Quasi-Steady Model of a Pumping Kite Power System
Rolf van der Vlugt, Anna Bley, Michael Noom, Roland Schmehl∗
Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629HS Delft, Netherlands

The traction force of a kite can be used to drive a cyclic motion for extracting wind energy from the atmosphere. This paper presents
a novel quasi-steady modelling framework for predicting the power generated over a full pumping cycle. The cycle is divided into
traction, retraction and transition phases, each described by an individual set of analytic equations. The effect of gravity on the
airborne system components is included in the framework. A trade-off is made between modelling accuracy and computation speed
such that the model is specifically useful for system optimisation and scaling in economic feasibility studies. Simulation results are
compared to experimental measurements of a 20 kW kite power system operated up to a tether length of 720 m. Simulation and
experiment agree reasonably well, both for moderate and for strong wind conditions, indicating that the effect of gravity has to be
taken into account for a predictive performance simulation.
Keywords: Airborne Wind Energy, Kite Power, Pumping Cycle, Traction Power Generation

1. Introduction
The pumping kite concept provides a simple yet effective
solution for wind energy conversion at a potentially low cost
[1]. Important aspects of the technology are the performance
characteristics of implemented concepts and how these depend
on the operational and environmental parameters. Various modelling frameworks have been proposed to predict the traction
force and power generated by a tethered wing, both for the
production of electricity [2, 3, 4, 5, 6] and for the propulsion
of ships [7, 8, 9, 10, 11]. The analysis presented in [7, 8]
has been validated experimentally, yet not assessed for its potential to predict the power generated over a full cycle of a
pumping system. Dynamic models have been proposed by
[12, 13, 14, 15, 16, 17] to address challenges in the field of
control. Recent studies have used measurement data from fullscale demonstrator systems to analyse the turning dynamics of
kites and to assess flight control algorithms [18, 19].
The current challenge is to formulate a model that does not
require advanced control algorithms, while accurately predicting the power generated over a pumping cycle. For this purpose it is important to critically revise commonly used simplifying assumptions, for example, regarding the wind velocity
gradient, the tether shape, the mass of tether and kite and the
aerodynamic properties of the wing. The model is intended for
optimisation of pumping cycle kite power systems and for predicting the achievable cost of energy. Section 2 first describes
the analytical framework assuming a massless system, which
is then extended to account for the effect of gravity on all airborne system components. An experimental setup, consisting
of a fully operational pumping kite power system is presented
author. Tel.: +31 15 278 5318.
Email address: r.schmehl@tudelft.nl (Roland Schmehl)

∗ Corresponding

Preprint submitted to Renewable Energy

in Sect. 3. To validate the described model, measured and computed results are compared in Sect. 4. The preliminary results
of this study had been presented at the Airborne Wind Energy
Conference 2015 in Delft [20].
2. Computational approach
For the theoretical analysis the pumping cycle is divided into
the three characteristic phases illustrated in Fig. 1: the retraction phase, from t0 until tA , the transition phase, from tA until
tB , and the traction phase, from tB until tC , closing the cycle.
A detailed presentation of the forces governing the flight op250
retraction, i
200

Zw [m]

arXiv:1705.04133v1 [] 10 May 2017

Abstract

tA

150

transition, x

100

vw

t0
tC
traction, o
tB

50

0

β
0

50

βo
100

150

200

250

300

350

400

Xw [m]

Figure 1: Idealised flight trajectory of a traction kite over a pumping cycle. The
trajectory segment in the traction phase does not resolve the crosswind flight
manoeuvres. Adapted from [21].

eration of a kite including the gravitational and inertial effects
is provided in [22, 23]. In the following we discuss several
assumptions that reduce the complexity of the computational
approach to achieve a substantial speed-up of the simulations.
12th May 2017

use a spherical coordinate system (r, θ, φ). As depicted in Fig. 2,
the position is described by the radial distance r, the polar angle
θ and the azimuth angle φ. The direction of flight in the local
tangential plane τ is described by the course angle χ.

Firstly, the study is limited to kites with relatively large
surface-to-mass ratio. For such kites the timescale of dynamic
processes is generally very short compared to the timescales
of typical flight manoeuvres or complete pumping cycles. As
consequence the flight operation is dominated by the balance
of aerodynamic, tether and gravitational forces and can be approximated as a transition through quasi-steady flight states.
The analysis is further limited to typical tether lengths during
pumping operation which are much larger than the geometrical
dimensions of the kite. At very short tether length, as occurring
during launching and landing, inertial forces such as centrifugal
forces, can contribute substantially.
Secondly, the tether is assumed to be inelastic. It is represented by a straight line although the effect of sagging due to
distributed gravitational loading is taken into account. Thirdly,
the aerodynamic properties of the kite are assumed to be constant throughout each phase. Lastly, the atmospheric properties
are assumed to be constant over time but varying with altitude.
This is taken into account by assuming altitude profiles for both
the wind velocity and the air density.

Zw
er
vk,r
va

θ

K
χ

eφ
vk

r
eθ

vk,τ
O

φ

Yw

Xw
Figure 2: Decomposition of kite velocity vk into radial component vk,r and
tangential component vk,τ , definition of apparent wind velocity va = vw − vk .
Course angle χ is measured in the tangential plane τ, spherical coordinates
(r, θ, φ) defined in the wind reference frame Xw , Yw , Zw , where Xw represents
the wind direction [22].

Conventional tower-based wind turbines have a constant hub
height and operate within a limited atmospheric layer close to
the ground. Pumping kite power systems on the other hand can
harvest energy from a much larger and variable altitude range.
Because the wind velocity vw increases substantially between
the minimum and maximum altitude of the kite it is important
to include the wind velocity profile in the simulation. In the
atmospheric boundary layer up to 500 m altitude the functional
dependency can be estimated by the logarithmic wind law [24]
ln(z/z0 )
,
ln(zre f /z0 )

τ
vw

2.1. Atmospheric Wind Model

vw = vw,re f

−vk

The apparent wind velocity describes the flow velocity relative to the kite
va = vw − vk .
(3)
This vector can be described in spherical coordinates as follows


 


 sin θ cos φ 
1
 0 


 


va = cos θ cos φ vw − 0 vk,r − cos χ vk,τ ,
(4)


 


− sin φ
0
sin χ

(1)

where vk,r and vk,τ represent the radial and tangential contributions to the kite velocity, respectively.
The straight tether implies that the radial kite velocity is
identical with the reeling velocity

where vw,re f is a known reference wind speed at a reference
altitude zre f and z0 is the aerodynamic roughness length. The
logarithmic profile suits best to model a neutral boundary layer,
which typically develops in overcast or windy conditions.
The decrease of air density ρ with increasing altitude can be
approximated by the barometric altitude formula for constant
temperature [24]
!
z
ρ = ρ0 exp −
,
(2)
Hρ

vk,r = vt .

(5)

Introducing the reeling factor
f =

vk,r
vw

(6)

and the tangential velocity factor

where ρ0 = 1.225 kg/m3 is the standard atmospheric density at
sea level at the standard temperature of T 0 = 15◦ C and Hρ =
8.55 km is the scale height for density.

λ=

vk,τ
,
vw

Eq. (4) can be formulated as


 sin θ cos φ − f 


va = cos θ cos φ − λ cos χ vw .


