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Control of tethered airfoils for a new class of wind energy generator

M. Canale, L. Fagiano, M. Ippolito, M. Milanese

Abstract— The paper investigates the control of tethered

airfoils in order to devise a new class of wind generators able to

overcome the main limitations of the present aeolian technology

based on wind mills. A model taken from the literature is

used to simulate the dynamic of a kite which can be controlled

by suitably pulling two lines. Energy is generated by a cycle

composed of two phases, indicated as the traction and the

recovery one. The control unit has two electric drives which act

as motors in pulling the lines for controlling the ﬂight or for

recovering the kite and as generators if the kite pulls the lines.

In each phase, control is obtained by “fast” implementations

of suitable NMPC designs. In the traction phase the control

is designed such that the kite pulls the lines, maximizing the

amount of generated energy. When the maximal length of the

lines is reached, the control enters the recovery phase and the

kite is driven to a region where the lines can be pulled by

the motors until the minimal length is reached, spending a

small fraction of the energy generated in the traction phase,

and a new traction phase is undertaken. Simulation results are

presented, related to a small scale prototype whose construction

is undergoing at our laboratory.

I. INTRO DUC TIO N

Energy generation has become an urgent, strategic issue at

the global scale. At present, about 80% of world electric

energy is produced from thermal plants making use of fossil

sources (oil, gas, coal). The economical, geopolitical and

environmental problems related to such sources are becoming

every day more and more evident. Nuclear plants have their

own problems related to security aspects and radioactive

waste management. For these reasons it is of primary im-

portance for the scientiﬁc community to explore and support

new renewable energy technologies, able to provide more

efﬁcient solutions than the existing ones to the severe energy

shortage of the planet.

Wind turbines are currently the largest source of electric

power produced with renewable energy (excluding hydro

power plants) [1]. However, they require heavy towers, foun-

dations and huge blades, which make a signiﬁcant impact on

the environment, require massive investments and long-term

amortization periods. All these problems reﬂect in electric

energy production costs that are not yet competitive, in strict

economic sense, with the ones of thermal generators, despite

recent large rises of oil and gas prices. Moreover, wind farms

have wide problems of social acceptance due their territory

Supported by Regione Piemonte under the Project “Controllo di aquiloni

di potenza per la generazione eolica di energia” and by Ministero

dell’Università e della Ricerca of Italy, under the National Project “Robust-

ness and optimization techniques for high performances control systems”.

M. Canale, L. Fagiano and M. Milanese are with Dip. di Automatica e

Informatica, and Respira Lab, Politecnico di Torino, Italy. M. Ippolito is

with Sequoia Automation and Respira Lab, Politecnico di Torino, Italy

Email: massimo.canale@polito.it, lorenzo.fagiano@polito.it,

m.ippolito@sequoiaonline.com, mario.milanese@polito.it

occupation, which is unacceptably higher than for thermal

plants of the same power (up to 200-300 times).

This paper illustrates a ﬁrst study conducted at Respira Lab,

a laboratory founded at Politecnico di Torino to design and

build a new class of wind generator, indicated as KiteGen,

aimed to overcome the above limitations. The key idea

(patented by one of the authors, [2]) is to capture wind

energy by means of tethered airfoils (e.g. power kites used

for surﬁng or sailing, see Fig. 1), whose ﬂight is suitably

driven by an automatic control unit. It is expected that wind

Fig. 1. A kite surfer

generator of this type will have a territory occupation much

lower than a wind farm of the same power (by a factor

up to 50-100) and much lower electric energy production

costs (by a factor up to 10-20). The reasons of such a

dramatic potential improvements are that for both generators

the energy captured from the wind approximately grows with

the cube of wind speed and linearly with the wind intercepted

front area. Assuming that the airfoils can ﬂy up to several

hundred meters of altitude, KiteGen can take advantage of

the fact that as altitude on the ground increases, wind is

stronger and more constant, see Fig. 2. For example, at 800

m the wind speed doubles with respect to 100 m (mean

altitude at which the largest wind mill operate) and, being

all other parameters constant, the generated power is eight

times greater. Moreover, with lines several hundred meters

long, the airfoils can intercept a wind front area much larger

than the intercepted area of wind mills, which for structural

limits cannot go beyond the size intercepted by 90 m rotor

diameters of big 5 MW towers (at present suitable only for

off-shore installations, see [3]).

