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The purpose of this paper is to determine the optimal size and number of tubes for a generic attenuator that is similar to Pelamis P2, the wave energy converter. Simulations using ANSYS Workbench, Design Modeler, and AQWA are performed to study the energy absorption at the nodes between the tubes. The analysis is limited to linearized hydrodynamic fluid waves loading on floating bodies by employing three-dimensional radiation/diffraction theory in regular waves in the frequency domain. Three sets of tests are conducted by varying total tube number, each tube length and the order of tubes with different lengths. After a systematic study in the frequency domain, the optimal size and number of the genetic attenuator is recommended.

Hydrokinetic wave energy is interchanging potential and kinetic energy, carried by the wave away from its origin as wave travels through space in time. Wind, creates the most common waves, surface wave and then sea swells. In order to convert the constant motion of ocean waves into usable energy such as electricity, the wave energy converter (WEC) must survive the hostile environment at extreme sea states, absorb the maximum wave power, and be cost-effective for commercial market. Considerable research and effort continue to focus on the following aspects: 1) design new wave energy converter [

The Pelamis is a particular type of ocean attenuator consisting of either four (P1 model) or five (P2 model) cylindrical sections linked together by universal joints that allow for motion with four degrees of freedom, as seen in

Pelamis Wave Power Corporation has published several papers [

The objective of this study is to determine the optimal design of a multibody self-referencing attenuator that is similar to the wave energy converter Pelamis P2. The software programs, ANSYS Workbench, Design Modeler, and AQWA [

The development of a WEC starts from concept and design, then numerical modeling, model testing in wave basin, testing under real sea conditions, and finally to commercial stage. It is a long, difficult, and expensive process. Numerical modeling has the advantage of providing quick and inexpensive evaluation and optimization

of designs. In this study, ANSYS AQWA, an industry standard hydrodynamic software package, is employed to capture the behavior of the attenuator by simulating the interaction between waves, the WEC device and the power take-off mechanism. The software ANSYS AQWQ can simulate linearized hydrodynamic fluid wave loading on floating bodies by employing three-dimensional radiation/diffraction theory in regular waves in the frequency domain. Frequency domain analysis is the first step in the hydrodynamic modeling process by assuming everything is linear. This analysis is especially useful in geometry optimization routines and thus it is employed in this paper. However, it should be fully aware the limitations of this analysis method compared to realistic conditions.

The resource, surface gravity waves can be linearized if a/λ << 1 and a/H << 1, where the variable, a, is the wave amplitude (m), λ is the wavelength (m) and H is the uniform water depth (m). Typically the surface gravity waves of oceans have wavelengths between 30 - 40 m, enabling the water surface tension to be neglected as it pertains to wavelengths of less than 5 - 10 cm. As the attenuator is deployed at water depths greater than 50 m it is classified as a deep-water device due to H > λ/3. It is also noted that as the frequency is much larger than the Coriolis frequency, the wave motion is unaffected by Earth’s rotation. Assumptions made here are that 1) the water is incompressible (constant density) and irrotational; 2) gravity is the only external force; and 3) viscosity can be neglected. For the regular wave based on linear theory, the power obtained from ocean per meter of wave front is the total energy of the system (kinetic plus potential) per unit horizontal area multiplied by the phase speed to yield the following relation:

where, P_{t} is power per unit length of wave front (W/m), ρ is density of seawater (kg/m^{3}), g is gravity (m/s^{2}), a is the wave amplitude (m), and ω is angular frequency (rad/s).

The European Marine Energy Center (EMEC) [

The multibody self-referencing attenuator generates electricity based on the relative motion between its tubes. The attenuator moves with two degrees of freedom to capture energy: pitch and heave, given that this study is restricted to the case in the x-z plane, with z as the vertical axis and x as the axis passing through the head and tail of a tube. Thus for each tube there are two dynamic equations, one for moment due to pitch and one for force due to heave. In order to find the potential power output from the attenuator, the forces and moments acting on the attenuator tube need to be defined. As shown in _{n}, on tube n is a resultant force of wave pressure. The pressure can be resolved from the Bernoulli’s equation after the velocity potential is determined. The reaction force, R_{n}, exerted on node n represents the power take-off mechanism. The hydrodynamic moment, M_{n}, about the midpoint of tube n can be expressed in terms of the angular displacement, Θ_{n}.