− sin φ − λ sin χ

2.2. Basic Modelling Framework
Starting point for the analysis is the wind reference frame
which has its origin O coinciding with the tether exit point from
the ground station and has its Zw -axis pointing vertically upwards and its Xw -axis aligned with the wind direction. The kite
is represented by a geometrical point. To describe its position
K and velocity vk we follow the approaches in [3, 22, 23] and

(7)

(8)

The meaning of the velocity variables in this expression can be
summarised as follows. The reeling factor f is controlled by
2

the ground station, the course angle χ is controlled by the steering system and the tangential velocity factor λ is a dependent
variable.
The integral aerodynamic force acting on the airborne system
components can be decomposed into lift and a drag vectors
Fa = L + D.

Fa

The lift and drag forces contributed solely by the wing are calculated as
1
L = ρ C L v2a S ,
(10)
2
and
1
(11)
Dk = ρ C D,k v2a S ,
2
where C L and C D,k are the aerodynamic lift and drag coefficients, respectively, and S the projected surface area of the
wing.
The aerodynamic drag of the tether is taken into account by
adding one fourth of the tether drag area to the kite drag area
as proposed in [3] and numerically validated in [25]. The total
aerodynamic drag D of the airborne system is then estimated as
D = Dk + Dt ,

r
β
φ

Ft

Yw

Figure 3: Geometrical similarity of the force and velocity diagrams. va and Fa
are decomposed in the plane spanned by the two vectors. D is aligned with va
by definition, whereas va,r is aligned with Fa when assuming a straight tether
and a negligible effect of mass. Adapted from [22].

On the other hand, inserting the radial and tangential velocity
components of Eq. (8) into Eq. (15) and solving for the tangential velocity factor λ results in
s
 L 2
(b − f )2 ,
(17)
λ = a + a2 + b2 − 1 +
D

(12)

1
ρ dt r C D,c v2a ,
(13)
8
with dt being the tether diameter, r the tether length, C D,c the
drag coefficient of a cylinder in cross flow and va the apparent
wind velocity at the kite. With the tether being subjected to a
relative velocity va,t between 10 and 30 m/s and a kinematic viscosity of ν = 1.47 × 10−5 , the Reynolds number Re = va,t dt /ν
is estimated to be between 2.7 × 103 and 8.2 × 103 . In this range
C D,c has a constant value of 1.1 [26]. As consequence a total
aerodynamic drag coefficient of the airborne system components can be defined as

with trigonometric coefficients
a = cos θ cos φ cos χ − sin φ sin χ,

(18)

b = sin θ cos φ.

(19)

The quasi-steady motion of a massless kite is governed by the
equilibrium of the tether force and the resultant aerodynamic
force
Ft + Fa = 0.
(20)
Inserting Eqs. (10) and (11) into Eq. (20) results in

(14)

Ft =

2.3. Analytic Model for Negligible Effect of Mass
For the massless case, the radial and tangential components
of the apparent wind velocity and the lift and drag components
of the aerodynamic force are related as follows
va,τ
L
= .
va,r
D

O

Xw

Dt =

1 dt r
C D,c .
4 S

K
D

va,τ

where

C D = C D,k +

Zw

θ

va,r

va

(9)

L

1
ρCR v2a S ,
2

with the resultant aerodynamic force coefficient
q
2
+ C L2 .
CR = C D

(15)

(21)

(22)

Using Eq. (16) to substitute the apparent wind velocity in
Eq. (21) gives the following equation for the normalised tether
force [3, Eq. (48)]
"
 L 2 #
Ft
= CR 1 +
(sin θ cos φ − f )2 ,
(23)
qS
D

The ratio of the relative velocity components is denoted as kinematic ratio and represented by the symbol κ. Equation (15)
can be derived from the geometrical similarity of the force and
velocity diagrams illustrated in Figs. 3 and 4. Starting from
the decomposition va = va,r + va,τ and using the radial component of Eq. (8) in conjunction with Eq. (15) to eliminate the
tangential component results in the following expression for the
nondimensional apparent wind velocity
s
 L 2
va
= (sin θ cos φ − f ) 1 +
.
(16)
vw
D

with the dynamic wind pressure at the altitude of the kite calculated as
1
q = ρv2w ,
(24)
2
with the air density and wind velocity described by Eqs. (2) and
(1), respectively.
3

The generated traction power is determined as the product of
tether force and reeling velocity
P = F t v t = F t f vw .

2.5. Effect of Gravity on the Tether Force
Equations (1) to (27) provide an analytic modelling framework for the operation of a kite in pumping cycles for the ideal
case of negligible gravity. However, a real system is subject
to gravitational and inertial forces which affect the flight behaviour and consequently also the traction power.
In the present modelling framework we assume that the tether
is long compared to the geometrical dimensions of the kite. Accordingly, the kite is represented by a point mass m and its gravitational force mg directly contributes to the quasi-steady force
equilibrium at point K. Because of the long tether the angular
velocities θ̇ and φ̇ are relatively small and the effect of inertial forces can be neglected. The tether, on the other hand, is
suspended between the ground station and the kite, its mass mt
is continuously distributed over its length and the distributed
loading by gravity and aerodynamic drag leads to sagging.

(25)

Expressing the tether force by Eq. (23) results in
ζ=

"
 L 2 #
P
f (sin θ cos φ − f )2 ,
= CR 1 +
Pw S
D

(26)

where Pw denotes the wind power density at the altitude of the
kite
1
Pw = ρv3w .
(27)
2
Equation (26) defines the instantaneous power harvesting factor
ζ as the normalised traction power per wing surface area.
2.4. Relative Flow Conditions at the Kite
The aerodynamic coefficients used in Eqs. (10) and (11) depend on the relative flow conditions that the kite experiences
along its flight path. For rigid and flexible membrane wings the
key influencing parameter is the angle of attack α, defined as
the angle between the chord line of the wing and the apparent
wind velocity vector va . The sketch in Fig. 4 illustrates this,
implying that the heading of the wing is in plane with the radial and tangential velocity components. This is generally the
case if the wing is not asymmetrically deformed due to steering
actuation and sideslip velocity components can be neglected.
L
r
er

va,r

Fa

D
va,τ

Figure 5: Strong sagging of the tether at low wind speed and static kite position
on 23 August 2012. The kite has a surface area of 25 m2 .

α va

Figure 4: Relative flow components va,τ and va,r as well as force components
L and D acting on the kite which is represented by the centre airfoil. The chord
line is indicated by dots.

The photo shown in Fig. 5 captures a moment of a particularly pronounced effect of gravity and aerodynamic drag. This
specific case was the combined result of low wind velocity and
low reel-in speed, both contributing to a reduced tension in the
tether. To calculate the force Ft that the kite exerts on the tether
and the force Ftg that the ground station exerts on the tether we
use the free body diagram illustrated in Fig. 6.
Because of its flexibility the tether can support only tensile
forces and no bending moment and as consequence the tether
force is always locally aligned with the tether, following its
curvature. This holds also for the tether suspension points,
as indicated by the corresponding reaction forces included in
Fig. 6. The sketch illustrates how the sagging induces the tangential reaction force components Ft,τ and Ftg,τ which are balancing the resultant tangential component of the gravitational
loading.
The reaction forces are calculated from the force and moment equilibria of the deformed tether. For small to moderate

It can be shown from Fig. 4 that the angle of attack does not
vary along the flight path of a massless kite if the angle between
wing and tether is constant. For flexible membrane wings this
angle is generally controlled by the bridle line system which has
the function of transferring the aerodynamic load to the tether.
On the other hand, Eq. (15) links the angle between the velocity
vector va and the tether to the lift-to-drag ratio L/D. A constant
L/D thus ensures a constant α, and vice versa. While the relative flow angle is constant along the flight path, the magnitude
of the relative flow velocity changes according to Eq. (16).
The effect of gravity induces variations of the flow angle
along the flight path because the aerodynamic force Fa is not
aligned anymore with the radial direction. In the following section this framework will be extended to include gravitational
forces.
4

