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energies Article

Numerical Investigation of a Tuned Heave Plate Energy-Harvesting System of a Semi-Submersible Platform Kun Liu 1,2, *, Haizhi Liang 3 and Jinping Ou 1,2,4 1 2 3 4

*

School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China; [email protected] Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education (Harbin Institute of Technology), Harbin 150090, China Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China; [email protected] State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China Correspondence: [email protected]; Tel.: +86-451-8628-2209

Academic Editor: Ling Bing Kong Received: 5 December 2015; Accepted: 20 January 2016; Published: 28 January 2016

Abstract: A novel tuned heave plate energy-harvesting system (THPEH) is presented for the motion suppressing and energy harvesting of a semi-submersible platform. This THPEH system is designed based on the principle of a tuned mass damper (TMD) and is composed of spring supports, a power take-off system (PTO) and four movable heave plates. The permanent magnet linear generators (PMLG) are used as the PTO system in this design. A semi-submersible platform operating in the South China Sea is selected as the research subject for investigating the effects of the THPEH system on motion reduction and harvesting energy through numerical simulations. The numerical model of the platform and the THPEH system, which was established based on hydrodynamic analysis, is modified and validated by the results of the flume test of a 1:70 scale model. The effects of the parameters, including the size, the frequency ratio and the damping ratio of the THPEH system, are systematically investigated. The results show that this THPEH system, with proper parameters, could significantly reduce the motions of the semi-submersible platform and generate considerable power under different wave conditions. Keywords: semi-submersible; heave plate; wave energy; wave energy converter; tuned mass damper (TMD)

1. Introduction Powerful waves are harmful to ocean structures but also have the great potential to provide clean energy at the same time. Utilizing this powerful renewable energy and protecting our structures at the same time is a meaningful and interesting topic. Although wave energy converters are still far from commercial use, with the development of energy harvesting technologies, especially the improvement of the permanent magnet linear generator, wave energy converters (WEC) employing PMLG as the PTO system have emerged in applications and research studies [1–3]. For deep water utilization, whose energy flux densities are much larger compared to near shore sites, multi-body floating WECs such as Wavebob [4] and IPS Buoy [5] have been systematically studied during the most recent decade. Nevertheless, building a power grid for transmitting the electronic power from a deep-water site would significantly increase the total cost of wave power. If the power generated by deep-water WECs is directly supplied for the operation of

Energies 2016, 9, 82; doi:10.3390/en9020082

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Energies 2016, 9, 82

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deep-water platforms, whose energy supply from the shore is difficult and expensive, it will make the WECs’ future brighter. Semi-submersible platforms are considered to be the earliest type and most widely used deep water platform for oil and gas exploration because of their large deck area and payload capacities. However, because of the small water plane area and the draft, the motions of semi-submersibles, especially heave motion, are much larger than other deep-water platforms such as Spars and the tension leg platforms (TLP). To improve the performance of semi-submersible platforms, many scholars have concentrated their efforts on inventing novel type semi-submersibles [6–9]. These concepts for improving the semi-submersible platform could be summarized as having a deeper draft, installing heave plates and using truss columns instead of conventional columns. In the author’s opinion, the best solution is converting the kinetic energy of the platform into electronic power, which would make the operation more safe and economical. The tuned mass damper (TMD) is an effective way for the vibration control of structures and has been fully developed since Frahm [10] proposed the first tuned mass damper in his patent in 1911. Since then, many adaptive, semi-active and active methods for improving the performance of the TMD have been developed [11–13] and realized in practical engineering projects [14–16]. The dampers are adapted in conventional TMDs, and the energy absorbed by a TMD system is converted to heat and dissipated in the damper. Recently, the absorbed energy could be converted into electrical power by different generators or the use of piezoelectric materials [17–19] applied in the TMD system. For reducing the heave motion of the semi-submersible platform, a novel tuned heave plate system was proposed by Kun Liu [20] and Hang Zhu [21] based on the same idea as the TMD. Compared to the traditional heave plate system, the plate is connected to the platform by spring and damping elements. In this way, the tuned heave plate system could be tuned to a certain period in order to achieve better control than a traditional fixed heave plate. In this paper, four movable heave plates are installed beneath the columns to control not only the heave motion but also the roll and pitch motions. The damping elements of the tuned heave plate system are replaced by PMLGs, and this tuned heave plate energy-harvesting system (THPEH) could not only suppress the motions of the platform but also produce power for the platform’s operation at the same time. In this paper, the effectiveness of the THPEH system with different parameters under different wave conditions is investigated systematically. First, the numerical model of the semi-submersible platform with the THPEH system is built in the Matlab & Simulink module. Second, the numerical model of the semi-submersible platform is modified and verified by testing a 1:70 scale model in a wave tank. Third, the parameter studies of the size, the tuned period ratio and the damping ratio of the THPEH system are carried out. Finally, the performances of the semi-submersible platform with the designed THPEH system under different waves are illustrated and analyzed. The results show that the THPEH system could significantly reduce motions and generate considerable power at the same time. 2. Conceptual Design of the Tuned Heave Plate Energy Harvesting System The illustration of the THPEH system installed on the semi-submersible platform and the details of the THPEH system are shown in Figure 1. Figure 1a shows the conceptual design of a semi-submersible with the tuned heave plate system. The THPEH system is composed of four truss-plate structures located under the columns. The plate below the pontoon is installed deep enough to avoid the wave exciting force. Figure 1b shows an illustration of a truss-plate structure. The permanent magnet linear generator is used and is placed on the top of the truss tube in the platform pontoon. Figure 1c illustrates the connection structure of the heave plate and the truss structure. The connection only allows the heave plate move along the vertical truss tube. The stator coil and iron are rigidly connected to the outer tube on the heave plate, and the tube wrapped by the permanent-magnets extended from the truss structure can move through the stator and the outer tube on the plate. This PMLG can harvest


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the related motion energy into electrical power. The spring support is placed between the truss tube and the heaveEnergies 2016, 9, 82 plate, which determines the natural period of the THPEH system. 3 of 22

(a)

(b)

(c)

Figure 1. (a) The illustration of the semi‐submersible platform with the tuned heave plate energy Figure 1. (a) The illustration of the semi-submersible platform with the tuned heave plate energy harvesting system; (b) the illustration of a truss‐plate structure of the THPEH system; (c) the diagram of the connection structure in the sleeve tube. harvesting system; (b) the illustration of a truss-plate structure of the THPEH system; (c) the diagram

of the connection structure in the sleeve tube. Assuming the force of PMLG is a linear damping force, Figure 2a shows the mechanical model diagram of the semi‐submersible platform with the THPEH system in heave motion. The whole system of the spring supports and PMLGs are the power take‐off system (PTO) in this model. The Assuming the force of PMLG is a linear damping force, Figure 2a shows the mechanical model plane layout of the plates is shown in Figure 2b. Under the small motion assumption, the motion of diagram of thethe THPEH system is governed by Equation (1). semi-submersible platform with the THPEH system in heave motion. The whole system

1 of the spring supports and PMLGs power take-off system (PTO) in this model.   (1) The plane layout + Cthe  m + a zare d S z  z = f   CPTO  z  T46 x   KPTO  z  T46 x   2 of the plates is shown in Figure 2b. Under the small motion assumption, the motion of the THPEH where m , a , C , and f are the matrices of the mass, added mass, the drag coefficient and wave system is governed by Equation (1). d

exciting force of the heave plates of the THPEH system, respectively; z , z and z are the heave displacement vector, velocity vector and acceleration vector of the four plates of the THPEH system; ˇ.ˇ . “ `. ‰ .˘ .. 1 C PTO and K PTO are the damping coefficient and stiffness coefficient of the THPEH provided by the ˇ ˇ

pm ` aq z` Cd ρS z ¨ z “ f ´ CPTO z ´ T46 x ` KPTO pz ´ T46 xq

2  is the density of sea water; S is the area of the plate; PMLGs and spring supports, respectively.

(1)

T46 and T64 are the displacement transformation matrices of the THPEH system, defined by

where m, a, Cd , and f are the matrices of the mass, added mass, the drag coefficient and wave exciting . .. force of the heave plates of the THPEH system, respectively; z, z and z are the heave displacement vector, velocity vector and acceleration vector of the four plates of the THPEH system; CPTO and KPTO are the damping coefficient and stiffness coefficient of the THPEH provided by the PMLGs and spring supports, respectively. ρ is the density of sea water; S is the area of the plate; T46 and T64 are the displacement transformation matrices of the THPEH system, defined by »

T64

— — — — “— — — –

0 0 1 ly lx 0

0 0 1 ly ´lx 0

0 0 1 ´ly ´lx 0

0 0 1 ´ly lx 0

fi » ffi 0 0 1 ffi — 0 0 1 ffi ffi — ffi , T46 “ — ffi – 0 0 1 ffi fl 0 0 1

ly ly ´ly ´ly

lx ´lx ´lx lx

0 0 0 0

fi ffi ffi ffi fl

(2)

0  1 T64 =  l y  lx   0


0

0

1 ly

1 l y

lx 0

lx 0

0  0 0  1  , T46 =  0 l y   lx  0  0 

ly ly

lx lx

0 1 l y 0 1 l y

lx lx

0 1 0 1

0 0 0  0

(2) 4 of 22

where lxlxand and are the distances between the centers of the platform and the plates along the x‐axis where ly lare the distances between the centers of the platform and the plates along the x-axis y and y-axis, respectively. and y‐axis, respectively.

