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Wind Energ. Sci., 9, 2195–2215, 2024
https://doi.org/10.5194/wes-9-2195-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
Power curve modelling and scaling of fixed-wing
ground-generation airborne wind energy systems
Rishikesh Joshi1, Roland Schmehl1, and Michiel Kruijff2
1Delft University of Technology, Faculty of Aerospace Engineering,
Kluyverweg 1, 2629 HS Delft, the Netherlands
2Ampyx Power B.V., Lulofsstraat 55, Unit 13, 2521AL The Hague, the Netherlands
Correspondence: Rishikesh Joshi (r.joshi@tudelft.nl) and Roland Schmehl (r.schmehl@tudelft.nl)
Received: 8 July 2024 – Discussion started: 5 August 2024
Revised: 18 September 2024 – Accepted: 24 September 2024 – Published: 14 November 2024
Abstract. The economic viability of future large-scale airborne wind energy systems critically hinges on the
achievable power output in a given wind environment and the system costs. This work presents a fast model for
estimating the net power output of fixed-wing ground-generation airborne wind energy systems in the conceptual
design phase. In this quasi-steady approach, the kite is represented as a point mass and operated in circular flight
manoeuvres while reeling out the tether. This phase is subdivided into several segments. Each segment is assigned
a single flight state resulting from an equilibrium of the forces acting on the kite. The model accounts for the
effects of flight pattern elevation, gravity, vertical wind shear, hardware limitations, and drivetrain losses. The
simulated system is defined by the kite, tether, and drivetrain properties, such as the kite wing area, aspect ratio,
aerodynamic properties, tether dimensions and material properties, generator rating, maximum allowable drum
speed, etc. For defined system and environmental conditions, the cycle power is maximised by optimising the
operational parameters for each phase segment. The operational parameters include cycle properties such as the
stroke length (reeling distance over the cycle), the flight pattern average elevation angle, and the pattern cone
angle, as well as segment properties such as the turning radius of the circular manoeuvre, the wing lift coefficient,
and the reeling speed. To analyse the scaling behaviour, we present a kite mass estimation model based on the
wing area, aspect ratio, and maximum tether force. The computed results are compared with 6-degree-of-freedom
simulation results of a system with a rated power of 150 kW. The results show the interdependencies between
key environmental, system design, and operational parameters. Gravity penalises performance more at low wind
speeds than at high wind speeds, and excluding gravity does not yield optimistic performance since it assists
in the reel-in phase by reducing the required power. Thin tethers perform better at lower wind speeds but limit
power extraction at higher wind speeds and vice versa for thick tethers. Upscaling results in a diminishing gain in
performance with an increase in kite wing area. The proposed model is suitable for integration with cost models
and is aimed at sensitivity and scaling studies to support design and innovation trade-offs in the conceptual
design of systems.
1 Introduction
Airborne wind energy (AWE) is an emerging technology
that uses tethered airborne devices to harness higher-altitude
wind resources that are inaccessible to conventional towered
wind turbines with potentially lower material usage. Fig-
ure 1 shows several implemented prototypes in the power
range of rated powers up to 600 kW. Makani and Kitekraft
use onboard ram-air turbines to convert the relative flow
at the aircraft into electricity, using a conductive tether to
transmit the electricity to the ground. SkySails Power and
Kitepower operate large soft-wing kites in pumping cycles,
using suspended cable robots for control and converting the
pulling force of the kite into electricity by means of a ground-
Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.
2196 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Figure 1. Implemented AWE systems (AWESs): (a) Makani
(2020), discontinued in 2020; (b) Kitekraft (2024); (c) SkySails
(2024); (d) Kitepower (2024); (e) TwingTec (2024); (f) Kitemill
(2024); (g) Enerkíte (2024); and (h) Mozaero (2024), formerly
Ampyx Power. Photos courtesy of the respective companies.
Figure 2. Operation schematic of the fixed-wing ground-generation
airborne wind energy concept: (a) reel-out phase and (b) reel-in
phase (image courtesy of Ampyx Power B.V.).
based drum-generator module. TwingTec and Kitemill use
fixed-wing kites that adopt the same conversion principle in
combination with the vertical takeoff and landing (VTOL)
subsystem. EnerKíte operates a lightweight fixed-wing kite
in pumping cycles, using three tethers controlled from the
ground station. A rotational mast on the ground station is
employed to launch the kite. Mozaero, previously known as
Ampyx Power, uses a catapult subsystem combined with on-
board propellers to launch the kite.
This work focuses on the fixed-wing ground-generation
(ground-gen) concept developed by Ampyx Power, which
has been continued by Mozaero since 2023. As illustrated in
Fig. 2, a fixed-wing kite analogous to a glider aircraft is con-
nected by a tether to a drum-generator module on the ground.
The kite flies in repetitive crosswind patterns, pulling the
tether with high force from the drum and driving the genera-
tor, as shown in Fig. 2a. During this reel-out phase, electricity
is generated. Once the tether has reached a certain length, the
kite is retracted toward the generator with minimum aerody-
namic drag and substantially lower force as shown in Fig. 2b.
A small fraction of the generated electricity is consumed dur-
ing this reel-in phase. An intermediate buffer storage is typ-
ically used for this purpose. The reel-out and reel-in phases
are repeated cyclically to generate a net power output.
Compared to flexible-membrane kites, fixed-wing kites
are characterised by better aerodynamic performance, a
higher lift-to-drag ratio, and a substantially larger mass-to-
wing surface ratio. While the first ratio stays roughly the
same when increasing the size of the kite, the latter ratio pro-
gressively increases. This increase affects fixed-wing kites
much more than it does soft-wing kites, rendering mass a cru-
cial parameter for designing large-scale fixed-wing AWESs.
The present work focuses on the gravitational effect of mass
and its interplay with the resultant aerodynamic force and the
tether force during quasi-steady flight operation. To under-
stand this effect, it is important to note that the gravitational
force is a constant contribution to the force equilibrium at
the kite. In contrast, the aerodynamic force depends on the
instantaneous flight speed and can thus vary greatly in the
different operational phases of the system. The tether force,
on the other hand, is a reaction force to the vectorial sum of
the two external forces.
Because of the fast crosswind manoeuvres in the reel-out
phase, the apparent wind speed is high, and the quasi-steady
force equilibrium is dominated by the aerodynamic loading
and the tether force – except for low wind speeds at cut-in
where the gravitational force contributes substantially. In the
reel-in phase, the gravitational effect is exploited to retract
the kite with a minimum tether force and thus reel-in power.
The force equilibrium is dominated by the aerodynamic and
gravitational forces, while the tether force plays only a mi-
nor role. Several different strategies exist for launching and
landing. For AWE systems pursuing a vertical takeoff and
landing (VTOL) strategy, the aerodynamic force generated
by the VTOL subsystem is used entirely to overcome gravity
because the wing is either perpendicular to the wind and thus
ineffective for lift (e.g. Makani or Kitekraft) or aligned with
the wind but providing an insufficient lift force (e.g. Kitemill
or TwingTec). For AWE systems pursuing a horizontal take-
off and landing (HTOL) strategy, the kite needs to be accel-
erated by an external mechanism to a certain minimum flight
speed at which the aerodynamic force can overcome grav-
ity. This can be done with a catapult and optional onboard
propellers (e.g. Ampyx Power) or a swivelling mast (En-
erKíte). Irrespective of how the fixed-wing kite is launched
or landed, it has to maintain a certain minimum flight speed
during crosswind operation to create an aerodynamic load
level sufficient to compensate for gravity’s effect and stay
airborne. This interplay between aerodynamic, gravitational,
and tether forces during the different operational phases re-
quires careful trade-off analysis when designing a fixed-wing
AWE system.
Several physical models with a broad spectrum of fidelity
and scope have been developed to understand and mathe-
matically describe the operation of AWE systems. Higher-
fidelity approaches based on dynamic models and system
control, such as in Licitra et al. (2019), Malz et al. (2019),
and Eijkelhof and Schmehl (2022), are computationally ex-
pensive and require the initialisation and tuning of many pa-
Wind Energ. Sci., 9, 2195–2215, 2024 https://doi.org/10.5194/wes-9-2195-2024
R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2197
rameters. Sommerfeld et al. (2022) investigated the scaling
effects of fixed-wing ground-generation AWE systems using
AWEbox (Schutter et al., 2023), which is an optimal con-
trol framework. Their simulation results do not reveal consis-
tent trends, which could indicate non-converged results. The
results of such models are highly dependent on the imple-
mented controller’s performance. Hence, they are not the best
option for understanding the fundamental principles of sys-
tems; the achievable energy output; and the interdependen-
cies between environmental, system design, and operational
parameters. Lower-fidelity approaches based on steady or
quasi-steady models can be used for this purpose and techno-
economic analysis, such as in Heilmann and Houle (2013),
Faggiani and Schmehl (2018), and Joshi et al. (2023).
