This chapter 3 assumes a linear hydrodynamic model for the WEC. The simulator used for testing also assumes a linear hydrodynamic model. For a heaving buoy, the Cummins’ equation of motion is [76]:
(m+ ˜a(∞))¨z+ Z ∞
0
hr(τ) ˙z(t−τ)dτ +kz =fe+u (3.1)
wheremis the buoy mass, ˜a(∞) is the added mass at infinite frequency,zis the heave position of the buoy’s center of mass with respect to the mean water level, k is the hydrostatic stiffness due to buoyancy,uis the control force,fe is the excitation force, and hr is the radiation force term (radiation kernel). The second term in equation 3.1 is affected by the present as well as past oscillations.
Consider the simple case of a regular wave where the excitation force has only one frequency (ωi); in such case it is possible to show that the radiation term can be
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quantified using an added mass and a radiation damping term, each being considered constant at frequency ωi only [49]. The equation of motion for this simple case becomes:
(m+ ˜ai)¨z+ciz˙i+kzi =fei+ui (3.2)
Where ˜ai and ci are constants for a given exciting frequency. The excitation force in this case is:
fei =Aeisin(ωit+φi) (3.3) If we assume no control, the system input-output transfer function in the Laplace domain becomes:
Gi(s)≡ Zi(s)
Fei(s) = 1
(m+ ˜ai)s2+cis+k (3.4) Where Zi(s) and Fei(s) are the Laplace transforms of zi(t) and fei(t) respectively.
Using this transfer function we can add a feedback control that uses measurements of the buoy’s position and velocity. Let the controller transfer function in the Laplace domain be Di(s), the the block diagram for this WEC control system can be con- structed as shown in Figure 3.1.
Figure 3.1: Block diagram of a WEC control system of a single frequency regular wave
LetHi(s) be the closed loop system transfer function in the Laplace domain; hence
Hi(s)≡ Zi(s)
Fei(s) = Gi(s)
1−Di(s)Gi(s) (3.5)
The exciting force fe for a practical converter in a realistic wave is band-limited (ap- proaching zero at high frequencies) [49], and can be assumed as a linear superposition of N different exciting forces at different frequencies [77], that is:
fe=
N
X
i=1
fei =
N
X
i=1
Aeisin(ωit+φi) (3.6)
In the case of a real wave, then, there will beN transfer functionsGi(s) because each frequency has its own value of ci and ˜ai. In other words, the system reacts differently to different input frequencies, and hence a different transfer function is needed for each input frequency. This can be represented by the block diagram shown in Figure 3.2. The resultant buoy motion will be the combined motion of the individually computed motions; that is the buoy positionz(t) is the summation of all individually
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computed positions zi(t).
Figure 3.2: Block diagram of the decomposed WEC control system
Implicit in this treatment is the assumption that each block Gi-Di responds to the particular frequency ωi only. That is, the overall converter response is modeled as a linear superposition of multiple blocks Gi(s)δ(siωi), where δ(siωi) represents the Dirac delta function centered at ωi. In the limit as N → ∞, one expects that such a superposition would provide a close approximation to the true converter response [77].
Figure 3.2 shows a control strategy that designs a separate controller for each fre- quency in the spectrum. This strategy can be implemented if it is possible to measure (or estimate) the position and velocity components associated with each frequency.
This can be achieved using a Fast Fourier Transform (FFT) approach, and it will be discussed in detail in Sections 3.3 and 3.4. For now, let us assume that it is possible to extract the individual positions zi(t) given the buoy position z(t) and the individual
velocities ˙zi(t) given the buoy velocity ˙z(t). Then the system block diagram can be presented as shown in Figure 3.3. In Figure 3.3 the individual transfer functions Gi represent the system’s hydrodynamics and are all gathered in one dotted box labeled
”Hydrodynamic Model”; similarly all the individual controllers Di are gathered in a dotted box called ”Controller”.
Figure 3.3: Block diagram of the WEC multi resonant control system
In implementing this controller, the summation of all the individual controllers is computed and it is the control that gets applied to the system. Hence:
U(s) =
N
X
i=1
Ui(s) =
N
X
i=1
Di(s)Zi(s) (3.7)
We can also combine all the individual excitation forces Fei in the total excitation forceFe, and the resulting block diagram for the system becomes as shown in Figure 3.4. In Figure 3.4 the system hydrodynamic transfer functions Gi are gathered in
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block called ”Model”. The WEC control problem is to design the controller D(s);
Figure 3.4: Block diagram of the proposed feedback WEC control system
the design criteria considered in this chapter 3 is to maximize the energy extraction.
In summary, the WEC control problem is decomposed intoN control sub-problems, as illustrated in Figure 3.2. In this case, N different controls, Di(s), are designed.
Using a proportional derivative control, two control gains are to be designed for each individual controller Di. One advantage of this approach is that the input is a single-frequency for each sub-problem and hence eliminating the need to evaluate a convolution integral. The other advantage is that each controllerDi can be optimized independently from other controllers to its input frequency.