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Design Optimization and Sizing for Fly-Gen Airborne Wind Energy Systems

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Traditional on-shore horizontal-axis wind turbines need to be large for both performance reasons (e.g., clearing ground turbulence and reaching higher wind speeds) and for economic reasons (e.g., more efficient land use, lower maintenance costs, and fewer controllers and grid attachments) while their efficiency is scale and mass independent. Airborne wind energy (AWE) system efficiency is a function of system size and AWE system operating altitude is less directly coupled to system power rating. This paper derives fly-gen AWE system parameters from small number of design parameters, which are used to optimize a design for energy cost. This paper then scales AWE systems and optimizes them at each scale to determine the relationships between size, efficiency, power output, and cost. The results indicate that physics and economics favor a larger number of small units, at least offshore or where land cost is small.
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Article
Design Optimization and Sizing for Fly-Gen
Airborne Wind Energy Systems
Mark Aull 1,* , Andy Stough 1and Kelly Cohen 2
1Windlift, Inc., Morrisville, NC 27560, USA; andy@windlift.com
2Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati,
Cincinnati, OH 45221-0070, USA; cohenky@ucmail.uc.edu
*Correspondence: mark@windlift.com
Received: 13 May 2020; Accepted: 10 June 2020; Published: 17 June 2020


Abstract:
Traditional on-shore horizontal-axis wind turbines need to be large for both performance
reasons (e.g., clearing ground turbulence and reaching higher wind speeds) and for economic reasons
(e.g., more efficient land use, lower maintenance costs, and fewer controllers and grid attachments)
while their efficiency is scale and mass independent. Airborne wind energy (AWE) system efficiency
is a function of system size and AWE system operating altitude is less directly coupled to system
power rating. This paper derives fly-gen AWE system parameters from small number of design
parameters, which are used to optimize a design for energy cost. This paper then scales AWE systems
and optimizes them at each scale to determine the relationships between size, efficiency, power output,
and cost. The results indicate that physics and economics favor a larger number of small units, at least
offshore or where land cost is small.
Keywords: airborne wind energy; levelized cost of energy; design optimization
1. Introduction
1.1. Airborne Wind Energy
Offshore wind energy has advantages in resource availability over onshore wind and has better
load matching than solar energy [
1
,
2
]. Offshore wind is also more expensive than either [
3
,
4
].
Airborne wind energy (AWE) is a technology with the potential to harvest abundant wind
resources located over deep water less expensively than current wind energy technologies [
5
]. It is
therefore a good candidate for economically assisting with decarbonization and helping to mitigate
global warming.
AWE uses tethered aircraft to harvest wind energy. The combination of aerodynamic forces
and tether tension propel the aircraft perpendicular to the wind, analogous to a wind turbine blade.
Higher altitude winds are faster and more reliable than surface-level winds [
6
], and using a tether
rather than a tower makes it easier to increase operating altitude. While the blades and tower
of a conventional wind turbine must be designed for significant bending and compressive loads,
AWE systems are anchored to the ground by a tether and can use a bridle to support the aircraft wing,
resulting in significantly less structural weight for the same power production. A lower weight system
with a simpler foundation promises logistical benefits such as lower capital costs, transportation costs,
maintenance costs, easier installation for offshore systems, and a lower cost of energy.
Two forms of AWE were proposed by Miles Loyd in 1980. Ground-gen AWE systems operate
by reeling the tether out under high load, producing power from regenerative braking on the winch,
and then reeling back in under lower load. Fly-gen AWE system use turbines on the aircraft to harvest
wind power while moving at high crosswind speeds and transmit that power to the ground via an
Automation 2020,10, 1–16; doi:10.3390/automation10100001 www.mdpi.com/journal/automation
Automation 2020,10 2
electrified tether [
7
]. The turbines on board the aircraft used for fly-gen AWE systems experience
an airspeed significantly higher than the wind speed, allowing smaller, faster-spinning turbines and
lighter, more energy-dense power systems than comparable conventional wind turbines.
For a fly-gen system of given wing area (
Aw
) and lift (
L
) with a rotor drag (
Drot
) defined
(Equation (1)) as a fraction of aircraft drag (
DUAV
) operating at a given wind speed (
Vw
), and applying
simplifying assumptions including a massless aircraft, small angle approximations on velocities and
forces, and the tether parallel to the wind vector, the crosswind speed (
Vc
) is given by Equation (2).
Assuming the system lift to drag ratio is relatively high, the airspeed (
Va
) is approximately
Vc
,
and the drag power (
Drot Va
) produced by the turbine, neglecting efficiencies, is approximated by
Equation (3). This expression for power is maximized when
rp=1
2
, i.e.,
Drot =DU AV/
2 (the solution
to
d P/d rp=
0). This result is substituted back into Equation (3) to obtain Equation (4), the maximum
power for a fly-gen system [7].
Drot =rpDU AV (1)
Vc=VwL
DUAV +Drot
=VwCL
CD,UAV
1
1+rp(2)
P=Drot Va1
2ρV3
w
C3
L
C2
D,UAV
rp
(1+rp)3Aw(3)
Pmax 2
27 ρAwV3
w
C3
L
C2
D,UAV
(4)
Vander Lind extended the Loyd performance analysis to cases where the tether is at an angle
θt
from the wind vector. Starting with a force balance (Equation (5)), then solving for power and
applying small angle approximations (Equation (6)), optimal crosswind speed (Equation (7)) and
maximum power (Equation (8)) are calculated. The paper also analyses cases where tension or power
are constrained, and optimizes Equation (8) for altitude given an expression wind speed vs. altitude
(θt0.36 for reasonable wind shear) [8].