Ft
Ft,r

Zw

with

Ft,τ
K
r

sin θ mt geθ
mt g

θ

g

O

Ftg

 q

− F 2 − 1 sin2 θ m2 g2 + cos θ m g
t
t
t

4

 .
F∗t = 
0



0

(33)

With the starred versions of the forces we have formally removed the gravitational contribution of the tether from the internal structural force Ft and lumped it to the gravitational force
mg of the kite. A similar approach was used with Eq. (12) to
lump the aerodynamic drag of the tether to the drag of the kite.
Regarding Eq. (32) it should be noted that unlike the contribution of the kite the contribution of the tether is not vertical
because of the sagging of the tether and the fact that it is attached to the ground station. The resulting tether force F∗t acts
in radial direction. Figure 7 illustrates the described lumping
approach and the effect on the steady force equilibrium of the
kite.

Ftg,r
Figure 6: Free body diagram of the deformed tether in the φk -plane. The reaction forces Ftg and Ft acting at the suspension points O and K, respectively,
are decomposed into radial and tangential components. The idealised straight
tether, which coincides with the radial coordinate, is included as dashed line.

sagging the centre of gravity of the tether is located halfway
between the suspension points in terms of ground plane distances. The tether force vector can be resolved in spherical coordinates (r, θ, φ) as a function of the tensile force Ft at the kite,
the tether mass mt and its orientation θ
 q

q
 F 2 − F 2   F 2 − 1 sin2 θ m2 g2 
t
t,τ 
t
t



4
 
 .
1


=
(28)
Ft = 


F
−
sin
θ
m
g
t,τ
t
2
 


0
0

Zw

g

K

F∗t
r
Ft

− cos θ mt ger
O

sin θ mt geθ
mt g

Figure 7: Steady force equilibrium of the kite K in the φ = const. plane showing
the original force triangle, Ft Fa mg, and the triangle resulting from the lumped
approach, F∗t Fa F∗g (shaded in blue).

(30)

The apparent wind velocity and the decomposition of the
aerodynamic force into lift and drag components is illustrated
in Fig. 8. Because the gravitational force F∗g causes a disalignment of the aerodynamic force Fa and the tether force F∗t , the
geometric similarity of the force and velocity diagrams does not
hold anymore. Consequently, the kinematic ratio κ = va,τ /va,r
can not be expressed by the lift-to-drag ratio L/D, as stated by
Eq. (15) which is valid for the limiting case of vanishing mass.
Starting from Eq. (8) the nondimensional apparent wind velocity can be formulated as

to quantify the relative importance of gravity. For small values
of γ the effect of gravitational forces will only be minor.
2.6. Analytic Model Including Effect of Gravity
The tether force Ft given by Eq. (28) describes the effect of
the kite on the tether. It reversely acts, but with opposite sign,
on the kite and implicitly includes the sagging-induced effect of
gravity, the tangential force component Ft,τ . The quasi-steady
force equilibrium is extended to
Ft + mg + Fa = F∗t + F∗g + Fa = 0,

mg
F∗g

with the sagging-induced tangential force component given by
Ft,τ = 1/2 sin θ mt g. If the tether mass is small compared to the
tensile force the sagging will be small and the tensile forces at
both suspension points will differ very little. We can introduce
the relative gravitational force
mt g
Ft

Fa
θ

For strong sagging the ground plane distance between the centre
of mass and the kite decreases and as a result Ft,τ increases
while Ftg,τ decreases. The tensile force Ftg at the ground station
can be calculated as
s
q
2
2
2
Ftg =
Ft2 − Ft,τ
− cos θ mt g + Ft,τ
,
(29)

γ=

(32)

and

− cos θ mt ger
Ftg,τ





− cos θ
− cos θ




F∗g =  sin θ  mg +  21 sin θ  mt g,




0
0

s
va
= (sin θ cos φ − f )
vw

(31)
5

1+

va,τ
va,r

!2
(34)

Fa

The process starts with setting the target value G∗ to the
given lift-to-drag ratio L/D of the kite and setting the initial
guess κ1 = G∗ based on Eq. (15). The following steps are performed to update κi in i = 1, . . . , n iterations: First, the spherical
components of the apparent wind velocity are computed from
Eqs. (34) and (35), then the respective components of the resultant aerodynamic force from Eqs. (36), (37) and (38). Using
Eq. (40) the value of the lift-to-drag ratio Gi corresponding to
the current value of κi is computed. From this we determine the
updated value of the kinematic ratio as
r
G∗
.
(41)
κi+i = κi
Gi

L

F∗g

Zw

g

va,r

va

θ
K

D

va,τ

r
O
F∗t

φ

Yw

This iteration loop is repeated until the lift-to-drag ratio Gi calculated from Eq. (40) is sufficiently close to the target value
G∗ .
The effect of gravity on the instantaneous traction power can
be significant depending on the kite course angle. Figure 9
shows computed isolines of the kite mass as functions of the
kite course angle and the kinematic ratio for a representative
example. For horizontal or upward flight (90◦ ≤ χ ≤ 270◦ )

Xw
Figure 8: Steady force equilibrium considering the effect of gravity. Adapted
from [22].

and the tangential kite velocity factor now takes the form
s
!
va,τ 2
(b − f )2
(35)
λ = a + a2 + b2 − 1 +
va,r

7

with the trigonometric coefficients a and b defined by Eqs. (18)
and (19). The magnitude of the resultant aerodynamic force Fa
can be formulated by using Eq. (34)

! 

va,τ 2 
Fa
 (sin θ cos φ − f )2 .

= CR 1 +
(36)
qS
va,r 

30

5 0

0

4.5

10

4

30
40

Fa · va
va

5.5 10

3.5
3
2.5

0

45

90

135
225
180
Kite course angle χ [◦ ]

270

315

360

Figure 9: Kite mass m as function of course angle χ and kinematic ratio κ for
β = 25◦ , f = 0.37, L/D = 5, C L = 1, S = 16.7 m2 , vw = 7 m/s and ρ = 1.225
kg/m3 [22].

the kinematic ratio is always smaller than the lift-to-drag ratio. The kinematic ratio can become zero when the component
of the gravitational force opposing the flight direction is larger
than the component of the aerodynamic force in flight direction.
In this case the forces in flight direction can not be in a quasisteady equilibrium and the algorithm fails to identify a physical
solution.
For downwards flight (χ < 90◦ or χ > 270◦ ) the kinematic
ratio can become larger than the lift-to-drag ratio and increases
with increasing mass of the kite. In specific cases, like an exceptionally heavy kite flying vertically downward while reeling
out fast, the kinematic ratio starts to approach infinity. Also in
this situation the model fails to identify a quasi-steady equilibrium. It is recommended to further investigate this situation in

(38)

using Eqs. (36) and (37) to resolve the forces on the right hand
side. Finally, the definition of the aerodynamic drag force
D=

20

20

Kinematic ratio κ

6

2.7. Iterative Solution Procedure
In the following we describe an iterative procedure to solve
for the unknown kinematic ratio κ. Maintaining a quasi-steady
motion requires a kinematic ratio for which the aerodynamic
force balances the tangential components of the gravitational
force. This is expressed by
!
1
∗
Fa,θ = −Fg,θ
= − mt + m g sin θ.
(37)
2
The radial component is determined by
q
2
Fa,r = Fa2 − Fa,θ
,