As a ﬁrst step, the KiteGen project will realize a small scale

prototype (see Fig. 3) to show the capability of controlling

the ﬂight of a single kite, by pulling the two lines which hold

it, in such a way to extract a signiﬁcant amount of energy.

In this paper we investigate such a capability in simulation,

Fig. 2. Variation in the wind speed, as a function of the altitude, based

on the average European wind speed (3 m/s at ground level). Source: Delft

University, prof. Wubbo Ockels

Fig. 3. KiteGen small scale prototype

employing the kite model used in [4]. The overall maneuver

is composed of two phases: the traction and the recovery

ones. The control unit (see Fig. 4) has two electric actuators

which act as motors in pulling the lines for controlling the

ﬂight or for recovering the kite and as generators if the lines

length increases when pulled by the kite. In the traction phase

the control is designed such that the kite pulls the lines,

so that a certain amount of energy is generated. When the

maximal length of the lines is reached, the control enters into

the recovery phase, where the kite is driven to a region where

the lines can be pulled by the motors until the minimal length

is reached, spending a small fraction of the energy generated

in the traction phase and a new traction phase is undertaken.

The control design is here carried on using a Fast imple-

mentation of a Predictive Controller (FMPC) as proposed

in [5] and [6]. Indeed, the design is formulated as an

optimization problem (power maximization in the traction

phase and minimization in the recovery phase) with state

and input constraints, since for example the kite height on

the ground cannot be negative and lines pulling and its rate

of variation cannot exceed given limits. From this point of

view, Model Predictive Control (MPC) appears to be an

appropriate technique. However, a “fast” implementation is

Fig. 4. KiteGen control unit

needed, in order to allow the real time control at the required

sampling time (of the order of 0.1 s). It can be noted that

MPC technique has been employed in [4] for kite control.

However, in [4], the concern was not on energy generation,

but the aim was to stabilize a given unstable periodic orbit

that can be obtained for constant line length. In this paper,

in order to investigate the energy extraction, the line length

is not supposed to be constant and no kite trajectory is

preassigned to be tracked. The derived kite trajectory is on

the contrary determined by the optimization of the generated

energy.

The paper is organized as follows. In Section II the used kite

model is presented. In Section III the MPC setup is formu-

lated and the control speciﬁcations are introduced; in Section

IV the used technique for the fast control implementation is

brieﬂy described. The simulation results related to the small

scale prototype whose construction is undergoing at Respira

Lab are presented in Section V. Some conclusions and future

developments are reported in Section VI.

II. KIT E MODEL

The model originally developed in [4] is employed to

describe the kite dynamics. A ﬁxed cartesian coordinate

system (X, Y, Z)is considered, with Xaxis aligned with

the nominal wind speed vector direction. Wind speed vector

is represented as ~

Wl=~

W0+~

Wt, where ~

W0is the nominal

wind, supposed to be known and expressed in (X, Y, Z )as:

~

W0=

Wx(Z)

0

0

(1)

Wx(Z)is a known function which gives the wind nominal

speed at a certain altitude Z. The term ~

Wtmay have

components in all directions and is not supposed to be

known, accounting for wind unmeasured turbulence.