There have been several attempts to model the power gen-
eration characteristics of AWE systems with lower fidelity,
but none of the theories account for all the relevant phys-
ical effects. The first mathematical foundation for estimat-
ing power extraction using tethered kites in a crosswind mo-
tion was laid by Loyd (1980). This analytical theory assumes
idealised flight states to estimate the mechanical power out-
put of a pumping AWE system, but it does not account for
losses due to elevation, retraction phase, gravity, vertical
wind shear, and hardware limitations.
Argatov et al. (2009) extended this crosswind theory to
spherical coordinates to compute the mean mechanical reel-
out power of ground-generation systems. The theory ac-
counts for the averaged effects of elevation and gravity on
the kite but does not account for the losses due to the retrac-
tion phase. Luchsinger (2013) extended Loyd’s ideal power
extraction theory for fixed-wing ground-generation systems
to account for the retraction phase, average pattern eleva-
tion, and hardware limitations such as the maximum tether
force and generator power rating. But, their study does not
account for the effect of gravity. Fechner and Schmehl (2013)
presented a model for soft-wing ground-generation systems
that accounts for losses due to elevation, the transition phase,
and the retraction phase but does not account for the effect
of gravity. They also accounted for various efficiencies of
the ground station in computing net electrical power output.
The model results were compared against measurements of a
4 kW prototype demonstrator.
Ranneberg et al. (2018) presented a model to compute the
net power output that accounts for the traction and retrac-
tion phases, including gravity and component efficiencies.
However, the paper does not provide explicit problem for-
mulation using gravity or highlight the relevance of differ-
ent force terms. The model results were compared against
measurements of a 5 kW prototype, which showed deviations
within their expected range. It was also compared to a full 6-
degree-of-freedom (6-DOF) simulation of a 30 kW system,
which showed good agreement between their models. Trevisi
et al. (2020) presented an analytical modelling framework to
compute net power output, accounting for the losses due to
elevation, gravity, and the retraction phase. Following this
work, Trevisi et al. (2023) proposed refining the power esti-
mation by including the effect of the far wake on the induced
drag. These losses were modelled as loss factors to calculate
the net power output. The model results were not compared
against higher-fidelity simulations or measurements.
A quasi-steady model (QSM) for soft-wing ground-
generation systems was developed by Schmehl et al. (2013)
and Van der Vlugt et al. (2019) and validated by Schelber-
gen and Schmehl (2020) using measurements from a 20 kW
prototype. This model accounts for the losses due to eleva-
tion, retraction phase, transition phase, and gravity. We have
extended this formulation to fixed-wing systems by incorpo-
rating changes in the retraction phase, angle of attack con-
trollability impacting the operational lift coefficient, induced
drag, effects of vertical wind shear, hardware limitations, and
drivetrain losses. The model is then used to formulate an op-
timisation problem to find the operational set points for max-
imising the system’s electrical cycle power. The study offers
several insights into the impact of gravity and system scaling
across low and high wind speeds. The findings highlight how
kite and tether mass scaling influences performance, reveal-
ing trade-offs between larger and smaller kite–tether config-
urations.
The paper is structured as follows. Section 2 describes
the theoretical framework, which begins with simplifying as-
sumptions and then builds up by relaxing them step by step;
Sect. 3 presents the model capabilities through a numerical
example and effects of gravity and scaling on performance;
and Sect. 4 presents the conclusions.
2 Model description
Consider a fixed-wing kite, analogous to a glider aeroplane,
flying on a circular flight trajectory, with a wing planform
area S; an aspect ratio AR; lift and drag coefficients CL
and CD, respectively; a tether of maximum allowable force
Ft,max; a turning radius Rp; a cone opening angle γp; and
a turning axis elevation angle βp, which is also referred to
as the pattern elevation angle in the following sections. The
pattern elevation angle is kept constant for one reel-out phase
and one reel-in phase. This is the first step of a system-level
performance analysis, so unsteady effects such as turbulence
or wind gusts are not considered. The kite kinematics and
the forces acting on the kite are formulated in a spherical
reference frame (r,θ, φ), defined with respect to the Carte-
sian wind reference frame (Xw,Yw, Zw). The horizontal axis
is aligned with the wind velocity vwand its Zaxis pointing
vertically upward, as shown in Fig. 3. The unit vectors er,eθ,
and eφdefine a right-handed local vector base. The kite’s po-
sition is represented by point K, and the ground station is
located at the origin O. The radial coordinate rspecifies the
geometrical distance between the kite and the ground station;
θis the polar angle, which is complementary to the tether el-
https://doi.org/10.5194/wes-9-2195-2024 Wind Energ. Sci., 9, 2195–2215, 2024
2198 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Figure 3. Decomposition of kite kinematics in a spherical reference
frame (Schmehl et al., 2013).
evation angle βmeasured from the ground (i.e. θ+β=90°);
and φis the azimuth angle.
The kite velocity vkcan be decomposed into radial and
tangential components vk,rand vk,τ , respectively. The direc-
tion of vk,τ is given by the course angle χmeasured in the
local tangential plane τfrom the unit vector eθ. The appar-
ent wind velocity can be expressed in spherical coordinates
(r,θ, φ) as
va=
va,r
va,θ
va,φ
=
sinθcos φ
cosθcos φ
−sinφ
vw−
1
0
0
vk,r
−
0
cosχ
sinχ
vk,τ .(1)
The final model simulating the reel-out and reel-in phases
of a system is formulated as an optimisation problem where
the net electrical cycle power is maximised for given wind
conditions. The fixed model inputs are the system design
parameters, such as the kite wing area, aspect ratio, wing
aerodynamic properties, tether properties, speed limits of the
drum-generator module, etc. The optimisation variables are
the operational parameters, such as the stroke length (reeling
distance over a cycle), pattern elevation angle, cone open-
ing angle, turning radius at the start of the cycle, kite reeling
speed, and operating lift coefficient. The optimisation prob-
lem is constrained by physical limits, such as the maximum
tether force, tether length limit, minimum ground clearance,
maximum operation height, etc.
Section 2.1 describes power extraction using the assump-
tion of a massless kite on a straight tether at an elevation. Sec-
tion 2.2 introduces a kite mass estimation function based on
prototype data and scaling laws. Section 2.3 incorporates the
effect of gravity on the power output. Section 2.4 describes
the impact of the retraction phase on the net power output of
the system. Section 2.5 introduces vertical wind shear, and,
finally, Sect. 2.6 describes the electrical cycle power of a sys-
tem considering all the effects together.
2.1 Massless kite at an elevation
Figure 4a illustrates the physical problem of a massless kite
with a straight inelastic tether flying in a circular pattern sym-
metrically around the Xwaxis. In the depicted instance along
the flight path, the kite just passes through the XwZwplane,
and the wind vector vwis orthogonal to the kite’s tangen-
tial motion component. At this analysis stage, we assume a
uniform and constant wind field; i.e. the wind speed and the
direction do not change in time and space. The rotation of the
wind reference frame is assumed to be so slow that the accel-
erations induced by this rotation are negligible. For any ar-
bitrary point on the circular flight manoeuvre, Fig. 4b shows
the decomposition of velocity and force vectors in the erva
plane.
Figure 4. Velocities and forces for the massless kite at an elevation.
(a) Side view illustrating the circular flight manoeuvre with the av-
erage pattern elevation angle βp, opening cone angle γp, reel-out
speed vo, and force equilibrium Ft+Fa=0 at the kite. (b) De-
composition of velocity and force vectors in the ervaplane of the
spherical reference frame for any arbitrary point on the trajectory.
