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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 Va≈1

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

(θt≈0.36 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 +6E−7Re (13)

Re =Vacw

ν=Va√Aw

ν√AR (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 ∝λ∝√AR

(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

m∝L3

,

m∝A3/2

,

or m/A∝m1/3, 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

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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

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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,w∝wca p tca p bw∝Tmax 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 (

Li≈Ixx ˙

p

). Roll rate (p) is inversely proportional to tether length (

λ=λb√AR 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 ˙

p∝mUAV 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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