Quasi-steady model of a pumping kite power system

Abstract

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.

Full text

Quasi-steady model of a pumping kite power system
Rolf van der Vlugt
a
, Anna Bley
a
,
b
, Michael Noom
a
, Roland Schmehl
a
,
*
a
Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Delft, Netherlands
b
Kitepower B.V., Kluyverweg 1, 2629 HS Delft, Netherlands
article info
Article history:
Received 9 May 2017
Received in revised form
10 May 2 018
Accepted 5 July 2018
Available online 7 July 2018
Keywords:
Airborne Wind Energy
Kite power
Pumping cycle
Traction power generation
abstract
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.
©2018 Elsevier Ltd. All rights reserved.
1. Introduction
The pumping kite concept provides a simple yet effective solu-
tion for wind energy conversion at a potentially low cost [1].
Important aspects of the technology are the performance charac-
teristics 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 [2e7] and for the propulsion of ships [8e13]. The anal-
ysis presented in Refs. [8,9] 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 pro-
posed by Refs. [14e19] to address challenges in the field of control
or by Ref. [20] for state estimation. Recent studies have used
measurement data from full-scale demonstrator systems to analyse
the turning dynamics of kites and to assess flight control algorithms
[21,22].
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 assump-
tions, 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 2first 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 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 [23].
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 t
0
until t
A
, the transition phase, from t
A
until t
B
, and the
traction phase, from t
B
until t
C
, closing the cycle. The depicted side
view of the idealised flight trajectory in the wind reference frame
includes the wind velocity v
w
in direction of the X
w
- axis and the
elevation angle as
b
. A detailed presentation of the forces governing
the flight operation of a kite including the gravitational and inertial
effects is provided in Refs. [25,26]. In the following we discuss
several assumptions that reduce the complexity of the computa-
tional approach to achieve a substantial speed-up of the
simulations.
Firstly, the study is limited to kites with relatively large surface-
*Corresponding author.
E-mail address: r.schmehl@tudelft.nl (R. Schmehl).
Contents lists available at ScienceDirect
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journal homepage: www.elsevier.com/locate/renene
https://doi.org/10.1016/j.renene.2018.07.023
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Renewable Energy 131 (2019) 83e99

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 aero-
dynamic 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.
2.1. Atmospheric wind model
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 v
w
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 esti-
mated by the logarithmic wind law [27]
v
w
¼v
w;ref
lnðz=z
0
Þ
lnz
ref
.z
0
;(1)
where v
w;ref
is a known reference wind speed at a reference altitude
z
ref
and z
0
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
r
with increasing altitude can be
approximated by the barometric altitude formula for constant
temperature [27]
r
¼
r
0
expz
H
r
;(2)
where
r
0
¼1:225 kg/m3 is the standard atmospheric density at sea
level at the standard temperature of T
0
¼15

C and H
r
¼8:55 km is
the scale height for density.
2.2. Basic modelling framework
Starting point for the analysis is the wind reference frame which
has its origin Ocoinciding with the tether exit point from the
ground station and has its Z
w
- axis pointing vertically upwards and
its X
w
- axis aligned with the wind direction. The kite is represented
by a geometrical point. To describe its position Kand velocity v
k
we
follow the approaches in Refs. [3,25,26] and use a spherical coor-
dinate system ðr;
q
;fÞ. As depicted in Fig. 2, the position is described
by the radial distance r, the polar angle
q
and the azimuth angle
f
.
The direction of flight in the local tangential plane
t
is described by
the course angle
c
.
The apparent wind velocity describes the flow velocity relative
to the kite
v
a
¼v
w
v
k
:(3)
This vector can be described in spherical coordinates as follows
v
a
¼2
4
sin
q
cos f
cos
q
cos f
sin f3
5v
w
2
4
1
0
03
5v
k;r
2
4
0
cos
c
sin
c
3
5v
k;
t
;(4)
where v
k;r
and v
k;
t
represent the radial and tangential contributions
to the kite velocity, respectively.
The straight tether implies that the radial kite velocity is iden-
tical with the reeling velocity
v
k;r
¼v
t
:(5)
Introducing the reeling factor
f¼
v
k;r
v
w
(6)
and the tangential velocity factor
Fig. 1. Idealised flight trajectory of a traction kite of a pumping cycle. The trajectory
segment in the traction phase does not resolve the crosswind flight manoeuvres.
Adapted from Ref. [24].
Fig. 2. Decomposition of kite velocity vkinto radial component vk;rand tangential
component vk;
t
,definition of apparent wind velocity va¼vwvk. Course angle
c
is
measured in the tangential plane
t
, spherical coordinates ðr;
q
;fÞdefined in the wind
reference frame Xw;Yw;Zw, where Xwrepresents the wind direction [25].
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9984

