combine their
effects on downstream turbines.
3.3 Computational performance
Table 3 contains the average computational time per time
step, which is equivalent to 4s simulation time. This can be
compared to SOWFA, which can take around 5.8×102to
5.4×103s per core per time step, depending on the setup
(van den Broek and van Wingerden, 2020). The FLORI-
Dyn measurements were performed for two and three con-
secutive turbines and a 2×2 and 3 ×3 turbines wind farm.
The times exclude plotting and the simulation setup time.
A setup can take up to 3 s, depending on how much data
need to be imported. The measurements were taken on a
MacBook Pro (2019) with a 2.3 GHz eight-core Intel i9
CPU, 32 GB of 2667 MHz DDR4 RAM, an SSD, and Ma-
cOS Catalina (10.15.7). The simulation environment is MAT-
LAB 2020a without the use of toolboxes, such as the parallel
computing toolbox, and without precompiled code, besides
what is built into the simulation environment. These results
naturally vary with the layout, atmospheric behavior, simula-
tion settings, etc. and are only meant to give an estimation of
the performance.
A first takeaway is that FLORIDyn simulates all cases
faster than real time: within 4 s, the simulation can perform
between 164 and 6.5 simulation steps, depending on the
number of turbines simulated. This results in 656 sSim in-
simulation time for two turbines and goes down to 26sSim
for nine turbines in 4 s of real time. This opens up the needed
computational headroom for a model-based real-time control
strategy and the necessary optimization. On the other hand,
the times also do not offer a large time window for opti-
mization. For instance, in the three-turbine yaw case, it takes
roughly 300 sSim in simulated time until the yaw changes
have propagated from the first turbine to the last turbine. To
optimize the control actions with this model for the near fu-
ture, parallel computing would be needed. With an increasing
number of turbines, this time window decreases.
Generally, the computational time increases quadratically
with n2
T−nT, as nTturbines need to determine if they are
in the wake of another turbine and calculate the influence.
This growth in computational effort is assumed to decrease
with larger wind farms: as the spacial dimension grows, not
every turbine needs to consider all other turbines for interac-
tion. Nevertheless, the simulation times will exceed what is
practical for a wind farm with 3 turbines.
There are multiple opportunities to improve performance
which have not been utilized so far. The main aspect which
increases computational effort is the interaction among the
turbines. A first step can be to calculate the turbine interac-
tions in parallel. A second step is to find a way to efficiently
determine if a turbine is influenced by a wake. Furthermore,
the number of OPs per turbine can be decreased and tuned:
not all chains need to be equally long, and OPs which wan-
der out of the domain can be disregarded. Then, there is the
fundamental question of whether the proposed structure of
FLORIDyn can be improved, for instance by using less but
more efficient OPs. Eventually, the programming platform
can be switched to a choice which allows more specific opti-
mization, e.g., C, C++ or Julia.
4 Conclusions and recommendations
In this paper, a new FLORIDyn model is presented and com-
pared to SOWFA simulations. This model utilizes a Gaus-
sian FLORIS model and concepts from the previously pub-
lished FLORIDyn framework by Gebraad and van Winger-
den (2014) to create a three-dimensional, dynamic and com-
putationally lightweight wind farm model. The new FLORI-
Dyn model is further capable of simulating its wakes under
heterogeneous and time-varying flow conditions. To achieve
this, we presented a mathematical approach to decouple
wake and flow characteristics into two coordinate system
which are connected by observation points. To simulate
https://doi.org/10.5194/wes-7-2163-2022 Wind Energ. Sci., 7, 2163–2179, 2022

2176 M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake
changing environmental conditions, a method to map sparse
flow field measurements to a finer grid was presented which
avoids the interpolation cost at runtime. The new FLORI-
Dyn model shows good performance compared to SOWFA
in terms of timing and is able to predict accurately when
a downstream turbine is experiencing influences from up-
stream turbines.
Despite the considerable advancements over the old
FLORIDyn implementation, there are still several aspect of
the model which can be improved. The central aspect is how
turbines influence wakes and how wakes are perceived by
turbines. In this work we have decoupled the OP propagation
speed from the effective wind speed, which effectively leads
to a simpler, lightweight model while the wake behavior is
still dynamically described. However, this way state changes
reach downstream turbines too soon and in a sudden manner.
Ideally this can be overcome by finding better, computation-
ally lightweight methods to model the influence of changing
turbine states on the wake needs and also how a turbine re-
acts to dynamic changes in the flow. Another aspect that can
be improved is related to the interface between FLORIDyn
and FLORIS. FLORIS has been subject to many develop-
ments and improvements, and FLORIDyn can utilize these
improvements if it improves the interface: with a generic in-
terface, newer developments can be included and existing
code can be used in a sustainable manner. The simulations
also show that parameter tuning has to be more accessible
and possibly needs to be performed online in some cases.