The assumption of a massless kite allows us to ignore the
effects of gravity and inertia on the kite’s motion. Since the
tether cannot support a bending moment, the radial force bal-
ance is
Ft=Fa,where (2)
Fa=1
2ρS qC2
L+C2
Dv2
a.(3)
Because of the assumed straight tether, the kite’s radial
speed vk,ris identical to the reel-out speed vo. For any point
on the flight trajectory, the apparent wind speed can be ex-
pressed by its radial and tangential components as
va=va,rsva,τ
va,r2
+1.(4)
The radial component is defined by Eq. (1) as
va,r=vwsinθcos φ−vo.(5)
The ratio of the tangential and radial components of the ap-
parent wind velocity is also known as the kinematic ratio as
Wind Energ. Sci., 9, 2195–2215, 2024 https://doi.org/10.5194/wes-9-2195-2024
R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2199
described in Schmehl et al. (2013). Because of the geomet-
ric similarity of the velocity and force triangles illustrated in
Fig. 4b, the kinematic ratio can be related to the lift-to-drag
ratio,
κ=va,τ
va,r=CL
CD
.(6)
From Eqs. (2), (3), (4), (5), and (6), the extractable me-
chanical power at the ground station can be computed as
Pm,o=Ftvo(7)
=1
2ρS qC2
L+C2
D(vwsinθcos φ−vo)2
"CL
CD2
+1#vo.(8)
From this equation, one can conclude that increasing val-
ues of the elevation angle and azimuth angles decrease the
magnitude of the apparent wind velocity vector. We simplify
our formulation and represent one circular flight manoeu-
vre by a single flight state. During one full circular flight
manoeuvre, the elevation angle βwill vary from βp−γpto
βp+γp, where γpis the cone opening angle. The geometric
average βpover one full manoeuvre can be considered a rep-
resentative elevation angle over the pattern. The effect of the
azimuth angle differs from that of the elevation angle. Using
the geometric average of the variation in the azimuth angle
as a representative angle will result in φp=0 due to the neg-
ative and positive signs of the azimuth angle on either side of
the symmetry plane XwZw. But in reality, the kite flying in a
circular manoeuvre is, on average, at a non-zero azimuth an-
gle (Van der Vlugt et al., 2019). Therefore, we regard the ge-
ometric centre of the semicircle as a representative azimuth
angle. The centroid ycof a semicircle with a radius Rin the
Cartesian reference frame is given by
yc=4R
3π.(9)
This translates to a specific azimuth angle in the spherical
reference frame. For a given cone angle γp, the azimuth angle
representing the centroid of a semicircle is
φp=sin−14sin γp
3π.(10)
Therefore, it can be approximated that a representative
point for the entire pattern, incorporating the average ef-
fect of elevation and azimuth, is with θ=π/2−βpand
φp=sin−1(4sin γp/3π). The mean pattern reel-out power
can now be estimated using Eq. (7). The reel-out speed is
an independent variable in our model, which is controlled
by the ground station, and the tangential velocity is a result
of the local force balance at the kite. The other dependent
properties of the system are the tether dimensions, the kite’s
operational envelope, and the effective drag coefficient. They
are computed as follows.
Figure 5. Side view illustrating the kite’s operational envelope (re-
gion shown by the dotted red lines) and the operational height range
(non-hashed region between hmin and hmax).
Tether dimensions
For a given turning radius Rpand opening cone angle γpdur-
ing operation, the required tether length is computed as
lt=Rp
sinγp
.(11)
For a given tether tensile strength σtand maximum allow-
able tether force Ft,max, the required tether diameter can be
calculated as
dt=s4Ft,max
πσt
.(12)
For ground-gen systems, the tether lifetime is driven pri-
marily by fatigue due to bending and creep (Bosman et al.,
2013). Hence, the tether will not be sized based on the mate-
rial’s ultimate breaking strength but on the optimal operating
stress levels for an extended fatigue life.
Kite’s operational envelope
From Eq. (7), one can see that the closer the tether is aligned
with the wind velocity, the higher is the power generated in
crosswind operation. However, in a practical operation sce-
nario, the kite must maintain a certain ground clearance hmin
and, in most cases, respect a maximum operating height hmax
for safety reasons and regulations. These limits must be con-
sidered when estimating the tether length and the operational
height during the cycle. Figure 5 shows this geometrical rela-
tionship where zk,min and zk,max are the bottom-most and top-
most operating points during the cycle, and lt,min and Rp,min
are the tether length and the turning radius at the start of the
cycle, respectively. The operational height range of the kite
is the non-hashed vertical region between hmin and hmax. The
region between the dotted red lines represents the actual op-
erational envelope of the kite. This envelope changes since
the optimal operating parameters change with respect to dif-
ferent wind conditions.
https://doi.org/10.5194/wes-9-2195-2024 Wind Energ. Sci., 9, 2195–2215, 2024
2200 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Determining the minimum tether length lt,min using
Eq. (11), the minimum ground clearance hmin can be en-
forced by computing the kite’s height at the bottom-most
point of its circular flight manoeuvre,
zk,min =Rp,min
sinγp
sin(βp−γp),(13)
such that zk,min ≥hmin. The maximum operational height
limit hmax can be enforced similarly by computing
zk,max =Rp,max
sinγp
sin(βp+γp).(14)
Wing lift coefficient
We assume complete control over the kite by modulating the
angle of attack to maintain the necessary wing lift coeffi-
cient CLfor optimal flight. Therefore, CLis a variable in our
model whose value is based on the lift polar. The maximum
lift coefficient will set the upper limit considering some stall
margins. This can be obtained from experimental measure-
ments or computational analysis such as in Vimalakanthan
et al. (2018).
Effective drag coefficient
In addition to the wing drag, the tether is responsible for drag
losses during operation. The tether drag contribution is as-
sumed to be lumped to the kite, resulting in an effective drag
of the combined kite and tether system (Houska and Diehl,
2006). To estimate this effective drag, it is assumed that the
generated moment at the ground station equals the sum of the
moments generated by the kite and the tether drag individu-
ally. The generated drag force is approximately perpendicu-
lar to the tether for high lift-to-drag ratios. We assume that
the apparent velocity of the topmost segment of the tether is
the same as that of the kite, and it uniformly drops to zero at
the ground station. For a given tether length lt, this moment
equality can be mathematically expressed as
ltDeff =ltDk+
lt
Z
0
l1
2ρv2
a,lCd,tdtdl, (15)
=lt
1
2ρv2
aCD,kS+1
2ρv2
a
l2
t
Cd,tdt
lt
Z
0
l3dl, (16)
where CD,kis the kite drag coefficient, va,lis the apparent
velocity of the tether element dlat a distance lfrom the
ground station, dtis the cross-sectional diameter, and Cd,t
is the cross-sectional drag coefficient of the tether.
This equation can be solved as follows to estimate the ef-
fective drag coefficient:
CD,eff =CD,k+CD,t,(17)
where
CD,t=1
4Cd,tdtlt
1
S.(18)
It is the effective drag coefficient of the tether lumped at the
kite. The total drag of a wing is the sum of parasitic drag and
lift-induced drag. Parasitic drag is comprised of a pressure
drag contribution due to flow separation and a skin friction
drag contribution. The induced drag is coupled to the gener-
ated lift (Anderson, 2016). For a given wing with an aspect
ratio AR and a wing planform efficiency (Oswald) factor e,
the total kite drag coefficient can be expressed as
CD,k=Cd,min +(CL−Cl,Cd,min)2
πARe,(19)
where Cd,min is the parasitic drag, CLis the wing lift co-
efficient, and Cl,Cd,min is the lift coefficient at Cd,min. As
stated earlier, the drag polar can be obtained from experi-
mental measurements or computational analysis such as in
Vimalakanthan et al. (2018).
2.2 Effective mass estimate
Equation (2) does not consider the effect of gravity in the
force equilibrium. This effect can generally be neglected dur-
ing the reel-out phase for smaller systems, especially for low-
mass soft-wing systems. This is because the gravitational
force is much lower than the traction force. The main impact
of weight for soft-wing systems is during the reel-in phase
since they typically fly to higher heights because the lift-to-
drag ratio is limited to a lower value. Gravity helps to reduce
this height and shorten the reel-in phase (Van der Vlugt et al.,
2019). This effect differs for larger and fixed-wing systems
with higher mass (Eijkelhof and Schmehl, 2022). Gravity re-
duces the attainable tether force and should be accounted for
in the power extraction.
Kruijff and Ruiterkamp (2018) and Bonnin (2019) devel-
oped a model for mass scaling at the part level based on the
150 kW prototype AP3 and the megawatt level concept study
AP4 developed by Ampyx Power B.V. The model uses the
prototype as a reference system and applies known scaling
laws for each structural part within the kite. The reference
prototype was designed to meet aviation standards with rel-
atively conservative safety factors. The prototype’s architec-
ture is scalable using a conventional design with ribs, spar
caps, webs, etc. It is, therefore, assumed to be a good rep-
resentation even for much larger fixed-wing kites. In the re-
sulting mass model, the kite mass mkis a non-linear function
of the kite planform wing area S, aspect ratio AR, and maxi-
mum tether force Ft,max, given as
mk=0.024 Ft,max
S+0.1S2+1.7Ft,max
S+32.5
S−50"0.46AR
ARref 2
−0.66AR
ARref +1.2#,(20)
Wind Energ. Sci., 9, 2195–2215, 2024 https://doi.org/10.5194/wes-9-2195-2024
R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2201
Figure 6. Kite mass as a function of wing area. Discrete data
points are from Joshi et al. (2023). MegAWES – Eijkelhof and
Schmehl (2022); Ampyx Power AP2, AP3, and AP4 – Kruijff and
Ruiterkamp (2018) and Ruiterkamp and Sieberling (2013); Ampyx
Power AP5 low and AP5 high – Hagen et al. (2023); Makani Power
M600, MX2, and M5 – Hardham (2012) and Echeverri et al. (2020);
Haas et al. (2019); aircraft wing scaling – Roskam (1989).
where ARref =12, Ft,max is in kilonewtons (kN), Sis in
square metres (m2), and mkis in kilogram (kg) The physical
meaning of the term Ft,max/S corresponds to the wing load-
ing for an aircraft. It states the maximum force it can han-
dle per unit wing area and varies around 1–10 kNm−2based
on the purpose and size of the aircraft. Figure 6 is a plot of
kite mass against wing area using the above equation and
fixing AR =12 and Ft,max/S =3.5 kNm−2. These fixed val-
ues represent the AP3 150 kW prototype values. The result is
compared against various data points compiled in Joshi et al.
