1. 1 Cesos-Workshop-March-2006 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. 2 Cesos-Workshop-March-2006 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. 3 Cesos-Workshop-March-2006 Control of low-frequency response level Response time

4. 4 Cesos-Workshop-March-2006 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. 5 Cesos-Workshop-March-2006 Principle: Measure of structural safety is the reliability index β which is related to the failure probability by β = Φ -1 (p f ) No activation, β > β Threshold Alert interval, increasing activation β Threshold > β > β Critical Full Activation, β < β Critical !!! ???

6. 6 Cesos-Workshop-March-2006 Definition of delta index (i.e. due to waves) is the mean breaking strength of the line is the standard deviation of the breaking strength

7. 7 Cesos-Workshop-March-2006 Computation of reliability index Failure probability (p f ) 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: p f =  (-δ)

8. 8 Cesos-Workshop-March-2006 Simplified quasistatic load/response model is applied for initial ”optimization study” k Tot ∙r = F E - F T where is total linearized stiffness of mooring lines, F E is external (low-frequency) excitation and F T is thruster force Conversely: r = (F E – F T )/k Tot

9. 9 Cesos-Workshop-March-2006 Two types of loss functions are considered: Typical LQG type of loss function: L( r ) = K T ·F T 2 + K F ·r 2 (r is response, F T is thruster force, K T and K F are constants) Loss function based on failure probability: L( r ) = K T ·F T 2 + K P ·Ф(-δ)

10. 10 Cesos-Workshop-March-2006 Two different types of control schemes are considered: PID control scheme: where e here is e= (r Target – r static, passive ) = (r Target - F E /k Tot ) which (by neglecting second and last term) simplifies into: F T = K p (r Target – r static, passive ) = K p (r Target - F E /k Tot ) LQG control scheme: F T = -Cr Normalized control factor is: x c = C/k Tot

11. 11 Cesos-Workshop-March-2006 First type of loss function versus vessel offset PID type of control scheme (K T ∙k Tot 2 )/K F = 1.0 and F E /k Tot = 2.0

12. 12 Cesos-Workshop-March-2006 First type of loss function versus vessel offset PID type of control scheme (K T ∙k Tot 2 )/K F = 0.01 and F E /k Tot = 2.0

13. 13 Cesos-Workshop-March-2006 Second type of loss function versus vessel offset PID type of control scheme (K T ∙k Tot 2 )/K F = 1.0 and F E /k Tot = 2.0

14. 14 Cesos-Workshop-March-2006 Second type of loss function versus vessel offset PID type of control scheme (K T ∙k Tot 2 )/K F = 0.01 and F E /k Tot = 2.0

15. 15 Cesos-Workshop-March-2006 Second type of loss function versus vessel offset PID type of control scheme (K T ∙k Tot 2 )/K F = 0.1 (intermediate value) and F E /k Tot = 2.0

16. 16 Cesos-Workshop-March-2006 First type of loss function expressed in terms of normalized control variable - LQG type of control scheme (K T ∙k Tot 2 )/K F = 1.0 and F E /k Tot = 2.0

17. 17 Cesos-Workshop-March-2006 First type of loss function expressed in terms of normalized control variable - LQG type of control scheme (K T ∙k Tot 2 )/K F = 0.01 and F E /k Tot = 2.0

18. 18 Cesos-Workshop-March-2006 Second type of loss function expressed in terms of normalized control variable -LQG type of control scheme (K T ∙k Tot 2 )/K F = 1.0 and F E /k Tot = 2.0

19. 19 Cesos-Workshop-March-2006 Second type of loss function expressed in terms of normalized control variable - LQG type of control scheme (K T ∙k Tot 2 )/K F = 0.01 and F E /k Tot = 2.0

20. 20 Cesos-Workshop-March-2006 Comparison of optimal offsets for different loss functions:

21. 21 Cesos-Workshop-March-2006 Example : Position control of turret moored vessel

22. 22 Cesos-Workshop-March-2006 Vessel data: Length of vessel: 175m Beam: 25.4m Draught: 9.5m. Displaced volume: 24 140m 3. Mooring lines are composed of a mixture of chains and wire lines. Representative linearized stiffness of the mooring system is 1.5∙10 4 N/m. Mean value of breaking strength of single line is 1.128∙10 6 N Standard deviation of the breaking strength is 7.5% of the mean value.

23. 23 Cesos-Workshop-March-2006 Numerical simulation model is the mooring force is the thruster force η = [p T, ψ] T = [x, y, ψ] T is the position and heading in earth-fixed coordinates ν = [w T, ρ] T = [u, v, ρ] T is the translational and rotational velocities in body-fixed coordinates M is the inertia matrix is the hydrodynamic damping matrix b is a slowly varying bias term representing external forces due to wind, currents, and waves

24. 24 Cesos-Workshop-March-2006 Feedback control law based on back-stepping technique

25. 25 Cesos-Workshop-March-2006 λ, γ and κ are strictly positive constants r j is the length of the horizontal projection of mooring line number j T j ’is the linearized mooring line tension in line j p j is the horizontal position of the end-point at the anchor for the same mooring line σ b,j is the standard deviation of the breaking strength of line number j The target value of the reliability index is designated by δ s. It can be shown that this controller is global exponentially stable Notation:

26. 26 Cesos-Workshop-March-2006 Time variation of water current velocity:

27. 27 Cesos-Workshop-March-2006 Time variation of resultant environmental force:

28. 28 Cesos-Workshop-March-2006 Time variation of vessel position in x-direction:

29. 29 Cesos-Workshop-March-2006 Time variation of thruster force:

30. 30 Cesos-Workshop-March-2006 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 (K T ∙k Tot 2 )/K P needs to be 10 -6, i.e. the failure cost needs to be very high compared to the ”unit thruster cost”.

31. 31 Cesos-Workshop-March-2006 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