## 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.

Figure 2.10: Energy contribution from various parts of the sea

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 2-1.

Max extension of hydraulics (m)

Max extension of hydraulics (m)

Figure 2.12: Effect of limiting hydraulic extension on maximum theoretical annual energy yield Table 2-1: Device characteristics for hydraulic ram with Amax= 4m

Force-peak (kN)

Velocity-peak (ms-1)

Power (kW)

Time period (s)

Maximum

4 430

3.1

4 290

13

Minimum

9.9

0.24

7.8

4

Average

375

0.94

403

The size of the plate is dictated by the drag force required for the power take off system to react against. The drag force, FD, associated with a cylindrical plate with cross-sectional area A may be expressed as (2.5).

Where CD = drag coefficient = 10 [57]

p = density = 1025 kg/m3 for sea water v = velocity (m/s)

From Table 2-1 the average velocity of the buoy is 0.94 m/s and the average required force is 375 kN. Specifying the area of the drag plate to give this drag force at 0.47 ms-1 ensures the buoy travels at least twice the speed of the cylinder and corresponds to a plate area of 360 m2, giving a diameter of 21m.

If the outer height of the plate is nominally taken as 0.2 m and it is made of aluminium with a density of 2700 kg/m3, a hollow design with only 50% material will weigh 94 tonnes. When submerged in sea water, the net buoyancy force downwards of the plate will be 220 kN. Size of buoy

The minimum size of the floating part of the device is calculated by the required up thrust to counteract the weight of the submerged plate. In order to simplify the oscillation model and use available mathematical models, e.g. [27], the same diameter as the plate may be used. Alternatively a diameter which puts the buoy's resonant frequency close to that of the predominant wave frequency may be selected.

### 2.6.2.2 Specific Hydraulic System Rams

Currently the largest commonly available cylinder [59] has a stroke of around 0.8 m. It is likely that this limit is due more to demand than any technical difficulties, so the characteristics of this cylinder are extrapolated here.

The maximum force provided by the cylinder is equal to the maximum drag provided by the plate, which itself is proportional to velocity. Assuming that the maximum velocity the plate can be expected to attain during operation is equal to half the maximum velocity of the buoy, 1.6 m/s from Table 2-1, results in a drag force of 4.7 MN. The cylinder design is hence based on a currently available cylinder with a maximum reactive force of 5118 kN [59], shown in Table 2-2.

Table 2-2: Parameters of cylinder

Maximum stroke - m

4 (extrapolated)

Average stroke - m

2.23

Outside diameter - m

0.4

Effective area push - m2

7.312 x 10-6

Effective area pull - m2

2.485 x 10-6

Max pressure - bar

700

Weight - tonnes

3.7 (extrapolated)

Max oil capacity m3

The force from the hydraulic piston and cylinder must be sufficient to keep the two bodies in the desired equilibrium positions. Due to the area displaced by the rod, the effective area of the piston is different depending on whether the cylinder is extending or contracting, as shown in Table 2-2. The orientation of the cylinder is such that if the smaller area is coupled to the high pressure accumulator, it is available to react against the residual buoyancy force plus the downward force from the low pressure acting in the push side. If the low pressure accumulator is 5 Bar, then the force required to overcome the residual buoyancy of the drag plate is reacted if the high pressure accumulator is set to 100 Bar.

### Accumulator

For the accumulator to provide some smoothing it must have enough capacity to smooth the power output for 30 seconds, equal to 4 average time periods. The equivalent flow into the high pressure accumulator would be equal to two pull and two push strokes during this time, 0.4 m3. Using (2.1), if it is specified that the high pressure accumulator must remain above 85 Bar during the loss of 0.4 m3, an initial gas volume of 3.2 m3 is suitable. If the total volume of the accumulator was set at 4 m3 this would make provision for ensuring the accumulators do not empty. A similar size can be expected for the low pressure accumulator.

## Renewable Energy Eco Friendly

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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