Power enhancement of pontoon-type wave energy convertor via hydroelastic response and variable power take-off system

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University of Groningen

Power enhancement of pontoon-type wave energy convertor via hydroelastic response and

variable power take-off system

Tay, Zhi Yung; Wei, Yanji

Published in:

Journal of ocean engineering and science DOI:

10.1016/j.joes.2019.07.002

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Publication date: 2020

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Citation for published version (APA):

Tay, Z. Y., & Wei, Y. (2020). Power enhancement of pontoon-type wave energy convertor via hydroelastic response and variable power take-off system. Journal of ocean engineering and science, 5(1), 1-18. https://doi.org/10.1016/j.joes.2019.07.002

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JournalofOceanEngineeringandScience5(2020)1–18

www.elsevier.com/locate/joes

Original

Article

Power

enhancement

of

pontoon-type

wave

energy

convertor

via

hydroelastic

response

and

variable

power

take-off

system

Zhi

Yung

Tay

a,∗

,

Yanji

Wei

b

a_Engineering_Cluster,_Singapore_Institute_of_Technology,₁₀_Dover_Drive,_Singapore_138683,_Singapore b_Faculty_of_Science_and_Engineering,_University_of_Groningen,_Nijenborgh_4,_Groningen_9747AG,_the_Netherlands

Received10April2019;receivedinrevisedform19July2019;accepted20July2019 Availableonline30July2019

Abstract

Waveenergyhasgaineditspopularityinrecentdecadesdue tothevastamount ofuntappedwaveenergy resources.Therearenumerous types of wave energy convertor (WEC) being proposed and to be economically viable, various means to enhance the power generation from WECshavebeen studiedandinvestigated.In thispaper,anovel pontoon-typeWEC, whichisformedby multipleplate-likemodules connectedby hinges,are considered.Thepowerenhancementof thispontoon-typeWECisachieved byallowingcertainlevelofstructural deformationand byutilizingaseriesofoptimalvariablepower take-off(PTO)system.Thewaveenergyisconvertedintousefulelectricity by attaching the PTO systems on the hinge connectors such that the mechanical movements of the hinges could produce electricity. In this paper, various structuralrigidity of the interconnected modulesare considered by changing the materialYoung’s modulus in order to investigate itsimpactonthe power enhancement.In addition,the geneticalgorithm optimizationscheme isutilized toseekforthe optimal PTOdampinginthevariablePTOsystem.Itisobservedthatundercertaincondition,theflexiblepontoon-typeWECwithlesserconnection joints is more effective in generatingenergy as compared to its rigid counterpart with higher connection joints. It is also found that the variablePTO systemisable togenerategreaterenergy ascomparedto thePTOsystem withconstant/uniformPTOdamping.

Thisisanopenaccess articleunderthe CCBY-NC-NDlicense.(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords:Powergenerationenhancement;Pontoon-typewaveenergyconvertor;Verylargefloatingstructure(VLFS);Hydroelasticresponse;Variablepower take-offsystem;Geneticalgorithmoptimization.

1. Introduction

In orderto mitigate the adverse effect of climate change, the International Panel onClimate Change(IPCC) hasurged various sectors to reduce their dependency on fossil fuels. Asaresult, researchersandengineers havelookedintoother cleanenergyoptionssuchaswind,wave,solaror ocean ther-mal energy conversion as alternatives to our energy source. Theglobal grosstheoreticalresourceforwaveenergyhasthe highest energy density among the renewable energy sources [1] and it is estimated to be 3.7 TW, which is in the same orderof magnitude as theglobal electricityconsumption [2]. Thus, this made it an attractive source of alternative energy

∗_{Corresponding}_author.

E-mailaddress:zhiyung.tay@singaporetech.edu.sg(Z.Y.Tay).

asareplacementtothefossilfuelsandresultedinthevarious ideationofwaveenergyconvertors(WECs)thatconvertwave energy toelectricityby using power take-off(PTO) systems. Thetraditional typesof WECssuch asthe pointabsorber, attenuator and terminator WECs generate energy via rigid bodymotion as waveshiton thestructures. Recently,the at-tenuatorWECthat isformedbyinterconnectingseveral float-ingmoduleswithaseriesofPTOsystemequippedin/between themoduleshasgainedpopularityduetoitshighratedpower and capture width ratio. The most well-known being the Pelamis WEC [3] which has a rated power of 750 kW and acapture widthratio of 7%. Otherattenuator WECssuch as theOceanGrazerWEC(www.oceangrazer.com)areproposed byresearchers from the University of Gronigen to maximize theenergy generation from wavevia aninterconnected array of floatingmodulesconnected byhinges.

https://doi.org/10.1016/j.joes.2019.07.002

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Researchers havebeenlookinginto numerouswaysto en-hancetheperformanceoftheWEC.Oneofthemostcommon methodsis to arrange the WECs inarrays [4–7] in order to maximizethe power generation from the wave farm.To fur-therenhancethe performanceof the wavefarm,optimization technique has been performed to seek for the optimal array configurationfor the wavefarm [8,9]. In addition,instead of using PTO systems with constant damping value, PTO with variabledamping value has been proposed and it appears to improvetheperformanceoftheWECarray.Thiswas demon-stratedby Wei et al.[10] where the authors investigated the OceanGrazerWECwithten-hingedconnectedfloating mod-uleswhere each module isconnectedby aPTO systemwith variablePTO damping. Their results showed that the perfor-mance of the WEC could be improved significantly through an appropriated PTO array configuration. Also, optimization onthe individualPTOsystemcouldbe performedinorderto improvethe overallpoweroutput,as suggestedbydeBacker etal.[11].

Another novel methodology to enhance the power output oftheWECistouseflexiblematerialfortheWEC that gen-eratesenergyviaitsflexiblemotionunderwaveactiontermed asthehydroelasticresponse.HarenandMei[12]wereamong thefirst topropose ananalytical model for atrainof slender pontoons in a channel with rotational PTOs attached to the connectinghinges.Another example of aflexible-type atten-uatorWEC isthe Wave Carpet[13]proposed byresearchers fromthe University of Berkeleyas meansto preventerosion andprotect the harbors byextracting energy from the waves togenerateelectricity.Other flexibletypeWECs are such as the AnacondaWEC [14], SBM S3WEC [15] and Bombora WEC [16] which utilize the structural deformation in gen-erating energy. Zhang et al. [17] has also demonstrated the effectofstructural flexibility onthe powergenerationof two interconnectedfloaters.

