Determination of Kite forces for ship propulsion

Determination of kite forces using three-dimensional flight trajectories

for ship propulsion

George M. Dadd*, Dominic A. Hudson, R.A. Shenoi

University of Southampton, Southampton, England SO17 1BJ, United Kingdom

a r t i c l e i n f o

Article history:

Received 15 November 2010

Accepted 23 January 2011

Available online 21 April 2011

Keywords:

Kite

Dynamics

Trajectories

Ship propulsion

Optimisaton

Experiment

a b s t r a c t

For application of kites to ships for power and propulsion, a scheme for predicting time averaged kite

forces is required. This paper presents a method for parameterizing figure of eight shape kite trajec-

tories and for predicting kite velocity, force and other performance characteristics. Results are

presented for a variety of maneuver shapes, assuming realistic performance characteristics from an

experimental test kite. Using a 300 m2 kite, with 300 m long flying lines in 6.18 ms!1 wind, a time

averaged propulsive force of 16.7 tonne is achievable. A typical kite force polar is presented and

a sensitivity study is carried out to identify the importance of various parameters in the ship kite

propulsion system. Small horizontally orientated figure of eights shape kite trajectories centred on an

elevation of 15" is preferred for maximizing propulsive benefit. Propulsive force is found to be highly

sensitive to aspect ratio. Increasing aspect ratio from 4 to 5 is estimated to yield up to 15% more

drive force.

! 2011 Published by Elsevier Ltd.

1. Introduction

Kite propulsion is an attractive means to reduce fuel

consumption on ships by assisting themain engine using the power

of the wind. Recent developments, such as in autopilot kite control

and in launch and recovery systems1 have enabled them to be used

commercially for trans-oceanic voyages, yielding financial savings

through reduced fuel costs as well as minimizing emissions that are

harmful to the environment.

The determination of drive forces using a kite performance

model is required for ship velocity prediction, for enabling design,

for synthesising fuel savings and for optimizing kite systems for the

best propulsive effect. In addition, a kite performance model can be

used to implement carefully considered kite trajectories for

a desired force output.

Kite performance prediction models have been previously

established by Lloyd, [1], Wellicome [2], Naaijen [3], Williams [4]

and Argatov [5,6] although only Wellicome’s zero mass theory

has received published experimental validation. Dadd et al. (2010)

previously used the zero mass kite manoeuvring theory [2] to

predict kite line tension and other performance parameters. These

results were compared with real kite trajectories that had been

recorded using a purpose-specific kite dynamometer. The results

were shown to agree favourably; that work focused on the

validation of performance prediction based on kite position only.

The onset velocity and resulting line tension were calculated

without directly knowing the kite velocity itself. This paper focuses

on the additional modelling required in order to determine kite

velocity theoretically, an essential feature to enable the kite

performance to be established as a function of time.

Section 2 in this paper discusses the assumptions made in the

kite performance model. Section 3 presents a method for creating

kite trajectory shapes theoretically [2] and extends previous

developments by allowing the parameterized kite trajectories to be

transformed to simulate different mean angles to the wind. Section

4 defines the mathematical model. Section 5 describes the imple-

mentation and presents results using a case study for a typical ship

kite propulsion system. A new kite force polar diagram is developed

showing the propulsive drive for different wind angles. The

investigations are carried out considering the influence of the

Earth’s natural boundary layer. Section 6 presents an optimization

and sensitivity study that shows how various parameters effect

system performance including elevation, kite aspect ratio, angle of

attack, maneuver pole separation and pole circle size. Section 7

provides validation by way of comparison between theoretical

and experimental results [7].

* Corresponding author. Tel.: þ44 7815 044873.

E-mail address: [email protected] (G.M. Dadd).1 Pamphlet “Skysails Technological Information” available from www.skysails.

com.

Contents lists available at ScienceDirect

Renewable Energy

journal homepage: www.elsevier .com/locate/renene

0960-1481/$ e see front matter ! 2011 Published by Elsevier Ltd.

doi:10.1016/j.renene.2011.01.027

Renewable Energy 36 (2011) 2667e2678

2. Assumptions in the kite force model

1. The zeromass theory [2] assumes that the kite and the lines are

weightless. This is reasonable provided that the real weight is

very small compared to the aerodynamic forces, as shown by

Dadd et al. [7].

2. The kite is assumed to maneuver on the surface of a sphere of

radius defined by the flying line.

3. The kite lift and drag coefficients are assumed to remain

constant. The aerodynamic lift and drag coefficients are given

by expressions of the form

CL ¼ f ðae;RnÞ (1)

and

CD ¼ gðae;RnÞ: (2)

This implies firstly that the dependence of the force coefficients on

Rn is negligible and secondly that the angle of attack is unchanging.

To explain the third assumption, the kite is in a condition of

force equilibrium during static flight, where the line tension is

equally opposed to the aerodynamic force (neglecting weight). The

kite assumes its position in the flight envelopewhere this condition

is met and the effective angle of attack (ae) is dependent on the

relative wind velocity and the angle of mount to the flying lines.

During dynamic flight, the kite seeks the same force equilibrium

condition. When an imbalance of force arises, the kite accelerates

almost instantaneously to achieve the apparent kite onset velocity

at which this equilibrium is again achieved. The angle of attack

remains the same as the static flight case where CL and CD are

constant.

The Reynolds number effects which can also influence CL and CDare not expressly included in the zero mass model, although it is

noted that from Dadd et al. [7] that Rn was seen to vary between

7'105 for static flight and 4.3'106 for dynamic flight using a small

3 m2 kite in light winds. These are above the critical Rn number

(w5'105) at which transition between laminar to turbulent flow

tends to occur and thus it can be expected that the flowwill remain

substantially turbulent during dynamic flight and expectedly more

so for larger kites or for stronger winds. Thus with transition

between laminar and turbulent flow being unlikely during normal

flying conditions, the Rn effects are very minor and safe to neglect

whilst maintaining good predictions for kite performance.

