reliability-based structural optimization for positioning of marine vessels

Cesos-Workshop-March-2006 1

RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR

POSITIONING OF MARINE VESSELS

B J Leira NTNU Trondheim Norway

P I B Berntsen NTNU Trondheim Norway

O M Aamo NTNU Trondheim Norway


Objective

bull To investigate the possibility of implementing structural response and design criteria into the Dynamic Positioning control loop

bull Use a simplified quasistatic response model to derive optimal reliability levels for PID and LQG control schemes in conjunction with two different types of loss functions

bull Implement a control algorithm that is capable of achieving a given target reliability level for a realistic and fully dynamic system


Control of low-frequency response level

Response

time


Possible strategies for control algorithm based on

reliability indices

1 Monitoring of reliability indices

2 Weight factors based on reliability indices

3 Derivation of optimal control criteria based on reliability indices


Principle Measure of structural safety is the reliability index β which is related to the failure probability by β = Φ-1(pf)

bull No activation β gt βThreshold

bull Alert interval increasing activation

βThreshold gt β gt βCritical

bull Full Activation β lt βCritical


Definition of delta index

(ie due to waves)

is the mean breaking strength of the line

is the standard deviation of the breaking strength


Computation of reliability index

bull Failure probability (pf) is probability that the extreme dynamic response will exceed critical level within a given reference duration

bull Failure probability is estimated for a stationary reference time interval of eg 20 minutes by application of a Gumbel distribution

bull Simplified relationship between delta-index and failure probability is expressed as

pf = (-δ)


Simplified quasistatic loadresponse model is applied

for initial rdquooptimization studyrdquo

kTot∙r = FE - FT

where is total linearized stiffness of mooring lines FE is external (low-frequency) excitation and FT is thruster force

Conversely r = (FE ndash FT)kTot


Two types of loss functions are considered

bull Typical LQG type of loss function

L( r ) = KTFT2 + KFr2

(r is response FT is thruster force KT and KF are constants)

bull Loss function based on failure probability

L( r ) = KT FT2 + KPФ(-δ)


Two different types of control schemes are considered

bull PID control scheme

where e here is e= (rTarget ndash rstatic passive) = (rTarget - FEkTot)

which (by neglecting second and last term) simplifies into

FT = Kp (rTarget ndash rstatic passive) = Kp (rTarget - FEkTot)

bull LQG control scheme FT = -Cr

Normalized control factor is xc= CkTot

t

0d)(eiKedKepKTF


First type of loss function versus vessel offset PID type of control scheme (KT∙kTot

2)KF = 10 and FEkTot = 20





Second type of loss function versus vessel offset PID type of control scheme (KT∙kTot







2)KF = 01 (intermediate value) and FEkTot = 20


First type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot






Second type of loss function expressed in terms of normalized control variable -LQG type of control scheme (KT∙kTot



Second type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot



Comparison of optimal offsets for different loss functions

Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m


Example Position control of turret moored vessel


Vessel data

bull Length of vessel 175m

bull Beam 254m

bull Draught 95m

bull Displaced volume 24 140m3

bull Mooring lines are composed of a mixture of chains and wire lines

bull Representative linearized stiffness of the mooring system is 15∙104 Nm

bull Mean value of breaking strength of single line is 1128∙106 N

bull Standard deviation of the breaking strength is 75 of the mean value


Numerical simulation model

τ

is the mooring force

is the thruster force

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

η = [pT ψ]T = [x y ψ]T is the position and heading in earth-fixed coordinates

ν = [wT ρ]T = [u v ρ]T is the translational and rotational velocities in body-fixed coordinates

M

is the inertia matrix D is the hydrodynamic damping matrix

b is a slowly varying bias term representing external forces due to wind currents and waves


Feedback control law based on back-stepping technique

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ


λ γ and κ are strictly positive constants

rj is the length of the horizontal projection of mooring line number j

Tjrsquois the linearized mooring line tension in line j

pj is the horizontal position of the end-point at the anchor for the same mooring line

σbj is the standard deviation of the breaking strength of line number j

The target value of the reliability index is designated by δs

It can be shown that this controller is global exponentially stable

Notation


Time variation of water current velocity


Time variation of resultant environmental force


Time variation of vessel position in x-direction


Time variation of thruster force


Time variation of delta-index

In order for a delta-index of 44 to be optimal for the present

case study the ratio of (KT∙kTot2)KP needs to be 10-6 ie the

failure cost needs to be very high compared to the rdquounit thruster costrdquo


Summaryconclusions

bull A simplified model is applied in order to study optimal offset values (and corresponding values of the delta-index) when considering both the cost and reliability level

bull Two different loss functions are compared The first type is quadratic in the response while the second is proportional to the failure probability

bull It is demonstrated for a particular example how structural reliability criteria can be incorporated directly into the control algorithm


Objective

bull To investigate the possibility of implementing structural response and design criteria into the Dynamic Positioning control loop

bull Use a simplified quasistatic response model to derive optimal reliability levels for PID and LQG control schemes in conjunction with two different types of loss functions

bull Implement a control algorithm that is capable of achieving a given target reliability level for a realistic and fully dynamic system



Response

time



reliability indices












(ie due to waves)








pf = (-δ)




kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions






Response

time



reliability indices












(ie due to waves)








pf = (-δ)




kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions






reliability indices












(ie due to waves)








pf = (-δ)




kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions












(ie due to waves)








pf = (-δ)




kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions









pf = (-δ)




kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions







kTot∙r = FE - FT


















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions



















t

0d)(eiKedKepKTF






























Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions

































Coefficient ratio

(KT∙k Tot2)KF

and

(KT∙k Tot2)KP

Quadratic loss

function

Loss function

based on failure

probability

100 100m 200m

001 002m 000m




Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions







Vessel data


bull Beam 254m

bull Draught 95m








τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions






τ



wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η



M





bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions






bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T T

r

T2

w

w S

j

jT2 r

ppJ









Notation















Summaryconclusions












Notation















Summaryconclusions


















Summaryconclusions




reliability-based structural optimization for positioning of marine vessels

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