Itistobenotedthatmostofthe aforementionedWECare longflexibleWECmodeledas beam[14,16–18]]whereasthe Wave Carpet WEC is a submerged plate-type WEC. So far, limited works on flexible plate-type WEC had been inves-tigated. For example, the Cyprus University of Technology has recently proposed the Water Level Carpet (WLC) WEC [19]whichconsists of four rectangularshaped floating mod-ules connectedflexibly intwo directions by connectors with PTO mechanisms where they found that the power produc-tion of the WLC obtains large meaningful values for wave frequenciesclosetotheresonanceof the generalizeddegrees of freedom. A novel type of WEC concept has been incor-porated in the very large floating structure (VLFS) for the useon oceanspace utilization. Zhanget al.[20]addressed a flexiblerunway supported by an array of circularbuoy with PTOandclaimed that an optimalbalance between maximiz-ingwaveenergy extractionandminimizingthe movementof the runwaycanbe achieved withproper stiffness and damp-ingcoefficientsofPTO.AnotherrecentworkbyTay[21]and Nguyenetal.[22]investigatedapontoon-typeVLFSwithan articulated plate that functions both as an antimotion device andaWEC. The author found that it is possible togenerate

an optimal amount of energy from the wave while keeping a highworkability of the articulated plate in minimizing the hydroelastic response.

In viewof the effectiveness inpowergeneration enhance-mentvia structural deformation for plate-typeWEC, this pa-peraimstofurtherstudyandunderstandtheeffectofallowing structural deformation in a pontoon-type WEC on the wave energy generation. The considered plate-type WEC is made up of multiple pontoon-type WECs floating on the surface of the water and comprises a grid of floating modules in-terconnected by line hinges where energy is generated via PTO systems. While a similar flexible raft-type WEC has been considered in [19], whichis made up of a two-by-two floating modules, our study shall consider the effectiveness of different moduleconfigurations in powergeneration. Four different configurations of the pontoon-type WECs are con-sidered anddescribed indetail inthefollowing section,with each made up of different numbers of interconnected mod-ules. The effect of structural rigidity of the WEC is simu-latedusingvariousYoung’smodulusandstructurallength.In addition, a PTO system with variable PTO damping values isconsideredinordertoquantifyitseffectiveness in enhanc-ing thepower generation as comparedtoits counterpartof a constant/uniform PTOdamping value. The genetic algorithm (GA) optimization scheme is used to seek for the optimal variable PTO damping that could maximize the power gen-eration of the WEC. The GA optimization technique [23], whichisasearch heuristicthat mimicstheprocessofnatural selection and involves techniques inspired by natural evolu-tion, such as selection, mutation and crossover, is used as it enhances the computational time in meeting the objective function. The GA will converge over successive generations towards the global optimum via the aforementioned process and has been proven to be a robust tool for optimizationin engineering problem [24]. To the knowledge of the authors, the flexiblepontoon-type WEC equipped with variable PTO systemsforconsiderationofpowerenhancementhasnotbeen investigated elsewhereandtheresults presentedshallprovide insight on the effectiveness of power enhancement via these two methods.

2. Problem definition

The paper considers a pontoon-type WEC which consists of a grid of N interconnected floating modules, where each module is connected to each other by using (N− 1) line hinge connectors (see Fig. 1), i.e. Fig. 1(a) for Type-A with N=12 (11 hinges), Fig. 1(b) for Type-B with N=6 (5 hinges), Fig. 1(c) for Type-C with N=4 (3 hinges) and Fig. 1(d) for Type-D with N =1 (0 hinges). Each pontoon-typemodulehasalength l,breadthB,depthhandisassumed to be made of an isotropic elastic material with a Young’s modulus Eandmassdensity ρp.Themodulefloatsinadraft

Tw andona constantwater depth of H.Whenconnected

to-gether, the pontoon type WEC has a total length dimension L=N× l, breadth B anddepth h. A totalof R numbers of PTO system is attached to the WEC to generate energy via

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Fig.1. Pontoon-typeWECwith(a)Type-A:N=12 modules(b)Type-B:N=6modules(c)Type-C:N=4 modules(d)Type-D:N=1module. Table1

Propertiesfordifferentcasestudiesconsidered.

Group Structuralproperties PTOsystem Waveproperties Waterdepth

H(m)

L(m) Type N E(GPa) ρp (_mkg 3) B(m) h(m) Tw(m) R BPTO(MN m ·s ) ω(rad s )

I 100 A 12 0.2, 2, 20, 200, 2000 and 20,000 512.5 30 2 1 99 0to662₃ (5/3 interval) 0.1to1.6 (0.025interval) 600 B 6 C 4 D 1 II 200 A 12 B 6 C 4 D 1 III 300 A 12 B 6 C 4 D 1

therotationalmotionsoftheWEC.ThePTOsystemisspaced atanequalintervalof ₁₂L inthehorizontaldirection(x− axis) and B₈ inthetransverse direction (y− axis) as showninFig. 1, therefore totaling a number of M =99 PTO systems as shown inFig.1(a)–(d).