Based on the above principles, Wellicome showed that the onset

wind velocity at the kite can be established in terms of its azimuth

and elevation spherical position angles, using the fundamental zero

mass equation [2].

U ¼ VAcos qcos f

sin 3: (3)

Here, q¼ 0 f¼ 0 defines the downwind direction.

Lloyd [1] had found that where the kite passes directly through

the downwind position, the onset velocity can be approximated by

Nomenclature

AK projected kite area, m2

AR aspect ratio

e aerodynamic planform efficiency factor (lifting line

theory)

f,g generic functions

F aerodynamic force magnitude, N

CL lift coefficient

CLa lift coefficient at a

CL0 lift coefficient at a¼ 0"

CD drag coefficient

CD0drag coefficient at a¼ 0"

D drag force magnitude, N

E rotation matrix

H pole of trajectory sphere

l aerodynamic lift force unit vector, N

L lift force magnitude, N

n exponent dependant on atmospheric and surface

conditions

n vector normal to great circle (right to left sweeps), m

n1,n2,n3 components of vector n, m

m vector normal to great circle (left to right sweeps), m

m1,m2,m3 components of vector m, m

O origin of trajectory sphere

P pole of small circle sweep

Q pole of small circle sweep

r kite position unit vector, m

ro small circle pole position vector, m

R kite position vector magnitude, m

R kite position vector, m

Re Reynolds number (Uck/n)

T time taken to traverse between twomaneuver points A

and B, s

u onset velocity unit vector, ms!1

U intersection node on trajectory

U onset velocity vector, ms!1

U onset velocity magnitude, ms!1

v apparent wind velocity unit vector, ms!1

V intersection node on trajectory

V apparent wind velocity magnitude at the kite when

static, ms!1

VT true wind speed, ms!1

VTreftrue wind speed at reference altitude, ms!1

W intersection node on trajectory

V apparent wind at the kite, as though it were static,

ms!1

x,y,z Cartesian position coordinates, m

X,Y,Z global Cartesian position coordinates, m

dt time step, s

f, g generic functions

ae effective angle of attack, "

a semi-vertex cone angle, "

a1 semi-vertex cone angle at P, "

a2 semi-vertex cone angle at Q, "

b azimuth angle of air onset velocity, "

d variable, "

3 aerodynamic drag angle, "

g elevation angle of air onset velocity, "

f azimuth angle, "

h1,2,3 transformation rotation angles about axis X, Y and Z, "

q elevation angle, "

ra density of air (1.19 kgm!3 at 20", 1 bar)

s variable, "

s variable, "

m substitution variable, (m¼ 1/2raAKCLsec3)

z variable, "

G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782668

the apparent wind speed multiplied by the lift to drag ratio of the

kite. Thus, with present day kites, such as the Flexifoil blade II para-

foil type test kite, used by Dadd [7], having lift to drag ratios of the

order of six, onset velocities six times wind speed are typical for

downwind dynamic kite flight and since aerodynamic force

increases with the square of the onset velocity, line tensions thirty

six times that of static kite flight are feasible.

Practical kite trajectories must remain downwind of the kite

tether. The preferredmaneuver shapes are those that do not result in

undesirable twisting of the flying lines or collision with the ground.

Consequently, vertical or horizontal figure of eight type motions are

commonplace for substantially downwind sailing courses, whilst

sine-wave (up and down) motions and static kite flight are

commonplace for upwind sailing courses. The manner in which

propulsive forces develop is largely dependent on the shape of

the kite trajectory adopted. Section 3 describes the way inwhich the

trajectories in thepresent paperwere parameterizedmathematically.

3. Parameterization of kite trajectory shapes

The trajectory parameterization follows the work of Wellicome

[2] and is further modified to allow vertically and diagonally

orientated trajectory shapes. The figure of eight flying maneuver

shape can be mathematically defined as comprising two small

circle sweeps for the ends of the maneuver and two great circle

sweeps which connect the small circular ends (see Figs. 1 and 2).

The great circle sweeps are along the intersections of a plane

passing through the origin with the sphere surface. The small circle

ends are defined using a semi-vertex angle for the cone swept out

by the radius and the position of the circle pole at its centre.

The spherical position angles, azimuth and elevation, are as

shown in Fig. 1. The position vector of the kite is given by

r¼ (cos q cos f, cos qsin f, sin q). The unit vector n¼ (1,2,3) is

chosen normal to the plane of the great circle sweep such that

n(r¼ 0.

rn1cos q cos fþ n2cos q sin fþ n3sin q ¼ 0 (4)

rtan q ¼ !

!n1n2

cos fþn2n3

sin f

"(5)

Hence Eq. (5) can be used to establish q, 4 ordinates on any great

circle defined by n.

