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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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