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Renewable Energy 35 (2010) 1052–1060

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Renewable Energy

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

Energy conversion efficiency of the pumping kite wind generator

I. Argatov*, R. SilvennoinenTampere University of Technology, Department of Mathematics, P.O. Box 553, 33101 Tampere, Finland

a r t i c l e i n f o

Article history:Received 27 February 2009Accepted 13 September 2009Available online 13 October 2009

Keywords:Wind energykite generatorCrosswind motionMechanical powerEnergy conversion efficiency

* Corresponding author.E-mail addresses: ivan.argatov@phpoint.fi (I. Arga

(R. Silvennoinen).

0960-1481/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.renene.2009.09.006

a b s t r a c t

This paper is devoted to analytical modelling and numerical simulation of the so-called pumping kitewind generator whose operating principle is to mechanically drive a groundbased electric generatorusing tethered kites. On the basis of the refined crosswind motion law, the mathematical model of steadycrosswind motion of the tethered kite is constructed. Necessary conditions for optimality of the meanmechanical power functional are derived. Optimal control of the tether length rate is studied for thecases of open-loop and closed-loop figure-of-eight trajectories. The performance coefficient of thepumping kite wind generator is introduced. Simple formulas for the mechanical power output areobtained.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

It is known [1] that the power generated by traditional windturbines does not merely increase linearly with wind speed, butrather by the cube of the wind speed. Hence, doubling the windspeed increases the available power by eight times. However, theEarth’s surface creates a boundary layer effect so that windsgenerally increase with altitude according to a power law at low-altitudes. It is this fact that has lead many researchers to proposevarious concepts for extracting electricity from high altitudes(1–10 km) by means of attempting to locate a wind turbine systemat high altitudes [2]. In recent years several designs have been alsoproposed to collect wind energy at high elevations by means ofkites [3,4].

In this paper, the so-called pumping kite generator concept forextracting the wind energy available in the high altitudes isconsidered (see Fig. 1). The concept’s operating principle is tomechanically drive a groundbased electric generator using a teth-ered kite, instead of attempting to locate a wind turbine system athigh altitudes. On the groundstation the lower portion of the tetheris wound around a drum connected to the generator. Energy isextracted from high altitude by letting the kite fly at a lying-eightorbit with high crosswind speed. During the fast crosswind motionthe kite develops a large pulling force, and thus the generatorgenerates electricity while the kite pulls the tether out of thegroundstation. Then the kite is controlled in such a way that the

tov), risto.silvennoinen@tut.fi

All rights reserved.

pulling force is reduced, and the lower part of the tether is woundback onto the drum using the generator as a motor. This cycle isrepeated, and thus the system is called a pumping kite generator.

The aim of this study is to develop an understanding of thegeneral form of optimal open-loop and closed-loop trajectoriesused in the crosswind power generation.

2. Refined crosswind motion law

The aerodynamic force acting upon the kite can be split into twocomponents known as the lift and drag forces and given by theformulas L¼ 0.5raACLjWej2 and D¼ 0.5raACDjWej2 [5]. Here,We ¼ W � _r is the effective wind vector, W is the wind speedvector, _r is the kite velocity, ra is the density of air, A is the char-acteristic area of the kite, CL and CD are the lift and drag coefficients.

In paper [6], the so-called effective glide ratio was introducedGe¼ L/(Dþ Ft), where, Ft is the drag of the tether acting on the kite.

The parameter Ge, describing the aerodynamic efficiency of thesystem kite – tether, is given by the formula [6,7]

Ge ¼CL

CD þ Ctrd4A

; (1)

where Ct is the normal reaction coefficient, r is the length of thetether, d is the diameter of the tether.

Let Wpe be the projection of the effective wind vector We onto

the plane orthogonal to the tether that is assumed to be a straightline. Then the refined crosswind motion law [7] is given by theformula jWp

e j ¼ GeVk, where Vk ¼er$W is the wind speed

Fig. 1. Pumping kite wind generator concept: 1) kite; 2) tether; 3) drum; 4) gearbox;5) electrical generator; 6) traction phase; 7) recovery phase; 8) trajectory; 9) steeringmechanism.