(2023). As seen from the aircraft scaling curve (Roskam,
1989), a conventional aircraft wing does not increase in mass
as drastically as an AWE system kite. Based on available in-
formation, the MegAWES kite has AR =12 and Ft,max/S =
11 kNm−2(Eijkelhof and Schmehl, 2022), AP2 has AR =
10, AP4 has AR =12 and Ft,max /S =3 kNm−2(Kruijff
and Ruiterkamp, 2018; Ruiterkamp and Sieberling, 2013),
AP5 has AR =9.6 and Ft,max/S =3.9 kNm−2(Hagen et al.,
2023), M600 has AR =20 and Ft,max/S =7.3 kNm−2, MX2
has AR =12.5 and Ft,max /S =4.6 kNm−2(Echeverri et al.,
2020), and Haas et al. (2019) has AR =26 and Ft,max /S =
12.4 kNm−2.
Figure 7 shows a 3D plot by varying all three parameters in
Eq. 20. As expected, the kite mass increases with increasing
values of wing area, maximum allowable tether force, and
aspect ratio.
The minimum wind speed required by a static kite to com-
pensate for the gravitational force and stay airborne, also
known as the static takeoff limit (STOL), will help us further
understand the effect of mass. The STOL can be calculated
as
vw,STOL =s2mkg
ρSCL,max
,(21)
Figure 7. A 3D plot illustrating the relationship between wing area
S, maximum tether force Ft,max, and aspect ratio AR, with the kite
mass mk.
Table 1. Static takeoff limits for different prototypes. AP2 – Krui-
jff and Ruiterkamp (2018) and Williams et al. (2019), AP3 – Vi-
malakanthan et al. (2018) and Kruijff and Ruiterkamp (2018),
MegAWES – Eijkelhof and Schmehl (2022), MX2 – Hardham
(2012), TU Delft V3 – Oehler and Schmehl (2019).
Kite Wing area Kite mass CL,max STOL
(m2) (kg) (–) (ms−1)
AP2 3 35 1.5 11.3
AP3 12 475 2.1 17.5
MegAWES 150.45 6885 1.9 19.8
MX2 54 1850 2 16.7
TU Delft V3 19.75 22.8 1 4.3
where CL,max is the maximum operable wing lift coefficient.
Table 1 shows the STOL for some of the prototypes. When
the wind speeds are below the STOL for the particular kite,
for example during takeoff, they need to be accelerated to
achieve a higher apparent wind speed that allows compensa-
tion for the gravitational force. This implies that less heavy
soft-wing kites, such as the TU Delft V3, could be launched
at lower wind speeds without additional accelerating mecha-
nisms for takeoff. On the other hand, the heavier fixed-wing
kites will always need to be accelerated to increase their ap-
parent speeds.
In addition to the kite mass, tether mass also affects power
extraction and is assumed to be lumped together with the kite
mass. To determine the equivalent lumped mass, we require
that the moment generated at the drum equals the sum of
the moments generated by the kite and the continuous tether
individually (Houska and Diehl, 2006). For a specific tether
length ltand elevation angle β,
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2202 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
ltmeffg cos β=ltmkgcos β+
lt
Z
0
lgcos βπ
4d2
tρtdl, (22)
where g is the gravitational acceleration, dtis the tether di-
ameter, ρtis the tether material density, and dlis the length of
the tether element at a distance lfrom the drum. This equa-
tion can be solved as
ltmeff =ltmk+π
4d2
tρt
lt
Z
0
ldl, (23)
meff =mk+1
2mt,(24)
where the tether mass varies with the deployed length of the
tether during the cycle as
mt=π
4d2
tρtlt.(25)
2.3 Effect of gravity
If we consider the top point of the pattern during operation,
shown in Fig. 8, the aerodynamic force vector Fahas to tilt
upward to compensate for the kite’s weight Fg. This tilt is
achieved by rolling the kite by an angle 9from the radial
direction. In this model, the roll and the pitch are defined as
orientation properties of the aerodynamic force vector rela-
tive to the radial direction. Along the manoeuvre, the aerody-
namic force vector will continuously roll and pitch to coun-
teract gravity. This effectively reduces the contribution of Fa
to Ft. Since Fgalways points downward, it does not have
a component in the eφdirection. For the top point of the
pattern, the quasi-steady force balance of Ft,Fg, and Fain
spherical coordinates is
−Ft
0
0
+
−Fgcosθ
Fgsinθ
0
+
Fa,r
Fa,θ
0
=0.(26)
Due to the tilting of Farelative to the radial direction, the
geometric similarity between the velocity and force triangles,
as used to formulate Eq. (6), is lost. Hence, the kinematic ra-
tio κcannot be substituted with the glide ratio. The mechan-
ical reel-out power, in this case, becomes
Pm,o=Ftvo=
s1
2ρS 2
(C2
L+C2
D)(vwsinθcos φ−vo)4κ2+12−(Fgsin θ)2
−Fgcosθvo.(27)
As explained in Schmehl et al. (2013) and Van der Vlugt
et al. (2019), while maintaining the force balance given in
Figure 8. Side view illustrating the forces and velocities, including
weight, during the reel-out phase of the kite at the top point of its
circular manoeuvre.
Eq. (26), there should be a solution for the kinematic ratio κ
for which the decomposition of Fain Land Dcomponents
corresponds to the glide ratio. By the definition of drag force,
D=(Fa·va)/va, and using Fa=√L2+D2, we get the equa-
tion
L
D=sFava
Fa·va2
−1.(28)
This equation is a consistency constraint that must be re-
spected for the solution of the kite speed within the force
balance. The value for κis solved numerically.
During the cyclic motion of the kite through the pattern,
the apparent wind speed varies due to the aerodynamic work
and the potential and kinetic energy exchange. The apparent
wind velocity will be the highest on the bottom and the low-
est on the topmost part of the pattern (Eijkelhof and Schmehl,
2022). This variation in velocity leads to oscillations in the
mechanical power. Although it should ideally not affect the
pattern average power during the cycle, it will demand over-
sizing of the drivetrain to be able to handle the oscillation
peaks. This will lead to increased costs and reduced overall
efficiency since the drivetrain will not operate near its rated
conditions most of the time. This undesired effect is more
extreme for larger kite masses. The oscillating mechanical
power must be capped if it exceeds the generator limit. This
can be done in multiple ways, for example, by modulating
the reeling speed or changing the angle of attack, which can
both be done relatively quickly, or by increasing the pattern
elevation angle, which takes more time. The work of grav-
ity during the upward and downward parts of the pattern is
conserved, but at the same time, there are non-conservative
forces, such as the drag force, which lead to energy dissi-
pation. We choose the same representative point to evaluate
the mean pattern reel-out power as discussed in Sect. 2.1.
The power is estimated using Eq. (27) with θ=π/2−βpand
φp=sin−1(4sin γp/3π).
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2203
2.4 Retraction phase
At the end of the reel-out phase, when the kite is at the top-
most point along its trajectory, it is assumed to be pulled back
in a straight line starting from the top of the pattern, cover-
ing the reeled-out distance. This is as shown in Fig. 8 but
with the difference that the kite does not have a tangential ve-
locity; i.e. the kite’s tangential velocity component vk,τ =0,
with θ=π/2−(βp+γp) and φp=0. It only has a velocity in
the negative radial direction. This is the reel-in velocity vi, an
independent variable in the model controlled by the ground
station.
A force balance similar to the one described in Sect. 2.3 is
solved to estimate the required mechanical reel-in power:
Pm,i=Ftvi=
s1
2ρS 2
C2
L+C2
D(vwsin θcos φ+vi)2+(vwcos θcos φ)22−Fgsin θ2
−Fgcosθvi.(29)
In contrast to the reel-out phase, gravity assists the kite
in the retraction phase by reducing the required tether force
for reeling in. When the reel-in speed is increased, the time
required for reel in can be decreased, but this increases the
apparent speed, consequently increasing the tether force. To
achieve this descent, the kite needs to modulate CLto a
lower value by pitching the kite. By doing so, the kite could
be reeled in faster without necessarily increasing the tether
force, hence minimising the required reel-in power. This
trade-off should be captured when optimising the system’s
performance.
2.5 Effect of vertical wind shear
Since the kite gradually climbs from lower to higher heights
during the reel-out phase, it is exposed to vertical wind shear.
The wind resource varies with the height from location to lo-
cation based on the ground surface roughness and other local
meteorological parameters (Bechtle et al., 2019). These ver-
tical wind distributions can be modelled using meteorologi-
cal data and exhibit significant diurnal and annual variations.