M + M a   F

Ks  Km

f

f

C + Cr   

m+a

m+a

1 Cd  S z 2

1 Cd  S z 2

(a)

(b)

Figure 2.2. (a) (a) The mechanical model diagram of semi-submersible the semi‐submersible platform with the THPEH Figure The mechanical model diagram of the platform with the THPEH system; system; (b) The horizontal arrangement and serial number of the plates in the THPEH system. (b) The horizontal arrangement and serial number of the plates in the THPEH system.

The second item on the right side of the Equation (1) is the force of the PTO system act on the The second item on the right side of the Equation (1) is the force of the PTO system act on the THPEH system, and it is the interaction force between the THPEH system and the semi‐submersible THPEH system, and it is the interaction force between the THPEH system and the semi-submersible platform. In this way, the motion of the semi‐submersible platform is governed by Equation (3). platform. In this way, the motion of the semi-submersible platform is governed by Equation (3).  M + M a     M p   x +  C + C r     x +  K s  K m  x = F  T64  C P T O  z  T4 6 x   K P T O  z  T4 6 x   “ ‰ .. “ `. ‰ (3) . .˘ M`Ma pωq ` M p x` rC`Cr pωqs x` pKs ` Km q x“F ` T64 CPTO z ´ T46 x ` KPTO pz ´ T46 xq (3) where M , M a are the matrices of the mass and the added mass of the platform and respectively. F are the matrices of the radiation damping, viscous damping, stationary water C r , C, M,KM where matrices of the mass and the added mass of the platform and respectively. Cr , s , and a are the x water C, Ks , and F are the matrices of the radiation damping, viscous damping, stationary stiffness stiffness and wave exciting force of the semi‐submersible platform, respectively; is the six‐DOF and wave exciting forceof ofthe the platform. semi-submersible platform, x is theis six-DOF displacement displacement vector The force of the respectively; mooring system accounted for in an vector of the platform. The force of the system is accounted for in an equivalent linear stiffness Mp is the mass contributed by the THPEH system equivalent linear stiffness matrix Km . mooring matrix Km . M p is the mass contributed by the THPEH system 2 2 2 2  ı M p  diag (4) ” 4m 4m 4m 4mh 4mh 4m´lx  l y ¯  2 2 2 2 M p “ diag 4m 4m 4m 4mh 4mh 4m lx ` ly (4) where m is the mass of the heave plate, h is the distance between the heave plate and the C.O.G of the platform. The second item on the right side of the Equation (3) is the force of the PTO system where m is the mass of the heave plate, h is the distance between the heave plate and the C.O.G of the u the PTO system act on act on the semi‐submersible platform system, defined as the control force platform. The second item on the right side of the Equation (3) is the force of the semi-submersible platform system, defined as the control force u u  T64 CPTO  z  T46x   KPTO  z  T46x  (5) “ `. ‰ .˘ u “ T64 CPTO z ´ T46 x ` KPTO pz ´ T46 xq (5)





3. Numerical Model of the Semi‐Submersible with the THPEH System 3. Numerical Model of the Semi-Submersible with the THPEH System 3.1. The State‐Space Model for Radiation Damping Force Calculation 3.1. The State-Space Model for Radiation Damping Force Calculation The added added mass mass and and radiation radiation damping damping coefficients relative motion frequency. The coefficients areare relative to to thethe motion frequency. A A convolution calculation method is introduced to calculate the radiation force of the platform [22]. convolution calculation method is introduced to calculate the radiation force of the platform [22]. The The radiation force of the semi‐submersible is composed of two parts: radiation force of the semi-submersible is composed of two parts: “ .. .‰ Fr “ ´ Ma pωq x`Cr pωq x

(6)


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Introducing the impulse response function for calculating the radiation forces in the time domain gives ż8 2 (7) Cr pωq cos pωtq dω hr ptq “ π 0

Therefore, the radiation force can be expressed as a convolution function of the velocity of the platform: żt .. . Fr ptq “ ´Ma p8q x ´ hr pt ´ τqxpτqdτ (8) ´8

Additionally, the convolution calculation could be replaced by a linear system whose inputs are the velocities and outputs are the convolution parts of the radiation forces, and the linear system in the state-space model is an auxiliary state vector for convolution [23]. #

.

.

w ptq “ar w ptq `br z3 ptq . F1r ptq “cr w ptq `dr z3 ptq

(9)

where ar , br , cr , and dr are the system matrices, and w is an auxiliary state vector for calculation. F1r ptq is the convolution part of the radiation force: żt F1r ptq “ ´

.

(10)

hr pt ´ τqxpτqdτ

´8

3.2. The State-Space Model for the Platform with the THPEH System The motion Equation of the semi-submersible platform could be transformed to the state-space Equation as . ξ “ As ξ ` Bs pF ` uq (11) where the state vector ξ is ”

.

x x

ξ“

w

ıT

(12)

By introducing the model of the radiation force model, the system matrices As and Bs are »

” ı A` 012ˆ6 Bdr ” ı As “ – 0nˆ6 br « ff B Bs “ 0nˆ2

fi Bcr fl ar

(13)

The displacements and velocities are set as the outputs, so the output matrices can be built as ” Cs “

ı I12ˆ12

012ˆn

(14)

Ds “ 012ˆ6

According to the motion Equation, Equation (1), the state-space model of the heave plate motion is «

.

z .. z

ff

« “ Ap

ff z . z

´ ˇ.ˇ . ” ` Bp f´Cd ˇzˇ z ` KPTO T46

CPTO T46

04ˆn

ı ¯ x

(15)


where Energies 2016, 9, 82 where

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«

04ˆ4 I4ˆ4 Ap “ ´KPTO pm ` aq´1 ´CPTO pm ` aq´1 « ff 044   04ˆ4 I 44 A p =  Bp “ 1 1  ´ 1  `CaqPTO  m + a    KPTO  m + apm

ff

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(16) (16)

 44 The plates are assumed to beBplaced deep enough to ignore the wave exciting force on the plates. 1 p= + a   matrices are constructed as To output the velocity of the plates, the  moutput 0

The plates are assumed to be placed deep enough to ignore the wave exciting force on the plates. C p “ I8ˆ8 , D p “ 08ˆ8 (17) To output the velocity of the plates, the output matrices are constructed as p =ITHPEH 8  8 , D p = 0 8System 8 3.3. The Simulink Model for the Platform withCthe

(17)

According to Equations (11) and (15), the state-space model of the semi-submersible and THPEH system, the diagram of the Simulink model, for calculating the performance of the system is shown in Figure 3.According to Equations (11) and (15), the state‐space model of the semi‐submersible and THPEH In addition, four gain matrices G1 , G2 , G3 and G4 are introduced to establish the calculation system, the diagram of the Simulink model, for calculating the performance of the system is shown model in Simulink for the state-space model of the platform with the THPEH system. 3.3. The Simulink Model for the Platform with the THPEH System

in Figure 3. In addition, four gain matrices G 1 , G 2 , G 3 and G 4 are introduced to establish the

”

ı

calculation model in Simulink for the state‐space model of the platform with the THPEH system. G1 “ K T C T 0 PTO «

PTO 46

46

4ˆ n

G1   K PTOTT46 C0PTO 46  nn 4ˆT 646 0044ˆ G2 “ 04ˆn  T 04ˆ0646 T046 4 n  ı G 2  ” 46  TT G3 “ 04K T 46 64 04C 6 PTO n PTO 64 ” ı G3  K T C T  PTO 64 PTO 64 G “ 0 I  4

G 4   044

ff

4 ˆ4

I 44 

(18) (18)

4ˆ4

In this way, the control force, relative displacements and velocities of the THPEH system In this way, the control force, relative displacements and velocities of the THPEH system are outputted. are outputted.

Figure 3. The diagram of the numerical model of the semi‐submersible platform in the Figure 3. The diagram of the numerical model of the semi-submersible platform in the Simulink module. Simulink module.


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Energies 2016, 9, 82 Energies 2016, 9, 82 4. Wave Tank Tests for Semi-Platform Numerical Model Modification and Validation

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4. Wave Tank Tests for Semi‐Platform Numerical Model Modification and Validation

4. Wave Tank Tests for Semi‐Platform Numerical Model Modification and Validation 4.1. Hydrodynamic Analysis of the Platform 4.1. Hydrodynamic Analysis of the Platform

4.1. Hydrodynamic Analysis of the Platform For the semi-submersible platform, the panel model of the platform is established in AQWA For the semi‐submersible platform, the panel model of the platform is established in AQWA software,For as shown in Figure 4. Fifty regular uniformly in the frequency interval the semi‐submersible platform, the waves panel model of the distributed platform is established in AQWA software, as shown in Figure 4. Fifty regular waves uniformly distributed in the frequency interval 0.1 rad/s~1.2 rad/s are adopted for the hydrodynamic parameter analysis. Only the hydrodynamic software, as shown in Figure 4. Fifty regular waves uniformly distributed in the frequency interval 0.1 rad/s~1.2 rad/s are adopted for the hydrodynamic parameter analysis. Only the hydrodynamic parameters of the heave degree of freedom in the frequency domain are given in Figure 5 because of 0.1 rad/s~1.2 rad/s are adopted for the hydrodynamic parameter analysis. Only the hydrodynamic parameters of the heave degree of freedom in the frequency domain are given in Figure 5 because of parameters of the heave degree of freedom in the frequency domain are given in Figure 5 because of limited space. limited space. limited space.

