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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 186056, 8 pages

http://dx.doi.org/10.1155/2013/186056

## Numerical Simulation of Section Systems in the Pelamis Wave Energy Converter

^{1}Fujian Province Key Laboratory of Energy Cleaning Utilization and Development, Jimei University, Xiamen 361021, China^{2}Cleaning Combustion and Energy Utilization Research Center of Fujian Province, Jimei University, Xiamen 361021, China

Received 28 June 2013; Revised 13 September 2013; Accepted 13 September 2013

Academic Editor: Fabrizio Marignetti

Copyright © 2013 Hongzhou He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The working principle of the Pelamis wave energy converter is described in this paper. The sectional size suitable for the outside sea of Xiamen Bay is redesigned according to the Froude and Strouhal similarity criteria. The swing angles, hydrodynamic coefficients, and wave exciting forces are calculated based on the AQWA hydrodynamic software and the average outside sea condition of Xiamen Bay. It is concluded that the Pelamis after redesigned by the Froude and Strouhal similarity criteria can run better in the outside sea area of Xiamen Bay. The three sections indirectly contacted with the fixed axis have larger swing angles. The wave period and height affect the speed of the response of the section and its swing angle range, respectively. The larger the total forces, the larger the swing angles. The wave circular frequency has a greater effect on the added mass and the wave exciting force than on the radiation damping; the heave added mass and the heave wave exciting force will become smaller when the heave radiation damping becomes larger with the increase of the wave circular frequency.

#### 1. Introduction

Power generation is the main wave utilization style. Many countries paid great attention to the wave utilization technologies. Britain has built oscillating-water column wave power station and Pelamis wave energy converter [1, 2]. Most of the oscillating-water column wave power stations are limited to be built off shore at the present stage. This technology is not stable enough to be utilized in deep sea conditions. Japan has focused on Mighty whale wave power boat and pendular type machines [3, 4]. Its Pendulum power generation device holds high energy conversion efficiency but low power efficiency. Meanwhile, it needs high initial investment and is easy to be damaged. Norway has studied Tapchan wave energy converter [5]. However, their device is highly related to the local geography. The Pelamis wave energy converter is of high stability and adapting to harsh environment and is concerned by many experts and scholars. Correspondingly, its generated electricity has been supplied to the grid network [6]. Pelamis wave energy converter has developed two generations of machines: the Pelamis P1 and the Pelamis P2. The Pelamis P2 is developed on the basis of the Pelamis P1. So a lot of significant design has been improved and it has higher efficiency than the Pelamis P1.

Considering the advantages mentioned above, it is of high significance to explore its performance in China’s sea conditions. So the purpose of this paper is to analyze whether the Pelamis after redesigning by the Froude and Strouhal similarity criteria can achieve better performance in the outside sea condition of Xiamen Bay and to study the factors affecting its performance. The size of the sections based on the average sea condition of the outside sea area of Xiamen Bay is designed and the operating conditions of Pelamis wave energy converter are described. Meanwhile, the four-section swing angles, hydrodynamic coefficients, total forces, and wave exciting forces are calculated, respectively.

#### 2. The Pelamis Wave Energy Converter

##### 2.1. The Constitute of the Pelamis Wave Energy Converter

Although the Pelamis P2 has more outstanding feature than Pelamis P1 in Europe, no research has been done on their performance in China’s sea conditions. The present research is based on the Pelamis P1 machine. The prototype was 120 m in length, 3.5 m in diameter, and comprised of four tubes sections linked by three short power conversion modules [7]. The power conversion modules are joined to the tubes, which are used to convert wave energy into hydraulic energy.

Each machine requires its own individual mooring spread consisting of some main moorings and a yaw restraint line. The main moorings consist of a number of anchors connected to a central point. The yaw restrain line is a simple single anchor and mooring line configuration. The latter make the Pelamis always in the optimum orientation. There is scope for neighboring mooring spreads to share anchor points, depending on the anchoring techniques employed at the site. The Pelamis mooring spread has been designed to minimize its footprint area, allowing the highest concentration of MW capacity to seabed space and reducing infrastructure costs. The power take-off system includes hydraulic cylinders, motors, generators, reservoirs, accumulators, and associated piping and wiring, which are assembled in the tube section and have capability to isolate various parts of the system remotely.