Schelbergen et al. (2020) proposed a method to identify char-
acteristic shapes of the wind profile using reanalysis data.
Such characteristic wind profile shapes can be used with this
model to evaluate the energy production of systems. Since
an in-depth wind resource characterisation is not the focus
of this work, the commonly used characterisation of a verti-
cal wind profile in neutral atmospheric conditions using the
power law given by
vw(z2)=vw(z1)z2
z1α
(30)
is used to describe the relationship between wind speed vw
and height zbased on the ground surface roughness parame-
ter α(Peterson and Hennessey, 1978).
Figure 9. The discretised reel-out phase experiences different wind
speeds as an effect of the vertical wind shear.
To account for the changing inflow during the system’s
operation, the reel-out phase is discretised into several seg-
ments as shown in Fig. 9. The tractive power is evaluated for
each segment using the corresponding wind speed and result-
ing force balance. The orange points represent the numerical
evaluation points during the reel-out and reel-in phases.
2.6 Electrical cycle power optimisation
The power represented by Eq. (27) is the average mechanical
reel-out power over one pattern for a given wind speed. This
does not yet account for the losses due to the power con-
sumed in the reel-in phase and the losses in the drivetrain.
The drivetrain is the chain of components between the drum
of an AWE system and the point of connection to the electric-
ity grid. Since the power output of a ground-gen AWE sys-
tem is cyclic, a storage component needs to be used to charge
and discharge during the cycle to maintain smooth power fed
into the grid (Joshi et al., 2022). The electrical cycle average
power is computed as
Pe,avg =Pe,oto−Pe,iti
to+ti
,where (31)
Pe,o=Pm,oηDT,and (32)
Pe,i=Pm,i
ηDT
.(33)
Here, tois the time duration of the reel-out phase, Pe,iis the
power required during reel in, tiis the time duration of the
reel-in phase, and ηDT is the drivetrain efficiency.
2.6.1 Drivetrain efficiency
In a typical electrical drivetrain, the generator is connected
to the drum using a gearbox. The generator is then connected
to an electrical storage module via a power converter and to
the grid via a power converter in parallel configuration (Joshi
et al., 2022; Fechner and Schmehl, 2013). Therefore, ηDT is
a combination of the individual component efficiencies given
as
ηDT =ηgbηgen ηpcηsto ηpc ,(34)
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2204 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Figure 10. Generator efficiency as a function of the ratio of its op-
erating speed with respect to its rated speed.
where ηgb is the gearbox, ηgen is the generator, ηpc is the
power converter efficiencies, and ηsto is the electrical stor-
age. We assume a value of 95 % for all three components ex-
cept the generator. The generator efficiency at its rated speed
could be as high as 95 % and drops steeply below about 40 %
of its rated speed. This non-linear relationship is based on
the supplier data received from Ampyx Power B.V. and is
modelled as
ηgen =0.671v
vrated 3
−1.4141v
vrated 2
+0.9747v
vrated +0.7233,(35)
where vis the operating speed of the generator. Figure 10
shows this generator efficiency plot.
2.6.2 Reel-out and reel-in times
The reel-out and reel-in times heavily influence the average
electrical cycle power of the system. They are dependent on
the reel-out speed vo, reel-in speed vi, stroke length 1l, and
given maximum drum acceleration ad,max. Figure 11 shows
a velocity–time graph for a representative cycle. The reel-out
phase starts with a reeling speed of zero. The kite achieves its
set reel-out speed by accelerating with ad,max, and the speed
remains constant until the kite covers the stroke length. The
kite then decelerates back to zero to begin the reel-in phase.
Similar to the start of the reel-out phase, the reeling speed
reaches its set value by accelerating with ad,max and then re-
mains constant until the end of the reel-in phase, after which
it again decelerates to zero to begin a new cycle.
If t1is the time taken by the kite to reach the maximum
reel-out speed, to,eff is the effective time during which the
kite is in traction and producing power, t2is the time taken
by the kite to reach its maximum reel-in speed, and ti,eff is
the effective time during which the system is out of traction,
then the cycle time is expressed as
Figure 11. Velocity–time graph for a representative cycle.
Table 2. Operational parameters which are optimised for given
wind conditions.
Design variable Unit Description
1l m Stroke length
βpdeg. Pattern elevation angle
γpdeg. Pattern cone opening angle
Rp,min m Initial turning radius
vom s−1Reel-out speed
CL,o– Reel-out wing lift coefficient
vim s−1Reel-in speed
CL,i– Reel-in wing lift coefficient
tcycle =to+ti=to,eff +t1+ti,eff +t2;i.e.(36)
tcycle =1l
vo+vo
ad,max +1l
vi+vi
ad,max
.(37)
2.6.3 Optimisation problem setup
The optimisation objective is maximising the electrical cy-
cle average power Pe,avg, given by Eq. (31), for given wind
conditions. Table 2 shows a list of the operational design pa-
rameters, which are the variables of the optimisation prob-
lem. In general, the upper and lower bounds on the variables
should be set such that the optimum is not restricted by the
bounds. This will change if component limitations are con-
sidered. For example, the limits on the kite speed vwill de-
pend on the drivetrain’s maximum allowable speed, and the
operational wing lift coefficient CLwill depend on the wing
aerodynamic properties. Since this is a steady-state model
without a controller, the lower limit on the pattern radius Rp
could be set based on studies considering a controller, such as
Rossi (2023) and Eijkelhof et al. (2024). To account for the
feasibility of trajectories in this model, we propose a lower
limit on the turning radius of 5b, where bis the wing span.
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2205
Table 3. Optimisation problem constraints.
Constraint Unit Description
hmin ≤zk≤hmax m Operation height limits
Pe,avg ≤Prated W Electrical rated power
Pm,o≤Pgen,rated W Peak mechanical power
lt≤lt,max m Maximum tether length
Ft≤Ft,max N Maximum tether force
Np≤Np,min – Minimum number of patterns per cycle
Table 3 presents a list of the constraints of the optimi-
sation problem. Constraints are enforced on the minimum
ground clearance, required electrical rated power, peak me-
chanical power (limiting the size of the generator), maximum
tether length, maximum allowable tether force, and minimum
number of patterns per cycle. At least one full pattern dur-
ing a cycle is imposed to account for the fact that inertial
effects are excluded, and it can be unrealistic to have fast
transitions between reel out and reel in without completing
at least one circular trajectory. Another important constraint
that must be respected during reel out and reel in is given by
Eq. (28). Since the design space is continuous and has non-
linear constraints, sequential quadratic programming (SQP),
a gradient-based optimisation algorithm, is implemented in
MATLAB to solve the problem. The results give the optimal
operation set points for the defined system with respect to the
given wind conditions.
The optimiser attempts to find a solution for the quasi-
steady force balance for every wind speed in the given
range. The cut-in wind speed vw,cut-in is the minimum speed
at which the system produces net positive electrical cycle
power. That is, for wind speeds below vw,cut-in, power will be
consumed to keep the kite in the air (Pe,avg <0). The rated
wind speed vw,rated is the speed at which the system produces
its nameplate rated electrical power. Therefore, the cut-in and
rated wind speeds are part of the solution. On the other hand,
the cut-out wind speed vw,cut-out is a design choice and will
most probably be a consequence of the structural lifetime and
design limits of the system components. Following conven-
tional wind turbines, the cut-out wind speed is assumed to be
25 ms−1at the operational height.
3 Results and discussion
The attainable power curve for a fixed-wing ground-gen
AWE system can be estimated using the presented model.
Results from simulating a system with a rated power of
150 kW are presented in Sect. 3.1, and some effects of scaling
are discussed in Sect. 3.3, showcasing the capabilities of the
model and its application for the conceptual design phase.
Figure 12. The first untethered flight of the AP3 demonstrator
aircraft in the Netherlands in November 2023 (Paelink and Rand,
2024). Photo courtesy of Mozaero.
3.1 Simulation results of a 150 kW system
The system’s parameters are based on the prototype AP3
(Fig. 12), originally developed by Ampyx Power (Kruijff and
Ruiterkamp, 2018; Vimalakanthan et al., 2018) and contin-
ued by Mozaero since 2023 (Paelink and Rand, 2024). Ta-
ble 4 lists the parameters and the limits used to define the
specific system. It is important to note that the system pa-
rameters are not optimised, and hence the power curve does
not characterise a commercial 150 kW system. A safety fac-
tor ηt,gust is applied on Ft,max, reducing the maximum allow-
able tether force value below the actual limit of the tether. To
account for 3D wing aerodynamic effects, an aerodynamic
efficiency factor is applied on the maximum airfoil lift coef-
ficient, setting an upper limit for the wing lift coefficient CL.
This is given as
CL,max =ηClCl,max,(38)
where ηClis the efficiency factor, and Cl,max is the maximum
airfoil lift coefficient. For the induced drag calculation using
Eq. (19), a wing planform efficiency factor e(Oswald effi-
ciency factor) is used.