40

6.5

(39)

is rewritten to obtain the following expression for the lift-todrag ratio
s
!2
L
Fa va
=
− 1.
(40)
D
Fa · va
This equation can be employed to iteratively determine the kinematic ratio [23].
6

future research. In the current work these extreme situations
do not occur and the kinematic ratio does not approach these
limits.
The effect of gravity on the average traction power generation can be significant, as for the upward flying regions where
the kinematic ratio becomes smaller, the quasi-steady flight velocity of the wing reduces. This means that the upward flying
regions of a closed-loop trajectory require more time than the
downward flying regions. As a result the time average course
angle can be expected to have an upward component as a result
of the mass.
In the following sections we adapt the developed theoretical framework to the specific flight manoeuvres in the different phases of the pumping cycle illustrated in Fig. 1. We start
the cycle with the retraction phase because at the start of this
phase is the only fix point of the trajectory determined by given
problem parameters βo and rmax .

force Ft,i over the entire retraction phase. This requires for each
retraction step the solution of Eq. (23) for the reel-out factor
v
u
t
Ft,i

f = sin θ cos φ ±
(45)
 2  .
qS CR 1 + DL
The larger value of f can be excluded because it describes the
unphysical case of compressive loading of the tether. Tensile
loading requires a radially outward pointing aerodynamic force
Fa which is linked to a positive value of va,r . According to
Eq. (8) this is only possible for f ≤ sin θ cos φ which can only
be fulfilled if the root is subtracted from sin θ cos φ.
For a constant and uniform wind velocity vw and constant
reeling factor the kite would asymptotically approach a steady
flight state which is characterised by a constant elevation angle
βi,∞ . This radial retraction state is generally not reached before
the minimum tether length rmin is reached and the retraction
phase is terminated at tA .

2.8. Retraction Phase

2.9. Transition Phase

The objective of the retraction phase is to pull the kite back
to the minimum tether length rmin at a minimal cost of energy,
while ensuring stable flight throughout this manoeuvre. The retraction energy is calculated as the integral of the instantaneous
traction power Pi over the retraction time ∆ti . This is conventionally achieved by reducing the angle of attack of the wing,
which reduces the aerodynamic coefficients but not the wing
reference area. A more aggressive, but also more risky manoeuvre, such as sideways flagging, substantially decreases also
the wing area [27]. Within the scope of the present analysis the
aerodynamic force is modified solely by means of the aerodynamic coefficients.
It is assumed that the aerodynamic coefficients C L,i and C D,i
are constant during the retraction phase. At the start of the
phase the tether is at its maximum length rmax and the elevation angle has still the constant value βo of the traction phase.
The course angle is set to a constant value of χi = 180◦ in order to fly in upwards direction during the complete retraction
phase.
The trajectory described by the kite is located in the φi = 0
plane. The position of the kite is updated by a finite difference
scheme
r(t + ∆t) = r(t) + vk (t)∆t.
(42)

As shown in Fig. 1, the retraction phase generally ends at
an elevation angle that is substantially larger than the constant
elevation angle βo of the traction phase. On the other hand, the
tether force Ft,i during retraction is much lower than the force
Ft,o during the traction phase. The objective of the transition
phase is to fly the kite back to the lower angle βo and to safely
increase the force in the tether to Ft,o .
To initiate the transition flight manoeuvre at tA the aerodynamic coefficients are set to the values C L,o and C D,o of the traction phase. At the same time the course angle is set to χ = 0◦ ,
which means the kite is flying in a downward direction. The
control algorithm generally aims to keep the tether at constant
length, but takes corrective action to ensure that the tether tension stays within a limited range during the transition phase.
Because the kite can overfly the ground station during the
retraction phase the described flight manoeuvre can result in a
sudden drop of the tether tension below the required minimum
value for the kite to ensure a stable operation. In such situation
the tether is reeled in further to restore the minimum tension.
For the simulation and the operation of the real system the target force Ft,i of the retraction phase is used as lower limit. As
consequence, the parameter rmin can only be regarded as a target value and the true minimum tether length can be less as a
result of the described minimum tension requirement.
On the other hand, flying to a lower elevation angle into the
so-called wind window increases the tether tension which could
exceed the value Ft,o set for the traction phase. In this situation
the reeling velocity is increased to stay below the value Ft,o .
The transition phase ends when the required elevation angle βo
for the traction phase is reached.

We define the characteristic time of the traction phase as
t∗ =

rmax − rmin
vw,re f

(43)

and use this together with a given nondimensional time step ∆T
to scale the integration time step ∆t to the physical dimension
of the system
∆t = t∗ ∆T.
(44)

2.10. Traction Phase

As the kite describes its path through the retraction phase,
a control strategy needs to be defined to determine the reelout factor f . Three principal strategies can be applied: velocity
control, force control and power control. We will use a constant

During the traction phase the kite is operated in crosswind
motion to maximise the traction force and thus also the generated mechanical power. A variety of different flight manoeuvres
7

3. Experimental Setup

are in use, of which circular and figure eight trajectories are
most frequently described in literature.
Instead of resolving the tangential motion component of the
manoeuvre we use a constant representative flight state to describe the average traction force and power of the kite. As consequence the angular coordinates β and φ as well as the course
angle χ have constant values during the traction phase. The proposed approach has the advantage to not only reduce the simulation times substantially but also to keep the model generally
applicable for a range of different crosswind manoeuvres. We
hypothesise that this constant representative flight state is best
determined as a time average of the real flight state, taking into
account that it is the predicted traction power that should match
the average traction power of the crosswind manoeuvre. The
constant representative flight state is a predefined experiencebased setting and can be evaluated on the basis of experimental
data or by means of a dynamic kite model.
According to Eq. (26) the traction power depends on the
product of cos β and cos φ and for this reason the time average
of the trigonometric functions is used to define the representative angular positions φo and βo by
cos φo = cos φ

and

cos βo = cos β.

The quality of the presented quasi-steady model is assessed
on the basis of measurement data retrieved from comprehensive
tests of a pumping kite power system. In this section we outline
the key features of the technology demonstrator and select two
specific representative test cases for comparison. Because the
aerodynamic characteristics of the kite in the different phases
of the cycle have a decisive influence on the computed power
output particular attention is devoted to this subject.
The common approach to determine the aerodynamic characteristics of rigid wings under controlled conditions are wind
tunnel measurements of scaled models or computational fluid
dynamics. Although these techniques have been applied to
tethered flexible membrane wings [28, 29] the practical usability of the data is limited. On the one hand, windtunnel measurements are costly and because of the strong fluid-structure
coupling the aero-elastic behaviour of scale models can generally not be extrapolated to the size of the real system. On
the other hand, reasonably accurate aerodynamic simulations
of deforming membrane wings are still a major challenge for
currently available computational methods.
Tow testing of kites has developed as an interesting alternative to determine the aerodynamic performance of kites [30]. Although cost-effective, this technique also imposes a clear limit
on the wing size.
We describe a procedure for estimating the aerodynamic
properties directly from available flight data. This approach has
the advantage that the aerodynamic loading and structural deformation of the wing during the specific flight manoeuvres of
the pumping cycle is taken into account.

