Analysis of a Wave Energy Converter with a Particular Focus on the Effects of Power Take-Off Forces on the Structural Responses

Transcription

1 Analysis of a Wave Energy Converter with a Particular Focus on the Effects of Power Take-Off Forces on the Structural Responses Andrew Zurkinden, Lars Damkilde Wave Energy Research Group Civil Engineering Department June 3, 2014

2 Outline Failures of WEC s Non-linear WEC model Control mechanism Unconstrained control: Passive and Active Constrained control Structural model Conclusion

3 Failure I: Motions, Power extraction A good wave energy converter is a good wave absorber! Figure: Detail of the connection joint between arm and float

4 Failure II: Wear and fatigue damage The devil lies in the detail! A Figure: Detail of the connection joint between arm and float

5 Failure III: Inadequate design, absence of safety system Safety system to reduce wave loads in extreme events Figure: Mooring line failure in a five years extreme event, January 2012

6 PhD Outline Wave Spectrum Jonswap, Pierson- Moskovitz. Directionality functions to account for different incident wave angles Linear Hydrodynamic Potential Flow Analysis AQWA-LINE / WAMIT CeSOS PhD guest stay Spectral-based fatigue Fatigue Analysis anlysis Detailed FEanalysis Stress Concentration Factors (SCF) Global FEanalysis Quasi-Static response Analysis Complex load cases Active Control Passive Control Paper 2 Spectral-based fatigue analysis Paper 6 Structural modeling, Quasi-static approach Frequency response analysis (RAO) Paper 7 Fatigue analysis, control Fatigue Analysis based on Rain-flow counting cyclic analysis Global FEanalysis Structural dynamic model Paper 5 Validation of the non-linear numerical model by experiments Hybrid frequency-time domain model Site specific scatter diagram Extended Abstract 1 Transient structural analysis Paper 8 Effect of nonlinear motions on structural response Paper 1 Non-linear hydrostatic moment, drag forces Paper 3 Validation of the linear numerical model by experiments Extended Abstract 2 Direct integration vs. statespace modeling Power Matrix and Annual Energy Production, AEP Paper 4 Power matrizes for different control strategies

7 Case study: Wavestar arm Structural response analysis: stress analysis in the arm. The stress amplitudes vary in function of the applied control mechanisms. a) b) Figure: Wavestar arm with float, a) Protection mode b) Operational mode.

8 Laboratory Model 1:20 Representing the Wavestar Prototype WEC Laser DM z Load cell Mc A φ y MR MFK +MD MB Md MA do d h

9 Wave Excitation Force/Moments 10 2 VI LARGE DRAG λ D = 0.25 m H 10 1 V INERTIA & DRAG h = 0.65 m H/D 10 0 SMALL DRAG III IRB5 LARGE INERTIA IRA5 NEGLIGIBLE DRAG IRB4 IRA4 IRA3 RC2 IRB3 IRB2 IRA2 DEEP WATER BREAKING WAVE CURVE RC1 IRB1 IV 10-1 I ALL INERTIA NEGLIGIBLE DIFFRACTION IRA1 II DIFFRACTION REGION πd λ

10 Power Take Off Mechanism Lever arm d a from the cylinder force F c to the moment M c is depending on the angular rotation θ(t) of the arm. A time domain analysis is needed to account for this effect. F c (t) = k c l(t)+c c l(t) = k c da(t)θ(t)+c c d a (t) θ(t) d 1 = 3.27m M c (t) = d a (t) 2 (c c θ(t)+k c θ(t)) P c =< θ(t)d a (t)f c (t) > d a (t) θ(t) F cx = F c sin(γ(θ(t)) F cy = F c cos(γ(θ(t)) d 2 = 2.55m

11 Variable lever arm d a (t) Lever arm in function of the pitch angle θ(t). Effect on the power production and the maximum control force. Sinusoidal wave H = 1.0m,T s = 10s da[m] Pave[kW] Fc[MN] θ[rad] d 1 /d 2 Figure: Lever arm d a as a function of the θ, left: : d 1 /d 2 = 0.5,. : d 1 /d 2 = 0.6, : d 1 /d 2 = 1.0, : static referential state.

12 Time domain model Below: equation of motion (EQM) in the time domain, θ(t): Pitch. EQM is solved with a Newmark numerical integration scheme. The hydrostatic coefficient K is treated non-linear. }{{} M θ(t)+ k(t τ) θ(τ)+k(θ(t)) = M ex (t) M c (t) (1) J + a M c (t) = d a (t) 2 (c c θ(t)+kc θ(t)) (2)

13 Non linear hydrostatic stiffness coefficient Undisturbed wave elevation is used for buoy position tracking A B A MB/R C B h C h φ [rad] h Figure: Hydrostatic force vs. pitching angle.

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15 Method Newmark scheme: t = 0.01 s, γ=1/2, β=1/6, (Linear acceleration method) x k(t) IRF Radiation force, Pitch, Float D =4.94m Corr X1,X γ=0.5 γ=0.4 γ=0.3 k(t) [Nm/rad] γ=0.2 γ= t [s] Time (s)

16 Power wave spectrum Jonswap type, parametrized Random phase method S η (f) = 1.4 γ 5 16 H2 s f p 4 f 5 γ α exp ( 5 4 (fp f )4 ) γ = peak shape parameter H s = 4 m 0 f p, T p = T z, T z = γ = 1.0 δ = 1 m2 1 m 0 m 2 m0 m 2, (3) S(f)[m 2 s] band width parameter δ f[hz] g=1 g=3 g=5 g= peak shape parameter γ

17 Outline Failures of WEC s Non-linear WEC model Control mechanism Unconstrained control: Passive and Active Constrained control Structural model Conclusion

18 Unconstrained control No limits on the control force F c Arbitrary large angular displacements θ(t) Irregular wave, H s =2.4m, T p =7.5s, simulation time t=10 800s (3h). Passive Control, k c /k hyst = 0 Active Control, k c /k hyst = P ave P ave F c 40 F c Pc[kW] Fc[MN] Pc[kW] Fc[MN] c c /c opt c c /c opt

19 Outline Failures of WEC s Non-linear WEC model Control mechanism Unconstrained control: Passive and Active Constrained control Structural model Conclusion

20 Constrained control Force constrained: Limits due to the saturation in the hydraulic actuator system: F c,min (t) F c (t) F c,max (t) (4) Amplitude constrained: Absorber is not allowed to jump out of the water. θ(t) min θ(t) θ max (5) Stress responses including force and amplitude constraints have to be calculated in the time-domain.

21 Structural Model

22 Analysis of the Connection Joints Connection 1 W3 SN-curve Connection 2 F SN-curve Connection 3 F SN-curve Connection 4 F SN-curve 3 rd SDWED Symposium Figure: Connection Joints Aalborg University, Denmark

23 Stress response functions Hσ(ω θ) [MPa/m] Hσ(ω θ) [MPa/m] T z [s] T z [s]

24 Accumulated Fatigue Damage in a Joint Accumulated fatigue damage in Connection Joint 3 DHs=2.0m [-] DHs=2.0m [-] T z [s] T z [s]

25 Accumulated fatigue damage in Connection Joint 4 DHs=2.0m [-] DHs=2.0m [-] T z [s] T z [s]

26 Conclusion Reactive control has a significant impact on the stress amplitudes on the Wavestar arm. By introducing reactive control (k c /k hyst = 0.13) the average power is increased by a factor of 2 whereas the nominal stresses are increased by a factor of 4.

27 Acknowledgements Danish Council for Strategic Research

28 Thank you very much for your attention!