##### 2.2. Working Principle of the Pelamis Wave Energy Converter

The working principle of the Pelamis wave energy converter is shown in Figure 1. The longitudinal direction of Pelamis is perpendicular to the traveling direction of the waves, which will be ups and downs with the wave when they have effect on the section. This motion will cause the hydraulic cylinder to produce water pressure and to pass through the valve. Then it will push the hydraulic piston in hydraulic cylinder to produce a reciprocating motion. And finally, power generator is driven. In addition, Pelamis will dive into the sea when encountering high intensity wave.

The principle of energy transfer process of the Pelamis wave energy converter is described in Figure 2. It assumes that the hydraulic cylinder is double acting. Since double rod cylinder passes oil regardless of the motion direction, the upper and lower double rod cylinder can both transmit force. The section pushes the top double rod cylinder, simultaneously pulling the bottom double rod cylinder. So the energy transfer equation can be expressed as where is the input mechanical energy of two double rod cylinders, is the reaction, is the velocity of double rod cylinders, is the piston area, is the output pressure, is the radius, is the angular frequency of wave, is the total hydraulic flow of two double rod cylinders, and is the efficiency of the double rod cylinders.

The reaction force generated by the bottom oil cylinder and the force generated by the top oil cylinder are equal in magnitude and opposite in direction. So the reaction force acting on axis is

#### 3. Similarity Criteria

The designed wave height and period of Pelamis wave energy converter are 5.5 m and 8 s as described in the literature [7], but the average wave height and period of the outside sea area of Xiamen Bay are 1 m and 3.8 s, respectively [8]. In order to analyze the performance of the converter based on the outside sea area of Xiamen Bay, the size of the Pelamis wave energy converter described in the literature needs to be adjusted by the Froude and Strouhal similarity criteria.

##### 3.1. Froude Similarity Criteria

Froude similarity criteria ensure the correct relationship of the gravity and inertia force between the prototype and the studied model [9]. Froude number (Fr) is defined as where is the acceleration of gravity, is the geometric length of the structure, and is the characteristic velocity.

By assuming the subscript represents the prototype and represents the studied model and taking linear dimensions ratio (lamita) as , then the comprehensive equations can be described as

##### 3.2. Strouhal Similarity Criteria

The Froude similarity criteria are applied to design the configuration model of the wave energy conversion device in [10, 11]. Reference [12] which indicates that the Strouhal number (St) is also an important similarity criteria number when considering the flow of having a frequency, and the rocking ship belongs to periodic motion. So the Strouhal similarity criteria must be considered when the device is redesigned. Strouhal number (St) is the ratio of the modification inertial force and the local inertial force; it ensures the period similarity of the motion and force between the prototype and the studied model [13]. Equal Strouhal number (St) of the prototype and the studied model can be expressed as So the comprehensive equation is where is the wave frequency and is the wave period.

As it is mentioned above, the prototype of pelamis can achieve better operation in Europe, but it may not be applied to China. Based on the outside sea condition of Xiamen Bay and according to the Froude similar criteria combined with Strouhal similarity criteria, the dimension of the Pelamis wave energy converter is redesigned and the length, diameter, and weight of each section of the Pelamis wave energy converter suitable for Xiamen sea conditions should be 6.8 m, 1 m, and 1102 kg, respectively.

In addition, the geometry shape of pelamis tube was also optimized. First, the section of four different tube shapes was selected according to the cylindrical shape, and then it was simulated and analyzed under the same sea condition. It seems from the simulation results that the quadrant has a better work performance than the three other tubes shapes. Therefore, the redesigned pelamis is studied on the basis of quadrant shape in this paper.

The shape and dimensions of the redesigned section are shown in Figure 3. The quadrant’s two sides change into vertical planes because they are inclined planes which are not conducive to the converter’s installation. Assuming the draft volumes and weights of the quadrant and the cylinder are equal, the draft depth, volume, and density can be calculated as 0.33 m, 6.95 m^{3}, and 158.6 kg/m^{3}, respectively.

