1.1.5.1 Oscillating bodies

The fundamental principle behind absorbing energy from a water wave is that energy must be removed from that wave. It therefore follows that the resultant wave, after passing the wave energy device, is either reduced or cancelled altogether.

A WEC, or indeed any object, oscillating in water will produce waves. It is the interaction of the waves produced by the device and the original wave that gives the resultant wave. For the device to remove energy from the wave, it is necessary for the resultant wave to be smaller than the incoming wave, and hence the two waves interfere destructively. If a device is to be a good wave absorber it must therefore inherently be a good wave maker.

A symmetrical body constrained such that it may only oscillate in one plane, either perpendicular or parallel to the water surface, is only able to absorb a maximum of 50% of the energy contained in an incident wave [8]. This is demonstrated in Figure 1.2.

The upper curve (a) represents a pure undisturbed sinusoidal wave, the ideal/theoretical sea state. In curve (b), an axisymetric body is heaving (oscillating vertically) in otherwise undisturbed water. Similarly, curve (c) shows the same body producing axisymmetric waves by rocking.

Curve (d) shows the effects of summing the previous three curves. It hence shows the effect of a WEC being allowed to move in two degrees of freedom. The wave approaching the device is unaltered from the original wave, as the effects of the body corresponding to its degrees of freedom cancel each other out. After passing the WEC, however, these effects are summated and equal in magnitude to the original wave. Thus the theoretical ideal of 100 % energy absorption from the approaching wave by a device with two degrees of freedom is shown.

ii iik.

Figure 1.3, a plan view of a symmetrical body oscillating in uniform waves, demonstrates that an oscillating point absorber affects a section of the approaching wave front greater than its width. The maximum energy which may be absorbed from an axisymmetric body equals the wave energy transported in an incident wave front of width equal to the wavelength divided by 2n [8]. This width is termed the absorption width.

1.1.5.2 Controlling a Wave Energy Converter (WEC)

Any buoyant body on the surface of a still liquid will start to oscillate and create waves if it is given an initial displacement. The equation of motion for the body given in (1.3) shows three forces acting on the body, corresponding to inertia (I), buoyancy (B) and drag (D), being equal to the radiation force required to create the waves, (FR).

Considering the homogeneous case, where the radiation force is set to zero, the body will behave in the familiar damped oscillatory motion common to any mass-spring-damper system as described by (1.4). A crucial characteristic of this type of system is that its natural frequency, i.e. that at which it oscillates when unconstrained, is the frequency in which it must be excited to have maximum amplitude oscillation, its resonant frequency, defined in (1.5).

C = coefficient of damping (Ns-1) k = spring constant (Nm-1)

Where f0 = resonant frequency (Hz)

If the body is to be used to extract power from its supporting liquid its efficiency will be a maximum when the frequency of the waves within the liquid equals the natural frequency of the floating body. In a liquid with regular sinusoidal waves the equation of motion becomes (1.6).

Where FE = the exciting force on the body (N)

Fe is equal to the force felt by the body held fixed in the incident waves. Furthermore it is possible to decompose FR into its constituent parts corresponding to the added mass (Ma) and the damping (C) terms, as in (1.7).

Where ro = frequency (rs-1)

Physically, the added mass is the mass of water which must also be accelerated to allow the device to accelerate, it is hence dependant on both frequency and device topology. Substitution of (1.7) into the equation of motion, (1.6), gives (1.8) (l + Ma (m))x + (D + C (m))x + Bx = Fe (1.8)

Comparison of (1.8) with (1.4) and (1.5) shows that the resonant frequency of a floating body in regular waves is given by (1.9).

In order to maximise the power absorption from a variety of different frequency waves it will be necessary to alter the resonant frequency. Inspection of equation (1.9) shows that a controllable external force, provided by the power take off mechanism, either in phase with the acceleration or the displacement of the device, could be used to influence the behaviour of a WEC by altering the denominator and numerator of the right hand side respectively. The device could hence be manipulated such that it was continually in a state of resonance. Alternatively, varying the actual buoyancy of the device alters 'B' in (1.9) and would hence have an identical effect.

Many control strategies have been suggested to control WECs e.g. [9, 10, 11, 12, 13, 14], encompassing the introduction of extra forces, equipment and buoyancy alteration.

The choice of strategy depends on the permitted complexity, natural characteristics of the device and the capabilities of the power take off mechanism. The simplest to execute is known as latching and involves deliberately restraining the device at the extremities of oscillation. The release of the device is delayed until a sufficient buoyancy force has built up to ensure that the velocity and force peak at the same time.

The converse of this concept, i.e. unlatching, also exists. The device is allowed to move freely for part of the cycle and the power take off mechanism is only engaged when the device has reached the desired velocity. Unlatching is utilised when the natural period of a device is longer than the most frequent wave spectrum. Both latching and unlatching are described as discrete control functions, because they are only operable at specific points in the oscillation cycle.

In a continuous control strategy the position and velocity of a device are measured continuously and adjustments to its motion are made in real time at any instant in the device's cycle. The ultimate control system for a wave energy device which allows the theoretical maximum amount of power to be removed from a wave is known as complex conjugate control. Appropriate mathematical models exist which require future knowledge of the sea state [10] but their complexity leaves them outside the scope of this thesis. Its rationale is analogous to maximising power output of a generator by connecting a resistive load equal to that of its own internal resistance [15]. All the spring and inertia forces are manipulated to cancel each other out by continuously adjusting the damping force.

On the right hand side of (1.6) the equation of motion was split into two distinct forces, Fr and FE, acting on the body. The radiation term FR is primarily dependent on the geometry of the device and the velocity at which it is oscillating. In a system of more than one degree of freedom this consists of a matrix containing both real and imaginary terms, represented in the form of a complex radiation impedance matrix. The second term FE is the force that would be required by a device to remain stationary in water with unit amplitude incident waves.

Equation (1.9) above demonstrated how these equations could be applied to a single degree of freedom device and used to calculate the frequency at which it would resonate. For more complex systems this could be done by taking the eigenvalue of the resultant force equation. A simplified version of this method approximates the eigenvalue by a causal control function, whereby prediction of the sea state is avoided [15,11].

Continuous control may require the return of energy to the sea for part of the oscillation cycle in order to keep the exciting force and body velocity in phase. The theoretical advantages of such a move are demonstrated in Figure 1.4 [9] which shows the cumulative amount of energy absorbed by a small scale wave energy converter in a 5 second period under three differing control strategies. The lower curve has no control and absorbs energy at a reasonably constant rate throughout the cycle. The middle curve, which shows a converter employing latching control, also shows a constant rate of energy accumulation, yet after 5 seconds its total yield is 4 times that of the uncontrolled oscillation.

The upper curve, showing the energy absorbed by the device under full complex conjugate control shows some interesting features. The cumulative energy absorbed is continually oscillating, demonstrating that for part of the cycle energy is being returned to the ocean. By the end of the 5 seconds, however, the average energy removed from the sea is more than double that of the latching control and almost 10 times that of no control. The long term impact of an effective control strategy on the energy yield of a device over its lifetime is clear.

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