The properties of the pontoon-typeWECs and wave con-dition are shown in Table 1. Three groups of pontoon-type WECs, each with different L are considered, i.e. Group I with L=100m, Group II with L=200m and Group III with L=300m. For each group of the WEC, four types of WEC, i.e. Type-A, -B, -C and -D, each with N differ-ent numbers of module, are further considered and depicted

in Fig. 1(a), (b), (c) and (d), respectively. It is to be noted that Type-AWEC hasthe highest numberof linehinge con-nectors, whereas Type-B and -C have a reduced number of line hinge connectors. The Type-D pontoon-type WEC is a single module WEC and behaves like a continuous mat-like verylargefloatingstructures(VLFS).Thesedifferenttypesof WECsare subjectedtodifferent elasticdeformation behavior due to their different module length l and Young’s modu-lus E, thus allowing the investigation of the effect of elastic deformation (i.e. structural rigidity) on the power generation of the WEC. In order to investigate the effect of structural rigidityonthe waveenergygeneration,the WECsinTable1

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Fig.2. Computationdomainforpontoon-typeWEC.

are modeled with six different E values, i.e. E =0.2GPa, 2 GPa, 20 GPa, 200 GPa, 2000GPa and20,000 GPa. Also, itistobenotedthatthetotalnumberofPTOsystemsarekept the same (R=99) for all the WECs considered in Table 1 in order to ensure a fair comparison of the performance of the WECs. Studies will be carried out to investigate the ef-fectofvariablePTOdampingsysteminenhancingthepower generation of the WEC as compared to its counterpart of a constantPTOdampingsystem.Ageneticalgorithm optimiza-tion technique will be applied to seek for the optimal PTO damping following the scheme presented in [8]. Two cases are considered in the GA optimization to seek for the op-timal variable BPTO, i.e. Case 1 where the GA optimization

schemeis applied toall the BPTO in the pontoon-type WEC

andCase2 where the GAoptimizationscheme isapplied to theBPTOattachedtothelinehinge connectoronly. Itisnoted

herethat forthe variablePTO system, theBPTOare assumed

tovary along the x−axis direction but are kept constant for each line connector (along the y-axis direction) in order to reduce the computation time. The scheme will be explained indetail inSection3.6.

The pontoon-typeWEC is subjected toaseriesof regular waveswithaconstantwaveamplitude2A.Thewave frequen-cies ω range from 0.1 rad/s to 1.6 rad/s with an interval of 0.025 rad/s where the regular waves approach the WEC at thehead sea direction. The WEC isassumed tooperate ina deepwatercondition where theeffect of seabedon the struc-tural motion is negligible. The particulars and properties of WEC models are given inTable 1. In order to facilitate the discussion,thepontoon-typeWECwillbereferredtobytheir Group and Type as summarized in Table 1. For example, a GroupIType-AWEC referstothe100-mlongpontoon-type WECinterconnected withN =12 floatingmodules.

3. Mathematicalformulation

Theresponseofthepontoon-typeWECiscomputedby us-ingthehybridboundaryelement–finite elementmethod(BE– FE)developedinMATLAB® wherethe WECismodeled as anisotropic platewhereasthe fluidis assumedtobe inviscid and incompressible and its flow assumed to be irrotational. The global x−,y−,z− axes have the positive direction ac-cordingtotherighthandruleandtheoriginislocatedonthe center of the WEC, with the x− y plane located at the free surface, i.e. z=0. Fig. 2 shows the computational domain ofthe WEC, wherethe WEC isassumed tofloaton thefree surfacewithaconstantwaterdepthH.IntheBE–FEmethod, the waterdomain is represented by, the wettedsurface of the WEC by the boundary SH, the seabed by SB, the free

surface by SF andthe surface at the distance far away from

theWECasS_±∞.Thegoverningequationsforthe waterand plateaswellastheformulationforgeneratedpowerfromthe WEC and GA optimization scheme will be presented in the subsequent sections.

3.1. Water domain

Basedonthepotentialflowtheory,thefluidmotionmaybe representedby avelocitypotential (x, y,z,t).Weconsider the waterto oscillateinasteady-stateharmonic motion with the circular frequency ω. The velocity potential (x, y,z, t) could be expressedinto the followingform:

(x,y,z,t)=Reφ(x,y,z)e−iωt, (1)

The velocity potential φ(x, y, z) can be expressed as the sum of the diffracted potential φD and radiated potential by

using the linear potential theory,i.e. φ(x,y,z)=φD+

P

l₌₁

ζlφl(x,y,z) (2)

wherethe secondtermintherighthandsideofEq. (2)isthe radiatedpotential expressedas aseriesof productof P num-bers of modal amplitudes ζl andthe unit-amplitude radiated

potential φl.

Thesinglefrequencyvelocitypotentialφ(x,y,z)must sat-isfy the Laplace equation [25],

∇2_{φ =}₀ _in _, ₍₃₎

and the boundary conditions on the surfaces as shown in Fig.1,whichare given as follows [25]:

∂φl ∂z = −iωw forl=1,2,...,P 0 forl =D onSH B (4) ∂φl ∂n =0 onSH S (5) ∂φl ∂z = ω2_φ g onSF (6) ∂φl ∂z =0 onSB (7)

where n is the unit normal vector to the surface S. The de-flection of the pontoon-type WEC w in (4) is described in

Section 3.2. It is noted that the hull wetted surface SH

pre-sentedinFig.2isdivided intothebottom hullwettedsurface SHB andthe side hull wettedsurface SHS,i.e. SH B∪SH S ∈SH.

ThewavevelocitypotentialmustalsosatisfytheSommerfeld radiation condition at the artificial fluid boundary at infinity S_∞ as (x, y)→∞ [25] lim |(x,y)→∞| |(x,y)| _∂ ∂|(x,y)|− ik (φl− φin)=0 onS∞ (8)

where kand φin are the standard wavenumber and incident

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Fig.3. Figureshowingwandψx ofMindlinplatetheory.

TheLaplaceEq.(3) togetherwiththeboundaryconditions (4)–(8) on the surface S are transformed into aboundary in-tegral equation (BIE) by using the Green’s Second Theorem via a free surface Green’s function. Since the Green’s func-tion satisfiesthe surfaceboundary conditionatthe freewater surface SF, the seabed SB and at the infinity S_∞, only the

wetted surface of the bodies SH need to be discretized into

panelsso thattheboundary elementmethodcouldbeused to solve for the diffractedandradiated potential. Fordetails on the Green’s function used insolving the BIE,refer to[26]. 3.2. Structure domain

The pontoon-type WEC on the other hand is modeled as asolid plateby usingthe Mindlin thickplatetheory [27,28]. Thesolidplateissimplifiedtobeperfectlyflatwithfreeedges and the plate material is commonly assumed to be isotropic and obeys Hooke’s Law. The WEC is restraint from mov-ing inthe horizontal x− y plane directions by station keep-ing system and only allowed to move vertically. Hence, the hydroelasticresponse of thepontoon-typeWECcould be de-scribedbythedeflectionw(x,y),therotationaboutthe y-axis ψx(x,y)andtherotationaboutthex-axisψy(x,y)asshownin