Figure 13 shows the chosen vertical wind shear profile rep-
resenting an onshore scenario and neutral atmospheric con-
ditions using a surface roughness coefficient αof 0.143. The
figure also shows wind profiles from Cabauw, an onshore lo-
cation, and IJmuiden, an offshore location in the Netherlands.
These two profiles were generated using the wind profile
clustering approach described in Schelbergen et al. (2020)
and were utilised in Eijkelhof and Schmehl (2022). For any
given location, several profiles exist based on the probability
of occurrence. The profiles with the highest probabilities in
the two locations are shown in the figure. The modelled pro-
file with α=0.143 is comparable to the empirical onshore
profile and is hence chosen to represent a generic onshore
location.
3.1.1 Power curve comparison with 6-DOF simulation
The 6-degree-of-freedom (6-DOF) simulation results were
generated using the simulation framework developed at
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2206 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Table 4. Model input parameter list.
Parameter Description Value
αWind shear coefficient 0.143
SWing surface area 12 m2
AR Wing aspect ratio 12
Cl,max Maximum airfoil lift coefficient 2.5
ηClAirfoil efficiency factor 0.80
Cl,Cd,min Lift coefficient at minimum drag coefficient 0.65
Cd,min Minimum drag coefficient 0.056
eWing planform efficiency factor 0.60
Ft,max Maximum allowable tether force 42 kN
ηt,gust Gust margin factor 0.90
σtTether material strength 7 ×108Nm−2
ρtTether material density 980 kgm−3
Cd,tCross-sectional tether drag coefficient 1.2
lt,max Maximum tether length 1000 m
hmin Minimum ground clearance 100 m
hmax Maximum operating height 1000 m
Prated Rated electrical power 150 kW
Pgen,rated Generated mechanical power limit 375 kW
vd,max Maximum tangential drum speed 20ms−1
ad,max Maximum tangential drum acceleration 5 ms−2
Np,min Minimum number of patterns per cycle 1
Figure 13. The chosen vertical wind shear profile with a sur-
face roughness coefficient of 0.143 compared against profiles from
Cabauw, an onshore location, and IJmuiden, an offshore location in
the Netherlands (Eijkelhof and Schmehl, 2022).
Ampyx Power B.V., as described in Ruiterkamp and Sieber-
ling (2013), Williams et al. (2019), and Licitra et al. (2019).
This model accounts for 3 degrees of freedom related to the
kite’s attitude and 3 related to its position. Unlike steady-
state modelling, it solves the equations of motion, including
the acceleration terms. The wing’s aerodynamic derivatives
are obtained through computational analysis, as described in
Vimalakanthan et al. (2018). The tether is modelled as a flex-
ible component with discretised segments. Flight and winch
feedback controllers are implemented to simulate the teth-
ered kite system during takeoff, reel-out, reel-in, transition,
and landing phases.
Figure 14 shows the computed power curve compared to
the ideal crosswind power extraction theory by Loyd (1980)
and the results of the 6-DOF simulation. The horizontal axis
describes the wind speed at 100 m height. Loyd’s ideal cross-
wind power is computed using
PLoyd =4
27
C3
L,max
C2
D
1
2ρSv3
w,(39)
where CL,max is the upper limit as defined by Eq. (38), CDis
computed as described in Eq. (19), and vwis the wind speed
at 100 m height. This ideal crosswind theory overpredicts the
power because it neglects the losses due to gravity, elevation
and azimuth angles, tether drag, cyclic operation, hardware
limits, and drivetrain efficiency.
The kite mass mkusing Eq. (20) equals 437 kg, which is
close to the indication received from the company about the
AP3 prototype, as seen from Fig. 6. The shape of the esti-
mated power curve using the developed model resembles the
curve generated by the 6-DOF simulations, but it is more op-
timistic. This is mainly because the developed model ignores
the losses due to control and inertial effects. It also does not
account for realistic takeoff or flight sustenance conditions
at low wind speeds, which is most likely the reason for the
earlier cut-in. The rated power is reached at a wind speed of
15 ms−1. As a design choice, the cut-out wind speed is cho-
sen to be 25 ms−1at the operational height. Due to the verti-
cal wind shear, this translates to a wind speed of 21 ms−1at
100 m.
The mean mechanical and electrical powers, reel-out and
reel-in powers, and electrical cycle average power are shown
in Figs. 15 and 16. The reel-out power has three regimes, as
described in Luchsinger (2013) and Kruijff and Ruiterkamp
(2018). The cubic regime I is above the cut-in speed (here,
6 ms−1), in which the reel-out power increases cubically un-
til 10 ms−1when the maximum allowable tether force (here,
34 kN, considering the gust margin factor) is reached. The
linear regime II starts when the maximum allowable tether
force is reached, in which the reel-out power increases lin-
early until the chosen rated electrical power (here, 150 kW)
is reached. The flat regime III starts when the rated power is
reached and continues until the cut-out speed. In this regime,
the mechanical reel-out power is capped to maintain the rated
electrical power. The power is capped by varying the opera-
tional parameters. These changes in operational parameters
also affect the reel-in power seen in Fig. 16.
3.1.2 Forces and operational parameters for the entire
wind speed range
Figures 17 and 18 show the resultant aerodynamic force, the
tether force, and the gravitational force during the reel-out
and reel-in phases, respectively. As specified in Table 4, a
gust margin factor of 0.9 is applied to the maximum allow-
able tether force. Once this upper limit is reached, the aero-
dynamic force has to be capped to avoid tether overload. In
our specific example, this limit is reached at 10 ms−1. The
aerodynamic force can be capped by reducing the wing’s lift
coefficient, modulating the reeling speed, or increasing the
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2207
Figure 14. Power curve comparison of the quasi-steady model
(QSM) with Loyd and 6-DOF simulation results.
Figure 15. Mean mechanical power Pm,o, electrical reel-out power
Pe,o, and electrical cycle average power Pe,avg as functions of the
wind speed.
elevation angle. The choice of a specific capping strategy de-
pends on multiple trade-offs. The optimisation objective is
the average electrical cycle power, including the reel-out and
reel-in phases. During reel in, the kite is flown such that the
maximum contribution of the generated aerodynamic force
is used to counter the kite’s weight, reducing the required
pulling force and, consequently, the reel-in power. This is
seen in Fig. 18. Intuitively, if the aerodynamic force com-
pletely balanced the weight, it would lead to an Ft,i=0 and
hence no requirement of reel-in energy. This would be the
case for a freely gliding kite. However, this could also in-
crease the reel-in time, which could lead to lower net cycle
power. Hence, the optimiser finds a solution to the reel-in
speed such that it creates a non-zero tether force but still ul-
timately reduces the net cycle loss.
Figures 19 and 20 show the lift and drag coefficients dur-
ing the reel-out and reel-in phases. As stated earlier, CLis a
variable in our model, and CDis calculated using Eq. (17).
The aerodynamic force during reel in only has to counter the
kite’s weight, which is achieved by decreasing the lift coef-
ficient during reel in. Because of the lift-induced drag con-
tribution, the kite drag coefficient is a function of the lift
coefficient and follows its trend. The total drag coefficient
is the summation of the kite drag coefficient and the tether
drag coefficient as described in Eq. (17). Figure 20 shows
Figure 16. Mean mechanical reel-in power Pm,iand electrical
power Pe,ias functions of the wind speed.
Figure 17. Mean resultant aerodynamic force Fa,o, tether force
Ft,o, and weight of the kite and the tether lumped together Fgdur-
ing the reel-out phase as functions of the wind speed.
that the tether drag coefficient contributes significantly to the
system’s total drag coefficient. It is almost equal to the kite
drag coefficient during the reel-out phase and is higher dur-
ing the reel-in phase.
The kite’s radial and tangential velocity components are
commonly non-dimensionalised with the wind speed, lead-
ing to the reeling factor f=vk,r/vwand the tangential ve-
locity factor λ=vk,τ /vw(Schmehl et al., 2013). Figure 21
shows the tether reeling factors during reel out and reel in
and the tangential velocity factor during reel out. The reel-
out speed peaks when the rated power is reached, i.e. at
15 ms−1of wind speed, and then gradually reduces, assist-
ing in power capping. The reel-in speed is kept at the drum’s
tangential speed limit of 20 ms−1. This is seen from the grad-
ual decrease in the reel-in speed factor. After the maximum
tether force is reached at 10 ms−1, the kite’s tangential veloc-
ity is gradually reduced, decreasing the aerodynamic force to
maintain the tether force at its maximum value.
Figure 22 shows the reel-out time, reel-in time, average
time the kite takes to perform one circular pattern during reel
out, and the number of patterns per cycle. The number of
patterns is calculated using the reel-out time, pattern radius,
and tangential kite speed as
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2208 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Figure 18. Mean resultant aerodynamic force Fa,i, tether force Ft,i,
and weight of the kite and the tether lumped together Fgduring the
reel-in phase as functions of the wind speed.
