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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
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
9
Kinematics
Kite Aerodynamics
Tether
10
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.
12
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)
13
Kinematics
Kite Aerodynamics
Tether
Rotors
14
“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
17
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
18
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
19
Kinematics
Kite Aerodynamics
Tether
Rotors
Power Conversions
Atmosphere
Economics
20
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
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
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
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
23
Assumption 12: ∀vw,href ∈ [0, vw,href,cut-out] : arg{maxu
Pa} ≈ arg{maxu
Pel}
24
Assumption 12: ∀vw,href ∈ [0, vw,href,cut-out] : arg{maxu
Pa} ≈ arg{maxu
Pel}
Assumption 13:�
C2L + C2
D,Σ ≈ CL
24
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:
27
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
29
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
31
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
38
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
[email protected],October 5th, 2017,Airborne Wind Energy Conference 2017, Freiburg, Germany