Fig.3.The governing equations for the Mindlinplate theory are givenas follows:

κ2_Gh ∂2_w ∂x2 + ∂2_w ∂y2 + ∂ψx ∂x + ∂ψy ∂y +ρphω2w=p(x,y) (9) D 1− ν 2 ∂2_ψ x ∂x2 + ∂2_ψ y ∂y2 +1+ν 2 ∂2_ψ x ∂x2 + ∂2_ψ y ∂x∂y − κ2_Gh ∂w ∂x +ψx +ρp h3 12ω 2_ψ x=0 (10) D 1− ν 2 ∂2_ψ y ∂y2 + ∂2_ψ x ∂x2 +1+ν 2 ∂2_ψ y ∂y2 + ∂2_ψ x ∂x∂y − κ2_Gh ∂w ∂y +ψy +ρp h3 12ω 2_ψ y=0 (11)

where G=E/[2(1+ν)] is the shear modulus, κ2 _the _shear

correction factor taken as 5/6, ρp the mass density of

the plate, h the thickness (i.e. depth) of the plate, D= Eh2/[12(1− ν2)] the flexural rigidity, E the Young’s mod-ulusand ν the Poissonratio. The pressure p(x, y) inEq. (9) comprises the hydrostatic and hydrodynamic pressure. The boundary conditions at the free edges of the floating plate are Mnn=D ∂ψn ∂n +ν ∂ψs ∂s =0 (12) Mns=D 1− ν 2 ∂ψn ∂s + ∂ψs ∂n =0 (13) Qn=κ2Gh _∂w ∂n +ψn =0 (14)

where Mnn, Mns and Qn are the bending moment, twisting

moment and shear force, respectively. s and n denote the tangential and normal directions to the section of the plate, respectively.

3.3. Continuityequations for hinge connectorwith PTO system

The continuity equations for the interconnected plate at the hinge connector with PTO damping BPTO located at

These continuity requirements can be implemented into plate elements along the line connection using the standard finiteelement method. Notethat (xc−)r and (xc+)r denotethe

location at the left and right hand side of the rth PTO sys-tem, respectively. Power take-off system is attached at the connectortoconvert the kineticenergy of the interconnected plate due to wave action to electricity. This is modeled as damperwith BPTO along the line hinge connector, i.e. y-axis

atx =(xc)r.

3.4. Equationofmotionfor water-platemodel

The coupled water-plate problem is solved by using the coupledBE–FEscheme,where theLaplaceequation together with the water boundary conditions are solved using the

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boundary elementmethod whereasthe plate equation andits boundary conditions using the finite element method. Due to space constraint, details of the solution scheme are not presented here, but interested readers can refer to the de-tailsin[26].ForNnumbersofinterconnectedmodulesinthe pontoon-typeWEC,the equation of motion of modulep due tomoduleq is written as ω2₍_M₊_M a)− iω (Ba)pp+BPTO +Kf +Ks+Kr f (¯w)p − N q=1 q=p ω2₍_M a)qp+iω(Ba)qp =(Fe)p (20)

where ¯w=(w,ψx,ψy), M is the mass, Kf the flexural

stiff-ness, Ks the shear stiffness and Krf the restoring force. The

added mass Ma, radiated damping Ba and exciting force Fe

can be found in [26] and will not be presented here due to its lengthy derivation. Eq. (20) can be further transformed into the matrix form to be solved using the finite element method. The typical BPTO matrix of an interconnected node

inthe hinge connectoris presented as follow

BPTO= w ψ¯x ψ+ x ψ− y ψ+ y w ψ_x− ψ_x+ ψ_y− ψ_y+ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +BPTO −BPTO 0 0 0 −BPTO +BPTO ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (21)

Itisnotedherethatforeachnodealongthelineconnector, therewillbefivedegreesof freedom,namelyw,ψx−,ψx+,ψy−

and ψy+.The positive(+) andnegative(−)signsdenotethe

right hand side and left hand side of the node in the line connector.

3.5.Generated power fromanti-motiondevice

The rotation of the hinge connector ψy calculated from

(21)canbeusedtocomputethetotalaveragegeneratedpower Paoftheanti-motiondeviceovertherangeofwavefrequency

ω considered in the regular wave by using the following expression Pa= 1 2ω 2 R i₌₁ (BPTO)i ψ2 y i (22)

The totalaveragegeneratedpowerPa isthenexpressedas

capturewidth(CW)[29]bynormalizingwiththewavepower resource Presource (24) in order to quantify the efficiency of

theanti-motion deviceingenerating wave energy.

CW = Pa

Presource

(23) The CW is the width of a wave crest that contains the same Pa as extracted by a WEC and the CW has to be as

largeas possible forthe anti-motiondevicetobeeffective in generatingwaveenergy.ThewavepowerresourcePresource in

(23)is given as [30] Presource=ρg

2_T_H2

64π (24)

where ρ isthedensityof seawater,g thegravitational accel-eration,T thewaveperiodandHthe significantwaveheight. 3.6. Seeking optimalBPTO using GA optimizationscheme

The optimal constant and variable PTO damping for the pontoon-typeWEC are sought by using an in-house GA op-timizationcodedevelopedinMATLAB®.Theobjective func-tion is to achieve maximum absorbed power Pa (22) or CW

(23) which indicates the maximum power absorption from the waves. The variables used to satisfy the objective func-tion are the PTO damping BPTO rangingfrom 0 MN s/mto

662₃ MN s/mwith40intervals,henceconsideringa possibil-ity of 41 PTO damping values for each PTO system. Note that the maximum range of the variables (BPTO) is set to

662₃ MN s/mintheoptimizationprocessas theeffectof the BPTO on the hydroelastic response becomes negligiblewhen

thevalueisverylarge. Itisnoted thatthemaximumvalueof 662₃ MN s/misobtained bynormalizing themaximumBPTO

of 20 GMNs bythe structural length L of 300 m.