In rotational systems, the power is derived as the product of angular velocity and torque, or in our case the product of angular frequency ω and the moment M_{n}. Thus the power absorbed by attenuator at hinge n is the product of the extraction rate at node n and the angular displacement of tubes n and n − 1. The time averaged power extraction derived by Flanes [

where _{n}/L_{n}(m/m) is the tube amplitude. The extraction rate _{n} that is extracted at a node from Equation (2) divided by the total power of the system P_{t} that is defined in Equation (1), yielding

In the simulation process using computer software ANSYS AWQA, a computer model of attenuator is first created by use of the ANSYS Design Modeler software to generate three-dimensional tubes as shown in _{n}, extracted from the attenuator using Equation (2) and the efficiency, eff, using Equation (3) is displayed in _{n}/L_{n}, are also determined from the ANSYS AQWA software via the RAOs (Response Amplitude Operators); the results are generated in terms of m/m. A large enough range for the frequency f is selected to encompass the range of efficient energy capture for all design models of attenuator.

To validate the numerical model, ANSYS AQWA is applied to the control model T5L36D4, an attenuator with 5 tubes, a 36 m tube length, and a 4 m tube diameter as defined in

Parameter | Value(s) | Units |
---|---|---|

Density, ρ | 1023.485 | kg/m^{3} |

Gravity, g | 9.80665 | m/s^{2} |

Amplitude, a | 1.1 | m |

Frequency, f | 0.05 - 0.30 | Hz |

Extraction rate, _{ } | Program generated | Nm/(m/s) |

Tube amplitude, B_{n}/L_{n}_{ } | Program generated | m/m |

Control model Pelamis P2 | Element size 1.5 tolerance of 0.6 | Element size 1.5 tolerance of 0.8 | Element size 1.0 tolerance of 0.5 | Element size 0.8 tolerance of 0.3 | |
---|---|---|---|---|---|

Model number | T5L36D4 | E1.5 T0.6 | E1.5 T0.8 | E1.0 T0.5 | E0.8 T0.3 |

Efficiency eff (%) | 73.27 | 71.15 | 70.84 | 71.84 | 72.05 |

The attenuator uses resonance to increase power capture of small waves. The default setting of the attenuator is non-resonant, allowing it to withstand large swells. However, the joints can be actively controlled by its power take off system to create a cross-coupled resonant response. Optimal design of attenuator is based on such a principle that the device must be designed to operate efficiently within the frequency range and power level. In an attempt to standardize the amount of raw materials required for each model, the overall length of 180 m and weight of 1350 tons are chosen to maintain the same overall length and weight as the current Pelamis P2 model.

A summary of all test models is provided in

Number of tubes | Tube length (m) | Tube diameter (m) | Model number | |
---|---|---|---|---|

Test set one | 3 | 60 | 4 | T3L60D4 |

4 | 45 | 4 | T4L45D4 | |

5 | 36 | 4 | T5L36D4 | |

6 | 30 | 4 | T6L30D4 | |

8 | 22.5 | 4 | T8L22.5D4 | |

Test set two | 3 | 60 | 3 | T3L60D3 |

4 | 45 | 3 | T4L45D3 | |

5 | 36 | 3 | T5L36D3 | |

6 | 30 | 3 | T6L30D3 | |

8 | 22.5 | 3 | T8L22.5D3 | |

3 | 60 | 5 | T3L60D5 | |

4 | 45 | 5 | T4L45D5 | |

5 | 36 | 5 | T5L36D5 | |

6 | 30 | 5 | T6L30D5 | |

8 | 22.5 | 5 | T8L22.5D5 | |

Test set three | 6 | 10:18:26:34:42:50 | 5 | T6D5 increasing |

6 | 50:42:34:26:18:10 | 5 | T6D5 decreasing | |

6 | 15:30:45:45:30:15 | 5 | T6D5 SBS | |

6 | 45:30:15:15:30:45 | 5 | T6D5 BSB |

L4 > L5 > L6). For these two models (T6D5 SBS and BSB), the ratio of 1:2:3:3:2:1 with a total of 12 units does divide evenly into the total length, resulting in the lengths of each tube to be, in meters, 15:30:45:45:30:15 for the model T6D5 SBS and 45:30:15:15:30:45 for the model T6D5 BSB.