(46)

This averaging implies a weighting factor that decreases from 1
from the centre of the wind window, when the tether is aligned
with the Xw -axis, to 0 at the side of the wind window, when
the tether is perpendicular to the Xw -axis. For a figure eight
trajectory the averaging results in φo and βo at the centre of
one of the figure eight lobes. Because the kite flies slower in
upward than in downward direction the average course angle χo
is expected to be larger than 90◦ . We leave it for further research
to find a relation between χo and the mass and aerodynamic
properties of the kite. The traction phase is terminated when
the maximum tether length rmax is reached.

3.1. Technology Demonstrator
The technology demonstrator employed for the present study
is in periodical test operation since January 2010. Figure 10

2.11. Complete Pumping Cycle
The mean mechanical power production during one pumping cycle is computed from the mean traction power and time
duration of each phase
Pm =

Po ∆to + Pi ∆ti + P x ∆t x
,
∆to + ∆ti + ∆t x

Sensor Kite
Unit
Wind Meter
Traction
Tether

Pm
,
Pw S

Bridle Line
System

Drum/Generator
Control Module
Battery Center
Module

(47)

where the indices o, i and x denote the reel-out, reel-in and
transition phases, respectively. Using Eq. (47), an average
power harvesting factor
ζm =

Kite
Control
Unit

Airborne
Components
Launch
Mast

Power
Electronics
Ground
Station

(48)
Figure 10: Kite power system with optional launch mast. Adapted from [31].

can be defined for the complete pumping cycle. To account for
the varying atmospheric conditions along the cycle trajectory,
the wind power density is evaluated at an average traction altitude
1
zmt = cos θ (rmin + rmax ).
(49)
2
The equivalent for a horizontal axis wind turbine would be the
hub height.

shows an overview of the system and its major components. A
detailed description of the hard- and software components, the
installed measurement equipment and statistical performance
data is provided in [31]. The retrofitted experimental launch
setup is described in [32, 33]. A photographic sequence of the
launch procedure is shown in Fig. 11 with video footage available from [34].
8

3.3. Resultant Aerodynamic Coefficient
To estimate the aerodynamic force coefficient CR of the kite
from available experimental data we start with the tether force
Ftg measured at the ground station. This value is then used to
derive the aerodynamic force components at the kite in radial
and tangential directions, Fa,r and Fa,τ , respectively. The radial
component is calculated from the radial force equilibrium of the
tether illustrated in Fig. 6 as

Figure 11: Experimental kite launch from upside-down hanging position on 23
August 2012.

r
Fa,r =

2
Ftg
−

1 2
sin θ m2t g2 + cos θ (mt + m) g,
4

(50)

3.2. Selected Test Cases
assuming that the tether is only moderately sagging. Combining this with the tangential component defined by Eq. (37) we
can compute the total aerodynamic force according to

Two different test cases have been selected to assess the quality of the derived modelling framework. Firstly, for the strong
wind analysis a single representative cycle was selected randomly from a dataset recorded on 23 June 2012. The experiment was performed on the Maasvlakte 2 of the Rotterdam Harbour in The Netherlands, on an open field near the beach (see
[35] and Fig. 12). The test conditions were favourable with an

Fa =

q

2 + F2
Fa,r
a,τ

(51)

as a function of system parameters and the measured tether
force at the ground.
Next to the aerodynamic force the estimation process also requires information about the apparent wind velocity. To measure va directly some flights of the test campaign were equipped
with a Pitot tube mounted in the bridle line system between the
wing and the kite control unit. However, the quality of this data
was insufficient and for this reason we resorted to use Eq. (3)
to determine va as difference of the wind velocity vector vw at
the kite position and the kite velocity vector vk . To determine
vw the wind speed vw,re f measured at the reference altitude was
extrapolated to the kite position using Eq. (1), whereas vk was
determined using the GPS sensor attached to the kite.
The resultant aerodynamic force coefficient can then be derived from Eq. (21) as
CR =

Figure 12: Composite photo of the 14 m2 kite flying a figure of eight manoeuvre
(∆t = 1s) on 23 June 2012 at the Maasvlakte 2 of Rotterdam Harbour [21].

2Fa
,
ρv2a S

(52)

which implicitly contains the drag contribution of the tether.
Because this is small it has not been taken into account.
Figure 13 shows the variation of the resultant force coefficient CR over representative pumping cycles. During the retraction phase CR is low and varies only within a narrow band,
while during the traction phase the value is three to four times
higher, showing also substantially larger variations. These variations can be explained as follows.
Firstly, we have shown in Sect. 2.4 that the angle of attack
is constant along the flight path of an idealised massless kite.
However, the effect of gravity on a real kite induces variations
of the angle of attack which in turn lead to variations of CR .
Secondly, by extrapolating wind data that is measured at ground
level it is not possible to account for local wind gusts and leads
to over- or underestimation of the instantaneous value of CR .
We account for this effect by determining CR,i and CR,o as time
averages over the retraction and traction phases, respectively, as
specified in Table 1.

undisturbed wind approaching from the sea at an average velocity of 9.9 m/s. For this test a reinforced production kite was
used, a Genetrix Hydra 14 m2 with a projected surface area of
S = 10.2 m2 modified to withstand the high wing loading occurring in this experiment. The mass of the kite is 5 kg and the
control unit including the used sensor unit has a mass of 10 kg
such that the total mass of the airborne system components is
set to m = 15 kg.
Secondly, the presented moderate wind data was obtained at
Valkenburg, a former military airfield in The Netherlands located at 3 km inland. A steady 5.9 m/s north-eastern wind, blowing parallel to the coastline on 7 May 2013, provided good testing conditions. During this test a scaled up and redesigned version of the Genetrix Hydra with a wing surface area of A = 25
m2 , a projected area of S = 18.6 m2 and a mass of m = 10.6 kg
was used. This kite is shown in Figs. 5 and 11.
9

Lτ
Lτ2

1.0

Resultant force coefficient CR [-]

Strong wind

Fa,τ

0.8

τ
0.6

δ

K

0.4

va,τ

0.2

χ

Dτ

eφ

Lτ1
vk,τ

vw,τ
0.0

0

20

40

60

80

100

120

eθ

140

1.8

F∗g,τ

Moderate wind
Resultant force coefficient CR [-]

1.6
1.4

Figure 14: Tangential velocity and force components acting on the kite. The
placement of the local tangential plane τ is shown in Fig. 2. The tangential
flight direction is given by the vector vk,τ and indicated by the dashed line.

1.2
1.0
0.8

direction τ2

0.6

!
1
mt + m sin θ cos χ − Dτ = 0,
2
!
1
Lτ2 − g mt + m sin θ sin χ = 0.
2
Lτ1 + g

0.4
0.2
0.0

0

20

40

60

80

100

120

140

Time [s]

(55)

The gravitational contributions are orthogonal projections of
F∗g,θ defined by Eq. (32) onto the τ1 - and τ2 -directions, using
the course angle χ.
Because for fast crosswind manoeuvres the lift force L is by
far larger than the gravitational force Fg∗ we can conclude from
Eq. (55) that L  Lτ2 and accordingly also L ≈ Lr + Lτ1 . To
determine Lτ1 and Dτ1 in Eq. (54) we orthogonally project L
and D onto the tangential plane using Eq. (15) and following
the illustration in Fig. 4. This projection is possible because for
fast crosswind manoeuvres the deviation of the resultant aerodynamic force Fa from the radial direction er can be neglected.
The resulting force equilibrium in τ1 -direction is as follows
!
L
1
κD
+ g mt + m sin θ cos χ − √
= 0.
(56)
√
2
2
1+κ
1 + κ2

Figure 13: Estimated resultant force coefficient CR over a full pumping cycle.
The retraction phase starts at t = 0, the grey regions indicate the transition
phases and the cycle is completed with the end of the traction phase.