The direction of the kite is defined by the value of f(

=q(

which can be determined by differentiation of Eq. (4) for

great circle sweeps as

ð ! n1cos q sin fþ n2cos q cos fÞf(

þ ð ! n1sin q cos f! n2sin q sin fþ n3cos qÞq(

¼ 0:

(6)

r

f(

q( ¼

sin qðn1cos fþ n2sin fÞ ! n3cos q

cos qðn2cos f! n1sin fÞ: (7)

For the small circle ends in Fig. 2, the pole P with position vector rodefines the centre of the small circle sweep that is inscribed by the

kite with semi-vertex angle a. Points which lie on the small circular

arc conform to

r ( ro ¼ cos a; (8)

where

r ¼ ðcos q cos f; cos q sin f; sin qÞ (9)

and

ro ¼ ðcos qocos fo; cos qosin fo; sin qoÞ: (10)

rcos a ¼ cos q cos qocosðf! foÞ þ sin q sin qo; (11)

rf ¼ fo ) cos!1

!cos a! sin q sin qo

cos q cos qo

": (12)

Hence Eq. (12) can be used to establish q, f ordinates on any small

circle sweep. The direction of flight for the small circle ends is

determined through differentiation of Eq. (11) as

f(

q( ¼

cos q sin qo ! sin q cos qocosðf! foÞ

cos q cos qosinðf! foÞ: (13)

Thus, to fully define the maneuver geometry, two poles (P(q1, f1)

and Q(q2, f2)) are required with corresponding semi-vertex angles

(a1 and a2) for each of the small circle ends, as well as calculation of

the two normal vectors n and m to define the great circle sweeps,

right to left and left to right respectively. It is necessary to establish

these maneuver segments as forming a continuous path by

ensuring tangency at the respective joins. For this, consideration is

given to the spherical triangle shown in Fig. 3, with a view to

finding the ratios n1/n3 and n2/n3 which can later be used to ensure

that tangency and continuity are correctly obtained. PH and QH in

Fig. 3 are great circles passing through the Z axis and PQ is another

great circle passing through O. The angle subtended between OP

and OH is 90! q1, the angle subtended between OQ and OH is

90! q2, the angle subtended between OP and OQ is z and the angle

subtended between HP and HQ is f1! f2.

Applying the standard spherical cosine formula to triangle HPQ,

z can be found.

Fig. 1. Great circle sweep geometry.

Fig. 2. Small circle sweep geometry.

G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2669

cos z ¼ cosð90! q1Þcosð90! q2Þ

þ sinð90! q1Þsinð90! q2Þcosðf1 ! f2Þ (14)

or

cosz ¼ sinq1sinq2 þ cosq1cosq2cosðf1 ! f2Þ (15)

Fig. 4 shows the intersection node points for consecutive great

circle and small circle sweeps at R, T, U and V. z1 and z2 are the

angles subtended between OP and OS and between OQ and OS

respectively. a1, a2, s1 and s2 equivalently represent their included

angles with the origin. The angle formed by the intersection of PQ

and RT is d. The angle RPQ is s1, the angle HOR is 90! qR, the angle

HOP is 90! q1 and the angle RHP is f1! 4R. HR, HP, RP, PQ, RT and

QT are all great circles. DPRS and DSTQ are right triangles.

Applying the spherical sine formula to DPRS,

sin a1sin d

¼sin z1sin 90

(16)

rsin d ¼sin a1sin z1

(17)

Similarly for DSQT,

sin d ¼sin a2sin z2

(18)

Through equating Eqs. (17) and (18) and using z2¼ z! z1, the

value of z1 can be found,

tan z1 ¼sin z sin a1

sin a2 þ cos z sin a1(19)

Applying the cosine formula to DPRS, s1 can be obtained with

cos z1 ¼ cosa1cos s1þsina1sin s1cos90 ¼ cosa1cos s1 (20)

rcos s1 ¼cos z1cos a1

(21)

Applying sine formula to DPRS, s1 is obtained using

sin s1 ¼sin s1

sin z1(22)

Employing the sine formula with DHPQ in Fig. 3 gives

sin#HbPQ

%¼

cos q2sinðf1 ! f2Þ

sin z(23)

then

HbPQ ¼ HbPQ ! s1: (24)

Applying cosine formula to DHPR enables calculation of qR,

cosð90! qRÞ ¼ cos a1cosð90! q1Þ

þ sin a1sinð90! q1Þcos HbPQ (25)

rsinqR ¼ cosa1sinq1 þ sina1cosq1cos#HbPQ

%(26)

Then, applying the sine formula to DHPR allows fR to be solved

using

sinðf1 ! fRÞ ¼sin a1sin

#HbPQ

%

cos qR(27)

The use of Eqs. (15)e(27) allows the locations the nodes qR and fR to

be solved for a given input set [q1, f1, a1, q2, f2, a2] that define the

maneuver shape. A similar set of equations can be developed to

establish the values of qT and fT. These values for points R and T can

be input in Eq. (4) and solved to establish the ratios n1/n2 and n3/n2.

Thus Eq. (5) may be used to define a series of data points along the

great circle sweep, RT. A similar process is repeated to establish

a series of positions along the other great circle sweep, UV. Points

which lie on the small circular end about the pole P are established

using Eq. (12) and similarly for points which lie on the small

circular end about pole Q. Data points on the surface of the sphere

which form a continuous and tangent curve around a figure of eight

shape have now been defined.

The parameterized trajectory may be transformed using Euler

rotations to achieve a vertically or diagonally orientated figure of

eight maneuver shape positioned at different angles relative to the

onsetwind. Rotations are applied to the position vector, r,first about

the X axis, then about Y and about Z respectively in that order as

r0 ¼ E3E2E1r: (28)

The rotation angles h1, h2 and h3 about X, Y and Z respectively

are user selected and the rotation matrices are E1, E2 and E3

given byFig. 4. Spherical triangles PRS, QST and HPR.

Fig. 3. Spherical triangle HPQ.


E1¼

2

41 0 00 cosh1 sinh10 !sinh1 cosh1

3

5;E2¼

2

4cosh2 0 !sinh2

0 1 0sinh2 0 cosh2

3

5andE3

¼

2

4cosh3 sinh3 0!sinh3 cosh3 0

0 0 1

3

5 (29)

f(

=q(

is then determined numerically in place of Eqs. (7) and (13).

4. Determination of kite forces, kite velocity and time

averaged quantities

The remainder of this section essentially follows the develop-

ment of Wellicome [2]. It is intended here to provide the necessary

background to Section 5.