P
Va
=ρV2
aAw
2(CL
Vw
Vacos(θt)CD,U AV
Vc
Va
)(5)
P=ρAw
2(V2
cVwCLcos(θt)V3
cCD,U AV)(6)
Vc=2
3
L
DUAV
cos(θt)(7)
P=2
27 ρAw(Vwcos(θt))3C3
L
C2
D,UAV
(8)
Using similar methods, ground-gen systems can be shown to have the same theoretical maximum
power output. Research and development has been pursued for both technologies, though most
academic and commercial organizations have focused on ground-gen systems, likely due to the
difficulty in developing light-weight power electronics (and the related higher mass of fly-gen
aircraft), heavier and higher drag tethers, and lower barriers to entry for building and testing flexible
wing aircraft.
Fly-gen AWE systems have significant potential advantages. Because ground-gen systems
require a cycle involving power produced while reeling out and power expended while reeling
in, a ground-gen system must reel-in in zero time with zero drag in order to reach the theoretical
maximum power over a cycle. Ground-gen systems also have complications involving launching,
landing, and operating during lulls in the wind. Fly-gen systems lack a requisite power-consuming
reel-in phase, and have the ability to send power to the aircraft for takeoff, landing, or staying aloft
through a lull in the wind.
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Rigid-wing aircraft also have advantages over flexible aircraft, which have lower aerodynamic
performance than rigid-wing aircraft, are more difficult to analyze, and wear out more quickly: SkySails
GmbH, a company which produces flexible wing aircraft for ship propulsion, estimates its operating
and maintenance costs at $0.06 per kWh [
9
], which is comparable to the average LCOE (levelized cost
of energy; includes installation, operation, transmission, distribution, and financing costs) for on-shore
wind installations [4].
Most of the modeling of airborne wind systems in the literature is focused on control methods
rather than performance analysis. Many methods have been explored for controlling airborne wind
energy systems, including PID control using a simplified model [
10
], Nonlinear Model Predictive
Control methods [
11
,
12
], sequential quadratic programming [
13
], Legendre pseudo-spectral optimal
control [
14
], and neural networks trained by genetic algorithm [
15
]. However, this work has largely
focused on flexible wing ground-gen systems, which have a variable tether length, and often use
flexible wings. Flexible wings, in which the structure supports a tensile, rather than bending load,
very high lift/weight ratios and very high accelerations are possible, making these vehicles significantly
more maneuverable than the rigid wing vehicles more typically used in fly-gen systems.
Previous AWE literature includes LCOE estimation for a ground-gen farm vs. the number of units
in the farm and the system scale [
16
], as well as estimating AWE system LCOE in order to compare to the
best available renewable alternatives onshore [
17
] and to compare to other options for microgrids [
18
].
LCOE optimization for traditional wind turbines has been performed, to optimize blade length and
hub height for systems in low wind speed areas using particle swarm optimization [
19
], to optimize
rotor radius and rated speed for offshore systems [
20
], and to optimize rotor radius and rated speed
for several wind conditions using a genetic algorithm [21].
There are many AWE systems in various states of research and development, including several
tested prototypes, but the technology has been slow to commercialize. Some airborne wind energy
companies have switched focus from energy to other applications; Altaeros has pivoted to providing
telecommunication platforms and Joby Energy has become Joby Aviation, focused on electric aircraft.
One of the few fly-gen focused companies, Makani Power, demonstrated electrical power output with
a small (20 kW) unit [
8
], but their scaled-up (600 kW) unit did not produce positive net power [
22
],
and the company has not released an update on power production improvement. Alphabet has
stopped funding Makani Power because “the road to commercialization is longer and riskier than
hoped” [
23
]. AWE systems are both novel and significantly more complex to design, analyze, and test
than traditional wind turbines. The AWE industry also has an issue with reliability. A ground-gen
company, Kitemill, reported their longest duration flight to date was 2 h long and produced zero net
power [
24
]. Typical AWE test flights, like Kitemill’s, last for minutes to hours, with a large gap to
operating autonomously for weeks to months. In 2019, Makani demonstrated its 600 kW unit on an
offshore platform, but lost the aircraft during its second flight [25].
Because AWE systems are both novel and significantly more complex to design, analyze, and test
than traditional wind turbines, many aspects of a mature system are currently unknown. One basic
example is what wing size or power rating is optimal for an AWE system. Reliable analysis tools are
important for both system and controller design which will enable the technology to mature.
1.2. Airborne Wind Farm Output
As the scale of an AWE system changes, if Reynold’s number effects are neglected,
CL
and
CD
remain constant and the optimal crosswind speed of the aircraft is dependent only on wind speed
(see Equation (7)). If wind shear is neglected such that wind speed doesn’t change with altitude,
then the simplified approximation for aerodynamically harvested power is proportional to wing area
(see Equation (8)) and not dependent on any other variables as the scale of a system changes.
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For wings with equivalent control authority, tether length (
λ
) is proportional to wingspan (
bw
) [
8
],
which is proportional to Aw(Equation (9)) for a constant aspect ratio (AR).