3.4. Lift-to-Drag Ratio
To estimate the lift-to-drag ratio L/D of the kite we analyse
the forces in the tangential plane. A similar, but more simplified
approach has been proposed in [3]. Starting point is the quasisteady force equilibrium given by Eq. (31). Noting that the
tether force F∗t defined by Eq. (33) has only a radial component
the equilibrium in the tangential plane τ reduces to
F∗g,τ + Fa,τ = F∗g,τ + Dτ + Lτ = 0,

(54)

(53)

We use Eq. (34) to determine the kinematic ratio from measured
data
s
!2
va
κ=
− 1.
(57)
vw (sin θ cos φ − f )

which is illustrated in Fig. 14 together with the tangential velocity components. For two specific flight modes the tangential
force equilibrium can be reduced to a scalar equation relating
force contributions in the tangential flight direction.
In the traction phase the kite is operated in crosswind manoeuvres. To generate a high tether force the kite needs to fly
substantially faster than the wind speed (vk  vw ), which is
the case for a high lift-to-drag ratio (L/D  1). This is quantitatively described by the tangential velocity factor defined by
Eq. (17). As consequence, the alignment of the velocity components vk,τ and va,τ = vw,τ −vk,τ increases with the flight speed,
the angle δ shown in Fig. 14 decreases until it practically vanishes for L/D  1. For this limiting case we consider the tangential force equilibria in flight direction τ1 and in orthogonal

In the retraction phase the kite moves in the φ = 0 plane with
a course angle of χ = 180◦ . Accordingly, the force components F∗g,τ , Lτ and Dτ are all aligned with vk,τ . Similar to fast
crosswind flight we can use Eqs. (57) and (56) to estimate the
lift-to-drag ratio.
Starting from an initial estimate G1 = κ, which is based on
Eq. (15), the lift-to-drag ratio G is determined iteratively using
the following equation
q
!
1 + G2i
√
1
2
Gi+1 = κ − 1 + κ g mt + m sin θ cos χ
, (58)
2
Fa
10

with i = 1, . . . , n and Fa calculated from Eqs. (37), (50) and
(51) as a function of system parameters and the tether force at
the ground station. Equation (58) is derived from Eq. (56) by
solving
√ for L/D and substituting the remaining drag force D by
Fa / 1 + G2 . Because the lift-to-drag ratio does not significantly change anymore after two iterations we use L/D = G3 as
solution.
This estimate still includes the effect of tether drag Dt according to Eqs. (12) and (13). To eliminate this we first recalculate
the total aerodynamic drag
Fa
D= q
 2
1 + DL

4. Results and Validation
The presented modelling framework is suitable to derive a
fast estimate of the system performance. Optionally, the mass
of the kite and tether can be taken into account at the the expense of extra calculation time required to iteratively determine the force equilibrium. In this section we compare simulation results and measured data for two representative test cases,
one for moderate and one for strong wind speed. The pumping
cycles are calculated on the basis of simulation parameters that
are as close as possible to the conditions of the experiment. The
comparison is based on kite position and velocity, tether tension
and generated mechanical power.

(59)

Environmental parameters

and from this determine the lift-to-drag ratio of the kite without
the tether
L
D
L
=
.
Dk
D D − Dt

Wind condition
Reference wind speed vw,re f
Reference height hre f
Roughness length z0
Average traction altitude zmt
Wind speed at zmt

(60)

Figure 15 shows how the lift-to-drag ratio at the different
stages of the described estimation process varies over a representative pumping cycle. Time averages for L/Dk can be determined for each phase of the cycle, as specified in Table 1.
It is important to note that the estimation quality crucially depends on the accuracy at which the wind velocity at the altitude
and time of flight can be determined.

Reel-out azimuth angle φo
Reel-out elevation angle βo
Reel-out course angle χo
Min. tether length rmin
Max. tether length rmax
Reel-out tether force Ft,o
Reel-in tether force Ft,i

Strong wind
κ, initial estimate
L/D, incl. tether drag
L/Dk , excl. tether drag

Lift-to-Drag Ratio [-]

Kite surface area A
Projected kite area S
Mass kite incl. control unit m
Traction phase L/Dk
Retraction phase L/Dk
Traction phase res. coefficient CR,o
Retraction phase res. coefficient CR,i
Traction phase lift coefficient C L,o
Retraction phase lift coefficient C L,i
Tether drag coefficient C D,t
Tether diameter dt
Tether density ρt

4

2

0

20

40

60

80

100

120

140

10

Moderate wind
κ, initial estimate
L/D, incl. tether drag
L/Dk , excl. tether drag

Lift-to-Drag Ratio [-]

8

10.6◦
26.6◦
96.4 ◦
234 m
385 m
3069 N
750 N

10.5◦
27.0◦
100.9◦
390 m
720 m
3008 N
749 N

Kite and tether parameters

6

0

strong
9.9 m/s
6m
0.07 m
252 m
18.2 m/s

Operational parameters

10

8

moderate
5.9 m/s
6m
0.07 m
139 m
10.1 m/s

25 m2
14 m2
2
19.8 m
10.2 m2
19.6 kg
15.0 kg
3.6
4.0
3.5
3.1
0.61
0.71
0.20
0.18
0.59
0.69
0.15
0.17
1.1
1.1
4 mm
4 mm
724 kg/m3 724 kg/m3

Simulation parameters
6

Nondimensional time step ∆T

4

0.01

0.01

Table 1: Model input parameters, representative for the two experimental datasets.

2

0

0

20

40

60

80

100

120

Table 1 shows the modelling parameters that are used for this
comparison. The required temporal discretisation of the cycle
by means of a nondimensional time step ∆T is determined in
Sect. 4.1. The angular positions φo and βo and the course angle
χo during the traction phase are determined by time averaging
the data as explained in Sect. 2.10. The minimum and maximum tether lengths are also determined from the data. Both,

140

Time [s]

Figure 15: Estimated lift-to-drag ratio over a full pumping cycle. The grey
regions indicate the transition phases.

11

in the experiment and in the model, the tether force during the
traction phase was controlled to a set value of Ft,o = 3000 N
and during the retraction phase to a set value of Ft,i = 750
N. The reference wind speed is measured at an altitude of 6 m
above the ground. The surface roughness length is estimated to
be 0.07 m. The material density of the tether listed in Table 1 is
lower than the material density of Dyneema® as a result of the
braiding process.

500

Kite height [m]

400

4.1. Convergence Study

300

200

Strong wind
Gravity incl.
Gravity excl.
Experiment

100

As explained in Sect. 2 the model equations are numerically
integrated in time. Figure 16 shows how the accuracy of the
integration result, the average power harvesting factor, is influenced by the constant integration time step. For a time step of
∆T < 0.1, the simulations converge to less than 3% deviation
from the reference solution. This holds for simulations excluding and including the effect of gravity as well as for the strong
and moderate wind cases. For this reason we use an integration step size of ∆T = 0.01. As a result, the gravity-including
simulation of the strong wind case requires 534 time steps to
complete an entire pumping cycle.

0

0

100

200

300

400

500

600

700

150

200

250

300

350

400
350

Kite height [m]

300
250
200
150
100

Moderate wind
Gravity incl.
Gravity excl.
Experiment

50
1.2

0

0

50

100

Horizontal kite distance [m]
1.1

Figure 17: Position of the kite over a full pumping cycle. The dotted line
represents the computed path neglecting gravity, the dashed line the computed
path accounting for gravity and the solid line the measured path.