l
¼
v
k;
t
v
w
;(7)
Eq. (4) can be formulated as
v
a
¼2
4
sin
q
cos ff
cos
q
cos f
l
cos
c
sin f
l
sin
c
3
5v
w
:(8)
The meaning of the velocity variables in this expression can be
summarised as follows. The reeling factor fis controlled by the
ground station, the course angle
c
is controlled by the steering
system and the tangential velocity factor
l
is a dependent variable,
which is determined by the force equilibrium.
The integral aerodynamic force acting on the airborne system
components can be decomposed into lift and drag vectors
F
a
¼LþD:(9)
The lift and drag forces contributed solely by the wing are calcu-
lated as
L¼1
2
r
C
L
v
2
a
S;(10)
and
D
k
¼1
2
r
C
D;k
v
2
a
S;(11)
where C
L
and C
D;k
are the aerodynamic lift and drag coefficients,
respectively, and Sthe 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 Ref. [3] and numerically validated in Ref. [28]. The total
aerodynamic drag Dof the airborne system is then estimated as
D¼D
k
þD
t
;(12)
where
D
t
¼1
8
r
d
t
rC
D;c
v
2
a
;(13)
with d
t
being the tether diameter, rthe tether length, C
D;c
the drag
coefficient of a cylinder in cross flow and v
a
the apparent wind
velocity at the kite. With the tether being subjected to a relative
velocity v
a;t
between 10 and 30 m/s and a kinematic viscosity of
n
¼
1:47 10
5
, the Reynolds number Re ¼v
a;t
d
t
=
n
is estimated to be
between 2:710
3
and 8:210
3
. In this range C
D;c
has a constant
value of 1.1 [29]. As consequence a total aerodynamic drag coeffi-
cient of the airborne system components can be defined as
C
D
¼C
D;k
þ1
4
d
t
r
SC
D;c
:(14)
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
k
¼v
a;
t
v
a;r
¼L
D:(15)
The ratio of the relative velocity components is denoted as kine-
matic ratio and represented by the symbol
k
. Equation (15) can be
derived from the geometrical similarity of the force and velocity
diagrams illustrated in Fig. 3.
Starting from the decomposition v
a
¼v
a;r
þv
a;
t
and using the
radial component of Eq. (8) in conjunction with Eq. (15) to elimi-
nate the tangential component results in the following expression
for the nondimensional apparent wind velocity
v
a
v
w
¼ðsin
q
cos ffÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þL
D
2
s:(16)
On the other hand, inserting the radial and tangential velocity
components of Eq. (8) into Eq. (15) and solving for the tangential
velocity factor
l
results in
l
¼aþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
þb
2
1þL
D
2
ðbfÞ
2
s;(17)
with trigonometric coefficients
a¼cos
q
cos fcos
c
sin fsin
c
;(18)
b¼sin
q
cos f:(19)
The quasi-steady motion of a massless kite is governed by the
equilibrium of the tether force and the resultant aerodynamic force
F
t
þF
a
¼0:(20)
Inserting Eqs. (10) and (11) into Eq. (20) results in
F
t
¼1
2
r
C
R
v
2
a
S;(21)
with the resultant aerodynamic force coefficient
C
R
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
2
D
þC
2
L
q:(22)
Using Eq. (16) to substitute the apparent wind velocity in Eq. (21)
Fig. 3. Geometrical similarity of the force and velocity diagrams. vaand Faare
decomposed in the plane spanned by the two vectors. Dis aligned with vaby defi-
nition, whereas va;ris aligned with Fawhen assuming a straight tether and a negligible
effect of mass. Adapted from Ref. [25].
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 85