λ=λbbw=λbpAwAR (9)
For multiple units in a wind farm to have the highest minimum clearance, all aircraft should
operate in the same phase on the same pattern. Because a farm contains a large number of individual
units in each row and column, it is not feasible to increase clearance by operating units in one row
at significantly different elevation angles from units in the next row; this would result in the units
in the row furthest downstream flying at an elevation angle (
θtnrows
) higher than the units in the
row furthest upstream. Also note that such a scheme would not respond well to changes in wind
direction. For units flying the same trajectory, the design clearance (
units
) is given by Equation (10),
where
pf
is a packing factor defined as the distance between anchor points as a fraction of tether
length. The required surface area (
Af arm
) for
nunits
in an isometric grid (Equation (11)) is proportional
to
λ2
, i.e., to wing area. Because a system with a wing area
Aw
produces a power proportional to wing
area and occupies a surface area proportional to wing area, power per surface area is independent of
system scale, as a first approximation (Equation (12)).
units =pfλsin(θt)b/2 pfλsin(θt)(10)
Af arm =3
4(pfλ)2nunits =3
4p2
fλ2
bAwAR nunits (11)
Panunits
Af arm
=8
33.5
ρV3
wcos(θt)3
p2
fλ2
bAR
C3
L
C2
D,sys
(12)
This simplified analysis neglects the effects of Reynold’s number and wind shear, which benefit
large units operating at higher altitudes, and also neglects mass effects which benefit smaller units,
so it is non-obvious what scale is optimal for power. Because square-cube scaling is unfavorable for
flying systems, it seems unlikely that bigger is always better, as seems to be true for horizontal axis
wind turbines.
Facilities for manufacturing, packaging, installation, and maintenance will all be sized around
that system, therefore using smaller units may lower costs at multiple points in the supply chain.
For transportation, shipping anything that can fit in a standard shipping container will be significantly
less expensive than shipping anything that requires special accommodations. And a farm composed
of a larger number of smaller systems will be more robust to failures, because losing a small unit
represents a smaller proportion of overall power output. A farm using small systems may also be
more robust to rare or unexpected environmental factors because a small system should be able to
go into a safe mode more rapidly after a farm-wide controller detects a problem. However a farm
with many small units requires more electrical interconnects between units, which will likely increase
the difficulty of installation, especially if the interconnects are buried. And there may be other issues
that are dependent of system scale that are not well understood, such as the ability to navigate ships
around or through the farm or effects on wildlife. Therefore it is of interest to determine whether AWE
systems have an incentive to be designed to megawatt scales like traditional horizontal axis wind
turbines, rather than to be small enough to be transported in shipping containers.
1.3. Objective
Developing AWE systems involves many interdependent design trade-offs, therefore for the
technology to mature, identifying and analyzing those tradeoffs is required. This study analyzes
trade-offs involved in AWE system design at several scales and evaluates both performance and
estimated LCOE vs. system size.
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2. Method
2.1. Model
Design trade-offs are difficult to analyze with traditional tools; simulations are undesirable for
this purpose due to the iteration required to ensure that a controller is achieving the desired trajectory
before iterating to optimize a trajectory subject to constraints, then iterating through parameters for
design optimization.
A non-linear model inversion based on simplifying assumptions appropriate for fly-gen systems
has been validated against a high-fidelity simulation and is used for this design optimization.
Rather than running a simulation and tuning a closed-loop controller to track a desired trajectory,
the inverse model calculates power produced from operation on a given trajectory in a given wind
environment directly. An optimizer applied to this model modifies the trajectory to maximize total
cycle power production [26].
For the inverse model, azimuth and elevation angles, inertial speed, and lift coefficient of the
aircraft are input as Fourier series coefficients, guaranteeing a closed, steady state cycle. The inertial
speed and derivatives of the angles are used to determine the inertial velocity vector, which is added to
the wind velocity to determine air velocity. The gravitational and inertial accelerations on the trajectory
and aerodynamic forces for the aircraft and tether are calculated, then the roll angle which balances the
lateral forces is determined. The pitch and yaw angles relative to the tether which produce the desired
angle of attack and sideslip angle are calculated, then tether tension and rotor force are calculated to
balance the longitudinal forces. The drag power at each point is given by airspeed and rotor force
(Equation (3)) and efficiencies are applied to obtain electrical power. Constraints are applied to tension,
electrical power, rotor force, roll angle, wingtip angle of attack induced by roll rate, etc. to ensure
that the trajectory is realizable. A quasi-Newton’s method optimizer modifies the Fourier coefficients
(for path, velocity, and lift coefficient) to maximize average cycle power production [26].
2.2. Design Optimization Overview
For this analysis, a reference design (
Aw =
4 m
2
) is scaled up to 9 m
2
, 16 m
2
, 25 m
2
, and 36 m
2
.
At each scale, 4 parameters (decision variables) are iterated to optimize the design: maximum
power (
Pmax
), maximum tension (
Tmax
), tether length per unit span (
λb
), and wing aspect ratio (
AR
).
Other aspects of the design are derived from these parameters and assumed constants. The objective
function optimized is the levelized cost of energy (LCOE, see Section 2.5).
The optimization is performed using a quasi-Newton method, which numerically approximates
the first and second derivatives of LCOE with respect to each decision variable. In order to improve
convergence, reasonable initial values of the decision variables are used, and if any points used to
calculate the derivatives have a lower LCOE than the result of an iteration, they are used as the starting
point for the next iteration.
