Development of a new wave energy harnessing system

Joao Andre de Faria Westwoodjoao.westwood@ist.utl.pt

Instituto Superior Tecnico, Lisboa, Portugal

October 2014

Abstract

This dissertation’s subject is a new system of harnessing wave energy that intends to be more eco-nomically viable and competitive than the current nowadays systems by having all the key componentsto work mechanically in traction. Its main aim is to deepen the knowledge about the dynamic behaviorof the system when subjected to different types of waves. In order to achieve this, it was designeda computer program capable of simulating the temporal heaving motion of the floater as well as thebehavior of all the components of this plant, using the Runge-Kutta method. It was also implementeda controller for electric generator of the Proportional-Integral type so as to maintain its constant speedwithin an acceptable range. Simulations were carried out for regular waves, which varied independentlythe incident wave’s Height and Period. It was also analysed a group of 14 spectra of irregular wavescorresponding to a wave climate of a point off the coast of Leixes. An analysis of the impact that thevariation in sizing of the three main components (ballast mass, cables diameter and generator inertia)would have on the behavior of the plant was made in a further qualitative way. It was possible tovalidate the operation of the program and its conformity to the theoretically predicted. It was alsopossible to determine the average powers and efficiencies for regular waves and the wave climate instudy, having been defined in detail the influence of each component in the overall system operation.Keywords: wave energy, hydrodynamic modeling, non-linear states model, control.

1. Introduction

This work has as its subject a new system of har-nessing wave energy described in the patent [1] cre-ated by Professor Jos Maria Andr, Supervisor, andby Joo Campos Henriques, Co-supervisor of thisthesis.

The aim of this work is to create a program capa-ble of performing a computer simulation of the be-haviour of the mentioned wave energy power plant,for different sea states. It is intended to vary sev-eral parameters either the sea states or the plantcomponents sizing, in order to find out what is thepower that is possible to extract from this plant,what will its efficiency be and also in what range ofsea states will this plant be able to operate.

Unlike the numerical method used in another dis-sertation [2], is intended to use the Runge-Kuttamethod to simulate the systems of equations thatdescribe the behavior of all parts of this plant.

Obtaining these data is of high importance to de-termine the feasibility of this system, both on thetechnical side, regarding the energy production, andon the economical side, regarding economic sustain-ability.

1.1. State of the art and economic viability

Currently it has been awakened high interest in off-shore applications, in waters with a few tens of me-ters, in floating systems for harnessing wave energy.Groupings of offshore devices deployed at depths of40 to 80 m are currently considered as the most suit-able configuration for extensive resource exploita-tion of wave energy.

The option for floating systems has its justifica-tion on the fact that the available power is muchhigher in areas of deeper waters. Allied with thatfact, we have the relative simplicity of conceptionand construction of a floater.

Here is the point where this system [1] enters,which is the field of study of this work. It is aimedwith this system to achieve improvements at twolevels relative to the systems previously proposed.Firstly, the system is designed so that all compo-nents work exclusively in traction and none in com-pression, thus avoiding excessive scaling that is re-quired for a component to support such compres-sion efforts. Secondly, the commonly used hydrauliccylinders are no longer required and a ballast, at-tached to the floater by a cable system, which seeksto accumulate energy over several wave cycles bybeing lifted by cables shall be used, thereby storingenergy in the form of gravitational potential energy

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and with this increased electrical output.This all combines that, with this new system, it

has been considerably reduced the complexity ofthe plant’s structure, as well as to the difficultiesof construction. This makes the system moreeconomically viable, this also being one of themain reasons that preclude the development ofother structures previously thought. The goal is todecrease to the minimum the equipment construc-tion price, maximizing the amount of energy it canproduce, thus hoping to achieve competitivenessagainst other types of renewable energy.

2. Description of the wave energy

2.1. Installation’s main elements

The main elements of the installation are repre-sented in the figure 1

Figure 1: Scheme of the installation’s main elements [1].

The station consists of a floater (1), which issubjected to hydrodynamic forces of waves and theforce of a set of cables (3). The ballast (2) is con-nected to the floater through a set of cables andhas a vertical movement that varies its directiondepending on the stage it is on. This movementis guided by rods (5) which are anchored into theseabed (6) at its lower end, and is connected to aballoon (4) at the upper end, and thus will alwaysbe subjected to traction and not to compression.

The ballast is connected to its brakes (21), whichallow it to be immobilized in a position without los-ing elevation. The cable assembly is then wrappedaround a drum (101) that is connected to the elec-trical generator (104) through a gear ratio that al-lows it to have a lower rotational speed than thegenerator.

Connected to the drum shaft is a brake disk (here-inafter referred to as drum brake (102)) which is ac-

tuated to reduce the rotational speed of the drumor immobilize it, and is also combined with a cablewinding mechanism (105). The latter is used whenthe cable is untensioned and needs to be woundaround the drum.

2.2. General principle of operation

The process of energy production in this plant isdivided into what are called energy cycles. Thesecycles consist of two different phases: ”1 - EnergyAccumulation” and ”2 - Energy Production”.