Test set one models vary the number of tubes for each structure while maintaining a diameter of 4 m. The peak efficiencies for each model, along with the corresponding frequencies are provided in

Given the restrictions of maintaining the overall length and weight of the attenuator structure, the efficiencies are highest with a smaller tube length (i.e., larger number of tubes). The T8L22.5D4 model would be the best choice from this test set at 1.88% more efficient than the control model. However, devices with more moving parts require more maintenance and can experience a greater loss in efficiency due to friction when converting the mechanical energy to electrical energy, therefore it may not be cost effective from this perspective to increase the amount of moving parts associated with additional nodes while only achieving a 1.88% increase in

Model number | T3L60D4 | T4L45D4 | T5L36D4 | T6L30D4 | T8L22.5D4 |
---|---|---|---|---|---|

Efficiency (%) | 24.93 | 50.93 | 73.27 | 72.51 | 75.14 |

Frequency (Hz) | 0.121 | 0.143 | 0.150 | 0.157 | 0.150 |

efficiency from the control model.

Test set two demonstrates the effects of tube length at different diameter of the structures. The peak efficiencies for each model, along with the corresponding frequencies are provided in

For the set of models with a diameter of 5 m, the model with tube length of 30 m (six tubes, model T6L30D5) has the highest efficiency. It is interesting to note that the efficiencies for T3L60D5, T4L45D5, and T5L36D5 are lower than their counterparts in Test Set One, yet the efficiencies for T6L30D5 and T8L22.5D5 are higher than those in Test Set One, by 6.39% and 2.80% respectively. Therefore, additional models are created for tube lengths of 30 m and 22.5 m to have a diameter of 6 m; these efficiencies are lower than those of the 5 m diameter models. From

For the effect of tube diameter, it is observed that for a given tube length, there is an optimum tube diameter. By making the structure narrower, the tube slices through the oncoming waves rather than the wave lifting the tubes to create sufficient angular rotation about the hinges. The narrower models experience less hull pressure from the water, and thus it creates less of a hydrodynamic moment and therefore it absorbs less energy. This is the same principle used when oceangoing vessels are designed for speed. The reason that an increase in diameter reduces the efficiency of the system is theorized here that the first tube(s) does experience a higher vertical pressure but then it experiences an effect called “slamming” [

Based solely on maximum efficiency, from Test Set One and Test Set Two, the model T6L30D5 with 6 tubes, a

T3L60D3 | T4L45D3 | T5L36D3 | T6L30D3 | T8L22.5D3 | |
---|---|---|---|---|---|

Efficiency (%) | 24.39 | 49.76 | 52.19 | 56.93 | 58.66 |

Frequency (Hz) | 0.163 | 0.143 | 0.143 | 0.143 | 0.163 |

T3L60D4 | T4L45D4 | T5L36D4 | T6L30D4 | T8L22.5D4 | |

Efficiency (%) | 24.93 | 50.93 | 73.27 | 72.51 | 75.14 |

Frequency (Hz) | 0.121 | 0.143 | 0.150 | 0.157 | 0.150 |

T3L60D5 | T4L45D5 | T5L36D5 | T6L30D5 | T8L22.5D5 | |

Efficiency (%) | 25.12 | 46.45 | 67.06 | 78.90 | 77.94 |

Frequency (Hz) | 0.129 | 0.150 | 0.157 | 0.157 | 0.157 |

-- | -- | -- | T6L30D6 | T8L22.5D6 | |

Efficiency (%) | -- | -- | -- | 75.06 | 72.86 |

Frequency (Hz) | -- | -- | -- | 0.164 | 0.164 |

30 m tube length, and a 5 m tube diameter, is the highest performer with an efficiency of 78.905% and is used for Test Set Three. As shown in

Model number | T6L36D5 | T6D5 increasing | T6D5 decreasing | T6D5 SBS | T6D5 BSB |
---|---|---|---|---|---|

Efficiency (%) | 78.90 | 85.41 | 79.81 | 94.72 | 85.10 |

Frequency (Hz) | 0.157 | 0.164 | 0.171 | 0.164 | 0.171 |

0.164 Hz and the two with the lower efficiencies occur at the same frequency of 0.171 Hz, yet all four models experienced peak efficiencies at higher frequencies compared to the model T6L3D5 with consistent tube lengths, f = 0.156 Hz.

