Report

Waves, WECs, and Arrays Cameron McNatt Coastal and Ocean Engineering MS candidate Outline 1) Background on waves, constructive and destructive interference 2) WECs – linear theory, motions and optimal motions, power, WEC wave field 3) Arrays interactions, Some interesting papers Waves! Velocity Potential • Inviscid, Irrotational Flow • – scalar, function of time and space • Velocity is the gradient of the velocity potential = = = • also used to compute surface elevation () and pressure () Wave Energy and Wave Energy Flux 1 2 = 2 = 1 0 0 −ℎ ∙ = (Power) Linear Wave Theory = Two Waves at Different Frequencies 2 2 1 1 = 1 1 + 2 2 2 2 Two Waves at Same Frequency and Direction 1 = 1 2 2 + 2 2 + 21 2 cos 1 − 2 ∙ Constructive Interference 1 = 2 1 = 1 + 2 2 2 ∙ Destructive Interference 1 = 2 + = 0 Two Waves, Same Frequency and Opposite Directions 1 = 1 2 2 − 2 2 ∙ Standing Waves Fully Standing Wave = 0 Wave Energy Converters! (WECs) WEC Hydrodynamics = 0 + + ≠ 0 + + 0 - Incident wave - Scattered wave - Radiated wave each DOF ( = 1 ) Wave fields from WAMIT Linear WEC Forces (heave) Hydro Forces =− Body Mechanical = = − − 0 + 3 = 3 = − − ℎ = − = − WEC Equation of Motion (heave) = + + ℎ + + + + + + = = and = −2 + + + + + = = + − 2 + + + WEC Power = = = + + ℎ ∙ Math… 1 1 ∗ = − 2 2 2 When you use the WEC equation of motion… 1 = 2 2 WEC Max Power and Optimal Motions Again power is… Math… 1 1 ∗ = − 2 2 1 = 8 2 2 1 1 − − 2 2 2 So… = 1 2 when 8 = 1 2 Rather that using equation of motion, prescribe the velocity ( is also the velocity of resonance) WEC and Waves = 0 + + When you superimpose all the velocity potential, you get partial standing waves − 1 = 0 ∙ ∙ = − 0 −ℎ Optimal Motions = 1 2 means = 1 − 2 Low frequencies (long waves), could be very large. Optimal velocity is in phase with diffraction force and optimal position is 90 degrees out of phase. Typically, diffraction force is maximum at the wave crest But for long waves, we expect the device to be somewhat of a wave-follower, (position in phase with the wave) Optimal Motions Wave Follower Optimal Motions NOTE: No actual calculations of forces were made here! Video just to visualize the phase difference between wave-follower and optimal motions Capture Width • Capture width has units of length • The actual width or the body is not relevant ℒ= ℒ = • For a point absorber (no scattered wave) operating in heave ℒ = 2 How is it possible to absorb waves that are outside the width of the body? • The radiated wave interacts with the diffracted wave to create standing wave patterns which reduce the wave energy flux! • For optimal motions, the amplitude can be very very large – this creates a larger radiated wave, which in turn reduces more of the wave energy flux Example of Real v. Optimal Motions • Cylinder – – – – – – radius = 5 m depth = 18 m 12 second period 1 m amplitude Infinite depth operating in heave • Computations done in WAMIT Diffracted Wave • Incident + Scattered • Body held fixed 6 Real Damping: = 10 / • Motions = + − 2 + + + = 0.68 • Power = = 63 kW • Radiated Wave ~10−3 Optimal Motions • Motions 1 =− 2 = 29. 9 • Power 1 2 = = 1.6 MW 8 • Radiated Wave ~0.1 Arrays! Array Computation = 0 + + • • • • But the DOF are now multiple devices Solution is a matrix Can be done in WAMIT Lots of mathematical tricks that I don’t quite understand (i.e. just compute for a single device…) q Factor • Key feature of an array is constructive and destructive interference ℒ = = × × ℒ • Nondimensional number that indicates the performance of an array = 1 – no gain from the array < 1 – net destructive interference > 1 – net constructive interference Maximum q Factor ℒ = × ℒ • Early papers made assumption of device under optimal motions. ℒ = for point absorber in heave. 