2.3. Trajectory and Power Curve Optimization
For this analysis, the aircraft fly figure-eight up-loops (Figure 1), rather than circles or figure-eight
down-loops. Down loops provide the most consistent power throughout a cycle, but down loops
(and circles) require flying directly downward, which is the worst case for robustness to gusts.
Circles additionally require a tether unwinder, involving additional hardware and complexity.
The path is defined by tether angles, such that it scales automatically with tether length.
The horizontal and vertical extents as well as the mean vertical location are modified by the trajectory
optimizer. Other path Fourier coefficients affect the shape of the figure-eight. The initial path used is a
figure-eight which maximizes the minimum turn radius.
Wind speed with altitude is modeled as a power law wind profile, which involves the speed
(
Vw,0
) at a reference altitude (
z0
), and a shear exponent (
Vw,ex p
), such that
Vw=Vw,0 (z/z0)Vw,exp
.
The wind shear exponent used is 0.11, which is reasonable for offshore environments, and the reference
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altitude is 90 m, comparable to existing traditional offshore wind systems (see Figure 2). This neglects
phenomena such as low-level jets which may have significant effects on power production in some
locations. The wind is assumed to have a probability density function given by a Weibull distribution
with an average of 9 m/s and a shape factor of 2.0 (Figure 3).
Figure 1. Closed trajectory types.
0 100 200 300 400 500
0
2
4
6
8
10
12
altitude (m)
wind speed (m/s)
exponential wind shear model, reference altitude = 90 m, exponent = 0.11
Figure 2. Exponential wind speed model.
0 5 10 15 20 25
0
0.02
0.04
0.06
0.08
0.1 Weibull distribution, average = 9 m/s, shape factor = 2.0
probability density (s/m)
wind speed (m/s)
Figure 3. Weibull wind speed probability density function.
Power vs. wind speed is cubic for unconstrained wind energy systems, however tension and
power constraints cause the power curve to taper off and reach a maximum at the rated wind speed [
8
].
In practice, fitting power curves with a linear approximation up to the rated speed and a constant above
the rated speed produces a good estimate with reduced computation. For this analysis, a nominal wind
speed of 13 m/s is used as the rated speed, which is comparable to traditional offshore wind systems.
Power curves are estimated by optimizing trajectories at two points, 8 m/s and 13 m/s, and assuming
a trapezoidal power curve which is linear up to 13 m/s and constant from 13 m/s to a cut-out speed at
25 m/s. The extrapolation below 8 m/s is used to find the cut-in speed. For the final iteration of each
optimization, full power curves are evaluated at each integer wind speed between 5 m/s and 25 m/s
in order to validate the use of the trapezoidal power curve estimates for the optimization. The cut-in
speed is typically between 5 m/s and 7 m/s, and speeds above 25 m/s only occur approximately
0.23% of the time given the assumed wind profile, so it is assumed that speeds outside this range can
be neglected.
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The maximum airspeed (
Va,max
) is constrained such that a 20% higher airspeed at the stalled lift
coefficient produces a lift of
Tmax
(see Equation (19)). The initial maximum airspeed in 42 m/s and the
minimum airspeed (below which attitude control may become unreliable) is 20 m/s.
Aspect Ratio and Reynold’s Number Effects
Airfoils can be designed to have higher maximum
CL
at higher Reynold’s numbers, however this
relationship is not well defined. For this analysis, it is assumed that an airfoil can be designed with a
maximum lift given by Equation (13) for
Re
350 k . Reynold’s number increases with increasing
wing area and decreases with increasing aspect ratio (Equation (14)).
CL,max =2.5 +6E7Re (13)
Re =Vacw
ν=VaAw
νAR (14)
Induced drag (Equation (16)) is inversely proportional to aspect ratio and is a major component
of system drag (Equation (15)), especially at high lift coefficients. Induced drag increases with
C2
L
,
therefore
C3
L/C2
D,sys
approaches zero as
CL
becomes very large. This parameter is also equal to zero
at
CL=
0, so there is an optimal
CL
(Equation (17)) and maximum
C3
L/C2
D,sys
(Equation (18)) for a
system. Because
CD,teth λAR
(Equation (43)), the benefit of increasing aspect ratio on system
performance appears slightly larger than the benefit of decreasing aspect ratio on packing units more
closely (from Equation (12)).
CD,sys =CD,i+CD,p+CD,teth (15)
CD,i=C2
L
πAR osw (16)
CL,opt =q(3πAR osw (CD,p+CD,teth )) (17)
C3
L/C2
D,sys (3πAR osw)3/2
16 p(CD,p+CD,teth )(18)
However, this neglects the effect of mass, the coupling between tether diameter and power,
and assumes that
CL,max CL,o pt
. Therefore aspect ratio is an independently varied parameter in
this analysis.
This analysis assumes constant, parasitic drag (
CD,p=
0.04) and Oswald Efficiency (
osw =
0.71).
Tether drag normalized to wing area,
CD,teth
, is typically around 0.15, resulting in
CL,opt
4.5.
Because this is larger than
CL,max
for Reynold’s Numbers in the range studied, higher lift always
increases C3
L/C2
D,sys.