ζm /ζm (∆T = 10−4 )

1.0
0.9
0.8

the tensile loading of the tether. In the extreme case of gliding flight towards the ground station the tensile loading can be
reduced to a very low value.
Secondly, the tangential component of the gravitational force
exerts a particularly strong effect during the first part of the retraction phase. In this period the kite flies upwards from a low
elevation angle, the tether tension is low and the tangential component of the gravitational force adds up to the drag force to decelerate the kite. This force effect keeps the kite from reaching
a high velocity and by that limits the generated traction force.
Because we adjust the reeling velocity to achieve a constant
set value of the tether force, Ft,i = 750 N, the kite can be retracted faster in the simulation accounting for gravity. This analysis is quantitatively supported by the reeling velocities shown
in Fig. 20.
During the transition phase the flight paths are all very similar. Because the aerodynamic coefficients of the traction phase
are used and the kite is flying a downward crosswind manoeuvre, a positive reeling velocity is required to not exceed the
constant set value of the tether force, Ft,o = 3000 N, in this
phase. This can be seen from the data presented in Figs. 17 and
20.
During the traction phase the computed flight paths do not
resolve the measured figure eight manoeuvres but only the average motion of the kite along the straight line segment defined
by the constant elevation angle βo and azimuth angle φo . This

0.7
0.6
0.5 −4
10

Gravity incl.
Gravity excl.

10−3
10−2
10−1
Nondimensional time step []

100

Figure 16: Convergence of the average power harvesting factor ζm normalised
by the power harvesting factor ζm (∆T = 10−4 ) for the smallest nondimensional
time step used in this convergence study. The dash-dotted lines indicate the 3%
convergence range. The convergence study is for the strong wind case.

4.2. Flight Trajectory
The computed and measured flight paths of the kite are depicted in the side views shown in Fig. 17. The horizontal distance
is measured from the ground station while the height is measured from the ground. Most obvious are the differences in the
retraction phase which indicates how important the consideration of the gravitational effect is. The lower flight path due to
gravity is the result of two different mechanisms.
Firstly, we note that during retraction the gravitational force
acting on the kite is of the same order of magnitude as the tether
force. As consequence, the radial component of the gravitational force significantly contributes to counterbalancing the
resultant aerodynamic force of the kite and by that alleviates
12

is also visible from the diagrams in Fig. 18 which complement
the side views of the flight paths. As discussed in Sect. 2.10 this

750
700
650

80

Tether length [m]

70

Strong wind
Gravity incl.
Gravity excl.
Experiment

Elevation angle [◦ ]

60
50

600
550
500
450

40

400

30

350

Strong wind
Gravity incl.
Gravity excl.
Experiment
0

20

40

60

80

100

120

140

160

450
20
10
−30

−10

0

10

20

400

30

Moderate wind
Gravity incl.
Gravity excl.
Experiment

Tether length [m]

Elevation angle [◦ ]

100

−20

80

350

300

60

250

40

200

Moderate wind
Gravity incl.
Gravity excl.
Experiment
0

20

40

60

80

100

120

140

Time [s]
20
−60

−40

−20

0

Figure 19: Tether length over a full pumping cycle. The grey regions represent
the transition phases in the experiment.

20

Azimuth angle [◦ ]

Figure 18: Angular coordinates of the kite over a full pumping cycle. The dot
at the centre of the figure eight lobe indicates the constant values φo and βo that
are used during the traction phase. The vertical line represents the computed
flight path during the retraction phase.

longer when neglecting the effect of gravity.
4.3. Kinematic Properties
Comparing the reeling velocity of the tether vt , the flight velocity of the kite vk , the wind velocity vw and the apparent wind
velocity va provides additional insight into the behaviour of the
quasi-steady model and the effect of gravity. The reeling velocity is illustrated in Fig. 20. The diagrams show that during
the retraction phase the tether is reeled in with continuously
increasing speed which is a consequence of the constant force
control. From Eqs. (23) and (36) it can be seen that as the elevation angle β increases the aerodynamic force Fa decreases. As
consequence the retraction velocity can be increased continuously to keep the tether force at its set value.
The flight velocity of the kite is illustrated in Fig. 21. In
the traction phase both simulation models exhibit a velocity
that is slightly decreasing during the traction phase. This behaviour is a result of the competing effects of wind velocity and
tether drag. On the one hand the wind velocity increases with
the flight altitude which for itself would lead to an increase of
the flight velocity according to Eqs. (7), (17) and (35). On the
other hand the aerodynamic drag of the tether increases with
the tether length. For the specific case the effect of tether drag
predominates such that the flight velocity slightly decreases.
The apparent wind velocity experienced by the kite and the
reference wind speed at 6 m altitude are depicted in Fig. 22.
Both simulations use a constant value of the reference wind

constant average flight state during the traction phase coincides
with the centre of one of the figure eight lobes.
In terms of angular coordinates β and φ the computed retraction and transition paths are straight and centred line segments.
However, the measured retraction paths show a significant deviation from the central line. For the strong wind case the kite
reaches an azimuth angle of φ = 25◦ during the transition phase
to smoothly connect to the first figure eight manoeuvre of the
traction phase. For the moderate wind case the measured retraction and transition paths go far through the side of the wind
window. This is an alternative technique of decreasing the traction force, which was used in this specific flight test.
Plotting the tether length over time puts the comparison into
a time perspective. From Fig. 19 it can be seen that the real
system and the simulation model accounting for gravity immediately start reeling in the tether, while the simulation model
neglecting gravity initially continues to reel out the tether. At
the same time the kite flies to a higher elevation angle which
allows retracting the tether at the set value of the tether force,
Ft,i = 750 N. As consequence, the retraction phase ends at a
higher elevation angle which means that the flight path in the
transition phase is longer. It can be concluded that the simulation of the retraction and transition phases takes substantially
13

45

Strong wind
Gravity incl.
Gravity excl.
Experiment

10

Strong wind
Gravity incl.
Gravity excl.
Experiment

40
35

Kite velocity [m/s]

Tether reeling velocity [m/s]

15

5

0

30
25
20
15
10

−5

5
−10

0

20

40

60

80

100

120

140

0

160

8

Moderate wind
Gravity incl.
Gravity excl.
Experiment

6
4

20

40

60

80

100

120

140

160

Moderate wind
Gravity incl.
Gravity excl.
Experiment

30
25

2

Kite velocity [m/s]

Tether reeling velocity [m/s]

0

35

0
−2
−4

20
15
10

−6
5

−8
−10

0

20

40

60

80

100

120

0

140

0

20

Time [s]

40

60

80

100

120

140

Time [s]

Figure 20: Tether reeling velocity over a full pumping cycle.