gives the following equation for the normalised tether force [3, Eq.
(48)]
F
t
qS ¼C
R
"1þL
D
2
#ðsin
q
cos ffÞ
2
;(23)
with the dynamic wind pressure at the altitude of the kite calcu-
lated as
q¼1
2
r
v
2
w
;(24)
with the air density and wind velocity described by Eqs. (2) and (1),
respectively.
The generated traction power is determined as the product of
tether force and reeling velocity
P¼F
t
v
t
¼F
t
fv
w
:(25)
Expressing the tether force by Eq. (23) results in
z
¼P
P
w
S¼C
R
"1þL
D
2
#fðsin
q
cos ffÞ
2
;(26)
where P
w
denotes the wind power density at the altitude of the kite
P
w
¼1
2
r
v
3
w
:(27)
Equation (26) defines the instantaneous power harvesting factor
z
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 influ-
encing parameter is the angle of attack
a
,defined as the angle
between the chord line of the wing and the apparent wind velocity
vector v
a
. The sketch in Fig. 4 illustrates this, implying that the
heading of the wing is in plane with the radial and tangential ve-
locity components. This is generally the case if the wing is not
asymmetrically deformed due to steering actuation and sideslip
velocity components can be neglected.
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 v
a
and the tether to the lift-to-drag ratio L=D. A constant L=Dthus
ensures a constant
a
, 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 F
a
is not aligned
anymore with the radial direction. In the following section this
framework will be extended to include gravitational forces.
2.5. Effect of gravity on the tether force
Equations (1)e(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 conse-
quently 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 mand its
gravitational force mgdirectly contributes to the quasi-steady force
equilibrium at point K. Because of the long tether the angular ve-
locities
_
q
and
_
fare relatively small and the effect of inertial forces
can be neglected. The tether, on the other hand, is suspended be-
tween the ground station and the kite, its mass m
t
is continuously
distributed over its length and the distributed loading by gravity
and aerodynamic drag leads to sagging.
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 F
t
that the kite exerts on the tether and the force
F
tg
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
Fig. 4. Relative flow components va;
t
and va;ras well as force components Land D
acting on the kite which is represented by the centre airfoil. The chord line is indicated
by dots.
Fig. 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 m
2
.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9986

corresponding reaction forces included in Fig. 6. The sketch illus-
trates how the sagging induces the tangential reaction force com-
ponents F
t;
t
and F
tg;
t
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 sagging the
centre of gravity of the tether is located halfway between the sus-
pension points in terms of ground plane distances. The tether force
vector can be resolved in spherical coordinates ðr;
q
;fÞas a function
of the tensile force F
t
at the kite, the tether mass m
t
and its
orientation
q
F
t
¼2
6
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
t
F
2
t;
t
qF
t;
t
0
3
7
5¼2
6
6
6
6
6
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
t
1
4sin
2
q
m
2
t
g
2
r
1
2sin
q
m
t
g
0
3
7
7
7
7
7
5
:(28)
The tensile force F
tg
at the ground station can be calculated as
F
tg
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
t
F
2
t;
t
qcos
q
m
t
g
2
þF
2
t;
t
s;(29)
with the sagging-induced tangential force component given by
F
t;
t
¼
1
2
sin
q
m
t
g. For strong sagging the ground plane distance
between the centre of mass and the kite decreases and as a result
F
t;
t
increases while F
tg;
t
decreases. 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
g
¼m
t
g
F
t
(30)
to quantify the relative importance of gravity. For small values of
g
the effect of gravitational forces will only be minor.
2.6. Analytic model including effect of gravity
The tether force F
t
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 F
t;
t
. The quasi-steady force equi-
librium is extended to
F
t
þmgþF
a
¼F

t
þF

g
þF
a
¼0;(31)
with
F

g
¼2
4
cos
q
sin
q
03
5mg þ2
6
6
6
4
cos
q
1
2sin
q
0
3
7
7
7
5
m
t
g;(32)
and
F

t
¼2
6
6
6
6
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
t
1
4sin
2
q
m
2
t
g
2
rþcos
q
m
t
g
0
0
3
7
7
7
7
5
:(33)
With the starred versions of the forces we have formally removed
the gravitational contribution of the tether from the internal
structural force F
t
and lumped it to the gravitational force mgof the
kite. A similar approach was used with Eq. (12) to lump the aero-
dynamic 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. Fig. 7 illustrates the
described lumping approach and the effect on the steady force
equilibrium of the kite.
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 F
a
and the tether force F