2.4. Aircraft Mass Scaling
Mass is normally expected to scale with length cubed. For aerodynamic applications,
loads increase with length squared while distances increase with length, therefore structural masses
increase with length cubed. This relationship, which can be expressed as
mL3
,
mA3/2
,
or m/Am1/3, is a close fit for data on both manufactured aircraft and flying animals [27].
Though the mass of an aircraft is expected to increase with
A3/2
w
, by changing elements like the
bridle (and therefore the load distribution on the structure) or the number of rotors with scale, it is
possible that AWE aircraft mass may scale with area to a lower exponent. To capture this uncertainty
in the mass of a scaled system, optimizations are performed on heavier and lighter systems at each
scale. For this analysis, the reference design is a 60 kg, 4 m
2
, 16 aspect ratio aircraft. The lightweight
scaled systems assume constant mass/area with a scaled reference mass (
mUAV,re f
) of 15 kg/m
2Aw
,
and the heavier systems assume mass is proportional to length cubed with a scaled reference mass of
Automation 2020,10 8
7.5 kg/m
3A3/2
w
. The maximum electrical power of each scaled reference design (
Pre f
, Equation (20))
assumes a rated wind speed (
Vw,rated =
13 m/s) and a reference tether drag (
CD,teth,r e f =
0.15).
The maximum tension of each scaled reference design (
Tref
, Equation (19)) assumes the lift coefficient
at stall at a maximum air speed of 42 m/s, with a 20% safety factor on airspeed. The reference tether
length is 20
bw
(
λb,re f =
20). The tether voltage (
Vteth
) is 800
p(Aw)
and does not change with the
optimization. These values are summarized in Table 1.
Tref =0.5 ρ(1.2 Va,max )2CL,max Aw(19)
Pre f =0.5 2
27 ρAw(Vw,rated cos(θt0))3C3
L0
(CD,UAV +CD,teth,re f )2(20)
Table 1. Scaled reference design parameters.
Aw(m2)4 9 16 25 36
CL,max 2.911 3.116 3.322 3.527 3.733
Pre f (kW) 42.0 99.3 183.3 294.5 432.9
Tref (kN) 18.1 43.6 82.7 137.2 209.1
lighter mUAV,r e f (kg) 60 135 240 375 540
heavier mUAV,r e f (kg) 60 202.5 480 937.5 1620
λ(m) 160 240 320 400 480
Vteth (V) 1600 2400 3200 4000 4800
For each scaled reference design, the component mass is calculated according to the distribution
given in Table 2(i.e.,
mre f ,struct =
0.5
mUAV,re f
). As the design parameters change, the changes in
component masses are calculated from the scaled reference design in order to compute a new aircraft
mass. The following sections derive laws for how the component masses change with changes in the
design parameters.
Table 2. Component mass distribution.
mre f ,struct Wing + Fuselage + Tail + Actuators 50%
mre f ,rot SFG+prop 15%
mre f ,mot Motor 15%
mre f ,pow Power electronics 10%
mre f ,RCU Roll Control Unit (RCU) 10%
2.4.1. Structural Scaling
The moment of inertia for an I or C section (
Isec
) where the cap width is
wcap
and the thickness
of the caps (
tcap
) is much smaller than the section height (
hsec
) is approximated by Equation (21).
The maximum bending stress (
σ
) on a beam of length
L
with a distributed load (
F
) is given by
Equation (22).
Isec =wcap tcap h2
sec
2(21)
σ=F L hsec
4Isec (22)
For a wing supported in the center, the bending equation (Equation (22)) applies to half the
maximum tension and half the span where the section height is the wing thickness (Equation (23)).
Assuming a constant airfoil section, wing thickness (
tw
) is proportional to chord length (
cw
).
Also assuming spar cap width is proportional to chord length, the proportionality for spar cap thickness
is given by Equation (24) and the spar mass proportionality is given by Equation (25). Note that
Automation 2020,10 9
assuming
Tmax Aw
makes the mass proportional to
b3
w
, which is consistent with square-cube scaling.
The structural mass of the wing, fuselage, and tail (
mstruct,w
) is modified by Equation (26) to account
for changes in maximum tension and aspect ratio.
σw=Tmax bw
16 wcap tcap tw(23)
tcap Tmax bw/c2
w(24)
mstruct,wwca p tca p bwTmax b2
w
cw
Tmax A1/2
wAR3/2 (25)
mstruct =mre f ,struct
Tmax
Tref
(AR
ARre f
)3/2 (26)
2.4.2. Rotor Sizing
The maximum power output (Equation (27)) and drag (Equation (28)) of a turbine are defined by
the Betz Limit [
28
]. This is used to calculate the total area of
nrot
rotors (Equation (29)) and radius of
each rotor (Equation (30)) from the maximum power. The maximum rotor rotational speed is limited
by a Mach number of 0.85 at the blade tips (Equation (31)).
Pmax 8
27 ρArot V3
a(27)
Drot 8
27 ρArot V2
a(28)
Arot =27 Pmax
8ρV3
a,max
(29)
Rrot =q(Arot
πnrot
) = q(27 Pmax
8πnrot ρV3
a,max
)(30)
ωrot 0.85 a
Rrot (31)
A constant efficiency of 70% is used for conversion from drag power to electrical power when
generating (Pe=ηPm) and from electrical power to thrust power (Pm=ηPe).