Figure 21: Flight velocity of the kite over a full pumping cycle.

speed and Eq. (1) to extrapolate to the wind velocity at the flight
altitude. The apparent wind velocity is evaluated according to
Eqs. (16) and (34). We can recognize that during the retraction
phase the computed apparent wind velocity increases slightly
while it levels to a constant value in the traction phase. This is
caused by the constant force control and the fact that the tether
force and the apparent wind velocity are directly linked by the
quadratic relationship given by Eq. (21).
A side effect in this equation is the resultant aerodynamic
coefficient CR which increases slightly with the tether length
as a result of the increasing drag contribution, as quantified by
Eq. (22) and (14). Because of this, imposing a constant tether
force during the retraction phase leads to a slightly increasing
apparent wind velocity. Gravity will enhance this effect. During
the traction phase these side effects are negligible and imposing
a constant tether force directly translates into a constant apparent wind velocity.

the control mechanism of the ground station can not fully compensate anymore. This leads to instantaneous force overshoots
of the set value by about 20% which is taken into account in the
system design of the technology demonstrator by defining the
set value of the tether force with a safety margin.
4.5. Traction Power
Figure 24 shows the instantaneous value of the traction power
delivered to the ground station over a full pumping cycle.
Tables 2 and 3 list the mean values for the cycle and its three
phases. Considering the retraction phase we note that the consumed power and the time duration are within 10% of the measured values when gravity is taken into account. We further note
that the effect of gravity reduces the retraction time from a significant overestimation to a slight underestimation of the measured value. This underestimation can be explained by noting
that the measured flight path is not perfectly straight as the computed paths, resulting in less efficient and thus slower retraction.
Also for the traction phase the generated power and time duration are closer to the measured values when accounting for
gravity. The effect of gravity is however not as strong as in
the retraction phase. Yet, even when accounting for gravity the
simulation overestimates the generated power and underestimates the time duration of the phase. A possible reason could
be an overprediction of the computed wind velocity at the operational altitude of the kite. Future research with more accurate wind measurements [36] and a comparison with a dynamic

4.4. Traction Force
Figure 23 shows the development of the tether force at the
ground, Ft,g , over a full pumping cycle. Because we apply force
control the computed tether force fits the measurements quite
accurately, as expected. The largest deviation between simulations and experiment occurs in the traction phase. As the kite
manoeuvres through the figure eight loops it is confronted with
turbulence and wind gusts as well as motion-induced variations
of the apparent wind velocity, as described by Eqs. (16) and
(34). As a result the tether tension experiences variations which
14

35

4000

Strong wind
Gravity incl.
Gravity excl.
Experiment

3000

25

Tether force [N]

Wind velocity [m/s]

30

Strong wind
Gravity incl.
Gravity excl.
Experiment

3500

Apparent wind
20

15

2500
2000
1500
1000

10

500

True wind
5

0

20

40

60

80

100

120

140

0

160

30

Moderate wind
Gravity incl.
Gravity excl.
Experiment

20

Wind velocity [m/s]

20

40

60

3000

15

Apparent wind

10
5

120

140

160

True wind

2000
1500
1000

−5

500

0

20

40

60

80

100

120

0

140

0

20

40

Time [s]

Parameter

Retraction

Pm [kW]
Time [s]
Pm [kW]
Time [s]
Pm [kW]
Time [s]
Pm [kW]
Time [s]

Transition

80

100

120

140

Figure 23: Tether force at the ground end of the tether over a full pumping
cycle.

model [16] is necessary to better understand the reason for this
difference.
We finally compare the computed and measured performance characteristics for the complete pumping cycle. For strong
wind conditions the measured data is between the two simulation results. For moderate wind conditions the simulation neglecting gravity is closer to the experiment. The close match can
be traced back to a coincidental, mutual compensation of the
modelling errors occurring in the different cycle phases. However, because the modelling errors per phase are generally lower
when accounting for gravity we recommended to further improve this more advanced modelling option.

Phase

60

Time [s]

Figure 22: True wind velocity at 6 m high (thin lines) and apparent wind velocity (thick lines) for a full pumping cycle.

Complete
cycle

100

2500

0

Traction

80

Moderate wind
Gravity incl.
Gravity excl.
Experiment

3500

Tether force [N]

25

−10

0

4000

Gravity
excluded

Gravity
included

Experiment

-2.46
103
17.90
8
24.72
36
5.37
148

-4.03
60
23.67
7
22.57
38
7.59
106

-3.64
67
8.50
9
19.12
52
6.48
128

Phase

Parameter

Retraction

Pm [kW]
Time [s]
Pm [kW]
Time [s]
Pm [kW]
Time [s]
Pm [kW]
Time [s]

Transition
Traction
Complete
cycle

Gravity
excluded

Gravity
included

Experiment

-1.73
66
5.08
14
9.20
42
2.84
123

-2.73
40
8.11
10
7.67
49
3.55
101

-2.60
43
3.44
12
6.23
66
2.79
122

Table 3: Simulated and measured performance characteristics of the pumping
cycle for moderate wind conditions.

5. Conclusion
The present study comprises a quasi-steady modelling framework for a pumping kite power system and a comprehensive
validation of this framework based on experimental data. The
objective of the model is to estimate the mechanical power output as a function of the wind conditions, the system design
and operational parameters. Part of the study is a technique
to estimate the aerodynamic properties of the kite using available measurement data. The validation reference data is derived
from two separate test campaigns of a technology demonstrator
using kites of 14 and 25 m2 surface area to generate 20 kW of
nominal traction power. The data for moderate and strong wind

Table 2: Simulated and measured performance characteristics of the pumping
cycle for strong wind conditions.

15

of the wind velocity at several altitudes, for example, by statically positioning the kite at these altitudes and using the onboard
wind sensor. Similarly, this wind sensor should be used for direct measurement of the apparent wind velocity [36].
The presented modelling framework is perfectly suited as a
basis for optimisation and scaling studies, also to predict the
power generation potential and the achievable cost of energy at
a specific deployment site [37]. The framework has been used
for designing and predicting the power output of a kite wind
park [38]. The quasi-steady analysis is not suited, for example,
for investigating peak loading during crosswind manoeuvres or
for fully dynamic flight behaviour. For such analyses a dynamic
system model needs to be used [16].

35

Strong wind
Gravity incl.
Gravity excl.
Experiment

30

Mechanical power [kW]

25
20
15
10
5
0
−5
−10

0

20

40

60

80

100

120

140

160

20

Moderate wind
Gravity incl.
Gravity excl.
Experiment

Mechanical power [kW]

15

Acknowledgements

10

The authors would like to thank Johannes Oehler for
proofreading the manuscript. Anna Bley and Roland Schmehl
have received financial support by the project REACH (H2020FTIPilot-691173), funded by the European Union’s Horizon
2020 research and innovation programme under grant agreement No. 691173, and AWESCO (H2020-ITN-642682) funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 642682.

5

0

−5
−10

0

20

40

60

80

100

120

140

Time [s]

Figure 24: Mechanical power over a full pumping cycle.

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conditions comprises instantaneous values, average values for
each of the three phases and for the complete cycle.
The computational effort to numerically integrate the flight
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two-dimensional idealisation of the cycle we find that three operational phases have to be distinguished: retraction, transition
and traction. When accounting for this partitioning the modelling framework provides valuable insight into the energy conversion mechanisms which can be used as a starting point for
systematic optimisation.
Per cycle phase the simulation is generally closer to the experimental data when accounting for gravitational effects. Especially during the retraction phase, when the gravitational
force is of the same order of magnitude as the other forces governing the flight motion of the kite, the effect of gravity is substantial and neglecting these contributions leads to pronounced
deviations between simulation and experiment. We thus recommended to always take the mass of the airborne components
into account.
The analysis clearly indicates that additional information
about the aerodynamic properties of the airborne system components and the atmospheric conditions will greatly improve
the prediction quality. The current calculation of the apparent
wind velocity from an extrapolated measured ground surface
wind velocity and the GPS-velocity of the kite should be regarded only as a first step. As consequence, we recommend
to include in future test campaigns also separate measurements

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