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–1060 1053

component in the radial direction orthogonal to the flight envelope,i. e., Vk is the wind speed component along the tether.

Neglecting the tether drag acting upon the kite and assumingthat the kite is moving perpendicular to the wind direction, wederive from the refined crosswind motion law the well knownresult obtained in [8].

3. Periodic trajectories

Suppose that the kite moves along a given closed-loop trajectorywith the variable length of the tether. Without loss of generality wemay assume that the trajectory is specified by the parametricequations r¼ r(s), q¼ q(s), f¼ f(s), where r, q, f are sphericalcoordinates, q is the angle that the tether forms with the vertical,the angle f describes the deflection of the tether in the crosswinddirection, [0,2p)Hs is a parameter. Then, we have

ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir02 þ r2ðq02 þ f02sin2 qÞ

qds, where s is the distance along the

trajectory, the stroke denotes the derivative with respect to theparameter s. The unit tangent vector is given by the formulatðsÞ ¼ ðr0er þ rðq0eq þ f0 sin qefÞÞðds=dsÞ, where er(s), eq(s), ef(s)are the unit basis vectors of the spherical coordinate system whichare functions of position.

If the trajectory is given, then at each its point we know the unittangent vector t such that the kite velocity _r ¼ j _rjt.

We can always decompose the velocity _r into a sum of the twovectors _rƒ and _rt, where _rƒ ¼ _rer is the kite’s longitudinal velocity,_rt is the projection of the vector _r onto the tangent plane spannedby eq(s) and ef(s). We have _rt ¼ j _rtjtt, where

ttðsÞ ¼q0eq þ f0 sin q efffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q02 þ f02 sin2 q

q : (2)

Under the condition j _rtjs0, the following relation holds:

_r ¼�� _rt

��r0er þ r�

q0eq þ f0 sin q ef

�rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02 þ f02 sin2 q

q :

Now, denoting by W the wind speed vector as seen from the Earth-fixed coordinate system, we introduce the effective wind vector

We ¼ W � _r. Let Wpe be the projection of the vector We onto the

tangent plane, i. e., Wpe ¼ We � ðer$WeÞer . In view of the formula

Wpe ¼ Wt þ Vƒer , we have Wp

e ¼ Wt � _rt.Assuming that the inequality jWp

e j > jWt � _rtj is satisfiedduring the motion, we can derive the following scalar equality:

j _rtj ¼ Ut þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2

t þ G2eðVƒ � _rÞ2�jWtj2

q(3)

We have for the sake of brevity introduced the notationUt ¼ tt$Wt.

Thanks to the refined crosswind motion law, we will havejWp

e j ¼ GeðVƒ � _rÞ, where Ge is the effective glide ratio, Vk ¼W$er isthe wind speed component in the longitudinal direction. Theperiod of the periodic motion is given by the formula

T ¼ #dsj _rj ¼

Z2p

0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02 þ f02sin2 q

qj _rtj

ds (4)

where j _rtj considered as a function of s.Finally, the time variable t ˛ [0, T) and the dimensionless

parameter s ˛ [0, 2p) are connected by the equality

t ¼Zs

0

rðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02ðsÞ þ f02ðsÞsin2 qðsÞ

qj _rtðsÞj

ds

where j _rtðsÞj considered as a function of the integral variable s isgiven by formula (3).

We emphasize that the resulting formula (4) also holds true inthe case of an open-loop trajectory if the ends of the trajectorycorrespond to the values s¼ 0 and s¼ 2p whereas the quantity Ttakes the meaning of the duration of motion along the trajectoryfrom one end of the trajectory to the other.

4. Mean mechanical power generated by the kite

The kite pulling the tether produces the mechanical energy PM

that is the product of the load force F and the longitudinal velocityof the tether VL. Thus, averaging over the period of open-looptrajectory, we obtain

PM ¼1T

ZT

0

FðtÞVLðtÞdt (5)

where T is given by (4).Taking into account weight of the kite and the tether, we have

F ¼ Faerr � Fgra

r �wƒc (7). Denoting the mass of the kite and the

linear density of the tether by the symbols m and m, we obtainFgra

r ¼ mg cos q and wƒc ¼ mrg cos q. The pulling force generated in

the tether by the kite is given by the formulaFaer

r ¼ 0:5raACLG2eðVƒ � _rÞ2.