Figure 4. The panel model of the semi‐submersible platform for boundary element method (BEM) Figure 4. The panel model of the semi-submersible platform for boundary element method (BEM) Figure 4. The panel model of the semi‐submersible platform for boundary element method (BEM) calculation in AQWA software. calculation in AQWA software. calculation in AQWA software. 7

x 10 6 x 107 6 Radiation damping [kg/s] Radiation damping [kg/s]

Added mass Added mass [kg][kg]

7

x 10 11 x 107 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 0 0

0.5 1 0.5Frequency [rad/s] 1 Frequency [rad/s]

(a) (a)

5 5 4 4 3 3 2 2 1 1 0 0 0 0

1.5 1.5


(b) (b)

Wave exciting force [N/m] Wave exciting force [N/m]

6

12 12 10 10

1.5 1.5

x 10 6 x 10

8 8 6 6 4 4 2 2 0 0


1.5 1.5

(c) (c)

Figure 5. (a) Added mass coefficients of the semi‐submersible platform in the heave motion direction; Figure 5. (a) Added mass coefficients of the semi‐submersible platform in the heave motion direction; (b) radiation damping coefficients of the semi‐submersible platform in the heave motion direction; (c) Figure 5. (a) Added mass coefficients of the semi-submersible platform in the heave motion direction; (b) radiation damping coefficients of the semi‐submersible platform in the heave motion direction; (c) wave exciting force amplitudes of the semi‐submersible platform under 1 meter amplitude regular (b) radiation damping coefficients of the semi-submersible platform in the heave motion direction; wave exciting force amplitudes of the semi‐submersible platform under 1 meter amplitude regular waves in the heave motion direction. (c) wave exciting force amplitudes of the semi-submersible platform under 1 meter amplitude regular waves in the heave motion direction.

waves in the heave motion direction.


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However, the heave plate is widely used in deep ocean structures for bringing large damping and added mass to the structure compared to its small volume. As the viscous effects are prominent because of its small thickness, the drag force of the Morison Equation is adapted for the heave plate’s radiation force. The drag force coefficient of the heave plates of different shapes have been studied both numerically [24] and experimentally [25,26] in the past decade. The drag coefficient of the heave plate is related to the KC number and the motion frequency. For a solid square heave plate, when the Energies 2016, 9, 82 8 of 22 KC number is between 0.2 and 0.4, the frequency is between 0.1 and 0.5 Hz, and the drag coefficient is approximatelyHowever, the heave plate is widely used in deep ocean structures for bringing large damping 5~10. In this paper, the drag coefficient is taken as 8. For the added mass calculation, the fitting and added mass to the structure compared to its small volume. As the viscous effects are prominent Equation from the results of Prislin’s experiment is adopted in this study [27]. because of its small thickness, the drag force of the Morison Equation is adapted for the heave plate’s radiation force. The drag force coefficient of the heave plates of different shapes have been studied 4.2. Introduction of the Experiment Setup both numerically [24] and experimentally [25,26] in the past decade. The drag coefficient of the heave

Because the viscous effects are ignored in the hydrodynamic analysis, a viscous damping matrix plate is related to the KC number and the motion frequency. For a solid square heave plate, when the should be KC number is between 0.2 and 0.4, the frequency is between 0.1 and 0.5 Hz, and the drag coefficient introduced to the numerical model of the semi-submersible platform to consider the effects. is approximately 5~10. In this paper, the drag coefficient is taken as 8. For the added mass calculation, The free-decay tests of the scale model are adopted to identify the viscous damping coefficients. the fitting Equation from the results of Prislin’s experiment is adopted in this study [27]. In this study, a 1:70 scale model of the semi-submersible platform is constructed for the flume tests. The specifications of the prototype and the model are shown in Table 1. According to the Froude 4.2. Introduction of the Experiment Setup ? similarity criterion, the force scale and the time scale are 1 : 703 and 1 : 70, respectively. The tests Because the viscous effects are ignored in the hydrodynamic analysis, a viscous damping matrix were carried out in the nonlinear flume water tank in the state key laboratory of coastal and offshore should be introduced to the numerical model of the semi‐submersible platform to consider the effects. The free‐decay tests of the scale model are adopted to identify the viscous damping coefficients. engineering of Dalian University of Technology. The pictures of the experimental model and the wave In this study, a 1:70 scale model of the semi‐submersible platform is constructed for the flume tank are shown in Figures 6 and 7. The arrangement of the test is illustrated in Figure 8. The wave tank tests. The specifications of the prototype and the model are shown in Table 1. According to the Froude is 60 meters long and 5 meters wide, and the maximum water 2 meters. A piston wave-maker 1 : 70 3 depth similarity criterion, the force scale and the time scale are and 1:is 70 , respectively. The tests is located at one end of the tank, and the wave absorber is on the other end. The model of the platform were carried out in the nonlinear flume water tank in the state key laboratory of coastal and offshore is located in the central part of the tank. Two cameras used capture the six degrees of freedom engineering of Dalian University of Technology. The are pictures of to the experimental model and the wave tank are shown in Figures 6 and 7. The arrangement of the test is illustrated in Figure 8. The (DOF) motion of the platform, and tension sensors are employed to monitor the tension of the mooring wave tank is 60 meters long and 5 meters wide, and the maximum water depth is 2 meters. A piston system. A taut truncated mooring system is designed to provide the total horizontal restoring stiffness wave‐maker is located at one end of the tank, and the wave absorber is on the other end. The model as a deep-water mooring system. the The mooringof line is setTwo up cameras are with steel used wireto and a spring of the platform is located in central part the tank. capture the six to achieve this goal. degrees of freedom (DOF) motion of the platform, and tension sensors are employed to monitor the tension of the mooring system. A taut truncated mooring system is designed to provide the total horizontal restoring stiffness as a deep‐water mooring system. The mooring line is set up with steel Table 1. The physical parameters of the prototype and scale model of the semi-submersible. wire and a spring to achieve this goal.

Value Table 1. The physical parameters of the prototype and scale model of the semi‐submersible. Parameters Prototype Model Value Parameters

Prototype Deck (mm) 77470 ˆ 74380 ˆ 8600 77470 × 74380 × 8600 Column (mm)Deck (mm) 17385 ˆ 17385 ˆ 21460 Column (mm) Pontoon (mm) 11407017385 × 17385 × 21460 ˆ 20120 ˆ 8540 114070 × 20120 × 8540 Draft (mm)Pontoon (mm) 19,000 Draft (mm) Displacement (kg) 5.17 ˆ 19,000 107 7 Displacement (kg) 5.17 × 10 Roll Radius of inertia (mm) 33300 Roll Radius of inertia (mm) Pitch Radius of inertia (mm) 3340033300 Pitch Radius of inertia (mm) Yaw Radius of inertia (mm) 3500033400 Yaw Radius of inertia (mm)

35000

Model 1107 ˆ 1063 ˆ 123 1107 × 1063 × 123 307 ˆ 248 ˆ 248 307 × 248 × 248 1630 ˆ 287 ˆ 122 1630 × 287 × 122 271 271 150.73 150.73 476 476 477 477 500 500

Figure 6. The picture of the scale model of the semi‐submersible platform.

Figure 6. The picture of the scale model of the semi-submersible platform.


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Figure 7. The picture of the scale model in the wave tank.

Figure 7. The picture of the scale model in the wave tank. Figure 7. The picture of the scale model in the wave tank.

Figure 8. The sketch of the layout of the wave tank tests of the semi‐submersible platform.

Figure 8. The sketch of the layout of the wave tank tests of the semi‐submersible platform. Figure 8. The sketch of the layout of the wave tank tests of the semi-submersible platform.