#### 4. Mathematical Model and Initial Conditions

##### 4.1. Mathematical Model

In order to satisfy AQWA software’s limitations, some assumptions are proposed as follows.(1)Seawater is considered as the ideal fluid, the near-field solution method is used, and the four-section interactions are considered.(2)The converter’s three-dimensional model is built by the Design Modeler based on Workbench platform. Meanwhile, surface unit is applied to the quadrant’s section.(3)The pelamis prototype is a mooring system consisting of some main moorings and a yaw restraint line. So it can not adjust itself with the winds and waves. Generally, the pelamis is arranged based on the local average wave direction. So the Pelamis device can be simplified to a fixed system, assuming the entire system is around a fixed axis, and there is no torque in axis. The simplified section model is shown in Figure 4; from left to right, the sections are named structure 1, structure 2, structure 3, and structure 4, respectively.

##### 4.2. Initial Conditions

The limitations of AQWA software and the operation conditions of the Pelamis are described below.

(1) The quadrant section needs to meet the conditions where is the distance between the underside of device and the sea bed, is the radius of the grid cell , and is the area of the grid cell.

(2) The minimum input circular frequency of the AQWA software can be calculated according to the following equation: where is the acceleration of gravity and is the water depth. Because the water depths of most outside sea area of Xiamen Bay are from 5 m to 20 m, we assume that the average water depth is 10 m in simulation. According to (8), the minimum input circular frequency can be calculated as 0.0495 rad/s.

(3) Assuming the wave is regular (the second order Stokes wave as standard), the sectional motion response under the outside sea wave condition of Xiamen Bay is analyzed by the AQWA software. Assuming the default wave direction is 0°, namely, the wave is perpendicular to the longitudinal direction of the pelamis. The sea condition parameters of the outside sea area of Xiamen Bay are shown in Table 1.

(4) Each sectional inertias moment is equal:

#### 5. Results and Discussion

##### 5.1. The Swing Angle of Four Sections

Figures 5 and 6 show the four-section swing angles in spring and summer. It can be seen from Figures 5 and 6 that the swing angle ranges of structure 2, structure 3, and structure 4 are greater than that of structure 1, which indicates that the work capacity of structure 2, structure 3, and structure 4 is better than that of structure 1. It is concluded that fixed shaft limits the swing angle of structure 1. The swing angle ranges of structure 2 and structure 3 are the biggest, which indicates that the work capacities of the two section in the middle area are the strongest. Each sectional positive swing angle range is not equal to its negative swing angle range, which means that the work capacity of upper and lower hydraulic cylinder is different. The wave peaks of swing angle curve of the structure 4 vibrate with the time, which indicates that it is not conducive to achieve stable acting.

Figures 7 and 8 show the four-section swing angles in winter and autumn season. It can be seen from Figures 7 and 8 that only the swing angle curve of the structure 1 is in a steady swinging case, while the wave peaks of swing angle curve of the remaining three structures have a vibrating situation, which reveals that the swing becomes unstable with the wave period increasing. Different from Figures 5 and 6, the swing angle ranges of structure 3, and structure 4 are the greatest as it is shown in Figures 7 and 8; this means that the farther away from the fixed shaft, the greater the sectional swing angle range with the increasing of wave parameters. Meanwhile, it indicates that the work capacity of structure 3 and structure 4 is the strongest in autumn and winter season. The swing angles of structure 2, structure 3 and structure 4 are greater than that of structure 1 in Figures 7 and 8, and the ranges of the positive and negative swing angles of each swing angle curve are different.

Figure 9 describes the swing angles of structure 1 in spring and summer season. As shown in Figure 9, the swing angles of structure 1 are from −6.1° to 5.2° and from −5.8° to 5.4°, respectively, which indicates that the wave period has little impact on the sectional swing angle in these two seasons. In addition, the swing angle’s curve in summer lags behind that in spring, indicating that the wave period has some effect on the sectional swing speed.

Figure 10 shows the swing angles of structure 1 in autumn and winter season. As it is shown in Figure 10, the swing angles in structure 1 are from −7.3° to 7.7° and from −6.7° to 7.1° in autumn and winter season, respectively, which indicates that the wave height has some effect on the swing angle. The greater the wave height is, the greater the swing angle will be. The wave height has no evident effect on the speed of sectional response, because both the heave curves have no leading or lagging phenomena with the extension of the time.