When assuming the uniform wind field over the altitude in theflight envelope in the direction of the x axis, the wind speed vectorbecomes W ¼ Vƒer þWt, where Vk ¼ Vwk, Wt ¼ Vwt,wk ¼ sin q cos f, wt ¼ cos q cos feq � sin fef. Notice that, ingeneral, the wind speed V is a function of altitude.

In the case of inelastic tether, we have

_rðtÞ ¼ VLðtÞ; rðtÞ ¼ r0 þZt

0

VLðxÞdx

Changing the integral variable in (5) and making use of thesubstitution VL(t)¼ Vv(s), we obtain

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–10601054

P ¼ 1Z2p

Fðs; vÞvðsÞrðsÞGðs; vÞds; (6)

M T0

where

Gðs; vÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02 þ f02sin2q

qut þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

t þ G2eðwƒ � vÞ2�jwtj2

q ; (7)

Fðs; vÞ ¼ 12

raACLG2eV2ðwƒ � vÞ2�ðmþ mrÞg cos q: (8)

We have introduced the notation ut¼ tt$wt.Moreover, in view of (4) we have

T ¼ 1V

Z2p

0

rðsÞGðs; vÞds;

rðsÞ ¼ r0exp

8<:Zs

0

vðsÞGðs; vÞds

9=;$

The previous formula shows (see also (1)) that the effective glideratio Ge is a function of the longitudinal velocity of the tether.However, considering only one loop, we can assume for the sake ofsimplicity that r and Ge do not depend on VL. Hence, assuming thatr(s) z r0 and Ge¼ const, we obtain

PMðvÞ ¼12

raACLG2eV3JðvÞ; (9)

where we introduced the notation

JðvÞ ¼

Z2p

0

f ðs; vÞvðsÞGðs; vÞds

Z2p

0

Gðs; vÞds

; (10)

where f(s, v)¼ (wk � v)2� c cos q. Observe that c¼ 2(mþ mr)g/raACLGe

2V2 is a dimensionless parameter.

4.1. Simplified mathematical model

In the case of the aerodynamically efficient kite with high liftover drag ratio and optimal relation between the parameters of thekite and the tether, the quantity Ge has a high value. Therefore, themathematical model of wind energy generation (9), (10) can befurther simplified as follows:

PMðvÞ ¼12

raACLG2eV3J0ðvÞ; (11)

J0ðvÞ ¼

Z2p

0

ðwƒ � vÞvhðsÞds

Z2p

0

hðsÞdswƒ � v

: (12)

We have used the notation hðsÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02 þ f02sin2 q

q.

We stress that the simplified mathematical model (11), (12)neglecting weight of the tether is not adequate for estimation of themean generated power in the case of relatively long tethers.

Further, if we take into account the expression for the pullingforce f(s, v), we come to the following formulas:

PMðvÞ ¼12

raACLG2eV3JcðvÞ; (13)

JcðvÞ ¼

Z2p

0

½ðwƒ � vÞ2�k cos q�vwƒ � v

hðsÞds

Z2p

0

hðsÞdswƒ � v

: (14)

It can be readily seen that formulas (13) and (14) turn into (11)and (12) as c goes to zero.

4.2. Periodicity condition

In the case of closed-loop periodic trajectories we shouldimpose the condition

R T0 VLðtÞdt ¼ 0, where T is the period of

periodic motion, or the following one:

Z2p

0

vðsÞrðsÞGðs; vÞds ¼ 0: (15)

In the case of mathematical model (9), (10), Eq. (15) takes theform

Z2p

0

vðsÞGðs; vÞds ¼ 0: (16)

In the case of the simplified mathematical model (11), (12), wehave

Z2p

0

vhðsÞwƒ � v

ds ¼ 0: (17)

Note that the periodicity condition r(T)¼ r0 (11, 10) is equivalentto the condition

R T0 VLðtÞdt ¼ 0, whereas Eqs. (16) and (17) repre-

sent approximate periodicity conditions which can be used forestimation of the mean mechanical power output.