4.3. Free‐Decay Tests for Damping Modification 4.3. Free‐Decay Tests for Damping Modification 4.3. Free-Decay Tests for Damping Modification The viscous damping of the platform could be identified according to the free decay test results The viscous damping of the platform could be identified according to the free decay test results because the viscous damping predominant to the radiation to damping natural The viscous damping of the is platform couldcompared be identified according the freein decay testlow results because the viscous damping is predominant compared to the radiation damping in natural frequency motion. The free decay tests for each DOF motion are carried out three times, and low the because the viscous damping is predominant compared to the radiation damping in natural frequency motion. The free decay tests for each DOF motion are carried out three times, and the low logarithmic decrement ratio method is adopted to identify the damping ratio and natural period. The frequency motion. The free decay tests for each DOF motion are carried out three times, and the logarithmic decrement ratio method is adopted to identify the damping ratio and natural period. The identified results of the scale model and the equivalent values for the full scale numerical model are logarithmic decrement ratio method is adopted to identify the damping ratio and natural period. The identified results of the scale model and the equivalent values for the full scale numerical model are shown in Table 2. The diagonal elements of the viscous damping matrix are estimated by shown in Table 2. The diagonal elements of the viscous damping matrix are estimated by identified results of the scale model and the equivalent values for the full scale numerical model are C i, i of  2the iviscous  Ma  i, i   matrix shown in Table 2. The diagonal elements are estimated by (1) i M  i, i  damping C i, i   2ii M i, i   Ma  i, i  (1) where i and  i are the identification damping ratio and natural frequency for each DOF motion, (19) C pi, iq “ 2ξ i ωi rM pi, iq ` Ma pi, iqs where i and  i are the identification damping ratio and natural frequency for each DOF motion, respectively. As the stiffness of the platform in the horizontal plane was completely contributed to respectively. As the stiffness of the platform in the horizontal plane was completely contributed to by system, the equivalent linear ratio stiffness of the mooring system where ξ the andmooring ωi are the identification damping andcoefficients natural frequency for each DOF are motion, by i the mooring system, the equivalent linear stiffness coefficients of the mooring system are calculated by respectively. As the stiffness of the platform in the horizontal plane was completely contributed to by calculated by 2 the mooring system, the equivalent stiffness coefficients of the mooring system are calculated Kmlinear  i, i    (2) by i M i, i   Ma  i, i    Ks  i, i  Km  i, i   i2 M i, i   Ma  i, i   Ks  i, i  (2) 2 The time history curves of the numerical simulation and scale model in surge and pitch motion (20) Km pi, iq “ ωi rM pi, iq ` Ma pi, iqs ´ Ks pi, iq The time history curves of the numerical simulation and scale model in surge and pitch motion are shown in Figure 9. Notice that all of the experiment results in this paper are transformed into are shown in Figure 9. Notice that all of the experiment results in this paper are transformed into results for a full scale model. The time history curves of the numerical simulation and scale model in surge and pitch motion results for a full scale model.

are shown in Figure 9. Notice that all of the experiment results in this paper are transformed into Table 2. The identification results of the natural period and the damping ratio in free‐decay tests. results for aTable 2. The identification results of the natural period and the damping ratio in free‐decay tests. full scale model. Scale Model

Full Scale Numerical Model

Scale Model Full Scale Numerical Model TableMotion 2. The identification results of the natural periodPeriod (s) and the damping ratio in free-decay tests. Period (s) Damping Ratio Damping Ratio Motion Surge Surge Sway Motion Sway Heave Heave Roll Roll Surge Pitch Pitch Sway Yaw Yaw Heave

Roll Pitch Yaw

Period (s) Damping Ratio Period (s) Damping Ratio 32.32 6.05% 270.41 6.05% 32.32 6.05% 270.41 6.05% Model 42.40 6.06% 280.74 Full Scale Numerical 6.06% Scale Model 42.40 6.06% 280.74 6.06% 2.44 4.48% 20.41 4.48% Period (s) Damping Ratio 20.41 Period (s) Damping Ratio 2.44 4.48% 4.48% 7.26 6.70% 60.74 6.70% 7.26 32.32 6.70% 6.05% 60.74 270.41 6.70% 6.05% 6.28 7.61% 52.54 7.61% 6.28 42.40 7.61% 6.06% 52.54 280.74 7.61% 6.06% 42.8 8.64% 358.09 8.64% 42.8 2.44 8.64% 4.48% 358.09 20.41 8.64% 4.48%

7.26 6.28 42.8

6.70% 7.61% 8.64%

60.74 52.54 358.09

6.70% 7.61% 8.64%


10 of 22 10 of 22

10

15

Simulation data Experiment data

Simulation data Experiment data

10

Pitch [deg]

Surge [m]

5

0

5 0 -5

-5

-10 -10 0

500

1000

1500 2000 Time [s]

2500

3000

3500

0

(a)

50

100

150 Time [s]

200

250

(b)

Figure (a) The The comparison of the surge free‐decay time histories of the modified Figure 9. 9. (a) comparison of the surge free-decay time histories of the modified numericalnumerical simulation simulation and the wave tank test; (b) The comparison of the pitch free‐decay time histories of the and the wave tank test; (b) The comparison of the pitch free-decay time histories of the modified modified numerical simulation and the wave tank test. numerical simulation and the wave tank test.

4.4. Numerical Model Validation by the Results of the Regular Wave Tests 4.4. Numerical Model Validation by the Results of the Regular Wave Tests To validate the veracity of the modified numerical model of the semi‐submersible platform, the To validate the veracity of the modified numerical model of the semi-submersible platform, the response amplitude operator (RAO) curves of the numerical model and the scale model are compared. response amplitude operator (RAO) curves of the numerical model and the scale model are compared. For the scale model tests, considering the capacity of the wave‐maker and the wave tank, 10 regular For the scale model tests, considering the capacity of the wave-maker and the wave tank, 10 regular waves are designed for regular wave tests to obtain the RAO curves. The parameters of these regular waves are designed for regular wave tests to obtain the RAO curves. The parameters of these regular waves are shown in the Table 3. For example, the experimental time history curves of the platform waves are shown in the Table 3. For example, the experimental time history curves of the platform under the R2 in the 0 degree direction are shown in Figure 10. According to the RAO definition, the under the R2 in the 0 degree direction are shown in Figure 10. According to the RAO definition, the RAO is calculated as the ratio between the wave frequency motion amplitude and the wave height. RAO is calculated as the ratio between the wave frequency motion amplitude and the wave height. However, the motions in the test contains the low frequency motion and wave frequency motion. As However, the motions in the test contains the low frequency motion and wave frequency motion. As ? . The wave frequency the ratio between the amplitude and STD value of the harmonic motion is the ratio between the amplitude and STD value of the harmonic motion is 22. The wave frequency motion amplitude in experiment RAOs calculations is calculated in frequency domain as shown in motion amplitude in experiment RAOs calculations is calculated in frequency domain as shown in Equation (21). Equation (21). ? ωşj 2j2 b RRij “ 2 SijSijpωqdω  d ij  2 ω j1 (21)  j1 ` ˘ Rij (3) RAOi ω j R“ ij Aj



RAOi  j  

A

j where, Rij and Sij are the wave frequency motion amplitude and response power spectrum on the st st i DOF motion under the j regular wave; and ω and A are the frequency amplitude of on the j ı amplitude j where, Rij and S ij are the wave frequency motion and response and power spectrum ” st j regular wave; the frequency stinterval to only cover the wave frequency ω j1 ω j2 is designed ist DOF motion under the the j regular wave; and  j and A j are the frequency and amplitude motion in the frequency domain. In this study, they are defined as of the j st regular wave; the frequency interval  j1  j 2  is designed to only cover the wave ω j1 “ 0.8ω j , ω j2 “ 1.2ω j (22) frequency motion in the frequency domain. In this study, they are defined as

 j , the 1.2 j (4) j 1  0.8and j 2 experiment The RAO curves of the numericalmodel model are compared in Figure 11. The results indicate that the precision of the numerical model is acceptable, which indicates the The RAO curves of the numerical model and the experiment model are compared in Figure 11. similar trends of each DOF motion RAO curve, although there are still differences between two curves, The results indicate that the precision of the numerical model is acceptable, which indicates the especially the pitch motion under high-frequency waves. similar trends of each DOF motion RAO curve, although there are still differences between two curves, especially the pitch motion under high‐frequency waves.


11 of 22 11 of 22 11 of 22 Surge [m] [m] Surge

Energies 2016, 9, 82 Energies 2016, 9, 82 2

Experiment

21

Experiment

10 0 -1 -1 0

200

400

0

200

400

600 Time 600 [s] Time (a) [s]

800

1000

1200

800

1000

1200

Heave [m] [m] Heave

(a)

Experiment

1

Experiment

10 0 -1 -1

0

200

400

0

200

400

600 Time 600 [s] (b) [s] Time

1000 1000

1200 1200

(b)

2 Pitch [deg] Pitch [deg]

800 800

Experiment

21

Experiment

10 0 -1 -1

0

200

400

0

200

400

600 Time 600[s] Time (c) [s]

800

1000

1200

800

1000

1200

(c) Figure 10. (a) Surge motion of the platform under the R2 in the 0 degree direction; (b) heave motion Figure 10. (a) Surge motion of the platform under the R2 in the 0 degree direction; (b) heave motion of Figure 10. (a) Surge motion of the platform under the R2 in the 0 degree direction; (b) heave motion of the platform under the RW2 in the 0 degree direction; (c) pitch motion of the platform under the the platform under the RW2 in the 0 degree direction; (c) pitch motion of the platform under the R1 in of the platform under the RW2 in the 0 degree direction; (c) pitch motion of the platform under the R1 in the 0 degree direction. the 0R1 in the 0 degree direction. degree direction. 0.8

RAORAO of Heave [m/m] of Heave [m/m]

RAORAO of Surge [m/m] of Surge [m/m]

