RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS

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

2
Objective

• To investigate the possibility of implementing
  structural response and design criteria into the
  Dynamic Positioning control loop
• 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
• Implement a control algorithm that is capable of
  achieving a given target reliability level for a
  realistic and fully dynamic system

3
Control of low-frequency response level

Response
time
4
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

5
Principle Measure of structural safety is the
reliability index ß which is related to the
failure probability by ß F-1(pf)

• No activation, ß gt ßThreshold
• Alert interval, increasing activation
• ßThreshold gt ß gt ßCritical
• Full Activation, ß lt ßCritical

???
!!!
6
Definition of delta index
is the mean breaking strength of the line
(i.e. due to waves)
is the standard deviation of the breaking strength
7
Computation of reliability index

• Failure probability (pf) is probability that the
  extreme dynamic response will exceed critical
  level within a given reference duration
• Failure probability is estimated for a stationary
  reference time interval of e.g. 20 minutes by
  application of a Gumbel distribution
• Simplified relationship between delta-index and
  failure probability is expressed as
• pf
  ?(-d)

8
Simplified quasistatic load/response model is
applied for initial optimization study

• kTotr FE - FT
• where is total linearized stiffness of
  mooring lines, FE is external (low-frequency)
  excitation and FT is thruster force
• Conversely r (FE FT)/kTot

9
Two types of loss functions are considered

• Typical LQG type of loss function
• L( r ) KT?FT2 KF?r2
• (r is response, FT is thruster force, KT and
  KF are constants)
• Loss function based on failure probability
• L( r ) KT ?FT2 KP??(-d)

10
Two different types of control schemes are
considered

• PID control scheme
• where e here is e (rTarget rstatic, passive)
  (rTarget - FE/kTot)
• which (by neglecting second and last term)
  simplifies into
• FT Kp (rTarget rstatic, passive) Kp
  (rTarget - FE/kTot)
• LQG control scheme FT -Cr
• Normalized control factor is xc C/kTot

11
First type of loss function versus
vessel offset PID type of
control scheme (KTkTot2)/KF
1.0 and FE/kTot 2.0
12
First type of loss function versus
vessel offset PID type of
control scheme (KTkTot2)/KF
0.01 and FE/kTot 2.0
13
Second type of loss function versus vessel
offset PID type of
control scheme
(KTkTot2)/KF 1.0 and FE/kTot 2.0
14
Second type of loss function versus
vessel offset PID type
of control scheme
(KTkTot2)/KF 0.01 and FE/kTot 2.0
15
Second type of loss function versus vessel
offset PID type of
control scheme (KTkTot2)/KF 0.1
(intermediate value) and FE/kTot 2.0
16
First type of loss function expressed in terms of
normalized control variable - LQG type of control
scheme (KTkTot2)/KF 1.0
and FE/kTot 2.0
17
First type of loss function expressed in terms of
normalized control variable - LQG type of control
scheme (KTkTot2)/KF
0.01 and FE/kTot 2.0
18
Second type of loss function expressed in terms
of normalized control variable -LQG type of
control scheme
(KTkTot2)/KF 1.0 and FE/kTot 2.0
19
Second type of loss function expressed in terms
of normalized control variable - LQG type of
control scheme (KTkTot2)/KF
0.01 and FE/kTot 2.0
20
Comparison of optimal offsets for different loss
functions
21
Example Position control of turret moored
vessel
22
Vessel data

• Length of vessel 175m
• Beam 25.4m
• Draught 9.5m.
• Displaced volume 24 140m3.
• Mooring lines are composed of a mixture of chains
  and wire lines.
• Representative linearized stiffness of the
  mooring system is 1.5104 N/m.
• Mean value of breaking strength of single line is
  1.128106 N
• Standard deviation of the breaking strength is
  7.5 of the mean value.

23
Numerical simulation model
M
is the inertia matrix
is the hydrodynamic damping matrix
? pT , ?T x, y, ?T is the position and
heading
in earth-fixed coordinates
is the mooring force
? wT, ?T u, v, ?T is the translational
and rotational
velocities in body-fixed coordinates
b
is a slowly varying bias term representing
external forces due to wind, currents, and waves
is the thruster force
24
Feedback control law based on back-stepping
technique
25

Notation
?, ? and ? are strictly positive constants rj is
the length of the horizontal projection of
mooring line number j Tjis 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 sb,j is the
standard deviation of the breaking strength of
line number j The target value of the
reliability index is designated by ds. It can
be shown that this controller is global
exponentially stable
26
Time variation of water current velocity
27
Time variation of resultant environmental force
28
Time variation of vessel position in x-direction
29
Time variation of thruster force
30
Time variation of delta-index
In order for a delta-index of 4.4 to be optimal
for the present case study, the ratio of
(KTkTot2)/KP needs to be 10-6, i.e. the failure
cost needs to be very high compared to the unit
thruster cost.
31
Summary/conclusions

• 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
• Two different loss functions are compared. The
  first type is quadratic in the response while the
  second is proportional to the failure probability
  
• It is demonstrated for a particular example how
  structural reliability criteria can be
  incorporated directly into the control algorithm