A spherical coordinate system is also considered, centered

where the kite lines are constrained to the ground. In this

system, the kite position is given by its distance rfrom the

origin and by the two angles θand φ, as depicted in Fig. 5,

which also shows the three basis vectors eθ,eφand erof

a local coordinate system. These basis vectors are expressed

in the ﬁxed cartesian system (X, Y, Z )by:

¡eθeφer¢=

cos (θ) cos (φ)−sin (φ) sin (θ) cos (φ)

cos (θ) sin (φ) cos (φ) sin (θ) sin (φ)

−sin (θ) 0 cos (θ)

(2)

φ

θ

r

X Y

Zer

eθ

eφ

Fig. 5. Spherical and local coordinate systems

Applying Newton’s laws of motion in the local coordinate

system, the following equations are obtained:

r¨

θ−rsin (θ) cos (θ)˙

φ2+ 2 ˙

θ˙r=Fθ

m

rsin (θ)¨

φ+ 2rcos (θ)˙

φ˙

θ+ 2 sin (θ)˙

φ˙r=Fφ

m

¨r−r˙

θ2−rsin2(θ)˙

φ2=Fr

m

(3)

where mis the kite mass. The external forces Fθ,Fφ

and Frinclude the contributions of gravitational force mg,

aerodynamic force ~

Faer and force Fcexerted by the kite on

the lines. Their relations, expressed in the local coordinates,

are given by:

Fθ= sin (θ)mg +Faer

θ

Fφ=Faer

φ

Fr=−cos (θ)mg +Faer

r−Fc

(4)

where forces are assumed positive in the direction of basis

vectors eθ,eφand er.

~

Faer depends on the effective wind speed ~

We, which in the

local system is computed as:

~

We=~

Wa−~

Wl(5)

~

Wais the kite speed, expressed in the local system as:

~

Wa=

˙

θr

˙

φr sin θ

˙r

(6)

Let us consider now the kite wind coordinate system, with

the origin in the kite center of gravity, ~xwbasis vector

aligned with the effective wind speed vector, ~zwbasis vector

contained by the kite longitudinal mirror symmetry plane and

pointing from the top surface of the kite to the bottom, and

wind ~ywbasis vector completing the right handed system.

In the wind coordinate system the aerodynamic force ~

Faer

wis

given by: ~

Faer

w=FD~xw+FL~zw(7)

where FDis the drag force and FLis the lift force, computed

as:

FD=−1

2CDAρ|We|2

FL=−1

2CLAρ|We|2(8)

where ρis the air density, Ais the kite characteristic area,

CLand CDare the kite lift and drag coefﬁcients. All of

these variables are supposed to be constant. ~

Faer can then

be expressed in the local coordinate system as a nonlinear

function of several arguments:

~

Faer =

Faer

θ(θ, φ, r, ψ, ~

We)

Faer

φ(θ, φ, r, ψ, ~

We)

Faer

r(θ, φ, r, ψ, ~

We)

(9)

Angle ψindicated in (9) is the control variable, deﬁned by

ψ= arcsin µ∆l

d¶(10)

with dbeing the distance between the two lines ﬁxing points

at the kite and ∆lthe length difference of the two lines.

Angle ψinﬂuences the kite motion by changing the direction

of the vector ~

Faer.

As the kite can exert on the lines positive forces only, force

Fcis such that Fc≥0. Moreover, it is considered that

Fcis measured and that, using a local controller of the

drives, it is regulated in such a way that ˙r(t)≈˙rref (t)

where ˙rref(t)is suitably chosen. It results that Fc(t) =

Fc(θ, φ, r, ˙

θ, ˙

φ, ˙r, ˙rref,~

We). Since the power is P= ˙rFc,

in the traction phase ˙rref (t)>0is chosen and a positive

power is generated, the kite getting farther from the origin

and the drives acting as generators. In the recovery phase,

˙rref(t)<0is required, since the lines length has to be

reduced: a negative power results, to be provided by the

drives acting as motors.

Thus the system dynamics are of the form:

˙x(t) = g(x(t), u(t), Wx(t),˙rref(t),~

Wt(t)) (11)

where x(t)=[θ(t)φ(t)r(t)˙

θ(t)˙

φ(t) ˙r(t)]Tand u(t) =

ψ(t). All the model states are supposed to be measured, to

be used for feedback control.