t
, the geometric
similarity of the force and velocity diagrams does not hold
anymore. Consequently, the kinematic ratio
k
¼v
a;
t
=v
a;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
Fig. 6. Free body diagram of the deformed tether in the fk- plane. The reaction forces
Ftg and Ftacting at the suspension points Oand K, respectively, are decomposed into
radial and tangential components. The idealised straight tether, which coincides with
the radial coordinate, is included as dashed line.
Fig. 7. Steady force equilibrium of the kite Kin the f¼const:plane showing the
original force triangle, FtFamg, and the triangle resulting from the lumped approach,
F
tFaF
g(shaded in blue). (For interpretation of the references to colour in this figure
legend, the reader is referred to the Web version of this article.)
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 87

v
a
v
w
¼ðsin
q
cos ffÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þv
a;
t
v
a;r

2
s(34)
and the tangential kite velocity factor now takes the form
l
¼aþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
þb
2
1þv
a;
t
v
a;r

2
ðbfÞ
2
s(35)
with the trigonometric coefficients aand bdefined by Eqs. (18) and
(19). The magnitude of the resultant aerodynamic force F
a
can be
formulated by using Eq. (34)
F
a
qS ¼C
R
"1þv
a;
t
v
a;r

2
#ðsin
q
cos ffÞ
2
:(36)
2.7. Iterative solution procedure
In the following we describe an iterative procedure to solve for
the unknown kinematic ratio
k
. Maintaining a quasi-steady motion
requires a kinematic ratio for which the aerodynamic force bal-
ances the tangential components of the gravitational force. This is
expressed by
F
a;
q
¼F

g;
q
¼
1
2m
t
þmgsin
q
:(37)
The radial component is determined by
F
a;r
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
a
F
2
a;
q
q;(38)
using Eqs. (36) and (37) to resolve the forces on the right hand side.
Finally, the definition of the aerodynamic drag force
D¼F
a
,v
a
v
a
(39)
is rewritten to obtain the following expression for the lift-to-drag
ratio
L
D¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
a
v
a
F
a
,v
a

2
1
s:(40)
This equation can be employed to iteratively determine the kine-
matic ratio [26].
The process starts with setting the target value G

to the given
lift-to-drag ratio L=Dof the kite and setting the initial guess
k
1
¼G

based on Eq. (15). The following steps are performed to update
k
i
in
i¼1;…;niterations: 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)e(38). Using Eq. (40) the value of the lift-to-drag ratio G
i
corresponding to the current value of
k
i
is computed. From this we
determine the updated value of the kinematic ratio as
k
iþi
¼
k
i
ffiffiffiffiffiffi
G

G
i
s:(41)
This iteration loop is repeated until the lift-to-drag ratio G
i
calcu-
lated 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. Fig. 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
+

c
270
+
) 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 quasi-steady equilibrium and the algo-
rithm fails to identify a physical solution.
For downwards flight (
c
<90
+
or
c
>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 recom-
mended to further investigate this situation in future research. In
the current work these extreme situations do not occur and the
kinematic ratio does not approach these limits.
Fig. 8. Steady force equilibrium considering the effect of gravity. Adapted from
Ref. [25].
Fig. 9. Kite mass mas function of course angle
c
and kinematic ratio
k
for
b
¼25+,f¼
0:37, L=D¼5, CL¼1, S¼16:7m
2
,vw¼7 m/s and
r
¼1:225 kg/m3 [25].
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9988

The effect of gravity on the average traction power generation
can be significant, as for the upward flying regions where the ki-
nematic 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 specificflight 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
b
o
and r
max
.
2.8. Retraction phase
The objective of the retraction phase is to pull the kite back to
the minimum tether length r
min
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 P
i
over the retraction time
D
t
i
. This is conventionally ach-
ieved 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 flag-
ging, substantially decreases also the wing area [30]. 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 r
max
and the elevation angle has
still the constant value
b
o
of the traction phase. The course angle is
set to a constant value of
c
i
¼180
+
in order to fly in upwards di-
rection during the complete retraction phase.
The trajectory described by the kite is located in the f
i
¼0
plane. The position of the kite is updated by a finite difference
scheme
rðtþ
D
tÞ¼rðtÞþv
k
ðtÞ
D
t:(42)
We define the characteristic time of the traction phase as
t

¼r
max
r
min
v
w;ref
(43)
and use this together with a given nondimensional time step
D
Tto
scale the integration time step
D
tto the physical dimension of the
system
D
t¼t