For a rotor, the bending equation (Equation (22)) applies to the rotor radius,
Rrot
, and the force,
Drot
(Equation (28)) divided by the number of blades on all rotors (Equation (32)). Assuming spar
cap width and blade thickness are proportional to blade chord (
cb
), the proportionality for spar cap
thickness is given by Equation (33), and blade mass (Equation (34)) is consistent with square-cube
scaling assuming a constant blade aspect ratio and
Drot R2
rot
. Note that the mass of the
pylon/sideforce generator supporting the rotor scales the same way because its length is approximately
the rotor radius and the force it supports is the rotor force. The structural mass of the rotor
blades, hubs, and pylon/sideforce generators is modified by Equation (35) to account for changes in
maximum power.
σrot =Drot Rrot
4nblades nrot wc ap tcap tb
(32)
tcap Drot Rrot/(nrot nblades c2
b)(33)
mblades nrot nbl ades Rrot cbtcap Dro t R2
rot
cb
(34)
mrot =mre f ,ro t (Pmax
Pre f
)3/2 (35)
Automation 2020,10 10
2.4.3. Electrical Scaling
It is assumed that tether voltage scales with length (Equation (36), see Section 2.4.5), and that
the mass (
mpow
) and volume of any power electronics scale proportionally with maximum power
(Equation (37)), which is equivalent to scaling with current at a constant wing size/voltage .
Vteth =rvpAw(36)
mpow =mre f ,pow
Pmax
Pre f
(37)
It is assumed that motor mass (
mmot
) scales with maximum motor torque (
Trot
). Motor torque is
generated by magnetic fields which are limited by inductor saturation, which is limited by the mass and
magnetic permeability of the motor magnetic core. Maximum shaft power (Equation (38)) is related
to maximum electrical power by an efficiency and maximum rotor speed is inversely proportional
to radius (Equation (31)), which is proportional to
Pmax
(Equation (27)), therefore motor torque is
proportional to
P3/2
max
(Equation (39)). Therefore motor mass is related to maximum power by the same
square-cube law (Equation (40)) as the rotor and sideforce generator (Equation (35)) .
Psh a f t =ωrot Trot =ηPmax (38)
Trot Pmax Rrot P3/2
max R3
rot (39)
mmot =mre f ,mot (Pmax
Pre f
)3/2 (40)
2.4.4. Roll Control Unit
The roll control unit connects to the tether and uses a motor to vary the length of the bridle lines,
setting the roll angle of the aircraft relative to the tether. The RCU structure is driven by maximum
tether tension, but the motor is sized by the roll moment required on the aircraft, i.e., the differential
bridle line tension rather than total tension. Roll torque is approximately roll inertia times roll
acceleration (
LiIxx ˙
p
). Roll rate (p) is inversely proportional to tether length (
λ=λbAR Aw
),
assuming the same roll angles are required to follow a path with the same tether angles at the same
speed, because the path size increases linearly with tether length, providing linearly more time to
roll. Roll acceleration (
˙
p
) is inversely proportional to tether length squared because of the lower roll
rate and the increased time to change roll rate. Roll inertia is proportional to mass and span squared,
and using the mass from Equation (26) yields Equation (41). Because this expression for the RCU
motor scaling is very similar to the structural mass scaling of the aircraft, it is assumed to be reasonable
for both motor and structural scaling for the RCU (Equation (42)) .
Li=Ixx ˙
pmUAV b2
w
λ2
bb2
w
Tmax AR3/2
λ2
b
(41)
mRCU =mre f ,RCU
Tmax
Tref
(AR
ARre f
)3/2 (λb,re f
λb
)2(42)
2.4.5. Tether Design
The tether adds mass and drag to the system. The tether drag is proportional to a 2D drag
coefficient (
Cd,teth
), diameter (
dteth
), length, and
kdv
(a factor related to the difference in air velocity
along the length of the tether and the fraction of tether drag reacted at the aircraft vs. at the ground,
Automation 2020,10 11
which is around
1
4
). The tether drag coefficient normalized by the wing area is given by Equation (43).
The reference tether length is 20×the aircraft span, or 20 AwAR.
CD,teth =kdv Cd,teth
dteth λ
Aw
dteth λ
Aw(43)
The tether is made up of a strength member, conductors, insulative layers, and an outer jacket.
For this paper, a tether design is performed for each vehicle, resulting in an appropriate tether drag and
lift to drag ratio for the UAV-tether combined system, and a tether mass determined by the proportions
of copper, strength member, insulation, and jacketing required. The cross-sectional area of the tether
strength member is proportional to maximum tension (and therefore to the wing area), the insulative
layer thickness is proportional to voltage, and the jacket is a constant thickness. Conductor area (
Ac
)
determines tether resistance (Equation (45)), which must be low enough to maintain a reasonable
voltage drop and keep heat generation below a critical value.
For systems with the same voltage maintaining the same voltage drop (Equation (44)),
tether resistance must be be inversely proportional to power (Equation (45)). Because power
is proportional to wing area and tether length is proportional to the square root of wing area,
conductor cross-sectional area must increase with wing area to the 3
/
2 power (Equation (46)).