III. KITE CONTROL USING MPC

Control problem and related objectives are now described.

As highlighted in the Introduction, the main objective is to

generate energy by a suitable control action on the kite. In

order to accomplish this aim, a two-phase cycle has been

deﬁned. The two phases are referred to as the traction phase

and the recovery phase. For the whole cycle to be generative,

the total amount of energy produced in the ﬁrst phase has to

be greater than the energy spent in the second one to recover

the kite before starting another cycle. In both phases, MPC

controllers are designed, according to their own functional,

state and input constraints and terminal conditions.

The control move computation is performed at discrete time

instants deﬁned on the basis of a suitably chosen sampling

period ∆t. At each sampling time tk=k∆t,k∈Z+, the

measured values of the state x(tk)and of the wind speed

Wx(tk), together with the chosen value of the reference

speed ˙rref(tk), are used to compute the control move through

the optimization of a performance index of the form:

J(U, tk, Tp) = Ztk+Tp

tk

L(˜x(τ),˜u(τ), Wx(τ),˙rref(τ))dτ

(12)

where Tp=Np∆t, Np∈Z+is the prediction horizon, ˜x(τ)

is the state predicted inside the prediction horizon according

to the state equation (11), using ~

Wt(t) = 0,˜x(tk) = x(tk)

and the piecewise constant control input ˜u(t)belonging to

the sequence U={˜u(t)}, t ∈[tk, tk+Tp]deﬁned as:

˜u(t) = ½¯ui,∀t∈[ti, ti+1], i =k,...,k+Tc−1

¯uk+Tc−1,∀t∈[ti, ti+1], i =k+Tc, . . . , k +Tp−1

(13)

where Tc=Nc∆t, Nc∈Z+, Nc≤Npis the control

horizon.

As it will be discussed later, the function L(·)in (12) is

suitably deﬁned on the basis of the performances to be

achieved in the operating phase the kite lies in. Moreover,

in order to take into account physical limitations on both

the kite behaviour and the control input ψin the different

phases, linear constraints of the form F˜x(t) + G˜u(t)≤H

have been included too.

Thus the predictive control law is computed using a receding

horizon strategy:

1) At time instant tk, get x(tk).

2) Solve the optimization problem:

min

UJ(U, tk, Tp)(14a)

subject to

˙

˜x(t) = g(˜x(t),˜u(t), Wx(t),˙rref(t)) (14b)

F˜x(t) + G˜u(t)≤H, ∀t∈[tk, tk+Tp](14c)

3) Apply the ﬁrst element of the solution sequence Uto

the optimization problem as the actual control action

u(tk) = ˜u(tk).

4) Repeat the whole procedure at the next sampling time

tk+1.

Therefore the predictive controller results to be a nonlinear

static function of the system state x, the nominal measured

wind speed Wxand the reference ˙rref :

ψ(tk) = f(x(tk), Wx(tk),˙rref(tk)) (15)

A. Traction phase

The aim of this phase is to obtain as much mechanical energy

as possible from the wind stream. The following initial state

value ranges are considered to start the traction phase:

θI≤θ(t)≤θI

|φ(t)|≤φI

rI≤r(t)≤rI

(16)

with

0<θI<θI<π/2

0<φI<π/2(17)

Roughly speaking, the traction phase begins when the kite

is ﬂying in a symmetric zone with respect to the Xaxis,

at an altitude ZIsuch that (rIcos θI≤ZI≤rIcos θI).

