# «J. Mar. Sci. Eng. 2015, 3, 1362-1381; doi:10.3390/jmse3041362 OPEN ACCESS Journal of Marine Science and Engineering ISSN 2077-1312 ...»

The optimization problem is solved using the GlobalSearch Algorithm in Matlab [17], which uses a scatter-search mechanism for generating start points. From the starting points, GlobalSearch examines the trial points and choose the ones that can generate a better solution. Then, the chosen points are evaluated by a local minimization solver. The process ends when all the trial-points have been evaluated [18].

**6. Truncated Mooring System Design for Windcrete**

The truncated mooring system was designed for a radius to anchor of 140 m. The objective function was evaluated for a surge excursion ranging from −40 to 25 m. Surge excursion is the main platform motion that depends on the mooring system response. The surge interval ensures a horizontal response for a 600 kN mean wind force. The asymmetry of the mooring system (Figure 9) produces an asymmetry response in surge.

J. Mar. Sci. Eng. 2015, 3 1373

** Figure 9. Mooring system sketch.**

The chosen characteristics of the prototype to be minimized in the multi-objective function are: the tension of one line, the horizontal mooring system response and the vertical mooring system response.

Therefore the multi-objective function can be described as Equation (15).

f ( X ) T fT ( X ) x,s fx,s ( X ) z,s fz,s ( X )) (15) where fT is the function to optimize the tension of a line; and fx, s and fz,s are the functions which describe the difference between the response prototype and truncated mooring system in x and z direction, respectively. Since the surge response is mainly depends on the mooring system, the largest weight factor x,s is given to it. Selected weights are consistent with the references [7,10]. The weighting factors used for the multi-objective problem are 0.2, 0.6 and 0.2 for T x,s and z, s respectively. A sensitivity analysis of the variation of the weight factors was performed. It confirmed that low variations of the weight factors lead to a similar solution of the optimization problem.

The constraints applied to the problem are the length of the line, which is defined between its taut (Equation (16)) and completely slack (Equation (17)) shape. These constraints are expressed as a function of the radius to anchor and the mooring depth. Other constraints are the minimum and maximum diameter of the lines, defined by Equations (18) and (19), respectively, which are fixed to get a feasible scalable chain for the tests.

dmax 250mm (19) The solution of the optimization problem yields to a truncated catenary system composed by lines of two different segments with different weight per unit length. The segment a is the lower one and is linked to the anchor, while the segment b is the upper one and is connected to the platform. The segment a is heavier and shorter than segment b. With this configuration, the stiffness of the restoring force of the catenary system is mainly provided by the segment a, while the segment b contributes to reduce the suspended weight of the mooring line. The properties of each segment of the mooring line as a result of the optimization problem are shown in the Table 3.

The response of the optimized catenary system is presented in Figures 10 and 11. Figure 10 shows the comparison between the horizontal and the vertical response on both mooring systems. The horizontal response of the truncated system fits well with the prototype response. However, for large offsets, the responses start to diverge. The mooring system vertical force component is larger for the truncated one.

The reduction of the radius to anchor implies an increment of the suspended weight on the platform for a similar horizontal force. Then, a deeper draft would be expected in the platform during the tests: of about 0.5 m. The line tension is well fitted along the whole surge excursion studied in the optimization problem (Figure 11).

6 5 4 3

7. Experimental Results The model is scaled at 1:100 factor using the Froude similitude. The platform model is made of aluminum to adjust the density of the material to be close to concrete, simplifying the fit of the rest of the platform parameters. The mooring lines are composed of two chain segments that adjusted to fit the weight per meter length computed in the optimization problem. The scale model placed inside the flume attached to the mooring system is shown in Figure 12.

-0.6

-0.8

-1

-1.2

-1.4

Dynamic tests were carried out in several sea states with regular and irregular waves, and an almost constant wind force on the top of the nacelle. The experimental results were measured from the nacelle motion by an optical system which can track the 6 DOF [19].

Figure 14 shows the surge and heave response comparison between the experiment, the simulation with the truncated lines and the simulation of the platform equipped with the prototype mooring system.

These responses correspond to a regular wave of 14 cm height and a period of 1.5 s, and a constant force on the top of the platform of 0.6 N to simulate the wind. The experimental results show good agreement between the test and the numeric simulation with the truncated lines in terms of mean offset, mean draft and also with the wave amplitude movement. However, some differences can be seen due to a low frequency movement that occurred during the test. This disturbance was produced by a long wave reflection in the longitudinal direction of the flume. In the transverse direction, no reflections were noticed. The simulation of the prototype mooring system shows a shorter total surge excursion than the truncated one. This difference is explained by two effects. First, the stiffness of the prototype mooring system for positive excursions is higher than the truncated ones, as is shown in Figure 10. Second, there is a loss of stiffness due to a draft increase of about 0.5 cm. In this situation, the lower depth of the fairlead position requires an increased excursion to achieve the same horizontal force. The draft increase can be noticed in the heave response (Figure 14b) as a decrease of the mean heave position of the truncated mooring compared to the prototype one.

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** Figure 14. Comparison surge (a) and heave (b) nacelle motion response.**

The results of a Fast Fourier Transform (FFT) of the surge and heave platform responses are shown in Figure 15. Both diagrams show clearly the peak motion due to the wave excitation at a frequency of

0.66 Hz (period of 1.5 s) and the amplitudes match very well.

The main differences between the simulations and the experimental results are the excitation of the low frequency surge motion of the platform during the experiments, as already discussed. This affects the heave response, which FFT (Figure 15b) presents as two small peaks: at 0.05 Hz, the natural surge frequency, and at 0.337 Hz, the heave natural frequency.

J. Mar. Sci. Eng. 2015, 3 1378 Figure 15. Surge (a) and heave (b) motion Fast Fourier Transform functions.

8. Conclusions

radius to anchor has to be reduced, an optimization problem based on the static mooring system response helped to fit the horizontal and vertical mooring responses and the traction line. The optimization problem is evaluated in the surge work range because is the main platform motion that depends on the mooring system response.

The truncated mooring line stiffness is obtained by using two different line sections: the bottom one is the heaviest and provides the horizontal mooring line stiffness. The upper section is lighter than the prototype mooring and reduces the vertical force over the platform.

The truncated catenary presents almost the same traction response as the completed prototype mooring system (differences less than 5%). On the other hand, the horizontal stiffness of the truncated system differs from the prototype, particularly for large excursions. In addition, the truncated mooring system presents a higher vertical force on the platform that lead to an increment of the draft.

Experiments are compared to numerical simulations with the real and prototype mooring system.

The experimental results show good agreement with the numerical simulations. Some differences are noticed in the mean surge excursion of the truncated mooring system, which is larger. This is a consequence of the lower surge stiffness and an extra loss of stiffness due to the increase of the draft.

Despite this, surge and heave responses due to wave loads are well predicted.

Acknowledgments The work presented in this paper has been developed during the KIC Innoenergy (EIT) project AFOSP (Alternative Floating Offshore Support Platform). Its financial support is greatly appreciated.

Author Contributions C.M. and P.T. conceived and designed the optimization problem, and wrote the article. A.C. and X.G.

contributed in the tests and in the text of the article.

Conflicts of Interest The authors declare no conflict of interests.

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