0.5

0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

Experiment RAO Simulation RAO RAO Experiment Simulation RAO

0.5

0.8 0.7

0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

0.4 0.4

0.5 0.5

0.6 0.7 0.8 0.9 Frequency [rad/s] 0.6 0.7 0.8 0.9 Frequency [rad/s]

1 1

1.1 1.1

1.2

0.4

1.2

0.4

(a) (a)

0.6 0.7 0.8 0.9 Frequency [rad/s] 0.6 0.7 0.8 0.9 Frequency (b) [rad/s]

Experiment RAO Simulation RAO Experiment RAO

0.6 RAORAO of Pitch [deg/m] of Pitch [deg/m]

0.5 0.5

0.6 0.5

1 1

1.1 1.1

1.2 1.2

(b)

Simulation RAO

0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

0.4 0.4

0.5 0.5

0.6 0.7 0.8 0.9 Frequency [rad/s] 0.6 0.7 0.8 0.9 Frequency (c) [rad/s]

1 1

1.1 1.1

1.2 1.2

(c)

Figure 11. (a) Surge RAO curves of the scale model test and numerical simulation of the platform; Figure 11. (a) Surge RAO curves of the scale model test and numerical simulation of the platform; (b) heave RAO curves of the scale model test and numerical simulation of the platform; (c) pitch RAO Figure 11. (a) Surge RAO curves of the scale model test and numerical simulation of the platform; (b) heave RAO curves of the scale model test and numerical simulation of the platform; (c) pitch RAO curves of the scale model test and numerical simulation of the platform. (b) heave RAO curves of the scale model test and numerical simulation of the platform; (c) pitch RAO curves of the scale model test and numerical simulation of the platform.

curves of the scale model test and numerical simulation of the platform.


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Table 3. Regular wave parameters used in the tests. The Real Wave States No. R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Amplitude Frequency (mm) (rad/s) 3500 3500 3500 3500 3500 3500 3500 1400 1400 1400

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

The Aim Wave States in Tests

Period (s)

Amplitude Frequency (mm) (rad/s)

20.94 15.71 12.57 10.47 8.98 7.85 6.98 6.28 5.71 5.24

50.00 50.00 50.00 50.00 50.00 50.00 50.00 20.00 20.00 20.00

2.51 3.35 4.18 5.02 5.86 6.69 7.53 8.37 9.20 10.04

Period (s) 2.50 1.88 1.50 1.25 1.07 0.94 0.83 0.75 0.68 0.63

The Real Wave States in Tests Amplitude (mm) 49.00 50.00 49.50 49.30 46.20 50.30 40.60 16.90 18.60 21.60

Period (s) 2.50 1.88 1.50 1.25 1.07 0.94 0.83 0.75 0.68 0.63

5. Parameter Study of the THPEH System Although the optimal parameters of the tuned mass damper for tall buildings has already been fully investigated [11], the parameter study of the THPEH is still needed to investigate the motion performances of the semi-submersible, which is quite different from tall buildings’ performance, and the environment loads. The key to designing the tuned heave plate system is to determine the size, tuned period and damping ratio of the THPEH system. The main aim of the THPEH system is reducing the motion of the platform. In this paper, as only the x-axis-direction wave is adopted, the motions in the surge, heave and pitch are considered in the study, and the motion reduction ratio is introduced to evaluate the performance of the THPEH system. ri “

std pxiu q ´ std pxic q , i “ 1, 3, 5 std pxiu q

(23)

where xic and xiu are the motion of the platform with and without the THPEH system, respectively. The operator std is the standard deviation calculation, and ri is the motion reduction ratio. Another feature of the THPEH system is generating power, and the capture width is introduced to represent the energy absorption efficiency, which is already widely used in the wave energy field. The capture width is the ratio between the absorption power and the wave energy per wave front length of the incident wave. For irregular waves, the average wave energy power per wave front length is estimated by Ptotal “ 490.6Hs2 Tp (24) where Tp and Hs are the peak period and the significant wave height of the irregular wave, respectively. The unit of Ptotal is watt. Thereby, the capture width and the absorption power of the THPEH system are defined as P λc “ (25) Ptotal where λc is the capture width under irregular waves, and P is the average capture power of the THPEH system under irregular waves, defined as żT P“

.T .

CPTO zr zr dt{T 0

.

where zr is the relative velocity between the platform and plates.

(26)


13 of 22

may experience in its designed service life. These wave parameters are presented in Table 4. For each wave condition, a 3000‐s‐long history of the wave elevation is generated for the motion calculation. Energies 2016, 9, 82 13 of 22 The wave exciting force of the platform includes 1st‐order wave forces and 2nd‐order wave forces and is calculated in AQWA. The modified numerical model in the last section is used for the numerical simulation. The wave exciting force on the THPEH system is ignored, assuming the heave In this parameter study, seven irregular wave conditions, including three extreme waves, two plates are placed deeply enough. Sixteen different side lengths between 5 and 35 meters, 28 different working wave conditions and two average conditions, in the Gulf of Mexico (GM), the South China tuned periods between 1 and 60 s and 28 different damping ratios between 1% and 60% are selected Sea (SCS) and the North Sea (NS) are adopted to represent all of the wave conditions the platform for this parameter study. may experience in its designed service life. These wave parameters are presented in Table 4. For each wave condition,Table 4. Characteristic wave parameters of seven irregular wave sea states. a 3000-s-long history of the wave elevation is generated for the motion calculation. The wave exciting force of the platform includes 1st-order wave forces and 2nd-order wave forces and Wave Condition is calculated in AQWA. The modified numerical model in i the last section isi used for the numerical  i H si (m)  p (rad/s) T p (s) Name Descriptions simulation. The wave exciting force on the THPEH system is ignored, assuming the heave plates are IRW‐1 100‐year in GM 12.20 0.45 5 and 35 meters, 14.00 28 different 2.00 tuned placed deeply enough. Sixteen different side lengths between IRW‐2 between 1100‐year in SCS 13.30 ratios between 0.41 1% and15.50 3.30 periods and 60 s and 28 different damping 60% are selected for this IRW‐3 study. 10‐year in SCS 11.10 0.46 13.60 2.00 parameter IRW‐4 Working condition in GM 3.96 0.70 9.00 2.00 Table 4. Characteristic wave parameters irregular wave sea states. IRW‐5 Working condition in SCS 6.00 of seven0.56 11.20 2.00 IRW‐6 Average condition in GM 2.40 0.82 7.70 2.00 Wave Condition i i i IRW‐7 Average condition in NS 1.88 Hsi (m) 0.86 7.28 2.00 ωp (rad/s) Tp (s) γ Name

Descriptions

5.1. Parameter Study of the Size of the Tuned Heave Plates in the THPEH System IRW-1 100-year in GM 12.20 0.45 14.00

2.00 IRW-2 100-year in SCS 13.30 0.41 15.50 3.30 The size of the heave plate determines the added mass of the THPEH system, which dominates IRW-3 10-year in SCS 11.10 0.46 13.60 2.00 the mass ratio between the THPEH and the platform. To investigate the effect of the heave plate’s IRW-4 Working condition in GM 3.96 0.70 9.00 2.00 IRW-5 Working condition in SCS 6.00 0.56 11.20 2.00 size, for each size plate under each wave condition, the motion performance for different damping IRW-6 Average condition in GM 2.40 0.82 7.70 2.00 ratios and tuned periods are calculated. For every size of the plates, the maximum motion reduction IRW-7 Average condition in NS 1.88 0.86 7.28 2.00

ratio and maximum capture width under different wave conditions are selected to represent the optimal condition with this size of the plate. The optimal motion reduction ratio in the surge, heave 5.1. Parameter Study of the Size of the Tuned Heave Plates in the THPEH System and pitch motion and the capture width of the THPEH system with different sizes of the plate under seven wave conditions are shown in Figure 12. The size of the heave plate determines the added mass of the THPEH system, which dominates The result indicates that the reduction ratio for the three motions increases with the size of the the mass ratio between the THPEH and the platform. To investigate the effect of the heave plate’s size, THP plate for each sea state. However, for the capture width, there is an optimal size under each for each size plate under each wave condition, the motion performance for different damping ratios wave condition, and the optimal side lengths are distributed in the interval of 15–20 m. If motion and tuned periods are calculated. For every size of the plates, the maximum motion reduction ratio reduction is adopted as the prime goal of the THPEH system, a large heave plate should be chosen. and maximum capture width under different wave conditions are selected to represent the optimal However, in this study, as the side of the column is 17 meters long, considering the installation and condition with this size of the plate. The optimal motion reduction ratio in the surge, heave and pitch the efficiency of capturing power, the size of the heave plates in the THPEH system in paper is set to motion and the capture width of the THPEH system with different sizes of the plate under seven wave 17 meters long. conditions are shown in Figure 12. 0.7

0.14

0.1 0.08

0.6 Heave motion reduction ratio

Surge motion reduction ratio

0.12

IRW-1 IRW-2 IRW-3 IRW-4 IRW-5 IRW-6 IRW-7

0.06 0.04 0.02 0 5

0.5 0.4


0.3 0.2 0.1

10

15

20 25 Side length[m]

30

35

0 5

(a)

10

15


(b) Figure 12. Cont.