The onset velocity can be expressed as

U ¼ Uu ¼ VAv ! Rr(

(30)

The aerodynamic force magnitude is given by

F ¼ L sec 3 ¼1

2raAKU

2CLsec 3 ¼ mU2 (31)

The Lift force magnitude is given by

L ¼1

2raAKU

2CL ¼ mU2cos 3 (32)

The drag force magnitude is given by

D ¼1

2raAKU

2CD ¼ mU2sin 3 (33)

By vector addition

F ¼ Fr ¼ Llþ Du (34)

where l is a unit vector perpendicular to u so that l(u¼ 0.

Substituting for FL and D,

mU2r ¼ mU2cos 3lþ mU2sin 3u (35)

or

r ¼ l cos 3þ u sin 3 (36)

Taking the scalar product with onset unit direction u,

r ( u ¼ l ( u cos eþ u ( u sin 3rr ( u ¼ sin 3

(37)

By taking scalar products with the position unit vector, Eq. (30)

becomes:

Uu ( r ¼ U sin 3 ¼ Vv$r! R _r ( r (38)

Since kite motion is confined to the surface of a sphere, its

velocity is tangential to the spheres’ surface such that ðr$r(

¼ 0Þ. It

follows that

U ¼Vv$r

sin 3(39)

A Cartesian right handed system is chosenwith the wind parallel to

the X axis. The position angles of the vectors R and U are as shown

in Fig. 5. The position angles of U are b and g. The unit vectors v, r

and u are expressed in terms of their position angles as

v ¼ f1;0;0g (40)

r ¼ fcos q cos f; cos q sin f; sin qg (41)

u ¼ fcos g cos b; cos g sin b; sin gg (42)

r(is then obtained by differentiation of r as

r(

¼n! cos q sin ff

(

! sin q cos fq(

; cos q cos ff(

! sin q sin f _q; cos q _qo

(43)

Substituting the expression for r(

into 15 yields

Ufcos g cos b; cos g sin b; sin gg

¼,V þ R cos q sin ff

(

þ R sin q cos f q(

;R sin q sin fq(

! R cos q cos ff(

;!R cos q q( -

(44)

Since r ( v ¼ cos q cos f, Eq. (39) can be expressed as

U ¼ VAcos q cos f

sin 3(45)

which is the same as Eq. (3), thus the onset velocity of air onto the

kite can be calculated according to any given kite position in terms

of azimuth and elevation, such as those positions which may be

defined using the parameterization of kite trajectory shapes

developed in Section 3 or alternatively, by using experimentally

recorded trajectories.

In real conditions, the wind speed can be expected to increase

with altitude due to viscous effects that slow the air in the Earth’s

atmospheric boundary layer. Calculation of Eq. (45) can therefore

be improved by employing [8]

VT ¼ VTref

Z

Zref

!n

(46)

to account for the difference in altitude relative to the wind refer-

ence height. The value of the exponent n may be expected to vary

depending on atmospheric conditions, however a value of 1/7 is

suitable for typical conditions at sea [8]. The lift force is calculated

using

L ¼1

2rAKU

2CL: (47)

The drag force is calculated using

Fig. 5. Flight envelope showing position angles for the kite and the onset velocity, U.


D ¼1

2rAKU

2CD: (48)

The total aerodynamic kite force, assumed here to be equal to

the line tension is obtained as

F ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ D2

p: (49)

or

F ¼1

2rAU2CF (50)

The established onset velocity in Eq. (45) does not require or allow

the calculation of the kite velocity. This is obtained through the

implementation of the following formulae, Eqs. (51)e(58). Then, it

becomes possible to integrate the kite force around the maneuver

in order to determine time averaged force, magnitude and direc-

tion, using Eqs. (59)e(63).

TheX,Yand Z components of Eq. (44) can bewritten separately as

X : U cos g cos b ¼ V þ cos q sin f Rf(

þ sin q cos f Rq(

(51)

Y : U cos g sin b ¼ !cos q cos fRf(

þ sin q sin fRq(

(52)

Z : U sin g ¼ !cos qR _q (53)

Combining Eqs. (53) with (45) yields an expression for R _q, for the

velocity of the kite in the q direction.

rRq(

¼ !Vcos f sin g

sin 3(54)

Substituting Eqs. (45) and (54) in Eq. (52) yields an expression

for R _f, for the velocity of the kite in the f direction,

Rf(

¼ !V

2tan q sin f sin gþ cos g sin b

sin 3

3: (55)

Eqs. (45) and (54) can be combined to yield an expression for the

direction of the kite from kite flight dynamic considerations,

f(

q( ¼

tan q sin fsin gþ cos gsin b

cos fsin g(56)

Eq. (51) combined with Eqs. (45), (54) and (55) and after manipu-

lation becomes

sin 3 ¼ sin q sin gþ cos q cos g cosðf! bÞ; (57)

or

b ¼ f) cos!1

2sin 3! sin q sin g

cos q cos g

3(58)

For a stationary kite, g¼ b¼ 0, so that substitution of these values

into Eq. (58) enables determination of azimuth and elevation values

along the line of static equilibrium that limits themotion of the kite.

The time taken for the kite to maneuver from a point qA to qB can

be obtained from

T ¼

ZqB

qA

dt

dqdq ¼

ZqB

qA

1

q( dq (59)

and the time averaged aerodynamic force F is given by

F ¼1

T

ZFdt ¼

1

T

ZqB

qA

Fdt

dqdq ¼

1

T

ZqB

qA

F

q( dq (60)

The angle of elevation for the time averaged force F is deter-

mined as

q ¼ tan!1

0

B@Fyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#

F2x þ F2y

%r

1

CA (61)

The azimuth angle for the time averaged force is determined as

f ¼ tan!1

!FyFx

"(62)

The magnitude of force acting in the horizontal plane for useful

ship propulsion is thus obtained as

Fxy ¼ F cos q (63)

5. Implementation and results

Using the procedure outlined in Section 3, a horizontally

orientated figure of eight maneuver was defined as the default

maneuver shape (q1¼!25, f1¼7, a1¼8, q2¼ 25, f2¼7, a2¼ 8).

