Simulink model

Simulink, developed by MathWorks, provides an interactive, graphical environment for

modeling, simulating and analyzing dynamical systems. Its primary interface is a graphical

block diagramming tool and a customizable set of block libraries. Examples of commonly used

blocks are integrators, gain and summation blocks all of which are depicted in Figure 4:15: The

figure can be thought of as a block diagram representation of the time domain based physical

equation (4.23) described previously.

The summation block takes all the relevant forces as inputs and outputs (after multiplication

with the mass gain block) the acceleration (labelled x’’ in the figure).

The integrator is the most basic for modelling dynamical systems, taking the acceleration signal

as input and outputting the velocity. This velocity signal is then used to as input to the state

space block to determine the radiation force as described in the previous sections. The velocity

signal is also used for the power capture, which can be resistive damping, Coulomb damping

or other non-linear approaches. The ability to model non-linear methods of power extraction is

an advantage of the time domain numerical model over the frequency domain one.

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The second integration block takes velocity as input and outputs the position of the device. The

position signal is then used to determine the hydraulic restoring force.

Figure 4:15: Simulink model including state space representation of radiation force. The input to the model is the excitation force, which is prepared separately before each simulation. The excitation force can be determined in two possible ways; the preferred method used here is to obtain the spectral information of the incident wave, multiply this by the force transfer function obtained using WAMIT (see Figure 4:7), and then perform an inverse Fourier Transform on the result to obtain the force signal. Alternatively, the excitation force impulse response can be used to perform convolution with the wave elevation signal to obtain the excitation force.

Simulations are carried out using both linear damping and Coulomb damping for the PTO force.

Chapter 6: discusses these simulations along with experimental results.

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Now that the time domain model is assembled, including an approximation of the radiation force, it could be validated against the radiation forces produced by using only the frequency domain procedures. However, since the floater displacement response is usually of more interest than radiation forces, a natural validation check at this stage is to compare predicted floater responses using both frequency and time domain methods.

Figure 4:16 shows such a comparison of RAOs obtained using both methods. The RAO has been calculated using the time domain model at 256 separate frequencies ranging from 0 to 4 Hz. Simulations are run for 600s with a sampling time step of 0.05s. Curves for all three shapes with a constant draft of 221 mm are plotted, with RAO results being plotted on the left hand y-axis and the difference between both methods being plotted on the right hand y-axis. The agreement between the curves for all shapes is good with the main difference being close to the resonance zone. Referring back to Figure 4:12 it can be seen that although the fit of the retardation function is very close, there is some error near resonance.

Figure 4:16 evaluates the RAOs with both the supplementary mass and external damping settings set to zero. As a result of this the rankings of highest RAO observed; cylinder hemisphere-cylinder and cone-cylinder are merely an artefact of the buoys intrinsic mass and hence the natural frequency. Upon tweaking the supplementary mass and external damping parameters towards the optimum setting, the superior excitation force and hydrodynamic damping of the lighter buoys can be exploited.

4.4.1.1 Effect of using a linear spring term in numerical model

Recall from chapter 2 that the hydrostatic restoring force Fhyd is equal to the Archimedes force FArch

minus the gravitational force Fg:

𝐹ℎ𝑦𝑑= 𝜌𝑔𝑉(𝑡) − 𝑚𝑔 (4.44)

Assuming the heave excursions are small then the hydrostatic force can be approximated by

𝐹ℎ𝑦𝑑= −𝑐𝑧 (4.45)

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The use of a constant spring term is permissible for a cylinder shape buoy except for two possible scenarios (illustrated in Figure 4:17 below): the first being where the buoy has fully emerged from the surface of the water, in which case the restoring force will be comprised solely of the buoys own weight; the second is whereby the buoy becomes submerged to a depth greater than the freeboard of the device.

Using a linear spring term, the magnitude of restoring force would increase with submerged depth (dashed red line) when, in reality, the hydrostatic restoring force would be limited: that limit being the additional buoyancy force given by the now submerged freeboard.

If a linear spring term is used for the cone-cylinder and hemisphere-cylinder shaped buoys, then the numerical model will overestimate device performance, especially near the resonance zone.

Clearly a more sophisticated approach to estimating the hydrostatic restoring force for the cone-cylinder and hemisphere-cylinder would be to calculate the instantaneous submerged volume which can then be used to calculate the buoyancy force. Determining the total submerged volume becomes a problem of finding the volume of cylindrical section and additional conical or hemispherical section.

Z<FB Z>d

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The volume of the cone is given by,

𝑉𝑐 = 1 3𝜋𝑟𝑐 2_ℎ 𝑐 = 1 3𝜋ℎ𝑐 3 _(4.46)

Since the cone extends an angle of 90° meaning ℎ𝑐 = 𝑟𝑐.

As the volume of the hemisphere section changes with depth the immersed volume is that of a spherical cap (a portion of a sphere cut off by a plane) given by,

𝑉𝑐𝑎𝑝= 𝜋ℎℎ 6 (3𝑎 2_{+ ℎ} ℎ 2_{) (4.47)}

Here 𝑎 is the radius of the base of the cap, and ℎℎ is the height of the cap. When the height

and radius of the spherical cap are equal to the initial radius R, then the volume of the hemisphere is

𝑉ℎ=

2𝜋 3 𝑅

2 _(4.48)

It may be of interest to note that for the same radius and height, the volume ratios of the cone, hemisphere and cylinder are 1/3, 2/3 and 3/3 respectively.

The spherical cap radius 𝑎 is related to the initial radius of the hemisphere R by hc

hh

a rc

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𝑎 = √2𝑅ℎℎ− ℎℎ2 (4.49)

Thus the volume can expressed in terms of known parameters ℎℎ and 𝑅 as

𝑉ℎ= 𝜋 6(6𝑅ℎℎ 2_{− 2ℎ} ℎ 3_{) (4.50)}

A schematic of the Simulink block code used to estimate the instantaneous volume of the cone-cylinder and hemisphere-cylinder is given in Figure 4:19 & Figure 4:20 respectively.

Figure 4:19: Hydrostatic restoring force for the cone-cylinder shaped buoy

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In document Laboratory scale modelling of a point absorber wave energy converter (Page 174-181)