Without the GA optimization scheme, the total possible combinationoftheBPTO isdenoted byNT,whereNT isgiven

as NT = (BPTO)max− (BPTO)min (BPTO)int erval +1 NL (25) where NL is the numberof PTO considered in the

pontoon-type WEC, i.e. NL=R=99. Eq. (25) will produce a total

of 4199 _possible _combinations _of _PTO _damping _for _the

con-sideredWEC, whichis obviouslyimpracticalduetoitslarge computationalcostinvolved.Inordertodeviseamore practi-caloptimizationprocess,theBPTO alongthey−axisdirection

is varied but keptconstant along the x− axisdirection, thus reducing NL in (25) to 11 for the pontoon-type WEC

con-sideredhere. Theassumptionmade hereispracticalfrom the engineering point of view, as having a constant BPTO along

the y−axis direction, i.e. along the line hinge, will result in symmetricalstructuraldeflectionaboutthex−axisunderhead sea condition, thus inducing lower stress resultants as com-pared toits asymmetricalcounterpart.Besides, having a uni-form BPTO value along the hinge is more practical as it will

easetheinstallation of thePTOsystem. WithNL=11,NT is

reduced to 4111_≈5.5_{× 10}17 _but _is _still _{computationally}

ex-pensive withoutthe GAoptimizationscheme.

In order toreduce the computational timeto seekfor the optimalBPTO, the GAoptimizationscheme is utilizedwhere

it is divided into twosteps as depictedin Fig.4, i.e. (i)Step1:Generating an initial populationof NR

(ii) Step2:ApplyingGAoptimizationschemetotheinitial populationto seekfor the optimalBPTO.

InStep1,theinitialpopulationof NR =500,000 different

combinationofBPTOisgeneratedbyusingabiasdistribution.

In Step2, NI individuals having the highest fitness valuesin

terms of the Pa or CW are then selected from the

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Fig.4. FlowchartofGAoptimizationscheme.

Next, NI sets of parents (father and mother) are randomly

selected from the NI individuals for the crossover and

mu-tation operations inorder to createNI offspring for the next

generation. In addition, the individual with the best fitness value in the current generation are kept for the next gener-ation, and this individual is known as the elite child. This processof crossover,mutation andelitismwill continue until the objective function is met, i.e. the maximum Pa or CW

hasconverged.The convergencecriteriafor themaximumPa

or CW is 0.01%. It is to be noted that although increasing NI increasesthe computational time, it ensures afaster

com-putational time in achieving convergence for the Pa or CW,

vice-versa. For the case study, NI = 50 is selected as this

number issufficient toensure afaster convergence based on the authors’computationalresource.It isalsonotedherethat the parent areknown as the DNAwhereas theR numbers of BPTO are knownas the chromosomesinthe GAoptimization

scheme.

The chromosomes will be converted into binary numbers of 20 bits for the crossover and mutation processes. The crossover Cc andmutation Cm probabilities are taken as 0.2

and 1.0, respectively. Thisimplies that the crossoverprocess between the two DNAs applies only to 20% of the chromo-somes whereas the mutation process only applies to all the chromosome of the best fit. The choice of Cm=1.0 will be

explained inSection4.2.

4. Validationof results

4.1. Hydroelasticresponsefor interconnectedstructure The solver for the equation of motion for the water-plate model (20) is developed in MATLAB®. The code for the hydroelasticanalysisisvalidatedwiththeresultsoftheVLFS presented by Yago and Endo [32] and Fu et al. [33] in the subsequent sections.

4.1.1.Convergence study

A convergence study is performed to investigate the ef-fectsoftheboundary discretizationontheconvergenceofthe

Table2

Detailsoffloatingmodelusedforconvergencestudy.

Parameter Symbol Unit Value

Totallengthofplatesystem L m 300

Totalwidthofplatesystem B m 60

Totalheightofplatesystem h m 2

Densityofplatesystem ρp kg/m3 256.25

Young’smodulus E GN/m2 11.9

Poisson’sratio ν 0.13

Waterdepth H m 58.5

Wavelength-to-platelengthratio λ/L 0.2to0.8

hydroelasticresponseofthe VLFSpresentedin[33].The de-tailsofthenumericalmodelusedarepresentedinTable2and thecomplianceχ of thehydroelasticresponseisdefinedas

χ = 1 ρgA2_BL L_/2 −L/2 B_/2 −B/2 |p|· |w| dy dx (26)

The number of elements per wavelength is used as the basisforthe discretizationofthe platesystem.In the present convergence study, the number of elements per wavelength  is taken as 10, 15, 20, and 25; and the number of plate natural modesNm istaken as 10,15,20,and25.The results

are summarizedin Table3 whichshows that the compliance of thetwo-floating platesystemfor severalcombinationof andNm.

As can be seen in Table 3, the compliance χ converges

when  =25 and Nm=20. It is noted that the convergence

criteriais 1.5%for  and1.0%for Nm.Therefore, =25and

Nm=20 are considered as the optimal combination of the

two parameters for all wavelengths considered, and will be used inthe subsequent analyses.

4.1.2.Comparisonwith existing results

Thevalidity andaccuracy of the presentmethod for solv-ing the floating plate problem with mechanical line joints is established by comparing the hydroelastic responses com-putedbythepresentmethodwiththeexperimentalresults ob-tained by Yago and Endo [32] and the numerical results by Fu et al.[33]. The input data for the floating plate problem

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Fig.5. HydroelasticresponsealongthecenterlineofVLFSinTable2under(a)λ/L=0.2 (b)λ/L=0.4 (c)λ/L=0.6 (d)λ/L=0.8.Headseacondition.

Table3

Complianceforseveralcombinationsof ANDNmforthenumericalmodel

presentedinTable1. λ/L Nm 10 15 20 25 0.2 10 80.3251 79.2741 79.2671 79.3101 15 72.5620 71.4070 71.3960 71.4618 20 70.8723 69.6753 69.6637 69.7332 25 69.9075 68.6846 68.6725 68.7465 0.3 10 110.1649 108.9089 108.9065 108.9100 15 107.7885 106.3295 106.3270 106.3681 20 106.5589 105.0325 105.0293 105.1020 25 106.0471 104.4847 104.4810 104.5686 0.4 10 158.7024 158.8295 158.7786 158.7543 15 156.8455 157.1072 157.0976 157.0865 20 156.3173 156.6538 156.6597 156.6603 25 156.3556 156.7330 156.7524 156.7664

used by the aforementioned researchers are given in Table 2where theVLFSconsidered isatwo 150-mlong intercon-nectedfloatingmodulesconnectedbyusingamechanicalline hingeto forma VLFSwith300 min totallength.

