2 nd SDWED Symposium Advances in Modelling of Wave Energy Devices - WP4

Transcription

1 2 nd SDWED Symposium Advances in Modelling of Wave Energy Devices - WP4 Andrew Zurkinden Wave Energy Research Group, Civil Engineering Department, Aalborg University, Denmark April 26, 212

2 Structural Design of Wave Energy Devices

3 Outline Motivation Mathematical model Structural design Outlook

4 Motivation I Cost of energy from renewable resources in general, and especially from waves, is still too high compared with other forms of power generation. WEC s are affected by demands for structural efficiency, fast production time and cost effectiveness. The overall safety requirement of a WEC can be lowered due to the relatively high acceptance in losses. It is unlikely that a system failure will lead to unacceptable consequences such as loss of life or uncontrolled outflow of oil and gas.

5 Problems Numerical models are needed to accurately predict the motions of a floating wave energy converter, in order to i) estimate the structural response of the system ii) calculate the annual energy output. Advanced control strategies are needed in order to optimize the power absorbtion. A general characteristic of numerical models is the assumption of linear-fluid structure interaction. Thus it is important to know within what amplitude of the waves the system has a linear response? - experiments can be very revealing.

6 Mathematical model Wave loads and responses are calculated based on linear potential flow. Time domain models are used to perform structural analysis. Case study on a point absorber, Wavestar laboratory model: t (M 44 +a44) φ 4 (t)+ K 44 (t τ) φ 4 (τ)dτ +C 44 φ 4 (t)+m c (t) = h eτ (t τ)η(t)dτ (1) M c (t) = m c φ 4 (t)+c c φ 4 (t)+k c φ 4 (t)+ h c φ 4 (t τ) φ 4 (τ) dτ (2)

7 Non-linear hydrostatic restoring force Experimental tests on a hemisphere have been carried out at AAU to approximate the non-linear hydrostatic behavior. Trilinear piecewise approximation of the non-linear hydrostatic behavior is introduced in the numerical model. 2 A D B 1 A H H D R44 R 44 [Nm] B -1 C Hydrostatic experiments Piecewiese linear approx φ [rad] 4 C H D R44

8 Approximation of the hydrodynamic parameters Marinecontrol toolbox, Matlab control toolbox. 4 th order approximation is accurate enough. H r φ4 (s) = B(ω)+iω(A(ω) A ) (3) H r φ4 (s) P(s) Q(s) = p s m +p 1 s m p m 1 s +p m s n +q 1 s n q n 1 s +q n (4) 1.9 Approx. 3th order Approx. 4th order Approx. 5th order FD calculation Approx. 3th order Approx. 4th order Approx. 5th order FD calculation.8 2 Imag(H rφ4 ).7.6 Re(H rφ4 ) ω [rad/s] ω [rad/s]

9 Replacement of the convolution integral t K 44 (t τ) φ 4 (τ)dτ [ p p 1... p n 1 ] I(t) (5) ǫ = φ φ FD φ FD 1 ǫ =.1.6% φ 4 (t) [rad] φ 4 (t) [rad] Nonlinear model Experiments t [s].3 Nonlinear model Experiments t [s]

10 Interfaces with WP1, WP2, WP3, WP5 The equation of motion is rewritten in a ODE form and solved with the ode-package in Matlab. Integration of non-linear forces in EQM is possible Analysis of multiple degree of freedom systems Ẋ Ẍ İ i (t) = (6) Ẋ (M +a ) 1 [ I i (t) K H X +F PTO +F M +F H +F o ] İ i (t) (7)

11 Validation with experimental tests Eigenperiod of the laboratory model: T n =.79sec Wave states H m T p P wave H m λ p ν [m] s] [W/m] [ ] [-] IRA IRA IRA IRA IRA IRB IRB IRB IRB IRB

12 IRA1 and IRA2: Nonlinear model Experiments.6 Nonlinear model Experiments φ 4 (t) [rad] φ 4 (t) [rad] t [s] t [s] IRA3 and IRA5: Nonlinear model Experiments.15.1 Nonlinear model Experiments.4 φ 4 (t) [rad].2 φ 4 (t) [rad] t [s] t [s]

13 IRB1 and IRB2: φ 4 (t) [rad] Nonlinear model Experiments t [s] IRB4 and IRB5: φ 4 (t) [rad] Nonlinear model Experiments t [s] φ 4 (t) [rad] φ 4 (t) [rad] Nonlinear model Experiments.1.15 Nonlinear model Experiments t [s] t [s]

14 ν ν Correlation factors: Non linear model Non linear model E xx.92.9 E xx T p Non dimensional performance index vs.t p : T p.6.55 ν Experiments ν Linear numerical model.6.55 ν Experiments ν Linear numerical model T p T p

15 Conclusion of the mathematical model A mathematical model based on linear-fluid structure interaction has been set up and validated by experimental tests on a point absorber wave energy converter. Good agreement between numerical and experimental tests was found for mild, regular and irregular waves H m /λ p.3. For steep waves, as well as overtopping waves, where the wave steepness is larger than H m /λ p.6. the linear model seems to underestimate the response and thus underestimate the power absorbtion of the device. Non-liner effects such as the non-linear hydrostatic restoring force become important and should be included in the numerical model.

16 References & Acknowledgment Perez, T. and Fossen, T. I. A Matlab Tool for parametric identification of radiation-force models of ships and offshore structures. Modeling, Identification and Control, MIC-3(1): Cummins, W.E. The impulse response function and ship motions. Schiffstechnik 9:11-19, Yu, Z. and Falnes, J. State-space modeling of a vertical cylinder in heave. Applied Ocean Research. 17: , Taghipour, R. Efficient Prediciton of Dynamic Response for flexible and Multi-Body Marine Structures. PhD Thesis, 28. Morten Kramer and Francesco Ferri for providing the experimental data.

17 Structural design Floating offshore structures are subjected to high fatigue loading, stress response from wave action shows typically 5mio loading cycles a year. For the majority of wave energy devices the structural analysis can be decoupled from the hydrodynamic calculation. The objective of my work is to produce a model which can be used to estimate the fatigue damage of a component by incorporating the unique features of a WEC (PTO).

18 Example: Long-term fatigue damage of a hydraulic cylinder subjected to internal fluid pressure induced by wave loads, Limin Yang, NTNU, 211

19 Structural design Stresses on the structure are obtained numerically for each short term condition by solving the mathematical dynamical model. Rainflow cycle counting will be applied to each obtained time series. Long-term distribution of stresses for each short-term condition are obtained by considering the probability of occurrence of each wave state. Long-term fatigue analysis is performed based on the Miner-Palmgren damage accumulation rule.

20 Structural analysis of the Wavestar prototype

21 Outlook Today Goal, 214 Thank you for your Attention:) Start, 211

22 Acknowledgements Danish Council for Strategic Research