The next aspect which could be improved is the coupling of
FLORIDyn with the turbulent environment of the real wind
farm (or its surrogate). Combined with the changes to the
OP propagation speed from this work, this can lead to a more
uneven OP distribution with dense areas where high wind
speeds decrease and sparse areas where low wind speeds in-
crease. An extension to the model could feature a method
to combine and generate OPs, depending on the density of
OPs. Although, this could also lead to undesired information
loss, depending on the implementation. To achieve better re-
sults, the wind field model has to be replaced or enhanced
by estimators. The latter would provide a more accurate es-
timate of wind speed, direction and ambient turbulence in-
tensity for the FLORIDyn simulation. In the long term, a dy-
namic description of the environment could become part of
the FLORIDyn model. This could also include effects like
induction zones and speed-up between the turbines. The last
aspect to consider for improvement is the performance. In
its current implementation, FLORIDyn delivers its results at
a low computational cost. This has to be maintained, if not
improved, to allow its use for dynamic real-time closed-loop
control algorithms in the future. The simulation also needs to
be structurally improved to keep its low computational cost
for wind farms with large numbers of turbines.
Concluding, the new FLORIDyn model is a promising
concept with unique strengths. With FLORIS in its core, it
utilizes an existing, successfully employed model and pro-
vides a new dimension in a challenging environment at a
low computational cost. The model can already be adapted
to work in a closed-loop control design and shows more po-
tential if the mentioned aspects are improved.
Appendix A: Additional plots and aspects of the
simulation results
A1 Unfiltered difference between yawed and baseline
case
Figure A1 shows the difference between the power generated
in SOWFA in the yaw case (Sect. 3.1.2) and the steady-state
baseline case (Sect. 3.1.1). Both simulations are performed
in the same turbulent environment, something which would
be impossible to achieve in realistic conditions. This way, the
difference allows for a clearer interpretation of the influence
of the yaw step, at least the timing. In comparison to Fig. 8,
Fig. A1 shows the unfiltered data as well as the filtered data
for all three turbines. T1 shows between t=312 and 329 s a
first reaction due to the changed wake of T0. T2 shows a first
reaction between t=426 and 442 s. The filtered data show
a slightly earlier influence due to the nature of a zero-phase
filter.
Figure A1. Difference between the yaw case in SOWFA and the
baseline case with unfiltered and zero-phase-filtered data. Filtering
was performed before calculating the difference. The dotted lines
mark the start of the yaw angle changes of T0.
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M. Becker et al.: The revised FLORIDyn model: implementation of heterogeneous flow and the Gaussian wake 2177
A2 Averaged velocity in the nine-turbine case
Figure A2 shows the averaged wind speed in the nine-turbine
case from t=500 to 600 s at hub height in SOWFA. A total
of 34 slices were used to average. The wind speed is binned
into 11 wind speed sections which are plotted as contours.
During the time of averaging the wind direction is constant.
The average wind speed of the incoming air is approximately
8 m s−1. However, between the turbine rows, the wind speed
increases to a higher level, up to 9.44m s−1in some places.
This could be explained by speed-up effects: the turbines act
as resistances in the flow field, and the wind speed in the
place of least resistance, between the turbines, increases. The
effect has been observed and described in Bastankhah et al.
(2021) as well for instance. Due to the speed-up, the turbines
further downstream experience higher wind speeds than the
ones in free stream and generate more energy. Figure 11
quantifies the effect, where T2, T4 and T5 generate signif-
icantly more energy than T6 for instance. After the wind di-
rection change, the effect leads again T2 and T4 to generate
more power. T1 is now in the situation T5 was in initially and
also generates more energy. T5, however, drops to a lower
level. Without an added model for effects like these, FLORI-
Dyn (and FLORIS) will not be able to accurately describe
the wind field.
Figure A2. Averaged wind speed from t=500 to 600s, divided
into 11 speed sections.
Code and data availability. The FLORIDyn code and the
SOWFA code are publicly available under the GPL-3.0 li-
cense. The FLORIDyn repository contains the entire MATLAB
code and the used measurements (power generated, blade
pitch angle, rotor speed) from the used SOWFA simulations
(https://data.4tu.nl/articles/software/Gaussian_FLORIDyn_
Matlab_implementation_belonging_to_the_paper_The_revised_
FLORIDyn_model_Implementation_of_heterogeneous_flow_and_
the_Gaussian_wake/19867846; Becker, 2022a). It is published
by the Delft Center for Systems and Control (DCSC). SOWFA
is published by the National Renewable Energy Laboratory and
written in C++ (https://doi.org/10.5281/zenodo.3632051, Sale et
al., 2020; https://github.com/dcsale/SOWFA, National Renewable
Energy Laboratory, 2020). The SOWFA files for the nine-turbine
case are available to rerun and validate the claims from this paper
(https://data.4tu.nl; Becker, 2022b).
Author contributions. MB developed and implemented the
model under the supervision of and in discussion with BR, BD,
DvdH, UK, DA and JWvW. The SOWFA simulations were set
up by DvdH, and BD provided the CTand CPlookup tables.
JWvW was responsible for the funding. The manuscript was writ-
ten by MB and corrected and proofread by BR, BD, DvdH, UK, DA
and JWvW.
Competing interests. The contact author has declared that none
of the authors has any competing interests.
Disclaimer. Publisher’s note: Copernicus Publications remains
neutral with regard to jurisdictional claims in published maps and
institutional affiliations.
Financial support. This research has been supported by the Ned-
erlandse Organisatie voor Wetenschappelijk Onderzoek program
“Robust closed-loop wake steering for large densely space wind
farms” (grant no. 17512).
Review statement. This paper was edited by Jennifer King and
reviewed by Jaime Liew and Paul van der Laan.
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