Therefore, to keep the conductor area to wing area ratio constant, system voltage must increase
with the square root of tether length.
rt=I2Rteth
Pmax
=I Rteth
Vteth
=Pmax Rteth
V2
teth
(44)
Rteth =rt
V2
teth
Pmax
=ρcλ
Ac(45)
Ac=ρcλPmax
rtV2
teth
AR Aw3/2
V2
teth
(46)
Given a heat limit on the power dissipated in the tether per unit tether length (
pheat
, Equation (47)),
a required conductor area can be computed (Equation (48)). Because power is proportional to wing
area and tether length is proportional to the square root of wing area, in order to keep the conductor
area to wing area ratio constant, system voltage must increase with the tether length. Therefore tethers
will usually be limited by heat dissipation rather than voltage drop, and tether voltage must scale with
tether length.
pheat λ=I2Rteth =P2
max ρcλ
V2
teth Ac
(47)
Ac=P2
max ρc
V2
teth pheat
Aw2
V2
teth
(48)
Keeping the conductor area to wing area ratio constant doesn’t keep the tether drag coefficient
exactly constant, but because the required insulation thickness increases linearly with voltage (and
tether length), CD,teth for typical tethers stays within a relatively narrow range close to 0.14.
2.5. Economic Assumptions
Levelized Cost of Energy (LCOE) is used as the metric for optimization (Equation (49)), defined by
annual costs (operations and maintenance for the year,
OpEx
, plus a “mortgage payment” on the
initial capital costs,
Ca pEx FCR
) divided by annual energy production (
AEP
). A recent ARPA-E FOA
for floating offshore wind turbines (DE-FOA-0002051 [
29
]) provides a method for calculating LCOE
which appears to be a rational compromise between good fidelity and practicality of implementation
for conceptual systems. The ARPA-E method assumes a fixed charge rate (FCR) of 8.2% and a fixed
Automation 2020,10 12
operations and maintenance cost (OpEx) of
$
86/kW. Annual electricity production (AEP) is determined
by summing the power curve for a proposed design weighted by a Weibull distribution with a shape
factor of 2.1 and a scale factor of 10.13. The RFP determines the total capital expenditure (CapEx) of
offshore wind systems with specified costs per kilogram for materials, manufacturing, and installation
for each component (Equation (50)).
LCOE =CapEx FCR +OpEx
AEP (49)
Ca pEx =
j=component
massj(ct,j+cm,j+ci,j)(50)
This analysis adapts the ARPA-E method for AWE systems: the values for the rotor blades are
applied to the AWE aircraft structure, the values for the nacelle (including generator, drive train,
and yaw bearing) are applied to the motor and power electronics, the values for the tower installation
and maintenance are applied to the tether while the tether material is assumed to be mostly copper
rather than steel, and the floating platform, mooring, and anchor are unaffected as they exist for
both traditional and airborne offshore systems. A summary of the cost values and the results for
the reference design are given in Table 3. Masses for the floating platform, mooring, and anchor are
assumed to be 20 kg/kW, 1.5 kg/kW, and 1.5 kg/kW, based on linear scaling with a reference design.
Electrical interconnects are neglected by the ARPA-E method, but are a larger fraction of the capital
expenditures for a more distributed farm, and are therefore included here. It is assumed that the
interconnects do not have to be buried, therefore the interconnects use the same cost values as the
tethers, which carry similar power over similar distances in similar environments. This analysis also
replaces the constant OpEx value by assuming an average lifetime for the aircraft (2 years) and for the
platform and underwater components (5 years) and dividing the CapEx for those components by their
lifetime to obtain an average replacement cost per year.
The two-year lifetime assumes AWE development will resemble aircraft development,
which shows a consistent relationship between hours flown and reliability (after a few hundred
flight hours, the system mean time between failures (MTBF) should be about a week of continuous
operation, after 1000 flight hours MTBF is about a month, and after 100,000 flight hours MTBF is
about a year) [
30
]. The most mature aircraft have MTBFs equivalent to around 2 years of continuous
operation. Smaller scale AWE systems will benefit from lower development costs. Flying 100,000 h
on aircraft that last from 100 h to 10,000 h requires building and crashing a lot of prototypes, and a
smaller scale is advantageous for this development process.
Table 3. Cost per kg values.
ct(Material $/kg) cm(Manufacture $/kg) ci(Installation $/kg) Unit Cost $
Structure 8 30.96 0.8 2982
Generator, Inverter 2 18.98 0.2 529.5
Tether, Interconnects 3 20 0.2 1887
Floating platform 2 4 0.26 12,520
Mooring system 2 0.28 1.04 498
Anchor system 0.6 4.02 2.088 1006.2
3. Design Optimization Results
A representative power curve (Figure 4) shows that the power curve estimates (blue) match the
full power curves (green) relatively well, as do the average power results (Figure 5) and the LCOE
estimates (Figure 6) for the estimated (blue) and full (green) power curves.
The design parameters (
AR
,
Pmax
,
Tmax
, and
λ
), as well as selected dependent parameters, for the
optimized systems are shown in Table 4.
Tmax
and
AR
are regularly lower than their reference values
while Pmax is regularly higher than Pref . Tether length doesn’t change as much.
Automation 2020,10 13
Table 4. Optimized design params.