This wasmanipulated using Euler rotation angles h1, h2 and h3 in Eq.

(29) to produce vertically orientated trajectories with 15 different

mean wind angles across the flight envelope. The Euler angles for

transformations are shown in Table 1 and the resulting maneuver

shapes are shown graphically in Fig. 6.

The theory of Sections 3 and 4 was implemented using Matlab

[9] to obtain performance results. The flow diagram in Fig. 7 shows

the order in which calculations were done.

The kite performance input parameters were previously

measured for an experimental test kite (Model: Flexifoil Blade III

3 m2) similar to the type of kite that is typically used for ship

propulsion. These data are shown in Table 2 fromDadd et al. [7]. For

this purely theoretical study, the kite area used is 300 m2 and

default line length used is 300 m. These values are in keeping with

typical kite sizes and line lengths that are presently used on

commercially available ship kite propulsion systems.

During the implementation, it was found that for some input

sets, more than one solution was feasible for the parameters b and

g. The correct solution selected was that which ensured continuous

Table 1

Euler rotation angles for vertical trajectory manipulation.

Trajectory h1 h2 h3

1 105 35 67

2 102 35 60

3 100 35 50

4 98 35 40

5 96 35 30

6 94 35 20

7 92 35 10

8 90 35 0

9 88 35 !10

10 86 35 !20

11 84 35 !30

12 82 35 !40

13 80 35 !50

14 78 35 !60

15 75 35 !67


results throughout the maneuver trajectory and satisfied the

requirement that the proper sense of the force vector was achieved.

Selected results arising for trajectory number 8 (highlighted in

Fig. 6) are shown in are in Fig. 8. These show the azimuth and

elevation angles that define kite position as an input, the kite

velocity together with the air onset velocity and the aerodynamic

force generated. These results show how the kite trajectory shape is

related to force output and provides a basis with which shapes can

be tailored to attain the desired output. This may be useful for

producing a control algorithm to regulate the force output. In

Fig. 8B it can be noted that the kite velocity closely follows the onset

velocity. This shows that the majority of the onset velocity is due to

the motion of the kite itself and explains why large force amplifi-

cation arises as a result of traversing the kite across the sky. Fig. 8C

shows that the line tension reduces significantly from 2'105 to

0.3'105 N as it is elevated during the maneuver. This is expected

since the kite is moved away from its location of maximum force

output which occurs directly downwind.

For predicting ship performance, it is useful to provide a single

time averaged force vector for a particular maneuver shape. The

time averaged aerodynamic forces were obtained using Eqs. (59)

and (60) for each of the trajectories in Table 1. To demonstrate

the effect of the wind increase with altitude due to the Earths

atmospheric boundary layer, kite performance was investigated

first using a wind gradient according to Eq. (46), and secondly with

no velocity gradient (see Fig. 9). The two cases have equal wind

speed at 10 m altitude (6.18 ms!1).

Fig.10A, shows the time averaged force for trajectories in Table 1,

not including the effect of the wind gradient, whilst Fig. 10B shows

the time averaged force in the presence of a wind gradient

(6.18 ms!1 at Zref¼ 10 m). The difference between the two

demonstrates the improved performance that arises as a result of

increasing wind speed with altitude. This is a significant added

benefit which is not realized by other wind propulsion systems,

such as conventional sails.

Fig. 10C shows the aerodynamic force which is realized

instantaneously as the kite is traversed along a straight path from

left to right at constant elevation, q¼ 15.0" (inc. Wind gradient).

This result can be viewed alternatively as the force realised from

a figure of eight trajectory centred at q¼ 15" with infinitely long

lines, such that the full extent of the trajectory may be considered

to remain at a single discreet value of azimuth and elevation. By

comparing Fig. 10B with Fig. 10C it is possible to see the extent by

which the propulsive drive force is reduced by traversing the kite

around the trajectories of Fig. 6. Fig. 10D shows the time averaged

aerodynamic forces from horizontally orientated trajectories that

have resulted from the optimization study in Section 6. These are

the maneuver shapes which are recommended for practical ship

propulsion.

6. Optimization and sensitivity study

Having defined a generic kite force polar diagram using a time

averaged result in both an even wind and in a wind gradient,

investigations have been undertaken to establish factors which

influence optimal trajectories. These factors include

1. The ideal elevation for maximizing line tension,

2. The ideal elevation to maximize the useful propulsive drive

force,

3. Kite aspect ratio,

Fig. 6. Vertical trajectory parameterization.

Fig. 7. Flow diagram for implementation.

Table 2

statically measured performance characteristics for the test kite (Model: Flexifoil

Blade III 3 m).

