DESIGN OPTIMIZATION DESIGN OPTIMIZATION FORMULATION FOR FORMULATION FOR

VERTICAL AXIS WIND TURBINEVERTICAL AXIS WIND TURBINE

AMANDEEP

SAVI TAKKAR LAXMAN RAVESH 1

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Schematic diagram

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Material Properties

• 𝞼 yield strength of mild steel = 250 MPa• 𝞼 yield strength of stainless steel = 210 MPa• 𝞼 yield strength of HDPE = 15 MPa• E stainless steel (young’s modulus) = 200GPa• E Mild steel (young’s modulus) = 210GPa• Density of mild steel = 7.85*10-6 kg/mm3

• Density of stainless steel = 8*10-6 kg/mm3

• Density of HDPE =0.93*10-6 kg/mm3

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Design Variables

=

Db

HLc

btc

Ds

ts

Ls

tb

X1X2X3X4X5X6X7X8X9

=

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OBJECTIVE FUNCTIONOBJECTIVE FUNCTION

Our aim is to minimize the Rotating mass of wind turbine for a required minimum power.

F(x)= Rotating mass of wind turbine

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• Rotating mass = mass of connectors + mass of blades+ mass of shaft

= {8*Lc*(2btc-tc2) * (7.85*10-6 )} + 4*(π*Db/2)*H*tb*{0.93*10-6}

+ (π*Ds *ts*Ls*(8*10-6 )

F(x)=

Then the objective function {8*X3* (2*X4*X5-X52) * (7.85*10-6 )}

+ 4*(π*X1/2)* X2*X9*{0.93*10-6} + (π*X6 *X7*X8*(8*10-6 )

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Design assumptions/considerations

• Minimum air velocity considered for the calculation of power generated by the wind turbine is taken as 1 m/s, so that in practical, minimum power requirement will be met at any wind speed higher than 1 m/s.

• Force exerted on blades by wind is considered to be UDL over the blade acting horizontally.

• Force acting at the connector end is taken as perpendicular to the end of connector.

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Design wind speed calculation• Standard followed – IS-875 (Part III).• V = Vb*k1*k2*k3*k4 = 26.7 m/s.

• Vb = 47 m/s for Delhi being 3rd basic wind zone region.

• k1 = 0.71 (considering mean probable design life of structure for 5 years).

• k2= 0.80 (for height less than 10 m under terrain category 4).

• k3 = 1 for flat contoured area.

• k4 = 1 for non-cyclonic Delhi region.9

CONSTRAINTSCONSTRAINTS

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Design of hollow shaft

R2=Lc2 – H2/4

Torque T=F*(R+0.5Ds)where F=ρairAV2

T = 0.5*ρAV2*[sqrt(4Lc2-H2)+0.5Ds]

J =(π/4)Ds3ts (Assuming Ds>>ts)

τ =0.39 X 10-9 Db*H*V2[sqrt(4Lc2-H2)+0.5Ds]/(Ds

2*ts)

τ = T*Ds /(J*2)

(τallowable stress / τ) - 1 ≥ 0

τallowable stress

[0.39 X 10-9 *X1*X2*7.13*108 *[sqrt{4(X3)2 –(X2)2+0.5*X6}]/(X6)2*X7

Where

τallowable stress =62.87N/mm2 ; FOS = 1.67

R

H

Lc

- 1 ≥ 0

2Ɵ

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(Due to wind)

Buckling load on shaft• Dead load on shaft = weight of connectors + weight of blades• P = 8*Ac*Lc *ρ(mild steel) *g + 4*π*0.5Db*tb*H*ρHDPE*g• P= 8*(2*X4*X5-X52)*X3*7.85*10-6*g + 4*π*0.5*X1*X9*X2*0.93*10-6*g

• End conditions are “Effectively held in position at both ends but not restrained against rotation”

Effective length = 1*Ls ; Pcritical load= π2EI/(Le2)

I = π/8*Ds3*ts = π/8*X63*X7 (Assuming Ds>>ts)