##### 5.2. The Total Force on Sections

The total forces on sections are the main factors that affect the swing angle. Figures 11 and 12, respectively represents the vertical forces on four sections in the spring sea condition and the vertical forces on structure 1 in the four seasons’ sea condition. As it is shown in Figure 11, the force on structure 1 is smaller than that of the other three situations. Considering Figure 5, it could be found that the greater the force on section is the greater the swing angle is. The force on structure 1 is smaller due to the offsetting effect of the fixed shaft. The forces on the remaining three structures are almost equal, indicating that the front section has a smaller wave-killing influence; the front section has a pulling force to the rear section and enhances the motion of the rear section. As shown in Figure 12, the forces on structure 1 are almost equal in spring and summer and also almost equal in autumn and winter. This indicates that the higher the sea level, the greater the force on section, and the more the work performance.

##### 5.3. Hydrodynamic Coefficients and Wave Exciting Force of the Section

The purpose of this part is to investigate the hydrodynamic parameters and wave exciting force on the swing angle of section, which will illustrate both values when ****acting on the work performance of section. When the section is acted under the wave loads, it will cause the movement of the surrounding fluid which produces radiation that generates reaction force (torque). Depending on the wave phase, the reaction force can be decomposed into corresponding modal oscillating movement velocity and acceleration, respectively, called radiation damping force and additional mass force. And their proportional coefficients are called radiation damping and added mass, respectively. The added mass and radiation damping can be described as follows under simple harmonic motion mode [14, 15]:
where , , and represent the component force of , , and directions, respectively; , , and are the component torque of rotating the , , and directions and are the added mass and radiation damping which both have 36 parts and usually expressed as a form of matrix with .

The calculation results of hydrodynamic coefficients in the vertical direction are shown in Figures 13 and 14. It can be seen from Figure 13 that the added four-section mass is almost equal when the wave frequency is less than 0.6 rad/s. The added mass of structure 2 and structure 3 differs greatly from that of structure 1 and structure 4 when the wave frequency is greater than 0.6 rad/s, which indicates that the wave circular frequency will affect the added mass under the same section geometry. The added mass decreases with the increase in the wave circular frequency while the changing rate differs greatly with different frequency band. Figure 14 shows that the heave radiation damping increases with the increasing of the wave circular frequency. The four-section heave radiation damping is almost equal, indicating that the wave condition has almost no effect on the heave radiation damping under the same section geometry. All of these show that the added mass has a great influence while the radiation damping has little influence on the swing angle of section.

For the regular wave, the unsteady fluid pressure can be divided into two parts: one is the unsteady pressure caused by undisturbed wave and the other part is diffraction force generated by changing fluid pressure spaces which are affected by the structure’s existence. The sum of both parts is the wave exciting force which acts on the buoy in regular wave [16, 17]: where Re is the Reynolds number, is the density of sea water, is the wave amplitude, is the wave circular frequency, is the unit normal vector on , and and represent the wave incident velocity potential and diffraction velocity potential, respectively.

Figure 15 shows the amplitude of heave wave exciting force of the four sections under a unit wave height. It can be seen from Figure 15 that the heave wave exciting force of structure 1 is the minimum and that of structure 4 is the maximum. And the heave wave exciting forces of the other two structures are in the middle, which indicates that the swing angle increases with the increasing of heave wave exciting force, and also means that the wave exciting force has a great effect on the swing angle of section.

#### 6. Conclusions

The characteristics of section system of a Pelamis wave energy converter were analyzed based on AQWA hydrodynamic software under four seasons’ average sea condition of the outside sea area of Xiamen Bay, and the following conclusion can be drawn from this study.(1)The pelamis after redesigned according to the Froude and Strouhal similarity criteria can run in the outside sea area of Xiamen Bay, and factors which affect its performance include the total force, the added mass, and the wave exciting force.(2)The swing angle range of each section is different. The swing angle ranges of the three sections which are away from the fixed axis are greater than that of the section near the fixed axis.(3)The larger the wave period, the slower the section response. And the larger the wave height, the greater the swing angle range of section.(4)Different wave frequency will lead to different wave heave added mass and has little effect on radiation damping under the same geometry shape of section. With the increasing of wave circular frequency, the heave added mass and the exciting force decrease while the heave radiation damping increases.(5)The produced power of the redesigned Pelamis wave energy converter in autumn and winter season is greater than that in spring and summer season.

#### Acknowledgments

This work was supported by the National Natural Science Foundation (no. 51209104) and Ocean Renewable Energy Development Foundation of the State Oceanic Administration (no. XMME2011BL02), which are gratefully acknowledged by the authors.

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