4.3. Refined mathematical model

Finally, we derive the refined mathematical model that takesinto account the variability of the effective glide ratio Ge and theinfluence of the tangential component of the wind speed vectorWt upon the kite velocity. Due to the alternating nature of theterm tt$Wt in motion of the kite along a periodic orbit, we replacethe expression (7) by the following one:

G1ðs; vÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02 þ f02 sin2 q

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtt$wtÞ2þG2

eðwƒ � vÞ2�jwtj2q $ (18)

Hence, in accordance with formula (6), c¼ 0, we obtain

PMðvÞ ¼12

raAV3J1ðvÞ; (19)

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–1060 1055

J1ðvÞ ¼

Z2p

0

CLG2eðwƒ � vÞ2vG1ðs; vÞds

Z2p

0

G1ðs; vÞds

: (20)

Correspondingly, Eq. (16) takes the form

Z2p

0

vðsÞG1ðs; vÞds ¼ 0: (21)

The refined mathematical model (18)–(21) can be applied forstudying the scenarios of power generating cycles assuming thevariability of the kite angle of attack (9) and (10).

5. Necessary conditions for optimality of the meanmechanical power functional

The functionals (10), (12), (14), and (20) can be represented as

JðvÞ ¼

Z2p

0

Fðs; vÞds

Z2p

0

Gðs; vÞds

: (22)

If v < wt, then the functions Fðs; vÞ and Gðs; vÞ have continuousfirst partial derivatives with respect to the variable v.

Consider the Gateaux variation of J at v defined as [12]

dJðv; hÞ ¼ lime/0

Jðvþ ehÞ � JðvÞe

:

We find, after some simple calculations, that

dJðv; hÞ ¼

Z2p

0

F0vðs; vÞhðsÞds� JðvÞZ2p

0

G0vðs; vÞhðsÞds

Z2p

0

Gðs; vÞds

:

Thus, if the functional J(v) has a local extremum at v, then theequation F0vðs; vÞ � JðvÞG0vðs; vÞ ¼ 0 must hold. Observe that theobtained optimality condition can be used in the case of open-looptrajectory only.

Now, considering the optimization problem in the case ofclosed-loop trajectory, we should additionally take into account Eq.(16) or Eq. (17), which can be written as

Z2p

0

Hðs; vÞds ¼ 0: (23)

Recall that equations of the form (23) are called isoperimetricconditions.

Employing the method widely used in variational calculus [13],we can derive the following necessary condition:

F0vðs; vÞ � JðvÞG0vðs; vÞ þ lH0vðs; vÞ ¼ 0: (24)

where l is the generalized Lagrange multiplier that must bedetermined from Eq. (23).

5.1. Necessary optimality conditions for the simplifiedmathematical model

In the case of aerodynamically efficient kite with high value ofthe effective glide ratio Ge, we have (see (12), (17), (22))

F ¼�

wk � v�

vhðsÞ; G ¼ hðsÞwk � v

; H ¼ vhðsÞwk � v

$

Substituting these expressions into Eq. (24), we come to thecubic equation

v3 �5wk

2v2 þ 2w2

kvþJ02�

w3k

2�

lwk2¼ 0$ (25)

Through the substitution v¼ yþ (5/6)wk, Eq. (25) is transformedto the reduced form v3þ pyþ q¼ 0 with the coefficients

p ¼ �w2k

12; q ¼

w3k

108�

lwk2þ J0

2$ (26)

It is known that the number of real roots of the reduced cubicequation depends on the sign of the discriminant D¼ (p/3)3þ (q/2)2. In view of (26) we have

16D ¼w3k

27

�J0 � lwk

�þ�

J0 � lwk�2

$

It is evident from here that D> 0 in the case of open-looptrajectory when l¼ 0. It can be shown that l< 0 in the case ofclosed-loop trajectory and, therefore, in both cases D> 0. Thus, Eq.(25) has one real and two complex conjugate roots.

