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

!orian.bauer@tum.de,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

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

2

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

3

Kinematics

4

Assumption 1: Loyd’s model (extended)

5

va = cos(ϕ) cos(ϑ)vw

�C2

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

CD,Σ

Assumption 1: Loyd’s model (extended)

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

Assumption 1: Loyd’s model (extended)

5

Kinematics

Kite Aerodynamics

6

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

8

– CFD result

8

– CFD result

cD = cD,0 + cD,2c2L

– Assumption 3: quadratic approximation

8

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

9

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

9

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

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

“Aerodynamic” power: Pa =1

2ρAv3aCD,rot

15

Frot,s = 2ρArot,sv2aa(1− a)

Prot,s = 2ρArot,sv3aa(1− a)2

Single rotor:

After conversions:

Assumption 7: actuator disk

“Aerodynamic” power: Pa =1

2ρAv3aCD,rot

CD,rot = 4nrotArot,s

A� �� �rrot

a(1− a) and ηa,+ = 1− a

15

Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

16

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

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

LCOE: kLCOE =k

Eel,yr

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

LCOE: kLCOE =k

Eel,yr

Yearly costs: k = kinv + kop

kinv = KinvI(1 + I)T/yr

(1 + I)T/yr − 1kop = IopKinv

Assumption 11: launch & landing energy consumption negligible

21

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

1 yr·

∞�

0

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

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:

Assumption 11: launch & landing energy consumption negligible

21

Kinematics

Kite Aerodynamics

Tether

Rotors

Power Conversions

Atmosphere

Economics

22

Outline

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

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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

25

wind speed

powe

r

0 =dPa

dCD,rot

25

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

25

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

25

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

25

II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

→ Pa ∼ v3w

25

II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ Pa ∼ v3w

25

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

→ CD,rot ∼ vw − const.

→ Pa ∼ v3w→ Pa ∼ vw − const.

25

II

wind speed

powe

r

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ CD,rot ∼ vw − const.

→ Pa ∼ v3w→ Pa ∼ vw − const.

25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ CD,rot ∼ vw − const.

→ Pa ∼ v3w→ Pa ∼ vw − const.

25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − const.

25

II

wind speed

powe

r I(a) I(b)

0 =dPa

dCD,rot

→ CD,rot =CD,eq

2

Fa = Fa,min

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − 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

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − 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

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − const.

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

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − const.

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

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

→ Pa ∼ v3w→ Pa ∼ vw − const.

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

→ CD,rot ∼ vw − const.

Fa=

Fa,m

ax

IV

→ Pa ∼ v3w→ Pa ∼ vw − const.

Pa = const.

25

Outline

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

26

27

Key ideas

27

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

as well as for mass: not possible/too hard

27

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

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

27

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

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 • rearrange equations into sequence of explicit analytical equations

27

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

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 • rearrange equations into sequence of explicit analytical equations• optimize free design parameters w.r.t. cost function w/ constraints w/ CMA-ES

27

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

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 • 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

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

28

29

≈ 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

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

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

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:

31

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

31

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

31

32

optimum

31

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

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

31

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

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

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

31

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

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

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

31

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

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

• 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

31

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

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

• 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

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

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

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

34

35

Achievements:

35

Achievements:•multidisciplinary steady drag power kite model

35

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

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

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

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• numerous parameter studies

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• numerous parameter studies

Outlook:

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• numerous parameter studies

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

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• numerous parameter studies

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

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• numerous parameter studies

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

36

37

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

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

39

Power Curve and Design Optimizationof Drag Power Kites

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

!orian.bauer@tum.de,October 5th, 2017,Airborne Wind Energy Conference 2017, Freiburg, Germany