Thecomparisonofthehydroelasticresponsealongthe cen-terline of the VLFS between the present numerical results

with those found in [32] and [33] are presented in Fig. 5. Thegoodagreement betweenexperimentalandnumerical re-sultsforthecontinuousstructurevalidatesthecorrectnessand accuracy of the presentmethod for evaluating the hydroelas-ticresponseoftheVLFSwithlineconnectors.Thenumerical resultsof thehinged-connected VLFSalsoshowgood agree-mentbetweenthepresentmethodandthosepublishedin[33].

4.2. GAoptimization scheme

In order to validate the reliability of the present GA op-timization scheme, the results obtained from the GA opti-mizationschemeiscomparedwiththeircounterpartsobtained fromtheparametricanalysis.Asthepontoon-typeWECs pre-sentedinTable1 aretoolargetorunontheparametric anal-ysis,amuch simplerpontoon-typeWECis consideredinthe validation exercise where the WEC is connected by only a one-linehinge connectorwithPTOattachedtoit.The length LandbreadthBofthepontoonWECaretakenas300mand 60m,respectively(i.e.GroupIIIinTable1).Twowave peri-odsare consideredhere, i.e.T =6.2026sandT =8.7668s, which corresponds to λ/L=0.2and0.4, respectively. In the parametric analysis, the BPTO is assumed to range from 0 to

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Table4

ComputationaltimeforparametricanalysisvsGAoptimisationscheme.

λ/L Parametricanalysis GA TimereductioninGA(%)

No.ofsimulation Computationaltime(h) No.ofsimulation Computationaltime(h)

0.2 500,000 28 5000(Step1a₎₊₆₀₀

(Step2a₎

3.5 87.5

0.4

Notes:

1.OptimalBPTOandPa obtainedfrombothconventionalandGAoptimizationschemearethesame.

2.PleaserefertoFig.6fortherespectiveBPTOvalues.

a _Refer_to_the_steps_for_GA_in_Section_3.6_.

Fig. 6. Evolution ofBPTO to achieve maximum absorbed powerPa usingGA optimization schemefor (a) λ/L=0.2 and (b)λ/L=0.4.Cm=1.0 and

Cs=0.2.

16662₃ MN s/mwithan intervalof ₃₀₀1 MN s/m,hence re-quires an executionof NT =500,000 operations. It is noted

that the normalized maximumBPTO=16662₃ MNs/m when

multipliedwiththelengthL=300m yieldsamaximumBPTO

of 500GNs at each line connector. The BPTO that produces

the highest absorbedpower Pa or capture width ratio CW is

taken as the optimalPTO damping.

Onthe otherhand,theGAoptimizationschemeisused to obtain the optimal dampingfollowing the two stepsgiven in

theSection3.6.Instep1,theinitialpopulationNR istakenas

1% of the NT, i.e.NR=5,000. These5000 initialpopulated

samples are then fed into the GA optimization scheme to obtain the optimalBPTO.

The computational time for both the parametric analysis and GAoptimization schemeis compared in Table4.It can be clearly seen that the number of simulations required for the GAoptimizationschemeisfar lesser thanitscounterpart for the parametric analysis. As a result, the computational time for the GA optimization scheme is 87.5% faster than the parametric analysis, when running on an Intel® CoreTM i7-5600U CPU@2.60 GHz machine. This presents a signifi-cant enhancement in the computation time by using the GA optimizationscheme andwill be usefulfor largerNT as

pre-sented in the following sections. It is to be noted that the optimal BPTO obtained from the parametric analysis and the

GA optimizationscheme are the same(refer to Fig.6),thus confirming the reliability of the present GA code in seeking

the optimal BPTO. The evolution of the BPTO to achieve the

maximumabsorbedpowerPa for bothλ/L=0.2 and0.4are

presentedinFig.6(a)and(b), respectively.Thevariouscolor tonesdenotethedifferentabsorbedpowerPaoftheWEC

cor-respondingtotherespectiveBPTOinthe y-axis.ThePa inthe

legends changesinascending orderdenotedby the lighterto darker tone. It canbe seen that the present GAoptimization scheme successfully achieves the optimal BPTO at the 86th

and95thiteration.

In the GA optimization scheme presented in Fig. 4, the crossover and mutation probability have to be set in order to accelerate the convergence of the GA process in obtain-ing the optimal BPTO. Therefore, a convergence test is

con-ductedinFig. 7 for the samepontoon-typeWEC considered inthe validation exercise for two different Cm,i.e. Cm=0.5

andCm=1.0. As can be seen clearly, the GA optimization

schemerunningatCm=1.0 acceleratestheconvergence

pro-cess in seeking the optimal BPTO. It is noted that Cm=1.0

implies that all the chromosome (i.e. BPTO) is selected to be

mutated ateach iterationin the GAscheme.

5. Results and discussion

5.1. Effectofhydroelastic response

Figs. 8–10 show the CW of Groups I (L=100 m),

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Fig.7. ConvergencetestforGroupIII(L=300)pontoon-typeWECunderGAoptimizationschemefortwodifferentCm.Cs=0.2.