Pcritical load= π2*E* π/8*X63*X7 X82

Allowable load = Pcritical load / FOS (Where FOS = 2)

(Pallowable load / P) -1 ≥0

π2*200*103* π/8*X63*X7X82*2*(8*(2*X4*X5-X52)*X3*7.85*10-6*g + 4*π*0.5*X1*X9*X2*0.93*10-6*g)

- 1 ≥ 0

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Design of connectors• Total bending moment= sqrt(M1

2 + M22 )

• M1 =0.5*ρAV2 *Lc = 0.5*1.225*10-9*X1*X2*X3*7.13*108

• M2 = weight of blade * radius of arm=(0.5)*(3.14/2)*Db*H*tb*ρHDPE*g*Lc*cosθ where cosθ=R/Lc

(R=sqrt{Lc2 – H2/4})

=(3.14/4)*X1*X2*X9*0.93*10-6*g*sqrt{X32-X22/4}

• M = sqrt(M12 + M2

2)

Max. M1 due to wind force & M2 due to weight of blades

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b

b

tc

σ1 = M*y/I where y=b+tc/4 =X4+X5/4 andI=(5/24)b3tc =(5/24)*X43*X5 (b>>tc)

σ1 = sqrt(M12+M2

2)*(X4+X5)(5/24)*X43*X5*4

Direct compressive stress on the connector = W*sinθAc

W = 0.5*0.5*π*Db*tb *H* ρHDPE *g Where sinθ = H/(2Lc) Ac = 2*b*tc – tc

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σ2 = 0.5*0.5*π*X1*X9*X2*0.93*10-6*g*X2(2*X4*X5-X52 )*2*X3

σ = σ1 + σ2

hence,(σallowable stress/ σ) -1 ≥ 0 σallowable stress = 150 N/mm2 ; FOS = 1.67 14

Design of blades

UDL on blade with F = ρair AV2

Maximum bending moment in case of U.D.L = F*H/8 (at centre)M = ρair AV2 * H/8 = 1.225* 10 -9 *X1*X2*7.13*108*X2

8σ=M*y/I y=(Db/2)-2Db/3π=0.29Db = 0.29*X1I=(0.11*128)Db3tb=(0.11*128) *X13 *X9 (Db>>tb )σ= 1.225* 10 -9 *X1*X2*7.13*108*X2* 0.29*X18*(0.11*128) *X13 *X9Hence (σallowablestress/σ) -1 ≥ 0 where σallowablestress =8.982 MPa FOS = 1.67

Centroid

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Power constraint

• POWER = Cp *½(ρair*Α*V3) P = 0.16*0.5*1.225*10-9*Db*H*1012

= 0.16*0.5*1.225*10-

9*X1*X2*1012

P ≥ 3*106

0.16*0.5*1.225*10-9*X1*X2*1012

3*106

- 1 ≥ 0

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Other constraints

• Ds/ts ≥ 15Ds / ts – 15 ≥ 0

X6 / X7 - 15 ≥ 0

• b/tc ≥ 15b/tc – 15 ≥ 0

X1/X5 – 15 ≥ 0

• Db/tb ≥ 15

X1/X9 – 15 ≥ 017

OPTIMIZATION PROBLEMOPTIMIZATION PROBLEMFind X =

Which minimizes the objective function

{8*X3* (2*X4*X5-X52) * (7.85*10-6 )} + 4*(π*X1/2)* X2*X9*{0.93*10-6} + (π*X6 *X7*X8*(8*10-6 )

min. F(X) =

Subjected to,

g1(X) :

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g2(X) :

g3(X) :

g4(X) :

g5(X) :

g6(X) : X6 / X7 - 15 ≥ 0 g7(X) : X1/X5 – 15 ≥ 0

Variable bounds : Xi ≥ 0 for all i = 1 to 9 19

g8(X) : X1/X9 – 15 ≥ 0

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Objective Function

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Constraints

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Function Tool

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Output

THANK YOU

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