2 • Just as with a single device, there is a maximum power for an array ∗ 1 ∗ − 1 1 − 1 − = − − − 8 2 2 2 = 1 ∗ − 8 when = 1 − 2 q Factor q 2 1.5 q is a function of both frequency and direction 1 0.5 kd Budal (1977) • Row of N evenly spaced point abosbers operating at optimal phase (equal amplitude) • Point absorber makes analytical computation tractable • q asymtotes to for infinite N Mavrakos and Kalofonos (1997) Fitzgerald and Thomas (2007) Also discovered that (for a point absorber) 1 2 = 1 2 0 Point absorber approximation and optimal motions Found optimal arrangements of 5 buoys using constrained nonlinear optimization = 2.77 Didn’t really have an explanation Child and Venugopal (2010) • Full analytical solution for an array of 5 cylinders in heave • Optimal motions and motions with real PTO • Looked at maximizing and minimizing q • Noticed the formulation of parabolas where the radiated or scattered wave is in and 90o out of phase with the incident wave 1) Place device 1 in the wave field 2) Place device 2 on a parabola from 1, so that the parabola from 2 goes through 1 3) Place device 3 on the intersection of parabolas 1 and 2 4) Repeat for devices 4 and 5 Parabolic Intersection Method Child and Venugopal Results • Parabolic intersection method – Real device: = 1.136 – Optimal motions: = 1.787 – Worst case: = 0.453 • Genetic Algorithm Optimization – Real device: = 1.163 – Optimal motions: = 2.010 – Worst case: = 0.326 Take Away… • Array interactions lead to constructive and destructive interference • Be wary of very high q factors – probably used optimal motions (maybe point absorber) • Read devices – q is much lower • q is highly sensitive to wave direction • There may be no net gain – whatever is gained from one direction is lost at another direction McIver (1994) • “…part of a practical strategy for the design of wave-power stations with large numbers of devices might be to reduce destructive interference effects,…, rather than attempt large increases in power absorption through constructive interference.” ? Haller et al (2011) • Laboratory Observations of Waves in the Vicinity of WEC-Arrays • Proc. Of the 9th European Wave and Tidal Energy Conf., Southampton, UK • Wave field measurements from the Columbia Power array tests in Hinsdale last winter. • Mick and Aaron Porter are using SWAN to try to reproduce the results • I am will use WAMIT to try and reproduce the results Thank you! Questions, comments, suggestions? References • • • • • • • • • Budal, K. (1977). Theory of absorption of wave power by a system of interacting bodies. Journal of Ship Research, 21, 248-253 Child, B.M.F and V. Venugopal. (2010). Optimal configurations of wave energy device arrays. Ocean Engineering, 37, 1402-1417 Cruz, J., R. Sykes, P. Siddorn, R. Eatock Taylor. (2010). Estimating the loads and energy yield of arrays of wave energy converters under realistic seas. IET Renewable Power Generation, 4-6, 488-497. Evans, D. V. (1980). Some analytic results for two and three dimensional waveenergy absorbers. Power from Sea Waves - Proc I.M.A. Conf., Edinburgh, 213-249 Falcao, Antonio F. de O. (2010). Wave energy utilization: A review of the technologies. Renewable and Sustainable Energy Reviews, 14, 899-918 Fitzgerald, C., and G. Thomas. (2007). A preliminary study on the optimal formation of an array of wave power devices. Proc. Of the 7th European Wave and Tidal Energy Conf., Porto, Portugal Haller, M. C., A. Porter, P. Lenee-Bluhm, K. Rhinefrank, E. Hammagren, H. T. OzkanHaller, & D. Newborn. (2011). Laboratory observations of waves in the vicinity of WEC-arrays. Proc. Of the 9th European Wave and Tidal Energy Conf., Southampton, England. Mavrakos, S. A., and A. Kalofonos. (1997). Power absorption by arrays of interacting vertical axisymmetric wave-energy devices. Applied Ocean Research, 19, 283-291. McIver, P. (1994). Some hydrodynamic aspects of arrays of wave-energy devices. Applied Ocean Research, 16, 61-69