30

35


14 of 22 14 of 22

0.7

0.5 0.4 0.3 0.2

25 20 15 10

0.1 0 5


30 Capture width[m]

Pitch motion reduction ratio

0.6

35


10

15


30

5 5

35

10

15

(c)


30

35

(d)

Figure 12. (a) The surge motion reduction ratio with optimal connection parameters with different Figure 12. (a) The surge motion reduction ratio with optimal connection parameters with different sizes sizes the (b) plate; the motion heave motion reduction ratio with connection optimal connection parameters with of theof plate; the(b) heave reduction ratio with optimal parameters with different different sizes of the plate; (c) the pitch motion reduction ratio with optimal connection parameters sizes of the plate; (c) the pitch motion reduction ratio with optimal connection parameters with plate; (d) the width capture width reduction ratio with optimal parameters connection with different of the (d) different sizes ofsizes the plate; the capture reduction ratio with optimal connection parameters with different sizes of the plate. with different sizes of the plate.

TPTO [s]

TPTO [s]

The result indicates that the reduction ratio for the three motions increases with the size of the 5.2. Parameter Study of the Tuned Period of the Tuned Heave Plate THP plate for each sea state. However, for the capture width, there is an optimal size under each wave Generally, the conventional TMDs in tall buildings for suppressing wind‐induced condition, and the optimal side lengths areused distributed in the interval of 15–20 m. Ifthe motion reduction vibration tuned to goal the 1st‐order natural frequency the structure, as be the natural frequency is adoptedare as the prime of the THPEH system, a largeof heave plate should chosen. However, in motion is dominated under wide‐band frequency environment loads. However, compared to wind this study, as the side of the column is 17 meters long, considering the installation and the efficiency of loads, the wave loads have a narrow band frequency, and the natural frequency of a well‐designed capturing power, the size of the heave plates in the THPEH system in paper is set to 17 meters long. semi‐submersible platform for all motion is set far away from the peak frequency of waves. Especially 5.2. Parameter of the Period of the Tunedplatforms, Heave Platethe wave frequency motion is always for the heave Study motion of Tuned the semi‐submersible dominated compared to the low frequency motion in the natural period. As a result, in order to Generally, the conventional TMDs used in tall buildings for suppressing the wind-induced achieve optimal control efficiency, the effect of the tuned period TPT O of the THPEH system needs vibration are tuned to the 1st-order natural frequency of the structure, as the natural frequency to be investigated. The side length of the plates is set to 17 meters. motion is dominated under wide-band frequency environment loads. However, compared to wind Figure 13 presents the contour maps of the heave motion reduction ratio under IRW‐1 and the loads, the wave loads have a narrow band frequency, and the natural frequency of a well-designed capture width of the THPEH system under IRW‐6. Motion reduction under severe wave conditions semi-submersible platform for all motion is set far away from the peak frequency of waves. Especially and energy absorption under average wave conditions are the two prime goals of the THPEH system. for the heave motion of the semi-submersible platforms, the wave frequency motion is always Figure 13a compared indicates to that for in heave motion period. suppression under inIRW‐1 is TPTO dominated thethe lowoptimal frequency motion the natural As a result, order to approximately 7–10 s, and the damping coefficient of the PTO system has a small effect on the heave achieve optimal control efficiency, the effect of the tuned period TPTO of the THPEH system needs to motion control. The maximum capture width of the THPEH system under IRW‐6 is achieved for the be investigated. The side length of the plates is set to 17 meters. parameters with and  PTOmaps . the heave motion reduction ratio under IRW-1 and the TPTO  the 7 s contour  0.1of Figure 13 presents capture width of the THPEH system under IRW-6. Motion reduction under severe wave conditions 60 22 60 0.18 and energy 0 absorption under average wave conditions are the two prime goals of the THPEH system. 20 0.16 heave motion suppression under IRW-1 is approximately Figure 13a indicates that the optimal TPTO for 0.02 50 0.04 0.06 18 50 0.14 7–10 s, and the damping coefficient of the PTO system has a small effect on the heave motion control. 16 2 the parameters 0.12 40 14 with The maximum capture width of the THPEH system under 40 IRW-6 is achieved for 0.08 0.1 12 TPTO “ 307s and ξ PTO “ 0.1. 30 10 0.08 0.1 4 To investigate the effect of TPTO on motion suppression and energy harvesting, the performance 8 0.06 20 20 of the THPEH system with ξ PTO0.12 “ 0.1 is analyzed. Figure 14 shows the diagrams 6 of the surge motion 6 0.14 0.04 8 reduction, and capture width under different 4 wave 10 heave motion reduction, pitch motion reduction 10 16 14 0.02 0.18 22 20 2 conditions with the0.16 tuned period ratio, defined as 0.1

0.2

0.3



PTO

(a)

0.4

0.5

0.6

0

0.1

λiT

T “ PTO Tpi

0.2

0.3



0.4

0.5

0.6

PTO

(b)

Figure 13. (a) The heave motion reduction ratio under IRW‐1 with different  PTO and TPTO ; (b) the capture width under IRW‐6 with different  PTO and TPTO .

(27)

to be investigated. The side length of the plates is set to 17 meters. Figure 13 presents the contour maps of the heave motion reduction ratio under IRW‐1 and the capture width of the THPEH system under IRW‐6. Motion reduction under severe wave conditions and energy absorption under average wave conditions are the two prime goals of the THPEH system. Energies 82 15 of 22 Figure 2016, 13a 9, indicates that the optimal TPTO for heave motion suppression under IRW‐1 is approximately 7–10 s, and the damping coefficient of the PTO system has a small effect on the heave motion control. The maximum capture width of the THPEH system under IRW‐6 is achieved for the where λiT is the tuned period ratio between the tuned period of the THPEH and the ist wave condition’s parameters with TPTO  7 s and  PTO  0.1 . peak period. 60

0.02

0.04

TPTO [s]

50

16

0.08

0.1 0.08

0.1

2

40

0.12

30

18

0.14

40


22 20

0.16

0.06

TPTO [s]

50

60

0.18

0

14 12 15 of 22

30

10

4

8 0.06 0.12 To investigate the effect of TPTO on motion suppression and energy harvesting, the performance 20 20

6

0.14

6

0.04 8 of the THPEH system with  PTO  0.1 is analyzed. Figure 14 shows the diagrams of the surge motion 4 10

10

0.02

20

14

16

0.18 22 2 reduction, heave motion reduction, pitch motion reduction and capture width under different wave 0.16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 conditions with the tuned period ratio, defined as  

PTO

 Ti

(a)



TP T O

PTO

(b)

(9)

T pi Figure 13. (a) The heave motion reduction ratio under IRW‐1 with different  PTO and TPTO ; (b) the Figure 13. (a) The heave motion reduction ratio under IRW-1 with different ξ PTO and TPTO ; (b) the st and capture width under IRW‐6 with different where the under tuned IRW-6 period ratio between tuned TPTO  PTOthe  Ti is capture width with different ξ PTO and TPTO . . period of the THPEH and the i wave

condition’s peak period. 0.04

0.03 0.025


0.16 Heave motion reduction ratio


0.18


0.035

0.02 0.015 0.01 0.005

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0 0

1

2 3 Tuned period ratio

4

5

0

1

(a) 30


0.16

5

0.14


25 Capture width [m]

Pitch motion reduction ratio

4

(b)

0.18

0.12 0.1

20 15 10 5

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2 3 Tuned period ratio

1

2 3 4 Tuned period ratio

(c)

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6

0

1

2 3 4 Tuned period ratio

5

6

(d)

Figure 14. (a) wave Figure 14. (a) The The surge surge motion motion reduction reduction ratio ratio with with the the tuned tuned period period ratio ratio under under different different wave conditions with  PTO  0.1 ; (b) the heave motion reduction ratio with the tuned period ratio under conditions with ξ PTO “ 0.1; (b) the heave motion reduction ratio with the tuned period ratio different wave conditions; (c) the (c) pitch reduction ratio with tuned ratio under under different wave conditions; themotion pitch motion reduction ratiothe with the period tuned period ratio different wave conditions; (d) the capture width with the tuned period ratio under different under different wave conditions; (d) the capture width with the tuned period ratio under different wave conditions. wave conditions.

