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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

Wind Energ. Sci., 7, 2163–2179, 2022 https://doi.org/10.5194/wes-7-2163-2022

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

https://doi.org/10.5194/wes-7-2163-2022 Wind Energ. Sci., 7, 2163–2179, 2022

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

https://doi.org/10.5194/wes-7-2163-2022 Wind Energ. Sci., 7, 2163–2179, 2022

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,OPr1,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

21−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,mαs

uz0,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.

Wind Energ. Sci., 7, 2163–2179, 2022 https://doi.org/10.5194/wes-7-2163-2022

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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