3 L=D CF CL CD

N samples 21 21 21 21 21

Mean 9.55 6.07 0.786 0.776 0.128

Standard deviation 1.48 0.88 0.115 0.116 0.012

95% Confidence limit 0.63 0.38 0.049 0.050 0.005

99% Confidence limit 0.83 0.50 0.065 0.065 0.006


4. Angle of attack,

5. Maneuver pole separation and Pole circle size.

6.1. Elevation for maximizing line tension

The wind speed increases with altitude in the Earth’s natural

boundary layer. Hence the onset velocity in Eq. (45) and the

resulting lift through Eq. (47) is also increased. However, the

elevation angle in Eq. (45) reduces the onset velocity due to its

influence on kite motions. There is an optimum elevation at which

the line tension is maximized. In order to find this Eqs. (45) and (46)

are combined with Z ¼ R sin q to obtain onset velocity (U) as

a function of its kite position angles,

U ¼ VTref

R sin q

Zref

!ncos q cos f

sin 3: (64)

The maximum onset velocity (and thus lift) with respect to

elevation is the solution to DU/Dq¼ 0, subject to D2U=Dq2< 0,

hence

n

R sin q

Zref

!n!1R cos2q

Zref! sin q

R sin q

Zref

!n

¼ 0 (65)

Taking n¼ 1/7 for typical conditions at sea [8], Zref¼ 10 m,

R¼ 300 m and 3¼ 9.55, the solution to Eq. (65) is numerically

obtained as q¼ 20.7". Therefore in the presence of a wind gradient

caused by the Earths boundary layer, the onset velocity and hence

line tension can be maximized by maintaining a trajectory that is

Fig. 8. Parameters arising through trajectory number 8. (A) Position angles azimuth and

elevation; (B) kite onset flow velocity and actual kite velocity; (C) aerodynamic force.

Fig. 9. Wind profiles.

Fig. 10. Horizontal component of kite force, FAxy[N/104] for different cases. (A) Time

averaged vertical trajectories, wind speed¼ 6.18 ms!1, wind gradient off. (B) Time

averaged vertical trajectories, wind speed¼ 6.18 ms!1 at Zref¼ 10 m, wind gradient on.

(C) Instantaneous kite force along straight horizontal trajectory q¼ 15" , wind

speed¼ 6.18 ms!1 at Zref¼ 10 m, wind gradient on. (D) Time averaged horizontal

trajectories, wind speed 6.18ms!1 at Zref¼ 10 m, wind gradient on.


centred closely to this value of elevation. This result is useful where

the absolute value of the line tension magnitude is important, such

as for producing electricity using kite power.

6.2. Elevation for maximizing propulsive drive force

For ship propulsion, only the horizontal component of the line

tension leads to useful propulsive force. The horizontal component

reduces with the cosine of elevation, so that the optimum elevation

is obtained by maximizing the value of U cos q.

This is given by

U cos q ¼ VTref

R sin q

Zref

!ncos2q cos f

sin 3(66)

Setting D(U cos q)/Dq¼ 0, subject to D2U cos q=Dq2< 0 to find

the maximum gives,

n

R sin q

Zref

!n!1 R

Zref

!$cos2q! 2 sin q

R sin q

Zref

!n

¼ 0 (67)

When solved numerically, Eq. (67) yields the maximum horizontal

component of drive force where q¼ 15.0". Hence for ship pro-

pulsion in the presence of a wind gradient, an optimal trajectory

should centre closely to this elevation. In this respect, horizontally

orientated figure of eight trajectories are preferential to vertical

ones for downwind trajectories, since elevation angles during the

maneuver remain closer to the optimum value. However, for

courses sailed close to the wind a vertically orientated trajectory

must be used in order to maintain a close winded time averaged

force, such as for trajectories 1 and 15 in Table 1.

Trajectories which reflect this optimization have been investi-

gated using the default parameters (q1¼!25, f1¼7, a1¼8, q2¼ 25,

f2¼ 7, a2¼ 8). These have been transformed using Eq. (28) and

Table 3. The resulting trajectory shapes are shown in Fig. 11. The

time averaged horizontal force output for each trajectory is shown

in Fig. 10D. By comparing Fig. 10B with Fig. 10D, horizontally

orientated maneuvers centred proximally to the optimum eleva-

tion are seen to make a significant improvement to the drive force

compared to vertical ones.

6.3. Kite aspect ratio

To investigate the effects of kite aspect ratio on aerodynamic

force, the kite drag angle is expressed as a function of the lift to drag

ratio using

3 ¼ tan!1

!1

ðL=DÞ

"(68)

Eqs. (45) and (50) are combined to yield an expression that relates

the drag angle to aerodynamic force as

FA ¼1

2rA

!VAcos q cos f

sin 3

"2

CF (69)

The drag coefficient of a kite with a dissimilar aspect ratio to that

used in the present study can be calculated using

C0D ¼ CD þ

C2L

p

!1

AR0 !1

AR

"; (70)

whereC0D corresponds to thedragcoefficientof awingof anewaspect

ratio AR0 [10]. It is assumed here that the kite is set so that the lift

coefficient remains unchanged by modified aspect ratio. The lift to

drag ratio of a wing with modified aspect ratio is therefore given by

L

D¼

CL

CD þC2L

p

!1

AR0 !1

AR

": (71)

To find the sensitivity of the kite system to aspect ratio, the

derivative dFA=dðARÞ is sought. This is given by

dFAdðARÞ

¼dFAd3

$

d3

dðL=DÞ$

dðL=DÞ

dðARÞ; (72)

for which Eqs. (68)e(71) are differentiated to obtain

dFAd3

¼ !rACFðVAcos q cos fÞ2cos 3

sin33; (73)

d3

dðL=DÞ¼ !

1

ðL=DÞ2þ1; (74)

and

dðL=DÞ

dðARÞ¼

C3L

pAR2hCD þ

C2L

p

!1

AR0 !1

AR

"#2 (75)

where 3 is in radians. Using r¼ 1.19, A¼ 300, VA¼ 6.18, q¼ 15",

f¼ 0", CL¼ 0.776, CD¼ 0.128, CF¼ 0.786 and AR¼ 4.86, Table 4 was

formulated using Eq. (68)e(75), for a range of values of aspect ratio

modified from the experimentally measured performance data in

Table 2.

Table 3

Euler rotation angles for horizontal trajectory manipulation.

Trajectory h1 h2 h3

1 0 15 !48

2 0 15 !40

3 0 15 !32

4 0 15 !24

5 0 15 !16

6 0 15 !8

7 0 15 0

8 0 15 8

9 0 15 16

10 0 15 24

11 0 15 32

12 0 15 40

13 0 15 48

Fig. 11. Horizontal trajectory parameterization.


