Power Curve and Design Optimization of Drag ... - floba.info

Power Curve and Design Optimizationof Drag Power Kites

Florian Bauer, R. Kennel, C. Hackl, F. Campagnolo, M. Patt, R. Schmehl

[email protected],October 5th, 2017,Airborne Wind Energy Conference 2017, Freiburg, Germany

Institute for Electrical Drive Systems and Power Electronics,Department of Electrical Engineering and Information Technology,Technische Universität München


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How does an optimal drag power kite look like?

2

How does an optimal drag power kite look like?

... and what are the sensitivities of design parameters?

2

Outline

1. Model Derivation2. Power Curve Optimization3. Design Parameter Optimization4. Parameter Studies5. Conclusions

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Kinematics

4

Assumption 1: Loyd’s model (extended)

5

va = cos(ϕ) cos(ϑ)vw

�C2

L + (CD,eq + CD,rot)2

CD,Σ


5

va = cos(ϕ) cos(ϑ)vw

�C2

L + (CD,eq + CD,rot)2

CD,Σ

Assumption 2: azimuth and elevation are constant “e!ective” values


5

Kinematics

Kite Aerodynamics

6

cf.: D. Vander Lind. “Airfoil for a !ying wind turbine”. US Patent 9,709,026. July 2017. 7

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– CFD result

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– CFD result

cD = cD,0 + cD,2c2L

– Assumption 3: quadratic approximation

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https://upload.wikimedia.org/wikipedia/commons/f/fe/Airplane_vortex_edit.jpg

CL =cL

1 + 2A

with A =b2

A/nmw

CD,k = cD +C2

L

πeA� �� CD,k,i

+CD,k,a + CD,k,o

cf.: J. Katz and A. Plotkin. Low-Speed Aerodynamics. Cambridge Aerospace Series. Cambridge University Press, 2001. ISBN: 9780521665520.

Assumption 4: thin airfoil

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CL =cL

1 + 2A

with A =b2

A/nmw

CD,k = cD +C2

L


+CD,k,a + CD,k,o



9


CL =cL

1 + 2A

with A =b2

A/nmw

CD,k = cD +C2

L


+CD,k,a + CD,k,o



Assumption 5: no interaction between wings and rotors

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Kinematics

Kite Aerodynamics

Tether

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Assumption 6: wind contribution onairspeed negligible along tether length

After conversions: CD,te =1

4

dteLte

AcD,te

11cf. e.g.: B. Houska, M. Diehl, Optimal control for power generating kites, in: Proceedings of the 9th European Control Conference,

Kos, Greece, 2007, pp. 3560–3567.

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Fte,max ∼ Ate,coreMechanical strength:

Electrical resistance: Rte,wire ∼Lte

Ate,wire

Rte =Rte,wire

nte,c,++

Rte,wire

nte,c,−

Dielectric strength:

Total diameter, mass: [straight-forward summation]

Feasibility condition: [no overlapping electrical cables]

Ete,ins = f(Ute,n, rwire, wins)

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Kinematics

Kite Aerodynamics

Tether

Rotors

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“Aerodynamic” power: Pa =1

2ρAv3aCD,rot

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Frot,s = 2ρArot,sv2aa(1− a)

Prot,s = 2ρArot,sv3aa(1− a)2

Single rotor:

Assumption 7: actuator disk


2ρAv3aCD,rot

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Frot,s = 2ρArot,sv2aa(1− a)

Prot,s = 2ρArot,sv3aa(1− a)2

Single rotor:

After conversions:

Assumption 7: actuator disk


2ρAv3aCD,rot

CD,rot = 4nrotArot,s

A� �� rrot

a(1− a) and ηa,+ = 1− a

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Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

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17

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ηa,+ = 1 − a

Pte-loss = RteI2tevia

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ηa,+ = 1 − a

Pte-loss = RteI2tevia

Assumption 8: constant e"ciency factors for others

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ηa,+ = 1 − a

Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

Atmosphere

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vw = vw,href

ln�

hz0

�

ln�

hrefz0

� with h = hto + Lte sin(ϑ)

p(vw,href) =vw,href

v2w,href

exp

�−

v2w,href

2v2w,href

�

Assumption 9: logarithmic wind shear

Assumption 10: Rayleigh distribution

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Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

Atmosphere

Economics

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Assumption 11: launch & landing energy consumption negligible

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Year energy yield: Eel,yr[Wh/yr] =8, 760 h

1 yr·

∞�

0

p(vw,href)Pel,+(vw,href)dvw,href


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1 yr·

∞�

0


LCOE: kLCOE =k

Eel,yr


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1 yr·

∞�

0


LCOE: kLCOE =k

Eel,yr

Yearly costs: k = kinv + kop

kinv = KinvI(1 + I)T/yr

(1 + I)T/yr − 1kop = IopKinv


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1 yr·

∞�

0


LCOE: kLCOE =k

Eel,yr

Yearly costs: k = kinv + kop

kinv = KinvI(1 + I)T/yr

(1 + I)T/yr − 1kop = IopKinv

Kinv = kdtPel,n-ins +Kinv,o&pTotal capital costs:


21

Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

Atmosphere

Economics

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Outline


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Assumption 12: ∀vw,href ∈ [0, vw,href,cut-out] : arg{maxu

Pa} ≈ arg{maxu

Pel}

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Assumption 12: ∀vw,href ∈ [0, vw,href,cut-out] : arg{maxu

Pa} ≈ arg{maxu

Pel}

Assumption 13:�

C2L + C2

D,Σ ≈ CL

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wind speed

powe

r

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wind speed

powe

r

0 =dPa

dCD,rot

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wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

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wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

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wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

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II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

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II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ Pa ∼ v3w

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II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ CD,rot ∼ vw − const.