Fig.8. CapturewidthCWforGroupIPontoon-typeWEC(L=100m)under differentYoung’sModulusE.

pontoon-typeWEC, respectively. Sixdifferent Young’s mod-uli,i.e. E =0.2,2,20,200,2000,20000GPa withaconstant BPTO=600 kNs/mareconsidered. Thepontoon-typeWEC

Fig. 9. Capture width CW for Group II Pontoon-type WEC (L=200m) underdifferentYoung’sModulusE.

issubjected to wavefrequenciesω rangingfrom 0.1rad/sto 1.6 rad/s with wave approaching from the headsea. Fig. 8 showsthat the 100-mpontoon-typeWEC isvery effectivein

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Fig.10. CapturewidthCWforGroupIIIPontoon-type WEC(L=300m) underdifferentYoung’sModulusE.

generating wave energy when it is connected withthe most number of hinges (Type-A). This results in shorter modules

rotating in rigid body motion and thus generating more en-ergy whensubjected to waveaction. Also,the CW or the Pa

do not change with respect to the varying structural flexural stiffness, which depends on the Young’s modulus E. How-ever, when the pontoon-type WEC gets longer as presented

in Figs. 9 and 10, i.e. L=200 m and 300m, respectively,

it can be seen that greater energy can be generated depend-ing onthe Young’smodulus E of the pontoon-type WEC. It is observedthat more energy can be generated by the WEC when each module in the pontoon-type WEC deforms flexi-bly,with the most energy generated when the module isthe most flexible (i.e. E =0.2 GPa). In addition, it is interest-ing to find out that the amount of energy being generated increases with the reduction in the number of hinges in the WEC.Thisis becauseareduced numberof hinges resultsin longer connectedmodules, and thusthe structures deform in aflexible manner under wave action. As aconsequence, the 200-m and300-m longpontoon-type WEC have the highest power generation when the WEC is connected by only one hinge withE =0.2 GPa.

ThecomparisonsofthehydroelasticresponseoftheGroup I pontoon-type WEC for different numbers of hinges, i.e.

Fig. 11. Hydroelasticresponsefor GroupI(L=100 m)pontoon-typeWECwith (a) Type-A:N=12 modules(b) Type-B:N=6 modules(c)Type-C:

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Fig.12. Hydroelastic responseforGroup II(L=200m) pontoon-typeWECwith (a) Type-A:N=12 modules(b) Type-B:N=6 modules(c) Type-C:

N=4 modules(d)Type-D:N=1module.

Type-A: 11 hinges, Type-B:5 hinges, Type-C:3 hinges and Type-D:nohingesare presentedinFig.11.Thedeflection is measured along the centerline of the WEC and the structure is subjected to a headsea condition. In each subfigure, the six differentlines represent the hydroelasticresponses of the WECwithdifferent Ewhen subjected totheir corresponding wave periods T that produce the maximum CW. It can be seenthat the influenceof the Young’s modulus is negligibly small for the WEC connected by large numbers of connec-tors,i.e.Type-AandType-B;butitseffectincreaseswhenthe pontoon-typeWECisconnectedbysmallernumbers of mod-ule, i.e. Type-C andType-D. It is also observed clearly that theCWforacontinuouspontoon-typeWEC(Type-D)isable togeneratemoreenergythanitscounterpartwithconnectors. This finding denotes that a significant cost saving could be achieved due tothe shorter installation timeand the smaller number of connectors needed. In addition, it also saves on materialcost as the WECcould be manufacturedwithlesser material to allow for flexible deformation provided that the stress resultants onthe structure are within the stipulated al-lowablestresslimittoensurethesafetyandrobustnessofthe structure.

Similarly, the hydroelastic responses for the 200-meter and 300-meter pontoon-type WEC are presented in Figs. 12 and13, respectively. As the length of the WEC becomes longer,the effectof theYoung’smodulus Eonthemotion of the WEC becomes significant. According to Suzuki’s et al. [34] definition of a VLFS, the hydroelastic response is only dominant when the followingtwo ratios are largerthan 1.0:

iStructural length/wavelength(l/λ)

ii Structural length/characteristiclength (l/λc)

The characteristiclength λc is givenas

λc=4

4π2D

kc

(27) where D isthe flexural rigidity givenas D=Eh3/12 and kc

is the spring constant of the hydrostatic restoring force. By using theratios givenin(i) and(ii) above,thiscouldexplain the reason that someof the pontoon-typeWECs behave like a rigid body under wave action whereas others ina flexible motion.Forexample, thiscanbe shownclearlyinFig.13(d) wheretheWECwithE =20,000GPa movesinarigidbody

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Fig.13. Hydroelastic responseforGroupII(L=300m)pontoon-typeWECwith (a)Type-A: N=12 modules(b) Type-B:N=6modules(c)Type-C:

N=4 modules(d)Type-D:N=1module.

motion. Therefore, it is always important to check on these two ratios when designing the pontoon-typeWEC.

5.2. Effectof non-uniformoptimized PTOdamping

In additiontoallowing forcertaindegreeof structural de-formation in the pontoon-type WEC, the energy generation fromthepontoon-typeWECcouldbefurtherenhancedby us-inganon-uniformlydistributedoptimalPTOdamping,termed also as variable PTO system. In order to seek for the opti-mal PTO damping, the GA optimization scheme as outline

inSection 3.6isused. The Group III(L=300 m)

pontoon-typeWECsubjected totwo differentwavelength-to-structural length ratios, i.e. λ/L=0.2 and0.4 isused todemonstrate the power enhancementof the pontoon-type WEC.

By using theGAoptimizationschemeas giveninSection 3.6, with an initial population of NR =500,000, Cc=0.2

andCm=1.0, the evolutionof BPTO for Group III

pontoon-type WEC toachieve the optimalBPTO is plottedinFig. 14.

EachsubfiguredenotedbyFig.14(a)–(f)representstheWEC connectedbydifferentnumbersof connectionjointsand

sub-jected to regular waves of two different λ/L. The optimal damping value ateach PTO systemthat results in the maxi-mumCWispresentedineachsubfigure.Itcanbeseenthatby using the GA optimizationscheme,the optimalnon-uniform distributedPTOdampingateachlineconnectorthatproduces themaximumpower couldbe obtained.Fig. 14showsthat a combinationof theminimumBPTO(i.e.0 MNs/m)and

max-imum BPTO (i.e. 662₃ MNs/m) for the pontoon-type WEC

could be used to achieve the maximum CW. However, the non-uniform distributed optimal BPTO could be in the range

between these minima and maxima values such as for the Type-Apontoon-typeWEC,where theWEC isconnectedby moreconnectionjoints.