Figure 14a shows that the tuned period has a small effect on surge motion suppression, as the major contributor of the THPEH to the surge motion is the additional mass of the plates. For heave motion reduction and pitch motion reduction analysis, Figure 14b,c indicate that for sever wave conditions IRW‐1~3, the optimal tuned period ratio is approximately 0.7, which is not far from 1. This


16 of 22


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Figure 14a shows that the tuned period has a small effect on surge motion suppression, as the its maximum value under wave isconditions while mass the tuned majorachieves contributor of the THPEH to the different surge motion the additional of theperiod plates.ratio Foris heave approximately 1, which means the THPEH system is tuned to the wave peak period. motion reduction and pitch motion reduction analysis, Figure 14b,c indicate that for sever wave safety the is the prime consideration, the is parameters should 0.7, be which chosen isaccording to the conditionsWhen IRW-1~3, optimal tuned period ratio approximately not far from 1. This maximum heave motion under severe wave conditions. In this paper, the tuned period of the THPEH result indicates that the THPEH system should be tuned near the wave peak period to achieve the is set to 9 s because of the maximum heave motion reduction under IRW‐1. maximum control effect under these harsh wave conditions. For smaller waves IRW-4~7, there are two peaks5.3. Parameter Study of the Damping Coefficient of the Power Take‐Off System (PTO) for the tuned period ratio; one is approximately 1, and another is approximately 2~3 for heave motion reduction and 4~7 for pitch motion reduction. This result indicates that the THPEH system can Although the damping ratio of the PTO system has a small effect on the motion control of the be tuned near the wave peak period or natural period to achieve the maximum control effect under platform, it is the key factor for energy absorption and relative motion between the heave plate and thesethe platform. The relations between the motion reduction ratio, the relative motion between Plate 1 calm wave conditions. For energy absorption efficiency analysis, the capture width achieves its maximum value under different wave conditions while the tuned period ratio is approximately 1, and the platform, the capture width and the PTO damping ratio are shown in Figure 15. which means the15a THPEH is tuned to the wave Figure shows system that the PTO damping ratio peak has a small effect on surge motion  PTO period. When safety is the prime consideration, the parameters should be chosen according to the suppression. For heave motion reduction and pitch motion reduction analysis, Figures 15b,c indicate maximum heave motion under severe wave conditions. In this paper, the tuned period of the THPEH that for wave conditions IRW‐1~5, the motion reduction ratios decrease with increasing damping. for the wave condition IRW‐6~7, the relations between is setHowever, to 9 s because ofaverage the maximum heave motion reduction under IRW-1.the reduction ratio and damping are opposite. The maximum relative displacement between Plate 1 and the platform is shown in Figure 15d, which illustrates that the amplitude of the relative displacement decreases with 5.3. Parameter Study of the Damping Coefficient of the Power Take-Off System (PTO) increasing PTO damping ratio. As shown in Figure 15e, there is an optimal PTO damping ratio for

Although the damping ratio of the PTO system has a small effect on the motion control of the each wave condition to achieve maximum energy absorption. Although the optimal value depends platform, it is the key factor for energy absorption and relative motion between the heave plate and on the wave condition, the optimal PTO damping ratio can be chosen according to the curve of the the platform. The relations between the motion reduction ratio, the relative motion between Plate 1 capture width and damping ratio under IRW‐6 and IRW‐7. In this paper, the optimal PTO damping and the platform, the capture width and the PTO damping ratio are shown in Figure 15. ratio is set to 0.2. 0.3 IRW-1 IRW-2 IRW-3 IRW-4 IRW-5 IRW-6 IRW-7

0.02

0.015

0.01

0.005

0

Heave motion reduction ratio


0.025


0.25 0.2 0.15 0.1 0.05

0.1

0.2

0.3

PTO

0.4

0.5

0

0.6

0.1

0.2

(a)

0.2 0.15 0.1 0.05

0.2

0.3

PTO

0.4

0.5

0.6

Maximum relative displacement [m]


0.1

0.5

0.6

(b)

0.25 Pitch motion reduction ratio

0.4

7

0.3

0

0.3

PTO


6 5 4 3 2 1 0 0

(c)

0.1

0.2

0.3



0.4 PTO

(d) Figure 15. Cont.

0.5

0.6

0.7


17 of 22 17 of 22 25


Capture width [m]

20

15

10

5

0

0.1

0.2

0.3

PTO

0.4

0.5

0.6

(e) Figure 15. (a) The surge motion reduction ratio with the PTO damping ratio under different wave Figure 15. (a) The surge motion reduction ratio with the PTO damping ratio under different wave conditions with TPTO “  99s; s ; (b) the surge motion reduction ratio with the PTO damping ratio under conditions with T (b) the surge motion reduction ratio with the PTO damping ratio under PTO

different wave conditions with s ; (c) the pitch motion reduction ratio with the PTO damping different wave conditions with TTPTO “ 99s; (c) the pitch motion reduction ratio with the PTO damping PTO  ratio under with TPTO the the maximum relative displacement of Plate ratio under different different wave wave conditions conditions with TPTO“ 9s;9 s(d) ; (d) maximum relative displacement of 1 with the PTO damping ratio under different wave conditions with TPTO “ 9s ;  (e)9the capture width Plate 1 with the PTO damping ratio under different wave conditions with TPTO s ; (e) the capture with the PTO damping ratio under different wave conditions with TPTO “ T9s.  9 s . width with the PTO damping ratio under different wave conditions with PTO

Figure 15a shows that the PTO damping ratio ξ PTO has a small effect on surge motion suppression. 6. Results and Discussion For heave motion reduction and pitch motion reduction analysis, Figure 15b,c indicate that for wave According to the parameter study of the THPEH system in the last section, the optimal conditions IRW-1~5, the motion reduction ratios decrease with increasing damping. However, for the parameters of the THPEH system are selected. The side length of the plate is 17 meters, the tuned average wave condition IRW-6~7, the relations between the reduction ratio and damping are opposite. period of the THPEH system is 9 s, and the damping ratio of the PTO system is 20%. Under these The maximum relative displacement between Plate 1 and the platform is shown in Figure 15d, which conditions, the connection parameters are illustrates that the amplitude of the relative displacement decreases with increasing PTO damping 5 K P15e, 6 .8 7 2is 3 an 1 0 optimal N / m , CPTO 3 .9 3 7 5  1ratio 0 5 k g for / s each wave condition ratio. As shown in Figure to (10) T O there P T O damping achieve maximum energy absorption. Although the optimal value depends on the wave condition, the The performance of the THPEH on the semi‐submersible platform under different wave optimal PTO damping ratio can be chosen according to the curve of the capture width and damping conditions is analyzed. The results include the heave motion response, the pitch response and capture ratio under IRW-6 and IRW-7. In this paper, the optimal PTO damping ratio is set to 0.2. power, shown in Table 5. The THPEH system could reduce up to 20% of the heave motion and 10% of the pitch motion under severe wave conditions and provide up to 40 kW power under average 6. Results and Discussion wave conditions. Moreover, the system could theoretically capture more than 1000 kW of power According to the parameter study of the THPEH system in the last section, the optimal parameters under 100‐year wave conditions in the South China Sea. of the THPEH system are selected. The side length of the plate is 17 meters, the tuned period of the THPEH system is 9 s, and of thethe damping of the PTO system is 20%. Under these conditions, Table 5. Performances THPEH ratio system working on the semi‐submersible platform under the connection parameters are different wave conditions. Heave Motion Response Wave Name

Pitch Motion Response KPTO “ 6.8723 ˆ 105 N{m, CPTO “ 3.9375 ˆ 105 kg{s

Energy Harvesting

(28)

without with without with c Reduction Reduction P THPEH THPEH THPEH THPEH Ratio Ratio The performance of the THPEH on the semi-submersible platform under different wave conditions (kW) (m) (m) (m) (Degree) (Degree) is analyzed. The results include motion4.34 response, 3.86 the pitch 11.03% response and capture1.02 power, IRW‐1 1.99 1.62 the heave 18.25% 1040.5 IRW‐2 1.98 system 13.96% 4.85 4.30 of the heave 11.43% motion 972.11 0.72 shown in Table2.31 5. The THPEH could reduce up to 20% and 10% of the IRW‐3 1.72 severe wave 1.42 conditions 17.24% and provide 3.86 910.27 1.11 pitch motion under up3.44 to 40 kW10.96% power under average wave IRW‐4 0.16 0.14 9.73% 0.75 0.70 7.20% 130.64 1.89 conditions. Moreover, the system could theoretically capture more than 1000 kW of power under IRW‐5 0.57 0.51 10.74% 1.67 1.51 9.42% 367.95 1.86 100-year in 0.05 the South China IRW‐6 wave conditions 0.05 5.26% Sea. 0.29 0.28 4.64% 39.63 1.82 0.04 the time 3.30% 0.20 power0.19 1.97 IRW‐7 Figures 16 0.04 and 17 illustrate histories and spectra of2.94% the heave24.88 motion and pitch

motion under IRW-1. They show that the THPEH could reduce the amplitude for both the heave Figures 16 and 17 illustrate the time histories and power spectra of the heave motion and pitch motion and the pitch motion. For the heave and pitch motions, the wave frequency motion and low motion under IRW‐1. They that the THPEH the could reduce the amplitude both the heave natural frequency motion areshow suppressed, although THPEH system is tuned to for the wave frequency. motion and the pitch motion. For the heave and pitch motions, the wave frequency motion and low natural frequency motion are suppressed, although the THPEH system is tuned to the wave


18 of 22

The reduction effect of the low natural frequency motion is more significant compared to the wave frequency motion. Table 5. Performances of the THPEH system working on the semi-submersible platform under different wave conditions. Heave Motion Response Wave Name

with THPEH (m)

without THPEH (m)

Pitch Motion Response

Reduction Ratio

without THPEH (Degree)

with THPEH (Degree)

Reduction Ratio

Energy Harvesting P (kW)

λc (m)

IRW-1 1.99 1.62 18.25% 4.34 3.86 11.03% 1040.5 1.02 IRW-2 2.31 1.98 13.96% 4.85 4.30 11.43% 972.11 0.72 IRW-3 1.72 1.42 17.24% 3.86 3.44 10.96% 910.27 1.11 IRW-4 0.16 0.14 9.73% 0.75 0.70 7.20% 130.64 1.89 Energies 2016, 9, 82 18 of 22 IRW-5 0.57 0.51 10.74% 1.67 1.51 9.42% 367.95 1.86 IRW-6 0.05 0.05 5.26% 0.29 0.28 4.64% 39.63 1.82 IRW-7 0.04 0.04 3.30% 0.20 0.19 2.94% 24.88 1.97 frequency. The reduction effect of the low natural frequency motion is more significant compared to

the wave frequency motion.