For the example input set, Table 4 shows that increasing aspect

ratio from 4 to 5 for example, results in a dramatically improved

instantaneousaerodynamic force.19.21'104 Nisachieved insteadof

16.67'104 N yielding 15%more aerodynamic force per unit increase

in aspect ratio. There are other factors that require consideration in

selectionofkiteaspect ratio, suchashandling characteristics, stability

and response ingusts.Anecdotal evidence [11] suggests that theseare

factors which would tend to favour lower aspect ratios.

6.4. Kite angle of attack

The lift coefficient at which the kite operates has a significant

influence on aerodynamic force; firstly through its direct influence

on the lift generated, but also through a secondary effect due to the

change in the operational L/D that accompanies a change in angle of

attack. The optimum angle of attack for maximizing aerodynamic

forcemay be determined by considering Prandtl lifting line theory to

establish a functional relationship that can be assessed analytically

when combined with the zero mass theory. If an increase in lift

coefficientoccurs togetherwithan increase in lift todrag, thepositive

effect of each will be compounded. If an increase in lift coefficient

accompanies a reduction in lift to drag, the effectwill be negated. The

optimum condition for maximized propulsive force occurs when

dFAda

¼dFAdCL

$

dCLda

þdFA

dðL=DÞ$

dðL=DÞ

da¼ 0: (76)

From Eqs. (50), (45) and (68)

dFAdCL

¼1

2rAU2 ¼

1

2rA

!VA

cos q cos f

sin 3

"2

; (77)

and through combining (50), (68) and (71),

dFAdðL=DÞ

¼

>! rACFðVAcos q cos fÞ

2cos 3

sin33

?$

(!

1

ðL=DÞ2þ1

):

(78)

The 3D lift coefficient from Prandtl lifting line theory is

CLa ¼ CL0 þ 2p

!AR

AR þ 2

"a: (79)

The induced drag coefficient from Prandtl lifting line theory is

CD ¼ CD0þ

C2L

pARe: (80)

From Eq. (79)

dCLda

¼ 2p

!AR

AR þ 2

": (81)

Combining Eqs. (79) and (80) gives

L=D ¼CL0 þ 2p

B AR

AR þ 2

Ca

CD0þ

BCL0 þ 2p

B AR

AR þ 2

CaC2

pARe

(82)

Differentiating Eq. (82) gives

Table 4

Effect of aspect ratio on aerodynamic force.

AR’ 3 ["] L/D FA [104 N] d(FA)/d(AR) [104 N]

1 19.86 2.77 4.34 5.25

2 13.37 4.21 9.36 4.60

3 11.11 5.09 13.46 3.62

4 9.97 5.69 16.67 2.84

5 9.29 6.12 19.21 2.26

6 8.83 6.44 21.24 1.83

7 8.50 6.69 22.91 1.51

8 8.25 6.90 24.29 1.27

9 8.06 7.06 25.46 1.08

10 7.90 7.20 26.46 0.92

dðL=DÞ

da¼

0

@CD0þ

BCL0 þ 2p

B AR

AR þ 2

CaC2

pARe

1

A$2p

!AR

AR þ 2

"!

!CL0 þ 2p

!AR

AR þ 2

"a

"$

0

@2CL0$2p

B AR

AR þ 2

C

pAReþ8p2

B AR

AR þ 2

C2a

pARe

1

A

0

@CD0þ

BCL0 þ 2p

B AR

AR þ 2

CaC2

pARe

1

A2

(83)

Fig. 12. Trajectories shapes with different pole circle sizes.

Fig. 13. Trajectory shapes with different pole circle separation.


Using AR¼ 4.86, CD0¼ 0:05, CL0 ¼ 0:337, ra¼ 1.19, A¼ 300,

V ¼ 6:18, q¼ 15", f¼ 0", and e¼ 0.9 Eq. (76) was found yield

a¼ 15.28". This is the theoretical optimum as determined by

Prandtl lifting line theory, though there are other factors which

might influence the best angle of attack for practical use. In

particular, the way that the kite recovers from off-design flight

conditions, such as stall, is likely to be significantly improved by

mounting the kite to operate at a smaller angle of attack.

6.5. Pole separation and circle size

The effects of maneuver pole separation and pole circle size

were investigated by comparing a series of different kite trajecto-

ries (See Figs. 12 and 13). Results were obtained through imple-

mentation of the flowchart in Fig. 7, including the effect of thewind

gradient as determined by Eq. (46). The effect of increasing pole

circle size and pole circle separation on time averaged force output

is shown in Figs. 14 and 15 respectively. These figures show that

increasing pole circle size and separation both lead to a marked

reduction of time averaged propulsive drive force since the

maneuver is moved away from its optimal position for maximized

propulsive effect. The limitation of these investigations is that no

account is taken of the potential reduction in lift coefficient asso-

ciated with canopy deformation and angle of roll during a tight

turn.

7. Experimental validation

Previously Eq. (45), for obtaining onset velocity as a function of

kite position, was validated by comparing predicted aerodynamic

force with measured kite line tension [7]. The kite forces, force

direction and wind speed had been recorded by flying a kite from

a dynamometer in natural winds. In the present paper, the

implementation of the theory has been developed to allow the kite

velocity and motions to be computationally predicted, as well as

the instantaneous force based on position. For experimental

validation of the present work, two trajectories are shown in

Fig. 16.