D
T:(44)
As the kite describes its path through the retraction phase, a
control strategy needs to be defined to determine the reel-out
factor f. Three principal strategies can be applied: velocity control,
force control and power control. We will use a constant force F
t;i
over the entire retraction phase, because this minimizes the total
retraction time for a tether with a given tensile strength, which, in
the first instance, also maximizes the net power output of the
system. This requires for each retraction step the solution of Eq. (23)
for the reel-out factor
f¼sin
q
cos f±ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
t;i
qSC
R
h1þ
L
D

2
i
v
u
u
t:(45)
The larger value of fcan be excluded because it describes the
unphysical case of compressive loading of the tether. Tensile
loading requires a radially outward pointing aerodynamic force F
a
which is linked to a positive value of v
a;r
. According to Eq. (8) this is
only possible for fsin
q
cos fwhich can only be fulfilled if the
root is subtracted from sin
q
cosf.
For a constant and uniform wind velocity v
w
and constant
reeling factor the kite would asymptotically approach a steady
flight state which is characterised by a constant elevation angle
b
i;∞
. This radial retraction state is generally not reached before the
minimum tether length r
min
is reached and the retraction phase is
terminated at t
A
.
2.9. Transition phase
As shown in Fig. 1, the retraction phase generally ends at an
elevation angle that is substantially larger than the constant
elevation angle
b
o
of the traction phase. On the other hand, the
tether force F
t;i
during retraction is much lower than the force F
t;o
during the traction phase. The objective of the transition phase is to
fly the kite back to the lower angle
b
o
and to safely increase the
force in the tether to F
t;o
.
To initiate the transition flight manoeuvre at t
A
the aerodynamic
coefficients are set to the values C
L;o
and C
D;o
of the traction phase,
i.e. the kite is powered. At the same time the course angle is set to
c
¼0
+
, 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 F
t;i
of the retraction phase is used as lower limit. As consequence, the
parameter r
min
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 F
t;o
set for the traction phase. In this situation the
reeling velocity is increased to stay below the value F
t;o
. The tran-
sition phase ends when the required elevation angle
b
o
for the
traction phase is reached.
2.10. Traction phase
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 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
b
and
f
as well as the course angle
c
have
constant values during the traction phase. The proposed approach
has the advantage to not only reduce the simulation times sub-
stantially 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 experience-based setting and can be
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 89

evaluated on the basis of experimental data or by means of a dy-
namic kite model.
According to Eq. (26) the traction power depends on the product
of cos
b
and cosfand for this reason the time average of the trig-
onometric functions is used to define the representative angular
positions f
o
and
b
o
by
cos f
o
¼cos fand cos
b
o
¼cos
b
:(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 X
w
- axis, to 0 at the side of the wind window, when the
tether is perpendicular to the X
w
- axis. For a figure eight trajectory
the averaging results in f
o
and
b
o
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
c
o
is expected to be
larger than 90

. We leave it for further research to find a relation
between
c
o
and the mass and aerodynamic properties of the kite.
The traction phase is terminated when the maximum tether length
r
max
is reached.
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
P
m
¼P
o
D
t
o
þP
i
D
t
i
þP
x
D
t
x
D
t
o
þ
D
t
i
þ
D
t
x
;(47)
where the indices o,iand xdenote the reel-out, reel-in and tran-
sition phases, respectively. Using Eq. (47), an average power har-
vesting factor
z
m
¼P
m
P
w
S;(48)
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
z
mt
¼1
2cos
q
ðr
min
þr
max
Þ:(49)
The equivalent for a horizontal axis wind turbine would be the hub
height.
3. Experimental setup
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 charac-
teristics 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 [31,32] 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 [33]. 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 deforma-
tion of the wing during the specificflight manoeuvres of the
pumping cycle is taken into account.
3.1. Technology demonstrator
The 20 kW technology demonstrator employed for the present
study is in periodical test operation since January 2010.
Fig. 10 shows an overview of the system and its major compo-
nents. A detailed description of the hard- and software compo-
nents, the installed measurement equipment and statistical
performance data is provided in Ref. [34]. The retrofitted experi-
mental launch setup is described in Refs. [35,36]. A photographic
sequence of the launch procedure is shown in Fig. 11 with video
footage available from Ref. [37]. Starting in 2016, the spin-off
company Kitepower B.V. is developing a commercial 100 kW
version of the technology demonstrator [38,39].
3.2. Selected test cases
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 Ref. [40] and Fig. 12). The test
conditions were favourable with an undisturbed wind approaching
from the sea at an average velocity of 9.9 m/s. For this test a rein-
forced production kite was used, a Genetrix Hydra 14 m
2
with a
projected surface area of S¼10:2 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 m¼15 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
Fig. 10. Kite power system with optional launch mast. Adapted from Ref. [34].
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9990