→ Pa ∼ v3w

25

II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


→ Pa ∼ v3w→ Pa ∼ vw − const.

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II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min



25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min



25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


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II

wind speed

powe

r I(a) I(b) III(a)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


25

II

wind speed

powe

r I(a) I(b) III(a)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


Pa = const.

25

II

wind speed

powe

r I(a) I(b) III(a)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


Pa = const.

25

II

wind speed

powe

r I(a) I(b) III(a) III(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax


Pa = const.

25

II

wind speed

powe

r I(a) I(b) III(a) III(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min


Fa=

Fa,m

ax

IV


Pa = const.

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Outline


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Key ideas

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Key ideas• serious estimates/better models for investment costs of other parts/pro!t margin

as well as for mass: not possible/too hard

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as well as for mass: not possible/too hard! INSTEAD: compute “maximum allowed” investment costs and

“maximum allowed” mass as result, which are requirements for detailed design

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“maximum allowed” mass as result, which are requirements for detailed design • rearrange equations into sequence of explicit analytical equations

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“maximum allowed” mass as result, which are requirements for detailed design • rearrange equations into sequence of explicit analytical equations• optimize free design parameters w.r.t. cost function w/ constraints w/ CMA-ES

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“maximum allowed” mass as result, which are requirements for detailed design • rearrange equations into sequence of explicit analytical equations• optimize free design parameters w.r.t. cost function w/ constraints w/ CMA-ES

• optimization problem:

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maxy

Kinv,o&pA

s.t. y ≤ y ≤ y

Outline


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≈ 90 kW/m2

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Parameter Valuenominal airfoil lift coe"cient 4.16tether length 370.67 melevation angle 19.36 °tether voltage 9.8 kVmaximum allowed investment costs (o&p)/wing area 56 k$/m2total maximum allowed investment costs 5.5 Mio. $energy yield per year 18.5 Mio. kWhtether mass 1.1 tmaximum allowed kite mass 11.7 tmaximum allowed kite mass/wing area 140 kg/m2wing loading 1.6 t/m2 30




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Summary of key results of further parameter studies:

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Summary of key results of further parameter studies:• tether material selection and nominal voltage have almost no e!ect

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Summary of key results of further parameter studies:• tether material selection and nominal voltage have almost no e!ect• high airfoil lift and high wing loading are required for high maximum allowed

cost and a high power density

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optimum

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•wide Region III(a) enables much lower wing loading and higher maximum allowed cost

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• a (low) tower might cover more than its own cost

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• a (low) tower might cover more than its own cost• for o!shore, the maximum allowed cost is more than the double

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• a (low) tower might cover more than its own cost• for o!shore, the maximum allowed cost is more than the double• the technology is scalable: the larger the system, the higher the power density

and maximum allowed cost

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• a (low) tower might cover more than its own cost• for o!shore, the maximum allowed cost is more than the double• the technology is scalable: the larger the system, the higher the power density

and maximum allowed cost• the model can reproduce measured data by Makani (model ver$cation, at

least in part)31



Source of left (bottom) "gure: Damon Vander Lind. “Analysis and Flight Test Validation of High Performance Airborne Wind Turbines”. In: Airborne Wind Energy. Ed. by Uwe Ahrens, Moritz Diehl, and Roland Schmehl. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. Fig. 28.12. 33

Source of left (bottom) "gure: Damon Vander Lind. “Analysis and Flight Test Validation of High Performance Airborne Wind Turbines”. In: Airborne Wind Energy. Ed. by Uwe Ahrens, Moritz Diehl, and Roland Schmehl. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. Fig. 28.12. 33

Outline


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Achievements:

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Achievements:•multidisciplinary steady drag power kite model

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Achievements:•multidisciplinary steady drag power kite model• covers dominant e!ects of all involved disciplines

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Achievements:•multidisciplinary steady drag power kite model• covers dominant e!ects of all involved disciplines• explicit, mostly analytical (white-box), based on $rst principles

35

Achievements:•multidisciplinary steady drag power kite model• covers dominant e!ects of all involved disciplines• explicit, mostly analytical (white-box), based on $rst principles• can reproduce measurements by Makani

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Achievements:•multidisciplinary steady drag power kite model• covers dominant e!ects of all involved disciplines• explicit, mostly analytical (white-box), based on $rst principles• can reproduce measurements by Makani• numerous parameter studies

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Outlook:

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Outlook:•my dissertation/our papers: all details and additional enhancements

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Outlook:•my dissertation/our papers: all details and additional enhancements• airfoil optimizations

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Outlook:•my dissertation/our papers: all details and additional enhancements• airfoil optimizations• veri$cations: wind tunnel, higher $delity models, tiny-scale (1 kW) kite

on the way, small-scale (20 kW) kite planned35

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Power Curve and Design Optimizationof Drag Power Kites

Florian Bauer, R. Kennel, C. Hackl, F. Campagnolo, M. Patt, R. Schmehl

[email protected],October 5th, 2017,Airborne Wind Energy Conference 2017, Freiburg, Germany

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Power Curve and Design Optimization of Drag ... - floba.info