Figure 19. Mean kite lift coefficients CL,oand CL,iduring the reel
out and reel in, respectively, as functions of the wind speed.
Np=to
2πRp/(vk,τ ).(40)
In a more realistic operation, the number of patterns should
be a whole number such that the reel-in phase always starts
from the top point of the pattern. However, since we are not
resolving the full trajectory in this model, the number of pat-
terns is allowed to be a fractional result. Moreover, since in-
ertial effects are ignored in this model, the full cycle time du-
rations are optimistic. Realistic cycle times will increase due
to the transition phase between reel out and reel in, which is
unaccounted for in this model.
Figure 23 shows the average pattern height, pattern radius,
stroke length, maximum tether length, and minimum tether
length. The average pattern height is the height of the centre
point of the pattern at half of the stroke length. The minimum
tether length and, consequently, the pattern radius and height
are primarily driven by the ground clearance constraint, pat-
tern elevation angle, and cone opening angle. In reality, they
will also be influenced by the effect of the centrifugal force,
which is ignored in the quasi-steady approach. As the eleva-
tion angle increases, the required minimum tether length is
reduced. The maximum tether length is driven by the opti-
mised stroke length.
Figure 20. Mean effective system drag coefficients CD,oand CD,i;
mean kite drag coefficients CD,k,oand CD,k,iduring the reel out
and reel in, respectively; and mean tether drag coefficient CD,tas
functions of the wind speed.
Figure 21. Mean kite tangential speed factor λ, reel-out factor fo,
and reel-in factor fias functions of the wind speed.
Figure 24 shows the roll, pattern elevation, and opening
cone angles. The roll angle is the deviation of the resultant
aerodynamic force vector with respect to the radial direction.
The pattern elevation angle increases with the wind speed.
This quasi-steady flight state results from the trade-off be-
tween an increase in incoming wind speed, an increase in
cosine losses due to gravity, and a decrease in reel-in power
with a higher elevation angle. The optimiser trades all these
factors to maximise the average cycle power at each wind
speed.
3.1.3 Forces and operational parameters over one cycle
The maximum convertible power is limited by the genera-
tor rated power, which in our specific example is 375 kW,
as given in Table 4. To enforce this hardware limit in the
third wind speed regime, the operational parameters have to
be modulated. Figure 25 shows the mechanical, electrical,
and electrical cycle power over a single pumping cycle at the
rated wind speed of 15 ms−1. The delivered rated power of
150 kW is the electrical cycle average power. The difference
in the instantaneous mechanical and electrical power is due
to the drivetrain efficiency.
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2209
Figure 22. Reel-out time to, reel-in time ti, average pattern time
tpatt,avg, and number of patterns per cycle Npas functions of the
wind speed.
Figure 23. Average pattern height hp,avg, average pattern radius
Rp,avg, stroke length 1l, maximum tether length lt,max , and mini-
mum tether length lt,min as functions of the wind speed.
The power profile during the cycle has a slight downward
trend during the reel out and an upward trend during the reel
in. This is explained using Fig. 26. For the quasi-steady-state
evaluation, the reel-out phase is discretised into five segments
arranged in sequence on the horizontal axes of the diagrams.
The cycle begins with a tether length of around 400 m, and
the reel-out phase ends with a tether length of around 700 m.
The average pattern height and the pattern radius increase
during the reel-out phase. Due to the gain in height, the kite
experiences a higher wind speed vwas it climbs up. Due to
the increasing tether length, the overall drag of the system
increases, and hence the glide ratio decreases. Hence, to re-
spect the relation given by Eq. (28), the kite speed has to
drop, reflected in the reeling and tangential velocity factors.
Since the tether force is already maximised, the overall power
decreases due to a lower reel-out speed.
3.2 Effect of gravity
Figure 27 shows the power curve comparison between two
simulations, one with gravity and one without, including the
effect of gravity (i.e. weight). Gravity has a negative im-
pact on operation at low wind speeds because this affects the
Figure 24. Mean roll angle 9p, average pattern elevation angle βp,
and opening cone angle γpas functions of the wind speed.
Figure 25. Instantaneous powers Peand Pmtogether with net pow-
ers Pe,avg and Pm,avg over one pumping cycle at a rated wind speed
of 15 ms−1.
attainable reel-out power substantially. Hence, the simula-
tion without gravity yields better performance at lower wind
speeds than the one including gravity. But for higher wind
speeds, this effect is superseded by its impact on the reel-in
phase. As explained in Sect. 2.4, the weight assists in the re-
traction phase, positively impacting the net cycle power out-
put. The kite is pitched in such a way that the resultant aero-
dynamic force vector balances the gravitational force vector,
hence reducing the tether force magnitude in the quasi-steady
force balance. The kite can be retracted faster without con-
suming a lot of energy.
Figures 28 and 29 detail the effect of gravity on a pump-
ing cycle for a lower wind speed of 6 ms−1and the rated
wind speed 15 ms−1. The reel-out power without the effect
of gravity is substantially higher at 6 ms−1than at 15 ms−1.
The net difference between the energy generated during reel
out and consumed during reel in leads to higher net average
power for the case without gravity at 6 ms−1than at 15 ms−1.
This shows that excluding gravity in the analysis does not
necessarily lead to optimistic results. In any case, including
gravity should always be the more realistic simulation for the
pumping cycle.
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2210 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Figure 26. Evolution of parameters over the discretised reel-out phase in five segments at a rated wind speed of 15 ms−1.
Figure 27. Power curve comparison with and without the effect of
gravity.
Figure 28. Cycle power comparison with and without the effect of
gravity for a low wind speed of 6 ms−1.
Figure 29. Cycle power comparison with and without the effect of
gravity for a higher wind speed of 15 ms−1.
3.3 Effect of scaling
One of the primary purposes of this model is to capture the
effects of scaling on the performance of fixed-wing AWE
systems. Due to the interdisciplinary nature of AWE sys-
tems, multiple trade-offs must be considered. The kite and
the tether are the primary aspects affecting the system’s per-
formance. The performance metric used is the power harvest-
ing factor ζdefined as
ζ=P
PwS,(41)
where Pis the extracted mechanical power, and PwSis the
available power in the wind. This metric is based only on the
reel-out power and does not consider the reel-in power. The
force a tether can withstand for a given material strength is
proportional to its diameter as shown in Eq. (12). The kite
should also be able to withstand this tether force; hence,
with increasing tether force, the structural mass of the kite
increases to support the increasing wing loading. Though in-
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2211
Figure 30. Effect of tether diameter on the performance of an AWE
system with a fixed kite wing area of 100 m2at a constant wind
speed of 12 ms−1.
Figure 31. Effect of wing area on the performance of a system with
a fixed tether diameter of 2.7 cm at a wind speed of 12 ms−1.
creasing the tether force will enable the extraction of more
power, the consequent increase in kite mass will decrease the
performance. Moreover, the contribution of tether drag will
increase with increasing diameter, consequently penalising
the extractable power.
Figure 30 shows the effect of tether diameter on the per-
formance of an AWE system. For a kite with the same wing
area, the mass increases with increasing tether force as given
in Eq. (20). This increase in kite mass negatively impacts
the attainable reel-out power. For this simulation, the perfor-
mance is maximum when using a tether with a diameter of
around 3.8 cm. Similarly, Fig. 31 shows the effect of scaling
the wing area on the performance of an AWE system. For the
chosen tether, the kite wing area which maximises power is
50 m2.
Figure 32 shows the effect of the tether diameter on the
performance of a system with a fixed kite wing area of
100 m2for the complete operational wind speed range. As
seen from Fig. 30, the kite mass of a system with smaller
tether tension is lower. Lighter kites will experience lower
gravitational loss and will hence perform better at lower wind
speeds. At higher wind speeds, the maximum tether force
limits the extractable power. Therefore, for a given wing area,
Figure 32. Effect of scaling the tether (diameter) on the perfor-
mance of a system with a fixed kite wing area of 100 m2for the
complete operational wind speed range.
Figure 33. Effect of scaling the wing area on the performance of
a system with a fixed wing loading of 3 kNm−2for the complete
operational wind speed range.
systems with thinner tethers, i.e. lower Ft,max, perform bet-
ter at lower wind speeds, and systems with larger tethers, i.e.
higher Ft,max, perform better at higher wind speeds.
Figure 33 shows the wing area’s effect on a system’s per-
formance with a fixed wing loading for the complete opera-
tional wind speed range. Fixed wing loading is used instead
of a fixed tether force since simulation results of a high tether
force coupled to a small kite and vice versa do not converge
for the entire operational wind speed range. A larger tether
force demands a stronger kite, resulting in a heavier one. This
configuration cannot produce positive net cycle power at low
wind speeds. Therefore, the choice of tether force for a given
wing area must fall in a certain range to have converged re-
sults for the entire operational wind speed range. The system
performs better with increasing wing area, but these gains
diminish since the penalising effect of the gravitational force
scales faster than the performance gain.