Heave motion [m]

10

Without THPEH With THPEH

5 0 -5 -10 0

500

1000

1500 Time [s]

2000

2500

3000

(a) Without THPEH With THPEH

PSD of heave [m2.s/rad]

35 30 25 20 15 10 5 0

0.5

1

1.5 Frequency [rad/s]

2

2.5

3

(b) Figure 16. (a) The time histories of the heave motion with and without the THPEH system under IRW‐ Figure 16. (a) The time histories of the heave motion with and without the THPEH system under 1; (b) the power spectra of the heave motion with and without the THPEH system under IRW‐1. IRW-1; (b) the power spectra of the heave motion with and without the THPEH system under IRW-1.

Pitch motion [Degree]

The traces of the 20 control force against the displacement and velocity in the heave and pitch THPEH motions are shown in Figures 18 and 19. This result indicates that theWithout control forces in the heave and With THPEH pitch motions both have a damping force characteristic and a slight negative stiffness characteristic. 10 The damping force characteristic is the reason why the motion reduction effect on the low natural frequency motion is more significant and is due to the power spectrum figures. The negative stiffness 0 forces make the natural period farther away from the peak period of the waves. -10 -20 0

500

1000

1500 Time [s]

2000

2500

3000

(a) 0.04 0.035


PSD of heav

20 15 10 5


0

0.5

1

1.5 Frequency [rad/s]

2

2.5

19 of 22

3

(b) Figure 20 shows the time histories of the relative motions of the plates, the instant power and the capture energy of the THPEH system under IRW-1. This result indicates that the THPEH system could Figure 16. (a) The time histories of the heave motion with and without the THPEH system under IRW‐ 1; (b) the power spectra of the heave motion with and without the THPEH system under IRW‐1. harvest considerable power while suppressing the motion of the platform.

Pitch motion [Degree]

20


10 0 -10 -20 0

500

1000

1500 Time [s]

2000

2500

3000

(a) Without THPEH With THPEH

0.04


PSD of pitch [rad.s]

0.035 0.03 0.025

19 of 22

0.02 0.015

The traces of the control force against the displacement and velocity in the heave and pitch 0.01 motions are shown in Figures 18 and 19. This result indicates that the control forces in the heave and 0.005 pitch motions both have a damping force characteristic and a slight negative stiffness characteristic. 0 0.5 1 1.5 2 2.5 3 The damping force characteristic is the reason why the motion reduction effect on the low natural Frequency [rad/s] frequency motion is more significant and is due to the power spectrum figures. The negative stiffness (b) The power spectra of the pitch motions forces make the natural period farther away from the peak period of the waves. Figure 17. (a) The time histories of the pitch motion with and without the THPEH system under IRW‐1; Figure 20 shows the time histories of the relative motions of the plates, the instant power and Figure 17. (a) The time histories of the pitch motion with and without the THPEH system under IRW-1; (b) the power spectra of the pitch motion with and without the THPEH system under IRW‐1. the capture energy of the THPEH system under IRW‐1. This result indicates that the THPEH system (b) the power spectra of the pitch motion with and without the THPEH system under IRW-1. could harvest considerable power while suppressing the motion of the platform. 7

7

1

x 10

1

x 10

Control force Control force in heave [N]

Control force in heave [N]

Control force 0.5

0

-0.5

-1 -6

-4

-2 0 2 4 Heave displacement [m]

0.5

0

-0.5

-1 -3

6

-2

(a)

-1 0 1 Heave velocity [m/s]

2

3

(b)

Figure 18. (a) The relation between the control force in the heave motion from the THPEH system and Figure 18. (a) The relation between the control force in the heave motion from the THPEH system and the heave displacement of the platform; (b) the relation between the control force in the heave motion the heave displacement of the platform; (b) the relation between the control force in the heave motion from the THPEH system and the heave velocity of the platform. from the THPEH system and the heave velocity of the platform. 8

7

x 10

Control moment 2

0

-2

1 ntrol moment in pitch [N.m]

ntrol moment in pitch [N.m]

4

x 10

Control moment 0.5

0

-0.5

-1 -6

-4

-2 0 2 4 Heave displacement [m]

-1 -3

6

-2

(a)

-1 0 1 Heave velocity [m/s]

2

3

(b)

Figure 18. (a) The relation between the control force in the heave motion from the THPEH system and Energies 2016, 9,the heave displacement of the platform; (b) the relation between the control force in the heave motion 82

20 of 22

from the THPEH system and the heave velocity of the platform. 8

7

x 10

1

Control moment

Control moment in pitch [N.m]

Control moment in pitch [N.m]

4

2

0

-2

-4 -0.2

-0.1 0 0.1 0.2 Pitch displacement [rad]

x 10

Control moment 0.5

0

-0.5

-1 -3

0.3

-2

(a)

-1 0 1 Pitch velocity [rad/s]

2

3

(b)

Figure 19. (a) The relation between the control force in the pitch motion from the THPEH system and

Figure 19. (a) The relation between the control force in the pitch motion from the THPEH system and the pitch displacement of the platform; (b) the relation between the control force in the pitch motion the pitchfrom the THPEH system and the pitch velocity of the platform. displacement of the platform; (b) the relation between the control force in the pitch motion from the THPEH system and the pitch velocity of the platform. Energies 2016, 9, 82 20 of 22

Relative Displacement [m]

6 Plate 1 Plate 3

4 2 0 -2 -4 -6 0

500

1000

1500 Time [s]

2000

2500

3000

(a) 7000 Capture instant power [kW]

Power 6000 5000 4000 3000 2000 1000 0 0

500

1000

1500 Time [s]

2000

2500

3000

(b) 6

3.5

x 10

Power

Capture energy [kJ]

3 2.5 2 1.5 1 0.5 0 0

500

1000

1500 Time [s]

2000

2500

3000

(c) Figure 20. (a) The relative displacement of Plate1 and Plate3 under IRW‐1; (b) the captured instant

Figure 20. (a) The relative displacement of Plate1 and Plate3 under IRW-1; (b) the captured instant power of the THPEH system under IRW‐1; (c) the captured energy of the THPEH system under IRW‐1. power of the THPEH system under IRW-1; (c) the captured energy of the THPEH system under IRW-1. 7. Conclusions This paper presents a tuned heave plate system (THPEH) used on a semi‐submersible platform for suppressing the platform’s motions and harvesting energy from the wave‐induced response. Based on the numerical model of the platform, which is modified and testified by a 1:70 scale model test, parameters including the size, the tuned period and the damping ratio of the THPEH system are studied systematically. The effects of the THPEH system on motion suppression and energy


21 of 22

7. Conclusions This paper presents a tuned heave plate system (THPEH) used on a semi-submersible platform for suppressing the platform’s motions and harvesting energy from the wave-induced response. Based on the numerical model of the platform, which is modified and testified by a 1:70 scale model test, parameters including the size, the tuned period and the damping ratio of the THPEH system are studied systematically. The effects of the THPEH system on motion suppression and energy harvesting under different wave conditions are investigated. The results show the following: 1. 2.

3.

4.

5.

6.

By introducing a viscous damping matrix to the numerical model, the precision of the numerical model is acceptable compared to the wave tank tests. For the reasonable plate size range, the control performance of the platform motion increases with the growth of the plate size. However, there is an optimal size of the plate under each wave condition for energy harvesting. The novel THPEH system could significantly reduce the motions of the semi-submersible platform. The optimal tuned natural period of the THPEH depends on the wave condition. For normal severe wave conditions in the South China Sea, the optimal tuned period is approximately 9 s. For different wave conditions, in order to achieve the maximum energy absorption, the damping ratio of the THPEH system is approximately 20%. The damping ratio has significant impacts on the relative motion range. Moreover, the optimal tuned natural period for energy harvesting is near the wave peak frequency. With optimal parameters, the control forces in the heave and pitch motions have both a damping force characteristic and a slight negative stiffness characteristic, which are all positive for motion suppression. The novel tuned heave plate energy harvesting system could reduce the motion of a semi-submersible platform and generate considerable power, which makes the THPEH system have broad application prospects.

Acknowledgments: The research was financially supported by the National Natural Science Foundation of China (the Grant No.: 50921001) and the National Major Fundamental Research Program (973 Plan, the Grant No.: 2011CB013705). Author Contributions: Kun Liu carried out most of the work presented here, Jinping Ou was the supervisors of this work, and Haizhi Liang had a relevant contribution with the wave tank experiment and the numerical simulations. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2.

3. 4. 5.

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