While the short end tubes of the T6D5 SBS model have a low damping value, the middle tubes act as sort of vertical mooring structure that allow for these shorter end tubes to generate large vertical elevation which increase the angular rotation at the hinges, resulting in larger power generation and higher efficiency. The model T6D5 BSB operates on an opposing principle; there is an even balance of high damping and low elevation with low damping and high elevation. The large end tubes create a large amount of damping and experience slight elevation, while the smaller tubes connected to them are able to increase their elevation and angular rotation, resulting in high power generation and efficiency values. The model T6D5 Decreasing has its largest tubes at the head of the structure; this causes a greater amount of wave damping, leaving less wave height available for energy extraction at later hinges. The very opposite of this principle leads to the model T6D5 Increasing absorbing a larger amount of energy, as the smaller front tubes create less wave damping and hence leave more of the wave to be extracted at later hinges.

Thusly, out of all three test sets, the best model simulation based solely on maximum efficiency would be the model T6D5 SBS, having six tubes, a diameter of five meters, and the tubes varying in size, with the middle tubes longer than the end tubes (L1 < L2 < L3 = L4 > L5 > L6). However, selecting a model based on overall maximum efficiency is not always the best decision. It is important to create the model with the characteristics of the environment in mind, specifically by knowing what frequency range is experienced at the proposed location.

Ocean locations have energy distributed over a range of wave heights and periods, or frequencies. Off the coast of Scotland where the actual Pelamis P2 model resides, the ocean state experiences a mean wave period between 8.1 s in winter and 6.3 s in summer which corresponds to frequencies of 0.123 Hz and 0.159 Hz, respectively, [

In this study, the wave energy converter, amultibody self-referencing attenuator that is similar to the actual Pelamis P2 is studied using numerical modeling to determine the optimal size and number of the attenuator structure. Three sets of tests are conducted by varying total tube number, each tube length and the order of each tube with different length. The analysis is based pm linearized hydrodynamic fluid waves loading on a floating body by employing three-dimensional radiation/diffraction theory in regular waves in the frequency domainand the conclusion is drawn as follows.

1) Given the restrictions of maintaining the overall length and weight of the attenuator structure, the efficiencies are highest with a smaller tube length (i.e., larger number of tubes).

2) For a given tube length, there is an optimum tube diameter corresponding to the maximum efficiency.

3) Out of all three test sets, the best model simulation based solely on maximum efficiency of 94.74% would be the model T6D5 SBS with six tubes, a 5 m diameter, and the tubes varying in size, with the middle tubes longer than the end tubes (L1 < L2 < L3 = L4 > L5 > L6).

4) Given the restrictions of 180 m overall length and 1350 tons overall weight, the optimum design of the attenuator to be deployed off the coast of Scotland is the model T5D4 SBS with five tubes of four-meter diameters, whose tube lengths increase then decrease (L1 < L2 < L3 > L4 > L5).

The future work after this paper will be to credit the optimal design in the time-domain analysis by introducing nonlinearities and viscous effect with cost-effective in mind. It is expected that the extraction power and efficiency will be decreased by 3% - 5% after introducing nonlinearities and viscous effect. However, the results and conclusion should hold for irregular waves if the range of wave frequencies is appropriate.

The research project was partially supported by 2013-2014 Provost’s Research Fellow program and Provost’s Research Incentive Fund Summer 2014.

DongmeiZhou,Jennifer A.Eden, (2015) Optimal Design of a Multibody Self-Referencing Attenuator. Journal of Power and Energy Engineering,03,59-69. doi: 10.4236/jpee.2015.39005