Aw(m2)4 9 9 16 16 25 25 36 36
AR 16 15.8 15.9 15.9 16 15.9 15.9 15.7 15.7
Pmax (kW) 44 106 106 194 192 307 308 448 449
Tmax (kN) 16.1 43.1 43.5 82.6 81.2 136 136 208 209
λ(m) 160 240 240 319 320 398 399 473 476
CL02.91 3.12 3.12 3.33 3.32 3.53 3.53 3.74 3.74
CD,UAV 0.278 0.315 0.313 0.353 0.348 0.391 0.391 0.439 0.439
CD,teth 0.133 0.137 0.137 0.139 0.138 0.14 0.14 0.141 0.142
L/D7.09 6.9 6.92 6.77 6.83 6.66 6.64 6.46 6.45
Va,max (m/s) 39.6 41.7 41.9 41.9 41.6 41.8 41.8 41.9 41.9
mUAV (kg) 57.4 137 209 245 488 380 952 543 1630
mteth (kg) 16.9 66 66.4 167 164 337 340 606 611
dteth (mm) 11 17.1 17.2 23.1 23 29.2 29.3 35.7 35.7
5 10 15 20 25
-250
-200
-150
-100
-50
0
50
wind speed (m/s)
power (kW)
estimated
full power curve
Figure 4. Power curve for light 25 m2.
Pavg/Aw(kW/m2)
0 10 20 30 40
3
3.5
4
4.5
5estimated
full power curve
wing area (m2)
Figure 5. Average power per unit wing area.
0 10 20 30 40
20
30
40
50
60
70
LCOE ($/MWh)
estimated
full power curve
wing area (m2)
Figure 6. Levelized Cost of Energy vs. System size.
The levelized cost of energy calculated for the optimal systems is shown in Figure 6, both for
the estimated and full power curves. The upper and lower curves represent the heavier (solid line)
Automation 2020,10 14
and lighter (dotted line) system assumptions, respectively. The vertical range between these curves
can be thought of as the uncertainty due to uncertainty in mass scaling. The contributors to LCOE,
namely average power and system cost, are shown in Figures 5and 7, respectively. Note that for
Figure 5, the upper curve represents the lighter systems (dotted line). For Figure 7, “airborne” includes
the aircraft structure, electronics, and tether while “marine” includes the floating platform, mooring,
anchors, and electrical interconnects. Again, dotted lines represent the light systems, and solid lines
represent the heavy systems.
0 10 20 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Cost ($/W)
airborne
marine
total
wing area (m2)
Figure 7. Cost per watt vs. wing area.
4. Discussion
As noted in the results, values computed from a power curve extrapolated from two points closely
matches a full power curve, which validates the two-point method and provides confidence that
optimizations performed using the estimated power curve are real optimums.
Based on the design parameters following optimization, it appears that the reference design
was underpowered and structurally overbuilt, and the optimizations corrected that for each scaled
reference design. The tether length doesn’t change as much, indicating that either the reference tether
length is close to the optimal tether length or LCOE is not very sensitive to tether length near the
reference value. Because the slope of LCOE vs. tether length is zero at the optimum, both are likely true.
LCOE values from different studies can’t necessarily be compared directly (cost estimates for
existing technologies can be substantially different between sources such as [
3
,
4
]). This study is more
difficult to compare because it estimates costs at a future development stage where AWE aircraft have
a 2-year MTBF and floating systems have a 5-year MTBF (systems with longer lifetimes would be
substantially cheaper and less reliable systems more costly). However, ignoring the details (such as the
learning rate for competing technologies or different interest rates for projects with different perceived
risks or different water depths available to different technologies, etc.) and assuming these results can
be directly compared to s $92/MWh value for offshore wind [3], the heavy 36 m2system would only
be competitive with traditional offshore wind when fairly mature. The smaller and lighter systems
would become competitive earlier (while less technologically mature and reliable).
The main takeaway is that there is a small economic benefit to scaling up AWE systems if they
can be scaled at a constant weight/area, otherwise it is very advantageous to build a larger number
of smaller units. The optimal size is likely to be close to wherever the lowest mass per unit area
can be achieved; this is unlikely to be smaller than 4 m
2
because for systems below a certain size,
requirements other than operational loads (such as handling or manufacturability requirements) will
drive mass. Further research, including more detailed design work to reduce uncertainty in mass,
is necessary to determine the optimal scale more precisely. Further research will also involve use of a
Monte Carlo method, genetic algorithm, or other method to determine if local minima are an issue for
this optimization.
Automation 2020,10 15
5. Conclusions
Airborne Wind Energy systems are lighter, easier to install, and potentially lower in cost than
comparable conventional horizontal axis wind turbines but are at a lower technology readiness level.
Though future work is necessary to further develop AWE designs at different scales and reduce
uncertainty around the optimal scale, the physics and economics of AWE systems are very different
from horizontal axis wind turbines which benefit from being as large as possible, and it appears that
optimized AWE designs will involve farms with significantly smaller individual units than traditional
wind farms.
Author Contributions:
Conceptualization, M.A. and A.S.; methodology, M.A.; software, M.A.; validation, M.A.
and A.S.; investigation, M.A.; writing–original draft preparation, M.A.; writing–review and editing, M.A., A.S.,
and K.C.; visualization, M.A.; supervision, K.C.; project administration, A.S. All authors have read and agreed to
the published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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... [95]). In the context of AWE, the optimal control problem is usually posed such the average power output of the system is maximized which is studied extensively in the literature [72,157,144,71,137,92,97]. Other approaches maximize the total energy that is generated [21,53,10]. Also the incorporation of other, for instance safety constraints, can be incorporated into the optimization problem [73,127,160]. ...
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Talk: https://www.youtube.com/watch?v=1CyKrL5gwXM Public defense: https://collegerama.tudelft.nl/mediasite/play/096730b4e9e0435398176d0fb82c35311d
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