The theoretically parameterized trajectory is defined according

to Section 3, whilst the experimentally determined trajectory was

recorded using a purpose built kite dynamometer described by

Dadd [7]. The theoretically determined shape was selected to

closely match that of the experimental one (see Fig. 16). Small

differences are apparent since the manually flown trajectory

exhibits irregularities. The aerodynamic force was determined in

three ways (see Fig. 17). First, the zero mass theoretical aero-

dynamic force was determined using the experimentally deter-

mined position coordinates as an input. Second, the actual

aerodynamic kite force was obtained using the dynamometer.

Comparing these two results serves to validate Eq. (45), which

determines kite onset velocity as a function of position, as

described by Dadd et al [7]. The third result in Fig. 16 is also based

on Eq. (45), but uses Eqs. (51)e(59) to establish the kite velocity

and hence to theoretically determine the results to a base of time,

by integrating around the maneuver. It can be seen that the

theoretical time for the kite to progress around the maneuver

closely matches that of the experimental kite trajectory. This

serves to validate Eqs. (51)e(59). The errors arising through

determination of kite force through each of these methods is

Fig. 14. Force reduction with circle size.

Fig. 15. Force reduction with pole separation.

Fig. 16. Experimentally and theoretically defined trajectories for comparison.

Fig. 17. Comparison between theoretical predictions and experimental measurements.


discussed in detail by Dadd et al [7]. The most significant error for

experimentally determined kite trajectories was shown to be due

to spatial and temporal differences in wind speed across the flight

envelope which was not properly accounted for.

8. Conclusions

This paper has presented a scheme for predicting time averaged

kite forces together with a method for prameterizing figure of eight

shape kite trajectories. The theoretical model wasmodified to allow

a high degree of control over input maneuver paths in order to

assess the effects of maneuver orientation, shape and true wind

angle. The influence of the changing wind strength in the Earth’s

natural boundary layer is included. The scheme has been imple-

mented to show variation of onset velocity, kite velocity, onset

angles and aerodynamic force for an example maneuver. In a case

study with a 300 m2 kite, 300 m long flying lines and 6.18 ms!1

wind speed (at 10 m), peak drive force up to 30'104 N was pre-

dicted. A typical vertical figure of eight maneuver in the same

conditions was shown to produce a time averaged horizontal drive

force component 16.7'104 N. A complete force polar showing the

horizontal component of aerodynamic force for different true wind

angles has thus been obtained.

The calculation of kite velocity and the propagation of the kite

around a theoretically defined maneuver have been validated by

comparing a real recorded trajectory to a simulated one. The time

scales for a theoretical and practical trajectory are shown to agree

favourably, as do the magnitudes of minimum and maximum kite

force. The increase in wind with altitude is found to double the aero-

dynamic force compared toahypothetical casewithnowindgradient.

Through considering thewind gradient in combinationwith kite

motions, the optimum mean kite elevation angle for ship pro-

pulsion during dynamic flight was found to be 15". Horizontally

orientated trajectories are shown to improve downwind perfor-

mance by a factor of 1.4 compared to vertical ones, since the

maneuver can remain closer to the optimum elevation angle for

propulsion. Vertical maneuvers are, however, preferred when

maintaining close winded performance is necessary.

By combining Prandtl lifting line theory with the zero mass

model, the derivative dFA=dðARÞ was calculated at a variety of

different aspect ratios. For the downwind dynamic flight condition,

increasing the kite aspect ratio from 4 to 5, for example, was found

to yield a 15% increase in drive force showing that force is highly

sensitive to aspect ratio through its effect on kite L/D. Low aspect

ratios may however be favourable to kite stability characteristics.

The angle of attack at which the kite operates modifies the

aerodynamic force generated through its influence on lift and drag

coefficients, but also through its effect on L/D. Optimizing the angle

of attack with respect to both of these yields a¼ 15".

Optimal trajectories were found to be those that are centred

most closely to the best region of the flight envelope. Increasing

maneuver pole circle size and pole separation both diminished the

time averaged propulsive drive force arising. If the potential effect

of reduced CL during tight turns is negligible, the best maneuver is

found to be a figure of eight shape with small circular ends and

small pole separation, centred closely to an elevation of 15".

References

[1] Loyd ML. Crosswind kite power (for large-scale wind power production). AIAAJournal of Energy 1980;4(3):106e11.

[2] Wellicome, JF, Wilkinson S. Propulsive kites e an initial study. University ofSouthampton, Department of Ship Science, Faculty of Engineering and AppliedScience. Tech. Rept. SSSU 19; 1984.

[3] Naaijen P, Koster V, Dallinga RP. On the power savings by an auxiliary kitepropulsion system. International Shipbuilding Progress 2006;53(4):255e79.

[4] Williams P. Optimal wind power extraction with a tethered kite. AIAAGuidance, Navigation, and Control Conference and Exhibit. AIAA, Keystone,Colorado; 2006.

[5] Argatov I, Rautakorpi P, Silvennoinen R. Estimation of the mechanical energyoutput of the kite wind generator. Renewable Energy 2009;34(6):1525e32.

[6] Argatov I, Silvennoinen R. Energy conversion efficiency of the pumping kitewind generator. Renewable Energy 2010;35(5):1052e60.

[7] Dadd GM, Hudson DA, Shenoi RA. Comparison of two kite force models withexperiment. Journal of Aircraft 2010;47(1):212e24.

[8] Claughton AR, Shenoi RA, Wellicome JF. Sailing yacht design theory. 1 ed.Harlow: Addison Wesley Longman; 1998. pp. 5e6.

[9] Matlab. Mathmatics laboratory, software package [Ver. 7.0.0]. Cambridge:MathWorks; 2004.

[10] Abbott IH, Doenhoff AE. Theory of wing sections. New York: Dover Publica-tions; 1959.

[11] Bryant, LW, Brown WS. Collected researches on the stability of kites andtowed gliders. National Phisics Laboratory, Reports and Memoranda 2303;1942.


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Determination of Kite forces for ship propulsion