Since p< 0 (see the first formula (26)), the real root of the cubicequation can be expressed by means of the trigonometric solutionas follows [14]:

y ¼ �2

ffiffiffiffiffiffiffi�p

3

rcosec 2a; (27)

where the trigonometric function is evaluated using the formulas

tan a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffitan

b

23

r; jaj � p

4; sin b ¼ 2

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��p

3

�3r

; (28)

where jbj � p/2 and the real value of the cubic root is used.In view of formulas (26) and (27), we obtain

v ¼ 56

wk

�1� 1þ tan2 a

tan a

�; (29)

where tan a is given by the first formula (28). Moreover, we have

sin b ¼ ð1þ 54w�3k ðJ0 � lwkÞÞ�1.

Formula (29) gives the parametrized solution to Eq. (25). Theparameters J0 and l should be determined from the system of twoequations obtained after substitution of the expression (29) intoEqs. (12), (17).

In the case of the refined simplified mathematical model (13),(14), (17), the necessary condition for optimality of the meanmechanical power functional also takes the form of reduced cubicequation with the coefficients

p ¼ �w2k

12; q ¼

w3k

108�

wk2ðl� ccos qÞ þ Jc

2; (30)

At that, only formula for sin b undergoes an evidenttransformation.

Fig. 2. Similar lying-eight orbits. Fig. 3. Optimal tether length rate and normalized tension force: similar open-looplying figure-of-eight trajectories.

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–10601056

5.2. Necessary optimality conditions for the refined mathematicalmodel

In the case of the variable effective glide ratio Ge, we have (see(20), (21))

Fðs; vÞ ¼ CLG2e

�wk � v

�2vG1ðs; vÞ; Gðs; vÞ ¼ G1ðs; vÞ; Hðs; vÞ

¼ vG1ðs; vÞ:(31)

Here, G1(s, v) is given by formula (18).Differentiating functions (31) with respect to the variable v, we

get

F0vðs; vÞ ¼ CLG2e

��wk � v

� �wk � 3v

�G1ðs; vÞ

þ�

wk � v�2

vvG1

vvðs; vÞ

; (32)

vG1

vvðs; vÞ ¼

G2e

�wjj � v

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02þ f02sin2q

q�ðtt,wtÞ2þG2

e

�wjj � v

�2�jwt

���2�3=2: (33)

H0vðs; vÞ ¼ G1ðs; vÞ þ vvG1

vvðs; vÞ: (34)

Substituting (32)–(34) into Eq. (24), we can derive the followingcubic equation:

v3 �5wk

2v2 þ

�2w2k þ

33

2

�vþ

J1 � lwk2CLG2

e�

w3k

2�

3wk2¼ 0 (35)

Here, 3 ¼ ððtt$wtÞ2 � jwtj2Þ=G2e.

Further, Eq. (35) can be transformed to the reduced form withthe coefficients

Table 1Energy efficiency of the similar open-loop lying-eight trajectories.

n 1 2 3

h 0.914 0.951 0.978102J0 8.792 9.146 9.408

p ¼ �w2k

12þ 33

2; q ¼

w3k

108þ

J1 � lwk2CLG2

eþ 23wk (36)

Since p< 0 and D> 0 for small values of the variable e (seeformula (36)), the solution of Eq. (35) in the case of (36) is againgiven by formulas (27)–(29) where we have

sin b ¼

1� 363

w2k

!3=2 1þ 54

w3k

�J1 � lwk

�CLG2

eþ 2163

w2k

!�1

(37)

Recall that formulas (29), (37) give the parametric solution to Eq.(35). The parameters J1 and l should be determined from thesystem of two equations obtained after substitution of theexpression (29) into Eqs. (20), (21).

6. Numerical results and discussion

In the literature [3,9,11,10], there are known two main scenariosof power generating cycles. Both power generating cycles consist ofthe traction phase in which the tether is reeled out and the recoveryphase in which the tether is reeled in. The traction phase of the firstscenario [3,10] represents the periodic open-loop helical trajectoryconsisting of several similar ‘‘lying-eight’’ orbits. The maneuveringof the kite due to periodic control variations in the kite roll angleinduces changes in the tether tension. When the traction phase iscompleted, the control enters into the recovery phase. In paper [3],it was suggested that the kite is driven to a region where the twotethers can be pulled back spending a small traction of the energygenerated in the traction phase. The recovery phase proposed in[10] assumes that when the tether is reeled in, the kite angle ofattack is reduced to a level that maintains the tension at its lowerbound.