Hydra with a wing surface area of A¼25 m2, a projected area of S¼
18:6 m2 and a mass of m¼10:6 kg was used. This kite is shown in
Figs. 5 and 11.
3.3. Resultant aerodynamic coefficient
To estimate the aerodynamic force coefficient C
R
of the kite from
available experimental data we start with the tether force F
tg
measured at the ground station. This value is then used to derive
the aerodynamic force components at the kite in radial and
tangential directions, F
a;r
and F
a;
t
, respectively. The radial compo-
nent is calculated from the radial force equilibrium of the tether
illustrated in Fig. 6 as
F
a;r
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
tg
1
4sin
2
q
m
2
t
g
2
rþcos
q
ðm
t
þmÞg;(50)
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
F
a
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
2
a;r
þF
2
a;
t
q(51)
as a function of system parameters and the measured tether force at
the ground.
Next to the aerodynamic force the estimation process also re-
quires information about the apparent wind velocity. To measure v
a
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 insuf-
ficient and for this reason we resorted to use Eq. (3) to determine v
a
as difference of the wind velocity vector v
w
at the kite position and
the kite velocity vector v
k
. To determine v
w
the wind speed v
w;ref
measured at the reference altitude was extrapolated to the kite
position using Eq. (1), whereas v
k
was determined using the GPS
sensor attached to the kite.
The resultant aerodynamic force coefficient can then be derived
from Eq. (21) as
C
R
¼2F
a
r
v
2
a
S;(52)
which implicitly contains the drag contribution of the tether.
Because this is small it has not been taken into account.
Fig. 13 shows the variation of the resultant force coefficient C
R
over representative pumping cycles. During the retraction phase C
R
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.
Fig. 11. Experimental kite launch from upside-down hanging position on 23 August
2012.
Fig. 12. Composite photo of the 14 m
2
kite flying a figure of eight manoeuvre (
D
t¼1s)
on 23 June 2012at the Maasvlakte 2 of Rotterdam Harbour [24].
Fig. 13. Estimated resultant force coefficient CRover a full pumping cycle. The
retraction phase starts at t¼0, the grey regions indicate the transition phases and the
cycle is completed with the end of the traction phase.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 91

However, the effect of gravity on a real kite induces variations of the
angle of attack which in turn lead to variations of C
R
. 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 un-
derestimation of the instantaneous value of C
R
. We account for this
effect by determining C
R;i
and C
R;o
as time averages over the
retraction and traction phases, respectively, as specified in Table 1.
3.4. Lift-to-drag ratio
To estimate the lift-to-drag ratio L=Dof the kite we analyse the
forces in the tangential plane. A similar, but more simplified
approach has been proposed in Ref. [3]. Starting point is the quasi-
steady 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
t
reduces to
F

g;
t
þF
a;
t
¼F

g;
t
þD
t
þL
t
¼0;(53)
which is illustrated in Fig. 14 together with the tangential velocity
components. For two specificflight 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 ma-
noeuvres. To generate a high tether force the kite needs to fly
substantially faster than the wind speed (v
k
[v
w
), 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 v
k;
t
and
v
a;
t
¼v
w;
t
v
k;
t
increases with the flight speed, the angle
d
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
t
1
and in orthogonal direction
t
2
L
t
1
þg1
2m
t
þmsin
q
cos
c
D
t
¼0;(54)
L
t
2
g1
2m
t
þmsin
q
sin
c
¼0:(55)
The gravitational contributions are orthogonal projections of F

g;
q
defined by Eq. (32) onto the
t
1
- and
t
2
-directions, using the course
angle
c
.
Because for fast crosswind manoeuvres the lift force Lis by far
larger than the gravitational force F

g
we can conclude from Eq. (55)
that L[L
t
2
and accordingly also LzL
r
þL
t
1
. To determine L
t
1
and
D
t
1
in Eq. (54) we orthogonally project Land Donto the tangential
plane using Eq. (15) and following the illustration in Fig. 4. This
Table 1
Model input parameters, representative for the two experimental datasets.
Environmental parameters
Wind condition moderate strong
Reference wind speed v
w;ref
5.9 m/s 9.9 m/s
Reference height h
ref
6m 6m
Roughness length z
0
0.07 m 0.07 m
Average traction altitude z
mt
139 m 252 m
Wind speed at z
mt
10.1 m/s 18.2 m/s
Operational parameters
Reel-out azimuth angle f
o
10.6