When the traction phase starts, a positive value ˙rof ˙rref is

set so that the kite ﬂies with increasing values of rwhile

applying a traction force Fcon the lines, thus generating

mechanical power. The value ˙ris chosen to get a good

compromise between obtaining high traction force actions

and high line winding speed values. Basically, the stronger

the wind, the higher the values of ˙rthat can be set obtaining

high force values. Control system objective adopted in the

traction phase is to maximize the energy generated in the

interval [tk, tk+TP], while satisfying constraints concerning

state and input values. Mechanical power generated at each

instant is P= ˙rF c, thus the following cost function is

chosen to be minimized in MPC design (14):

J(tk) = −Ztk+Tp

tk

( ˙r(τ)Fc(τ))dτ (18)

During the whole phase the following state constraint is

considered to keep the kite sufﬁciently far from the ground:

θ(t)≤θ(19)

with θ< π/2. Actuator physical limitations give rise to the

constraints:

|ψ(t)| ≤ ψ

|˙

ψ(t)|≤˙

ψ(20)

To complete the traction phase description, ending condi-

tions have to be introduced. Each kite line is initially wrapped

around a pulley and unrolls while the kite gets farther. When

rreaches a value rit is necessary to wrap the lines back,

in order to make the KiteGen prototype able to start a new

cycle. Therefore, when the following condition is reached

the traction phase ends and the recovery phase can start:

r(t) = r(21)

B. Recovery phase

During this phase the kite lines must be wrapped back using

the least amount of energy, in order to maximize the net

energy gain of the whole cycle. The recovery phase has been

divided into three sub-phases.

In the ﬁrst sub-phase, ˙rref(t)is chosen to smoothly decrease

towards zero from value ˙r.The control objective is to move

the kite in a zone with low values of θand high values of |φ|,

where effective wind speed ~

Weand force Fcare low and

the kite is ready to be recovered with low energy expense.

Positive values θIIand φIIof θand φrespectively are

introduced to identify this zone. The following cost function

is considered:

J(tk) = Ztk+Tp

tk

θ2(τ) (|φ(τ)| − π/2)2dτ (22)

Once the following condition is reached:

|φ(t)| ≥ φII

θ(t)≤θII

(23)

the ﬁrst recovery part ends.

When the second recovery sub-phase begins, ˙rref(t)is chosen

to smoothly decrease from zero to a negative constant value

˙r.Such a value is chosen to give a good compromise between

high winding back speed and low Fcvalues.

During this second recovery sub-phase, control objective is

to minimize the energy spent to wind back the lines, thus

the following cost function is considered:

J(tk) = Ztk+Tp

tk

|˙r(τ)|Fc(τ)dτ (24)

The second sub-phase ends when the following condition is

satisﬁed:

rI≤r(t)≤rI(25)

which means when ris among the possible traction phase

initial state values. Then, the third recovery sub-phase begins

and ˙rref(t)is chosen to smoothly increase towards zero from

the negative value ˙r.Control objective is to move the kite

in the traction phase starting zone, expressed by (16). Cost

function J(tk)is set as follows:

J(tk) = Ztk+Tp

tk

(|θ(τ)−θ1|+|φ(τ)|)dτ (26)

where θ1= (θI+θI)/2.Ending conditions for the whole

recovery phase coincide with starting conditions for the

traction phase.

During the whole recovery phase the state constraint ex-

pressed by (19) and the input constraints (20) are considered

in the control optimization problems.

IV. “FAST” MPC IMP LE M EN TATIO N

For any of the MPC controller previously described, control

ψ(tk)results to be the nonlinear static function given by

(15), which can be rewritten as:

ψ(tk) = f(w(tk))

Where w(tk) = (x(tk), Wx(tk),˙rref(tk))T. For a given

w(tk), the value of the function f(w(tk)) is typically com-

puted by solving at each sampling time tkthe constrained

optimization problem (14). However, an online solution of

the optimization problem at each sampling time cannot be

performed at the sampling period required for this appli-

cation, of the order of 0.1 s. For example, the average

computation time of MPC implementation proposed in [4],

for the ﬂight control of the same kite model in a simpler

situation (periodic orbit with constant r), resulted to be about

0.45 s on a workstation. An approach to overcome this

problem is to evaluate off line a certain number of values

of f(w)to be used to ﬁnd an approximation ˆ

fof f, suitable

to be used for online implementation.

To be more speciﬁc, consider a bounded region W⊂R8

where wcan evolve.