The second scenario [9,11] as a whole represents the closed-loopfigure-of-eight trajectory inclined to the wind direction. In paper[9], the symmetric lying-eight orbit with smooth variation in theangle of attack was adopted. The corresponding control input, the

Table 2Energy efficiency of the vertically stretched open-loop lying-eight trajectories.

n 1 4 5

h 0.914 0.921 0.928102J0 8.792 8.858 8.925

Fig. 4. Inclined trajectories.

Table 3Energy efficiency of the horizontally stretched open-loop lying-eight trajectories.

n 1 6 7

h 0.914 0.945 0.968102J0 8.792 9.090 9.316

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–1060 1057

length rate, and the tension in the tether show two equal peaksduring the period. The nonsymmetric figure-of-eight orbit withnon-smooth control of the kite lift coefficient was used in [11]. Thekite lift coefficient is maintained at the maximum value when thetether is reeled out in the traction phase. In the recovery phasecorresponding to the outer region of the orbit, the kite lift coeffi-cient is reduced to the minimum value. Thus, the length rate showsonly one peak during the period.

6.1. Open-loop trajectories

Let us consider three lying-eight orbits (see Fig. 2) defined bythe equations q¼ q0þDqn sin 2s, f¼Dfn sin s (n¼ 1, 2, 3). Weassume that they are similar, i. e., one can be obtained from theother by uniformly stretching. In particular, we take q0¼ 60

�,

Dq1¼10�, Df1¼20

�. Below, we shall use the angle of inclination of

the tether with respect to the horizon w ¼ 90� � q, where q is the

angle that the tether forms with the vertical.Recall that the value (1/3) Vk represents the instantaneous

optimal value of the tether’s longitudinal speed obtained in (8) inthe case of maximizing the instantaneous value of the mechanicalpower generated by the deploying kite performing crosswindmotion. It can be shown that the optimal solution v*(s) tends to thevalue (1/3) wk evaluated at the point q¼ q0, f¼ 0, if the trajectoryshrinks to this point. Notice that wk ¼ sin q0 at q¼ q0, f¼ 0. Thus,representing formula (11) in the form

PM ¼2

27raACLG2

eV3cos3 w0h;

We introduce the performance coefficient of the kite generatorh. For arbitrary trajectory, we have h< 1. Observe that if we puth¼ 1, then this formula turns into the corresponding resultobtained in [7].

Table 1 contains numerical results obtained on the basis of thesimplified mathematical model (11), (12). Fig. 3 shows the optimaltether length rates corresponding to the three similar open-looptrajectories (2 and 3) depicted in Fig. 2.

Thus, we see that the maximum mechanical power generated bythe deploying kite decreases if the range of the trajectory increases.Similar trends in the results occur if only one of the trajectory’ssizes is varied as in the cases of vertically stretched trajectories (4and 6) and horizontally stretched trajectories (6 and 7). Physically,the observed dependence means that the deploying kite mustoperate in the so-called power zone in order to achieve themaximum value of power production. This is quantified in Table 1,which shows the normalized energy generated per cycle, J0, and theopen-loop trajectory performance coefficient, h.

Fig. 3 also shows the results of calculations of the normalizedtension force for the varying parameters of lying-eight trajectory. Itis interesting to note that the maximum value of f*(s) remains

Table 4Energy efficiency of the inclined open-loop figure-of-eight trajectories (Fig. 4).

n 1 8 9

h 0.914 0.927 0.946102J0 8.792 8.916 9.107

constant whereas its minimum value changes significantly inaccordance with variation in minimum of the tether length rate.

A parametric study of the effects of the kite trajectory’s shapeupon the optimal length rate was undertaken (see Tables 2 and 3). Itis shown that the performance coefficient h appears to be slightlyeffected by the mean angle of inclination of the tether w0. It is alsonoticeable that the optimal tether length rates for the threedifferent cases of horizontally stretched trajectories (1, 6, and 7) arevery similar.

Fig. 5 and Table 4 represent results of numerical calculationscorresponding to the three trajectories (1, 8, and 9) depicted inFig. 4. It is interesting that an inclination of the figure-of-eighttrajectory leads to greater power generation. Observe that thevertical figure-of-eight trajectory (n¼ 9) possesses only onemaximum of the optimal length rate function in contrast to theinitial horizontal trajectory (n¼ 1) having two equal maxima perperiod.