3.4 Discussion
The results show that the proposed model captures all rele-
vant dependencies between the system components, allow-
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2212 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
ing for the evaluation of different trade-offs at play. Since the
model is based on a quasi-steady flight motion, the results are
expected to be optimistic predictions of a real system’s per-
formance. Since the model relies on a limited set of input pa-
rameters defining an AWE system, it is suitable for coupling
with similar-fidelity cost models, as proposed in Joshi and
Trevisi (2024). The present modelling approach does not ac-
count for the various effects of inertia responsible for losses
in the different operational phases.
For example, centrifugal acceleration is important during
sharp turning manoeuvres, and its effect needs to be balanced
by an aerodynamic side force component. A fixed-wing kite
generates this side force component by rolling toward the
turning axis, reducing the aerodynamic force available for
conversion into electricity, thus representing a loss. On the
other hand, the path-aligned acceleration introduces a history
term in the kite’s equations of motion, affecting the tempo-
ral progress along the flight path. For relatively lightweight
soft-wing kites, these two inertial effects have only a minor
effect on the power output and are thus generally neglected.
However, with an increasing mass-to-wing surface ratio the
contribution of inertia becomes more important. An accom-
panying study by Van Deursen (2024) showed that for heav-
ier kites, the different acceleration terms can lead to quite
complex superposition effects. Because it is not possible to
account for all acceleration terms in a quasi-steady modelling
framework, we decided to neglect all inertial effects and in-
stead use existing 6-DOF dynamic simulation results for val-
idation.
Moreover, the transition times between the reel-out and
reel-in phases are not considered in the present modelling
framework. Because of the alternating loading and unload-
ing of the airborne subsystem, the tether alternates between
straightening and sagging. Straightening and sagging take
time during which no work is performed. The kite moves ra-
dially away from or toward the ground, while the winch does
not reel. This hysteresis effect represents a loss for the pump-
ing cycle operation. Eijkelhof and Schmehl (2022) found that
the kite needs to be slowed down at the start and end of the
transition phase to avoid tether rupture due to the change in
the magnitude of forces. This effect is also known as the
“whiplash effect”. The present model cannot estimate the
reel-out power oscillations due to the acceleration and de-
celeration of the kite when it follows the prescribed flight
path.
The model and simulation results in this paper are not val-
idated against measurement data. SkySails Power GmbH, a
German company, released a certified power curve of their
PN-14 system based upon the IEC 61400-12-1 standard used
for conventional wind turbines (Bartsch et al., 2024). They
reported good agreement between their measurements and
their simulation results. Figure 34 shows an overlay of their
measurements against the 150 kW simulation results from
Sect. 3.1. This is not a performance comparison since the
technologies and their system characteristics differ signifi-
Figure 34. Overlay of the power curve of the 150 kW system over
SkySails Powers’ validated power curve of SKS PN-14 (Bartsch
et al., 2024).
cantly, but the systems do have similar power ratings. The
IEC standard requires multiple changes considering the dif-
ferences between conventional wind turbines and AWE sys-
tems, such as the definition of the reference height, the wind
range, the method of averaging over time, the incorpora-
tion of the number of cycles in averaging, etc. The reference
height proposed by SkySails Power is 200 m, which is closer
to the average operational height of their system. Alignment
with the reference height for wind speed measurements while
communicating the power curve is essential for fair com-
parisons. As the AWE sector advances rapidly, there is an
increasing need for dedicated IEC standards to validate the
power curve of AWE systems.
4 Conclusions
The quasi-steady model presented in this paper enables fast
power curve computations based on a limited set of input
parameters. It is useful for understanding the fundamental
physical behaviour of fixed-wing ground-generation AWE
systems and is suitable for sensitivity analysis and estimat-
ing AWE systems’ theoretical potential. The model can eas-
ily be coupled to systems engineering tools, cost models, and
larger-scale energy system models. It may thus help to create
technology development roadmaps, investigate scaling po-
tential, and define research targets to validate assumptions.
Kite mass is a key parameter influencing the performance
of systems, primarily at lower wind speeds. A higher mass
leads to a larger component of the generated aerodynamic
force needed to compensate for the gravitational force, re-
ducing the usable mechanical power. On the other hand,
gravity positively impacts performance at higher wind speeds
by reducing the required energy during reel in. The tether di-
ameter and the kite’s structural mass are coupled to design an
optimised system. The maximum force-bearing capacity of
the tether is directly proportional to the diameter of the tether,
and a higher tether force requires a structurally stronger kite.
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R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs 2213
Hence, choosing a tether with a larger force-bearing capacity
also increases the kite mass, negatively impacting the perfor-
mance at low wind speeds but enabling higher power extrac-
tion at higher wind speeds. This trade-off becomes critical
for choosing the optimal tether–kite combination based on
site-specific requirements. Upscaling results in a diminish-
ing gain in performance with an increase in kite wing area.
Integrating the prescribed model in a system design optimi-
sation framework provides a computational design tool that
accounts for the multiple trade-offs for site-specific design.
The system design parameters, such as the kite wing area,
generator rating, tether diameter, etc., can be optimised to
maximise the annual energy production for a specific wind
resource. Moreover, annual energy prediction alone will not
give the right indication for system design since this metric
lacks the influence of costs. To include this important aspect,
the presented model can be coupled to a cost model to find
the system design that minimises the levelised cost of energy.
Since the presented model is based on the assumption of
quasi-steady flight motion, it does not account for inertial
effects. These will be significant for larger AWE systems;
hence, the model is likely too optimistic in estimating their
performance. This approach occupies a middle ground be-
tween ideal power extraction and fully resolved dynamic
simulations. The outcomes of this analysis define the upper
limits that practical systems might approach. Consequently,
these models are valuable for determining whether and under
what conditions AWE could benefit the entire energy system.
Appendix A: Nomenclature
Greek letters
γCone opening angle
βElevation angle
θPolar angle
φAzimuth angle
τTangential
χCourse angle
σMaterial strength
ρMaterial density
9Roll angle
αWind shear coefficient
ηEfficiency
λTangential velocity factor
ζPower harvesting factor
Latin letters
SWing area
AR Aspect ratio
CLLift coefficient
CDDrag coefficient
FForce
RRadius
vVelocity
LLift
DDrag
dDiameter
Latin letters
lLength
z Z-axis co-ordinate
hHeight
mMass
tTime
PPower
aAcceleration
NNumber
fReel-out factor
eWing planform efficiency factor
Subscripts
p Pattern
t Tether
max Maximum
min Minimum
w Wind
k Kite
r Radial
a Apparent
g Gravity
o Reel out
i Reel in
eff Effective
ref Reference
e Electrical
m Mechanical
avg Average
DT Drivetrain
gen Generator
sto Storage
pc Power converters
gb Gearbox
Code availability. The model is implemented in MATLAB and
is available on GitHub under the name AWE-Power at https://
github.com/awegroup/AWE-Power (Joshi, 2024a). It contains a pre-
defined input file, which can be used to run the model and reproduce
the results presented in the paper. An instance of the repository is
deposited and published on Zenodo and is available open source at
https://doi.org/10.5281/zenodo.13842298 (Joshi, 2024b).
Data availability. The company-specific data sets are not publicly
available since they are regarded as confidential information.
Author contributions. Conceptualisation: RJ, MK, and RS;
methodology: RJ, MK, and RS; software: RJ; validation: RJ and
MK; writing (original draft preparation): RJ and MK; writing (re-
view and editing): RS; supervision: MK and RS; funding acquisi-
tion: MK and RS.
Competing interests. At least one of the (co-)authors is a mem-
ber of the editorial board of Wind Energy Science. The peer-review
process was guided by an independent editor, and the authors also
have no other competing interests to declare.
https://doi.org/10.5194/wes-9-2195-2024 Wind Energ. Sci., 9, 2195–2215, 2024
2214 R. Joshi et al.: Power curve modelling and scaling of fixed-wing ground-generation AWESs
Disclaimer. Publisher’s note: Copernicus Publications remains
neutral with regard to jurisdictional claims made in the text, pub-
lished maps, institutional affiliations, or any other geographical rep-
resentation in this paper. While Copernicus Publications makes ev-
ery effort to include appropriate place names, the final responsibility
lies with the authors.
Acknowledgements. The authors thank Ampyx Power B.V. for
providing reference data and inputs. The development of the asso-
ciated code repository was supported by the Digital Competence
Centre, Delft University of Technology.
Financial support. This research has been supported by the Ned-
erlandse Organisatie voor Wetenschappelijk Onderzoek (grant no.
17628) and Interreg North-West Europe (MegaAWE project).
Review statement. This paper was edited by Jonathan Whale and
reviewed by two anonymous referees.
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