A number νof values of f(w)may be derived by performing

ofﬂine the MPC procedure starting from a set of values

Wν={˜wk∈W, k = 1, . . . , ν } , so that:

˜

ψk=f( ˜wk), k = 1, . . . , ν (27)

The aim is to derive, from these known values of ˜

ψkand ˜wk

and from known properties of f, an approximation b

fof f

and a measure of the approximation error. Neural networks

have been used in [7] and [8] for such approximation.

The problems with neural networks are the trapping in

local minima during the learning phase and the difﬁculty of

handling the constraints in the image set of the function to be

approximated. Moreover, no measure of the approximation

error is provided. In order to overcome these drawbacks,

a Set Membership approach has been proposed in [5] for

an MPC formulation involving linear models. Basic to this

approach is the observation that in order to derive a measure

of the approximation error achieved by any method, the

knowledge of f( ˜wk), k = 1, . . . , ν is not sufﬁcient, but

some additional information on fis needed. In this paper it

is assumed that f∈ Fγ, where Fγis the set of all Lipschitz

functions on W, with Lipschitz constant γ. Note that stronger

assumptions cannot be made, since even in the simple case

of linear dynamics and quadratic functional, fis a piece-

wise linear continuous function [9], [10]. An additional

information to be used in the approximation is the input

saturation condition giving |f(w)| ≤ ¯

ψ. These information

on function f, combined with the knowledge of the value of

the function at the points ˜wk∈W, k = 1, . . . , ν, allows

to conclude that f∈F F S, where the set F F S (Feasible

Functions Set), deﬁned as:

FFS ={f∈ Fγ:|f(w)| ≤ ¯

ψ;f( ˜wk) = ˜

ψk, k = 1, . . . , ν }

(28)

summarizes the overall information on f. Making use of

such overall information, Set Membership theory allows

to derive an optimal estimate of fand its approximation

error, in term of the Lp(W)norm p∈[1,∞], deﬁned as

||f||p.

=£RW|f(w)|pdw¤1

p,p∈[1,∞)and ||f||∞.

=ess-

supw∈W|f(w)|.For given b

f≈f, the related Lpapprox-

imation error is °

°

°f−ˆ

f°

°

°p. This error cannot be exactly

computed, but its tightest bound is given by:

°

°

°f−ˆ

f°

°

°p≤sup

˜

f∈F SS T°

°

°˜

f−ˆ

f°

°

°p

.

=E(b

f)(29)

where E(b

f)is called (guaranteed) approximation error.

A function f∗is called an optimal approximation if:

E(f∗) = inf

ˆ

f

E(b

f).

=rp

The quantity rp, called radius of information, gives the

minimal Lpapproximation error that can be guaranteed.

Let us deﬁne:

f(w).

=min ·¯

ψ, min

k=1,...,ν ³˜

ψk+γkw−˜wkk´¸

f(w).

=max ·−¯

ψ, max

k=1,...,ν ³˜

ψk−γkw−˜wkk´¸(30)

It results that the function:

f∗(w) = 1

2[f(w)+f(w)] (31)

is an optimal approximation for any Lp(W)norm, with p∈

[1,∞][6].

Moreover, the approximation error of f∗is pointwise

bounded as:

|f(w)−f∗(w)| ≤ 1

2|f(w)−f(w)|,∀w∈W

and is pointwise convergent to zero:

lim

ν→∞ |f(w)−f∗(w)|= 0,∀w∈W(32)

Thus, evaluating supw∈W|f(w)−f(w)|,it is possible to

decide if the chosen νis sufﬁcient to achieve a desired

accuracy in the estimation of for if νhas to be increased.

An estimate bγof γcan be derived as follows:

bγ= inf

γ:f(˜wk)≥˜

ψk, k=1,...,ν

γ(33)

Such estimate is convergent to γ:

lim

ν→∞ bγ=γ(34)

Note that convergence results (32) and (34) hold if

limν→∞ d(Wν, W )=0, where d(Wν, W )is the Hausdorff

distance between sets Wνand W. Such a condition is

satisﬁed if, for example, Wνis obtained by uniform gridding

of W.