If we take into account weight of the tether, we can suggest thefollowing formula:

PM ¼2

27raACLG2

eV3cos3 w0

�1� 9k

4sin w0

cos2 w0

�h:

Here, h is the performance coefficient of the kite generator (weuse the same notation as before). Note that this formula with h¼ 1

Fig. 5. Comparison of the optimal tether length rates: inclined open-loop figure-of-eight trajectories.

Fig. 6. Optimal tether length rate for the lying figure-of-eight open-loop trajectories inthe case of the heavy tether.

Table 5Energy efficiency of the open-loop lying-eight trajectory in the case of the heavytether.

c 0 0.05 0.1 0.2

h 0.914 0.915 0.920 0.949102J0 8.792 8.140 7.525 6.395

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–10601058

represents the corresponding result obtained in (7). This explainsthe paradoxical fact that the performance coefficient h increaseswith increasing value of the parameter c (Figs. 6 and 7).

Table 5 represent results of numerical calculations corre-sponding to the trajectory depicted in Fig. 2 (n¼ 1).

6.2. Closed-loop orbits with constant effective glide ratio

It is clear from Table 6 that cycles with the greater tether lengthrate variation (corresponding to the orbits with the greater range)are better from the point of view of energy generation in the case ofclosed-loop trajectory with constant effective glide ratio.

Numerical results obtained on the basis of the mathematicalmodel (11), (12), (17) are summarized in Table 6.

6.3. Closed-loop orbits with variable effective glide ratio

Finally, we consider the case of variable lift and drag coefficientsalong the closed-loop trajectory. First, we define the lift and drag ofthe kite in terms of its angle of attack as follows (9): CL¼ CL0þ CLaa

and CD¼ CD0þ KCL2. Here, CL0 is the zero-angle-of-attack lift coef-

ficient, CLa is the lift curve slope, CD0 is the zero-lift drag coefficient,

Fig. 7. Optimal tether length rate: inclined figure-of-eight closed-loop trajectories withconstant effective glide ratio.

K is the lift-induced drag coefficient. In numerical calculations weput CL0¼ 0, CLa¼ 5 rad�1, CD0¼ 0.02, K¼ 0.08.

Second, in order to calculate the effective glide ratio by means offormula (1), we need to specify the tether-kite system parameters:r¼ 1060 m, d¼ 5 mm, A¼ 50 m2, Ct ¼ 1:0 (9). It is important tonote that the precise values of r, d, and A are not important.

Third, we introduce the following longitudinal wind-inducedcontrol law: a(s)¼ A0wk(s)� A1, where the coefficients A0 and A1

are given by the formulas

A0 ¼amax � amin

wmaxjj �wmin

jj; A1 ¼

amaxwminjj � aminwmax

jj

wmaxjj �wmin

jj:

Here, amin and amax are prescribed minimal and maximal valuesof the angle of attack (these values are determined by the kite’sconstruction), wkmin and wkmax are minimal and maximal values ofthe longitudinal wind speed (these values are determined by theshape and sizes of the orbit).

Fig. 8 shows the variation in the effective glide ratio for lying-eight orbit. Observe that the quantity Ge(s) has two platforms ofmaximal values in contrast to the lift coefficient CL(s) having twospike maxima. Fig. 8 corresponds to the following parameters of thelying figure-of-eight orbit: q0¼ 49

�, Df¼ 17.5

�, Dq¼ 4.5

�(9). We

emphasize that the relative variation in the tension force in Fig. 8 ismuch less than that obtained in (9). This fact is a consequence of thesimple approximation for the tether length rate a priori adopted in(9) whereas Fig. 8 shows that the optimal tether length rate curve isfar from being sine-shaped.

Notice that in the case of similar lying-eight trajectories depic-ted in Fig. 2, the almost identical behavior of the optimal lengthrates curves was observed. This fact can be correlated with thevariation in angle of attack of the kite to show that the tether isreeled out when the angle of attack is high. Observe that there isa significant drop in the length rate curves when the tether is reeledback in. In fact, the absolute value of the minimum of v*(s) is twotimes less than the maximum of v*(s).

