Sizing the WEC for a given sea state
Wave data given in [56], cited in [57] and displayed in Figure 2.9 gives the probability of each sea state of a given point off the Norwegian coast in terms of the zero upcross time and significant wave height. If the float is assumed to follow exactly the surface of the water, these datum can be used as the basis of a simple design model. In reality, oscillation of two coupled bodies in water is mathematically more complex,
Designing for the most 'likely' sea state, with a probability of 6.73 %, the significant wave height is 2m and the zero upcross time is 6 seconds. If it is assumed that the sea in this state contains waves of one frequency all in phase, a highly idealised sea state, the zero upcross time becomes the time period and hence the frequency 0.16 Hz. Using this simplification, it is possible to speculate about the component sizes for the WEC and time average energy contained in the sea. Using equation (1.2) in Chapter 1, the energy present per m wave front in this sea state is equal to 23 kW/m, the wavelength of which is 56 m. If the buoy were to act as a point absorber and remembering that this means it is capable of absorbing power equivalent to that contained in a front width equal to the wavelength divided by 2n, the total energy incident on the buoy would be equal to 204 kW. As explained in section 1.1.5.1, any WEC having only one degree of freedom will have a maximum capture efficiency of 50%, giving the maximum power e.g. [58].
available to the buoy as 102 kW. On this assumption, one might expect a 100 kW power take off system to be making best use of the available sea and the rated output of the device in one hour to be 100 kWh. However, due to the nature of energy contained within waves, namely its increase with the square of the amplitude, the actual power which a device may capture over a 'typical' hour is 488 kWh. To obtain this value the relative probability of each sea state and the energy contained within it has been accounted for using ( 2.3).
Figure 2.9: Scatter table of wave data [56]
X PiProbi
Figure 2.9: Scatter table of wave data [56]
X PiProbi
A ave 7
X Probi
Where
power available at sea state i (W) probability of sea state i occurring Still assuming the idealised sine wave which fulfils the zero upcross time and now looking at the behaviour of the device over one year allows the relative importance of each sea state to be compared, in terms of kWh Figure 2.10. Despite their rare occurrence, waves of large amplitude and time period can be seen to make a significant contribution to annual yield.
2.6.2.1 Power take off Maximum extension
Hydraulic rams have a maximum permissible extension to avoid damage. Varying the magnitude of this extension will impact on both the power yield of the device and its cost. Figure 2.10 does not show a clear cut off point, where waves above a particular amplitude no longer contribute significantly to the annual yield. It is necessary to make further assumptions about the behaviour of the buoy and sea. Stipulating that during large waves the drag plate follows the oscillation of the float when the maximum cylinder extension is reached effectively limits the amplitude of oscillation. Take for example the graphs of Figure 2.11. The upper graph shows a constant frequency sea state, with a successive amplitude of two, three and four metres. The second graph shows the displacement of a submerged plate which must always be within a 4 metre envelope of the sea surface, due to its coupling with a floating buoy. The final graph shows the extension of the hydraulic ram.
amplitude 2m amplitude 3m sea sul~face amplitude 4m amplitude 2m amplitude 3m sea sul~face amplitude 4m
A 
/ \ 
/ \  
/ \ 
/ \  
\ I  
\ ■■' I V 
\J 
M  
0 1 
! 3 
plate 
depth 
5 
6 
7 8  
I  
0 1 
Extension o 
f hydraulics 
6 
7 8  
Figure 2.11: Effect of limiting amplitude of oscillation to 2 m From these graphs it is clear that when limiting the maximum excursion of the hydraulic rams, the plate will oscillate such that the profile of the ram extension is at the same frequency as the surface but with limited amplitude. If the shape of the resultant extension graph in large waves is approximated to a sine wave, the power captured by the device can be calculated using formula (1.2). The addition of the conditional statement given in (2.4) limits the amplitude of extension to Amax, the size of the hydraulic ram and can be used in combination with (1.2) to calculate the power take off at any sea state. The effect of using alternative amplitude rams on annual power yield may now be investigated, as given in Figure 2.12. Limiting the extension of the cylinder to 4 m ensures the capture of around 80 % of the available energy and gives the device characteristics shown in Table 21. Max extension of hydraulics (m) Max extension of hydraulics (m)

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