Thus, the MPC control can be approximately implemented

online, by simply evaluating the function f∗(wtk)at each

sampling time:

ψtk=f∗(wtk)

Increasing νthe approximation error decreases at the cost

of increased computing time. In the numerical case reported

below, using ν= 10000 the error in computing ψtis less

than 1% in all operating conditions. The corresponding mean

time required for the computation of control move f∗(wtk)

implemented on dSpacerMicroAutobox (800 MHz clock) is

about 0.01 s, largely lower than the required sampling time.

V. SI MUL ATI ON R ESU LTS

The results of three simulations are presented here. The

values of model and control parameters are reported in Table

I. Table II contains the state values which identify each phase

starting and ending conditions and the values of state and

input constraints.

Note that since the lines unroll at a speed of 0.5 m/s for

about 180 m, the traction phase lasts about 360 s. In the

recovery phase the lines wrapping back speed is -2.3 m/s,

thus the second recovery sub-phase lasts about 80 s. The

nominal wind speed is given as (1):

Wx(Z) = ½0.02Z+ 4 if Z≤100m

0.0086(Z−100) + 6 if Z > 100mm/s

(35)

TABLE I

MOD EL A ND C ON TRO L PAR AM ET ER S

m2.5 kite mass (kg)

A5 characteristic area (m2)

ρ1.2 air density (kg/m3)

CL1.2 lift coefﬁcient

CD0.15 drag coefﬁcient

˙r0.5 traction phase reference ˙rref (m/s)

˙r-2.3 recovery phase reference ˙rref (m/s)

Tc0.1 sample time (s)

Nc1 control horizon

Np25 prediction horizon

TABLE II

STATE AND INPUT CONSTRAINTS,CYC LE S TARTI NG A ND E N DI NG

CONDITIONS

θI35◦Traction phase starting conditions

θI45◦

φI5◦

rI95 m

rI105 m

r280 mTraction phase ending condition

φII30◦2nd Recovery sub-phase starting conditions

θII60◦

θ85◦State constraint

ψ4◦Input constraints

˙

ψ20 ◦/s

Nominal wind speed is 4 m/s at 0 m of altitude and grows

linearly to 6 m/s at 100 m and up to 7.7 m/s at 300 m

of height. In the ﬁrst simulation, no wind turbulence was

considered, so Wt(t)=0. In the second simulation a

lateral sinusoidal wind turbulence Wt,y(t)along Yaxis was

introduced:

Wt,y(t) = 3 sin(ω0t)m/s (36)

with ω0= 2π/10 rad/s. Finally, in the third simulation the

following vertical wind turbulence Wt,z(t)along Zaxis was

introduced:

Wt,z(t) =

0if t≤100s

−1.5if 100 < t ≤200s

0if 200 < t ≤350s

−1.5if t > 350s

m/s (37)

The disturbance described by (37) includes two vertical

wind steps: the ﬁrst lasts 100 s during the traction phase,

the second is applied over the whole recovery phase. Note

that the Wt,y and Wt,z amplitudes are equal to 50% and

25% of nominal wind speed at 100 meters of altitude, thus

introducing quite strong perturbations into the system.

Fig. 6 shows the trajectory of the kite in nominal conditions.

Fig. 7 depicts some orbits traced by the kite during the

traction phase: it can be seen that the kite follows “lying

eight” orbits in this phase, with a period of about 5 s; about

75 orbits are thus completed in a single traction phase. Power

generated during the cycle is reported in Fig. 8: the mean

value is 1.58 kW, which corresponds to 733 kJ per cycle.