10.5

Reel-out elevation angle
b
o
26.6

27.0

Reel-out course angle
c
o
96.4

100.9

Min. tether length r
min
234 m 390 m
Max. tether length r
max
385 m 720 m
Reel-out tether force F
t;o
3069 N 3008 N
Reel-in tether force F
t;i
750 N 749 N
Kite and tether parameters
Kite surface area A25 m
2
14 m
2
Projected kite area S19.8 m
2
10.2 m
2
Mass kite incl. control unit m19.6 kg 15.0 kg
Traction phase L=D
k
3.6 4.0
Retraction phase L=D
k
3.5 3.1
Traction phase res. coefficient C
R;o
0.61 0.71
Retraction phase res. coefficient C
R;i
0.20 0.18
Traction phase lift coefficient C
L;o
0.59 0.69
Retraction phase lift coefficient C
L;i
0.15 0.17
Tether drag coefficient C
D;t
1.1 1.1
Tether diameter d
t
4mm 4mm
Tether density
r
t
724 kg/m
3
724 kg/m
3
Simulation parameters
Nondimensional time step
D
T0.01 0.01
Fig. 14. Tangential velocity and force components acting on the kite. The placement of
the local tangential plane
t
is shown in Fig. 2. The tangential flight direction is given by
the vector vk;
t
and indicated by the dashed line.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9992

projection is possible because for fast crosswind manoeuvres the
deviation of the resultant aerodynamic force F
a
from the radial
direction e
r
can be neglected. The resulting force equilibrium in
t
1
-direction is as follows
L
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ
k
2
pþg1
2m
t
þmsin
q
cos
c

k
D
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ
k
2
p¼0:(56)
We use Eq. (34) to determine the kinematic ratio from measured
data
k
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v
a
v
w
ðsin
q
cos ffÞ
2
1
s:(57)
In the retraction phase the kite moves in the f¼0 plane with a
course angle of
c
¼180

. Accordingly, the force components F

g;
t
,L
t
and D
t
are all aligned with v
k;
t
. Similar to fast crosswind flight we
can use Eqs. (57) and (56) to estimate the lift-to-drag ratio.
Starting from an initial estimate G
1
¼
k
, which is based on Eq.
(15), the lift-to-drag ratio Gis determined iteratively using the
following equation
G
iþ1
¼
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ
k
2
pg1
2m
t
þmsin
q
cos
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þG
2
i
qF
a
;(58)
with i¼1;…;nand F
a
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=Dand
substituting the remaining drag force Dby F
a
=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þG
2
p. Because the
lift-to-drag ratio does not significantly change anymore after two
iterations we use L=D¼G
3
as solution.
This estimate still includes the effect of tether drag D
t
according
to Eqs. (12) and (13). To eliminate this we first recalculate the total
aerodynamic drag
D¼F
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ
L
D

2
q(59)
and from this determine the lift-to-drag ratio of the kite without
the tether
L
D
k
¼L
D
D
DD
t
:(60)
Fig. 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=D
k
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.
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 expense of extra
calculation time required to iteratively determine the force equi-
librium. 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.
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
D
Tis determined in Sect. 4.1.
The angular positions f
o
and
b
o
and the course angle
c
o
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, in the experiment and in
the model, the tether force during the traction phase was controlled
to a set value of F
t;o
¼3000 N and during the retraction phase to a
set value of F
t;i
¼750 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 density of the tether listed in Table 1
is lower than the material density of the fibre material Dyneema
®
as a result of the braiding process.
4.1. Convergence study
As explained in Sect. 2the model equations are numerically
integrated in time. Fig. 16 shows how the accuracy of the integra-
tion result, the average power harvesting factor, is influenced by the
constant integration time step. For a time step of
D
T<0:1, the
simulations converge to less than 3% deviation from the reference
Fig. 15. Estimated lift-to-drag ratio over a full pumping cycle. The grey regions indicate
the transition phases.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 93