Byfocusingonthepontoon-typeWECofGroupIII Type-A,the evolutionof the BPTO ispresented in Fig.15 for four

differentλ/L,i.e.λ/L=0.2,0.4,0.6 and 0.8. It canbe seen fromFig.15(a)thatthemaximumCWcanbeachievedby us-ing adifferent combination of BPTO where the optimalBPTO

for each PTO system does not necessary be the minimum BPTO (0 MN s/m) or maximumBPTO (662₃ MNs/m)

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Fig.14. EvolutionofPTOdampingBPTOforGroupIII(L=300.m)pontoon-typeWECunderGAoptimizationscheme.

valuecouldbeacombinationof theminimumandmaximum BPTO to achieve the maximum power generation when the

WECis subjected tolarge wavelength.

In order to visualize the hydroelastic response of the pontoon-type WEC under uniform and non-uniform optimal BPTO,the deflectionalong thecenterline of theWEC is

plot-ted in Fig. 16 for the Group III Types-A–C pontoon-type

WEC. A comparison of the deflection is made between the WECconnectedbyvariablePTOsystemwiththatbyuniform PTOsystem. ItcanbeseenclearlyfromFig.16thatthe CW of the non-uniform optimal BPTO counterpart is greater than

that predictedby the uniform counterpart. However, the dif-ference is more obvious for Type-A pontoon-type WEC due to the greater magnitude of rotational motion in the WEC.

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Fig.15. EvolutionofPTOdampingBPTOforGroupIII(L=300m)Type-A(N=12 modules)pontoon-typeWECunderGAoptimizationscheme.

Similarly, thesameresults areplottedfor λ/L=0.4,0.6 and

0.8 in Figs. 17, 18 and 19, respectively, and similar

obser-vation can be made on the CW of the WEC. An interest-ing point to make when comparing the results presented in

Figs. 16–19 is that the difference of the CW for the WEC

connected by variable PTO system with its uniform coun-terpart becomes greater when the wavelength increases. This suggeststhat theoptimalvariablePTO systemis more effec-tive when the WEC is subjected to regular waves of longer wavelengths due to the larger magnitude of hydroelastic re-sponse.In addition,theCWfor theWECisfoundtoincrease withthe increase inwavelength.

In Fig. 20, a comparison of the hydroelastic response of the pontoon-type WEC with Types-A, -B and -C pontoon-type WEC with the continuous pontoon-type WEC (Type-D) is presented. TheseWECs are subjected tofour different λ/L ranging from 0.2 to 0.8, with an interval of 0.2. It is interesting to notethat the continuouspontoon-type WEC is the mosteffective amongthe fouringenerating waveenergy when it is equipped with optimal variablePTO system. The optimal BPTO obtained from the GA optimization scheme in

thiscase falls inthe higher range of the BPTO considered in

Table1 to achievethe maximumCW.

6. Conclusion

The pontoon-type WEC was considered and the coupled finite element-boundary element method was used to solve forthe hydroelasticresponseof theWEC.The GA optimiza-tion scheme was utilized inseeking the optimal PTO damp-ing of the variable PTO system, where the objective func-tion was to achieve a maximum capture width CW or the absorbed power Pa of the WEC. Three WEC lengths were

considered, i.e. L=100 m, 200 m and 300 m, which were categorizedas GroupI,IIandIII,respectively. Fourdifferent numbers of interconnected modules, i.e. N=12,6,3 and 1 were then considered for each group of WEC, which were termed asTypes-A, -B,-Cand-D,respectively. Thetwo ob-jectivesof thepaperweresuccessfullyachieved,thatisto en-hancethepowergenerationoftheWECviacertainallowance of structural deformation and installation of variable PTO system.

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Fig.16. HydroelasticresponseforGroupIIIpontoon-typeWECunderλ/L=0.2 withuniformandnon-uniformoptimizedPTOdamping.

Fig.17. HydroelasticresponseforGroupIIIpontoon-typeWECunderλ/L=0.4 withuniformandnon-uniformoptimizedPTOdamping.

Fig.18. HydroelasticresponseforGroupIIIpontoon-typeWECunderλ/L=0.6withuniformandnon-uniformoptimizedPTOdamping.

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Fig. 20. Comparison of hydroelastic response forGroup IIIcontinuous pontoon-typeWECwith 4-, 6- and 12-connected modulesunder their respective optimizedBPTOfor(a)λ/L=0.2 and(b)λ/L=0.4(c)λ/L=0.6(d)λ/L =0.8.

In the investigation on the effect of structural deforma-tion, i.e.structural rigidity,on thepower enhancementof the pontoon-type WEC, it was found that greater energy could be generated when the interconnected module in the WEC is allowedto deform underwave action. The effectof struc-tural deformation is even larger when the structural length increases and when the number of interconnected hinge re-duces.Itisinterestingtonotethatacontinuouspontoon-type WEC without any hinge connector is able to generate the greatest amountof energy amongthecasesconsideredinthe study. The effect of Young’s modulus was found to be sig-nificant when the Group II and III pontoon-type WECs are connected with smaller amount of hinge connectors. This is becauselessernumberofhingeconnectorsimplylonger inter-connected floating modulesand thus higher structural defor-mations underwaveaction. Theresultspresentedshowthat a significant saving incost, material andconnector installation couldbeachievedforthepontoon-typeWECandatthesame time producing high amount of electricity, provided that the structural integrity of the WEC is preserved.

In theinvestigation onthe effectof utilizingvariablePTO system on the power enhancement of pontoon-type WEC, it was found that in some cases, the maximum power gener-ation could be achieved by merely utilizing a combination

of minimumandmaximumBPTO that wereconsideredinthe

casestudy. However,it wasalso foundthat the optimal non-uniformBPTOfortheType-Apontoon-typeWECcomprisesa

combinationof BPTO that falls inbetween the minimum and

maximumBPTO considered.TheseoptimalBPTOwas

success-fullysoughtbyusingtheGAoptimizationscheme.Theeffect of variablePTO systemwas foundto besignificant withthe increaseinthe numberofhinge connectorsandwavelengths. Thisis due to the greater magnitude of rotational motion in the WEC.

In summary, it was proven from the study that the power generation of the pontoon-type WEC can be enhanced by allowing the structure to deform flexibly under wave action aswellasbyinstallingaseriesofvariablePTOsysteminthe pontoon-typeWEC.Thisoutcome willbeusefultoacademia andindustryworkingonachievingthepowerenhancementof WEC.

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