Fig. 9 and 10 show the trajectory of the kite and the power

generated in presence of lateral wind disturbances described

−100 −50 050 100 150 200 250 300 −140

−120

−100

−80−60−40−20020 40

0

50

100

150

200

250

300

Y(m)

X(m)

Z(m)

Fig. 6. Kite trajectory with nominal conditions: traction phase(solid) and

recovery phase(dashed)

0

50

100

150

−30

−20

−10

0

10

20

30

0

10

20

30

40

50

60

70

80

X(m)

Y(m)

Z(m)

Fig. 7. Some traction phase orbits

by (36): the cycle is completed and the generated energy

value, 727 kJ, is almost the same that was obtained without

disturbances, showing the good tolerance of the control

system to lateral wind turbulence.

Simulation results in presence of vertical turbulence de-

scribed by (37) are reported in Fig. 11 and 12: the ﬁrst

wind step leads to a lower value of generated energy (681

kJ) but the cycle was completed anyway, showing good

system robustness also in presence of severe vertical wind

disturbances.

VI. CO N CL USI ONS A ND F U TU RE DE VEL OP M EN TS

The paper has presented a ﬁrst study aimed to investigate the

capability of controlling tethered airfoils in order to devise

a new class of wind generators able to overcome the main

limitations of the present aeolian technology based on wind

mills.

The obtained results appear to be very encouraging, but are

based on simulations carried on a kite model taken from

the literature, which certainly can give only approximate

description of involved dynamics. Indeed, accurate modeling

the dynamic of non rigid airfoils is well known to be a

quite challenging task and it can be expected that the control

design based on this model may not perform in a satisfactory

0 50 100 150 200 250 300 350 400 450

−5

−4

−3

−2

−1

0

1

2

3

4

time (s)

power (kW)

Fig. 8. Instant (solid) and mean (dashed) power generated in a single cycle,

nominal conditions

−100 −50 050 100 150 200 250 300 −50

0

50

100

150

200

0

50

100

150

200

250

300

Y(m)

X(m)

Z(m)

Fig. 9. Kite trajectory with lateral wind turbulence

way on the prototype under construction. Thus, advanced

methods for the identiﬁcation of complex nonlinear systems

such as [11], [12] are planned to be applied to measurements

obtained from the prototype under construction, in order

to derive more accurate models, sufﬁcient to obtain good

performances from the NMPC design.

It must be remarked that the results reported in the present

paper are related to the small kite (5 m2characteristic area)

planned for the prototype under construction. With a kite area

of 50 m2, simulations give about 200 kW power generated

with 12 m/s wind speed. A wind turbine of the same power

is 40 m high, weights about 62 t and costs about 900.000,00

euros. The expected KiteGen weight and cost are about 8

t and 60.000,00 euros respectively. These values, even if

only partially conﬁrmed by experiments, demonstrate the

potentialities of using controlled airfoils for the generation of

electric energy. Indeed, even greater potentialities are offered

by the more efﬁcient conﬁguration that will be investigated at

Respira Lab in case of success of the ﬁrst prototype. In this

conﬁguration, the airfoils are connected to a vertical axis

rotative turbine (see Fig. 13), and the control is designed

to maximize the power transmitted by the airfoils to the

turbine arms, suitably connected to the electric generators.

According to our preliminary evaluations, it is expected that

0 50 100 150 200 250 300 350 400 450

−6

−5

−4

−3

−2

−1

0

1

2

3

4

time (s)

power (kW)

Fig. 10. Instant (solid) and mean (dashed) power generated in a single

cycle, lateral wind turbulence

−50 050 100 150 200 250 300 −50

0

50

100

150

0

50

100

150

200

250

Y(m)

X(m)

Z(m)

Fig. 11. Kite trajectory with vertical wind turbulence

wind generators of this type may have much lower electric

energy production costs than actual wind farms (by a factor

up to 10-20) and could generate up to 250 MW/km2, vs. 3

MW/km2of wind farms.

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0 50 100 150 200 250 300 350 400 450

−4

−3

−2

−1

0

1

2

3

4

time (s)

power (kW)

Fig. 12. Instant (solid) and mean (dashed) power generated in a single

cycle, vertical wind turbulence

Fig. 13. A more powerful and effective conﬁguration of KiteGen

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