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
D
T¼0:01. As a
result, the gravity-including simulation of the strong wind case
requires 534 time steps to complete an entire pumping cycle.
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 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, F
t;i
¼750 N, the kite can be retracted faster
in the simulation accounting for gravity. This analysis is quantita-
tively supported by the reeling velocities shown in Sect. 4.3.
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, F
t;o
¼3000 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
b
o
and azimuth angle f
o
. This 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 constant
average flight state during the traction phase coincides with the
centre of one of the figure eight lobes.
In terms of angular coordinates
b
and
f
the computed retraction
and transition paths are straight and centred line segments. How-
ever, the measured retraction paths show a significant deviation
from the central line. For the strong wind case the kite reaches an
azimuth angle of f¼25
+
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 specificflight 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
Fig. 16. Convergence of the average power harvesting factor
z
mnormalised by the
power harvesting factor
z
mð
D
T¼104Þfor 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.
Fig. 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.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9994

at the set value of the tether force, F
t;i
¼750 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 longer when neglecting the effect of gravity.
4.3. Kinematic properties
Comparing the reeling velocity of the tether v
t
, the flight velocity
of the kite v
k
, the wind velocity v
w
and the apparent wind velocity
v
a
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
b
increases the
aerodynamic force F
a
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 ac-
cording to Eqs. (7), (17) and (35). On the other hand the aero-
dynamic 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 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 con-
stant 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 co-
efficient C
R
which increases slightly with the tether length as a
result of the increasing drag contribution, as quantified by Eqs. (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
Fig. 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 foand
b
othat are used during the
traction phase. The vertical line represents the computed flight path during the
retraction phase.
Fig. 19. Tether length over a full pumping cycle. The grey regions represent the
transition phases in the experiment.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 95

force directly translates into a constant apparent wind velocity.
4.4. Traction force
Fig. 23 shows the development of the tether force at the ground,
F
t;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 experi-
ment 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 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
Fig. 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 thatthe 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 over-
estimation 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 simu-
lation overestimates the generated power and underestimates the
time duration of the phase. A possible reason could be an over-
prediction of the computed wind velocity at the operational alti-
tude of the kite. Future research with more accurate wind
measurements [41] and a comparison with a dynamic model [18]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 re-
sults. 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 er-
rors occurring in the different cycle phases. However, because the
Fig. 20. Tether reeling velocity over a full pumping cycle.
Fig. 21. Flight velocity of the kite over a full pumping cycle.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e9996

modelling errors per phase are generally lower when accounting
for gravity we recommended to further improve this more
advanced modelling option.
5. Conclusion
The present study comprises a quasi-steady modelling frame-
work for a pumping kite power system and a comprehensive vali-
dation 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 aero-
dynamic 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 m
2
surface area to generate 20 kW of nominal traction power.
The data for moderate and strong wind 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 path
over a pumping cycle substantially reduces by not explicitly
resolving the transverse crosswind manoeuvres. Using a two-
dimensional idealisation of the cycle we find that three opera-
tional 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 exper-
imental 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 of the wind velocity at several alti-
tudes, for example, by statically positioning the kite at these alti-
tudes and using the onboard wind sensor. Similarly, this wind
sensor should be used for direct measurement of the apparent wind
velocity [41].
The presented modelling framework is perfectly suited as a basis
Fig. 22. True wind velocity at 6 m high (thin lines) and apparent wind velocity (thick
lines) for a full pumping cycle.
Fig. 23. Tether force at the ground end of the tether over a full pumping cycle.
R. van der Vlugt et al. / Renewable Energy 131 (2019) 83e99 97

for optimisation and scaling studies, also to predict the power
generation potential and the achievable cost of energy at a specific
deployment site [42]. The framework has been used for designing
and predicting the power output of a kite wind park [43]. 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 [18].
Acknowledgements
The authors would like to thank Johannes Oehler for proof-
reading the manuscript. Anna Bley and Roland Schmehl have
received financial support by the project REACH (H2020-FTIPilot-
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.
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Table 2
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Phase Parameter Gravity
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Gravity
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Experiment
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Time [s] 42 49 66
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cycle Time [s] 123 101 122
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