Table 7 contains numerical results obtained on the basis of therefined mathematical model (19)–(21). In order to get a possibilityto compare these results with the corresponding results in the caseof open-loop trajectories, we introduce the following normalizedvalue of energy generated per cycle: J0* ¼ J1/CL

*(Ge*)2, where CL

* is themaximum of the variable lift coefficient CL(s), Ge

* is the value of theeffective glide ratio corresponding to CL

*. Thus, the values CL* and Ge

*

correspond to the same moment of time.Introducing the performance coefficient of the kite generator h*

in the case of closed-loop orbit, we represent formula (19) in theform

PM ¼2

27raACLG2

eV3cos3 w0 h*:

Table 6Energy efficiency of the similar open-loop lying and inclined (Fig. 4) figure-of-eightorbits with constant effective glide ratio.

n 1 2 3 8 9

h 0.015 0.008 0.004 0.036 0.057102J0 1.494 0.831 0.367 3.622 5.672

Fig. 10. Angle of attack a, lift and drag coefficients CL and CD, effective glide ratio Ge forinclined figure-of-eight orbit.

Fig. 8. Angle of attack a, lift and drag coefficients CL and CD, effective glide ratio Ge,optimal tether length rate, and normalized tension force: lying-eight orbit with vari-able effective glide.

Table 7Energy efficiency of the similar closed-loop lying (Fig. 2) and inclined (Fig. 4) closed-loop figure-of-eight orbits with variable effective glide ratio.

n 1 2 3 8 9

h* 0.203 0.191 0.177 0.211 0.288102J0

* 1.951 1.833 1.706 2.031 2.767

I. Argatov, R. Silvennoinen / Renewable Energy 35 (2010) 1052–1060 1059

Fig. 9 and Table 7 show the results corresponding to the threeorbits depicted in Fig. 4. It is apparent that the trends observed inthe configurations with the constant effective glide ratio are fol-lowed for the case of variable angle of attack, i. e., the power outputincreases with inclination of the figure-of-eight due to thereplacement of the functional dependencies (for the length rateand tension force) with two equal maxima by the functionaldependencies with only one maximum.

Finally, we consider a nonsymmetric inclined figure-of-eightorbit (see trajectory 10 depicted in Fig. 4), which is analogous tothat considered in (11) in the case of non-smooth control of the kitelift coefficient. In the case of smooth control of the kite’s angle ofattack in accordance with the longitudinal wind-induced control,

Fig. 9. Optimal tether length rate: inclined closed-loop trajectories with variableeffective glide ratio

the variation in the effective glide ratio is shown in Fig. 10. Observethe existence of the almost flat platform in the Ge-curve.

In the case where the angle of attack is assumed to be variable,the normalized tension force ~f *ðsÞ is given by

~f *ðsÞ ¼CLðsÞðGeðsÞÞ2

C*L

�G*

e

�2

�wjjðsÞ � v*ðsÞ

�2:

Note that the actual value of the tension force is proportional tothe square of the relative wind speed approximated by the formulaG2

eV2ðwk�v*Þ2.

Fig. 10 also shows that the duration of the recovery phase istwice less than the duration of the traction phase. Our calculationsgive the results 102J0* ¼ 2.164 and h*¼ 0.171.

7. Conclusions

Optimal control of the tether length rate for the pumping kitewind generator using crosswind motion along the given kite’strajectory was considered for optimizing the mean mechanicalpower output. Open-loop and closed-loop figure-of-eight trajec-tories were studied on the basis of the constructed mathematicalmodels of steady crosswind motion of the kite along a prescribedspatial trajectory.

The obtained results are summarized in the three formulas forthe mechanical power output of the pumping kite generator PMgiven in the previous section. These formulas show that the poweroutput increases with the cube of the wind speed at the kiteoperating altitude.

The performance coefficient of the pumping kite wind generatorwas introduced. For representative solutions we have h¼ 0.9 (in thecase of open-loop trajectory we have in average) and h*¼ 0.2 (in thecase of closed-loop orbit with smoothly variable angle of attack).

Acknowledgments

The authors would like to express their gratitude to the anon-ymous referee for his/her helpful comments.

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