In the first phase the ballast is risen from a xmin

to a xmax to take advantage of the hydrodynamicforce exerted on the floater and in times when thecable is without tension, it is wound so as to ap-proximate the ballast to the floater.

The second phase begins at the moment in whichthe ballast has reached its maximum height andthus maximum power energy. At that moment it islet down, and thus gain speed to accelerate the thedrum and the generator, thereby producing electri-cal energy.

Hereinafter will be referred to as a ”energy cycle”to the set of accumulation phase and a productionphase in a row. While in the phase of accumulationexist a sequence of stages that are repeated cycli-cally until the xmax is reached, in the Productionthere is only one sequence of stages that is executedonly once.

2.2.1. Energy accumulation phase: rising the bal-last

The primary objective of this phase is to raise theballast from a minimum to a maximum elevation inthe shortest possible time so as to accumulate thewaves energy in the form of gravitational potentialenergy of the ballast.

Figure 2: Scheme of the energy accumulation phase [1].

We will consider its starting point with the floaterin the cave of the wave and the ballast in the its min-imum level (xmin), where the ballast is locked andthus remains fixed to rods. At the instant at whichthe floater starts to rise due to hydrodynamic forceof the wave, the brake drum is actuated, therebypreventing the cable from unrolling and it starts togain tension as the floater moves away from the bal-

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last. When the force exerted on the cables exceedsin magnitude, the ballast apparent weight (weightout of the water subtracted by the impulsion onthe surface of the ballast) we have that the resul-tant force has a positive direction and the brakes ofthe ballast will be disabled, allowing it to rise.

When the float reaches the crest of the wave andbegins to descend, the force that the cables exert onthe ballast also decreases causing it to stop climb-ing, and at that moment its brakes are actuated, be-fore it starts to descend and lose elevation. Then, asthe floater moves down the tension on the cable willalso decrease until the point at which it vanishes.Here is then triggered the winding mechanism as-sociated to the drum to wind the excess cable thatcame into existence due to the ballast being movedup, which takes advantage of the absence of tensionin the cable or a value below the one this can exert.Once that, by descending, the floater is found againin the cave of the wave, it will be repeat again allthe steps above until the ballast reaches the desiredmaximum level and reach the maximum value of thegravitational potential energy of the energy cycle.

2.2.2. Energy production phase: Ballast descending

The main objective of this phase is to convert asmuch of the gravitational potential energy accumu-lated in the previous phase, into kinetic energy ofrotation of the electrical generator and its rotationalinertia. The rotary inertia of the generator is whatwill keep it running for a longer period of time andfrom where a constant electrical power will be re-moved, thereby allowing continuous production ofelectricity, and not just only in the period in whichthe ballast is descending.

Figure 3: Scheme of the energy production phase [1].

Thus begins when the ballast reaches its high-est point (xmax) and both brakes from the ballastand the drum are disabled, allowing it to start theballast to move down and gain speed. With thismovement is generated tension on the cable whichwill cause the drum to rotate and to increase its ro-tational speed, and as soon as it reaches the value

of the generator’s rotational speed affected by thetransmission relation, the clutch will be activated,connecting the gearbox to the generator. This con-nection will cause the downward movement of theballast to be directly connected to the rotation ofthe drum and to the ballast, causing the accelera-tion of the first to generate an angular accelerationin the drum and also the electrical generator.

In the final meters of the descent of the ballastit needs to be slowed down so that it can stop atthe elevation of xmin. For this, a few meters beforethe required depth to stop it is triggered the drumbrake until the ballast stops completely. In thatinstant is then triggered the ballast brake to holdit the rods and can then be initiated a new energycycle with the beginning of a energy accumulationphase.

3. Numeric Modelling of the central

The equations that rule the physical phenomena ob-served in this plant [4] are of three types: secondorder equations, first order equations, and algebraicequations. Although there is not one system ofequations that can correspond to the behaviour ofthe central during all the energy cycle, so that thisis only able to be modelled by stages. Therefore,it will be here shown which are the equations thatmodel the behaviour of each main component of theplant, and the stages in which these are applicable.

3.1. Presentation of the equations thatmodel the power plant

In order to better describe and understand themovements that are verified at the central, here iswritten the variables used throughout this study.

• x1 – Floater vertical position,

• x2 – Ballast vertical position,

• θ – Drum angular position,

• ω – Electric generator angular velocity,

• f – Total force in the cables.

For a question of nomenclature and to simplify, allthe variables with the lower index ”1” are relatedto the floater, and all of those with the lower index”2” are related to the ballast.

3.1.1. FloaterThe floater oscillates in heave, subjected to a hy-drodynamic force Fwave produced by the incidentwave and the radiated wave, and is also subjectedto f transmitted by the cables that connect it tothe ballast. It’s general equation of motion is:

m1x1 = Fwave − f (1)

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The force Fwave will be approximated as a verticalforce only, just studying the motion of heave andnot the movements in other directions. Accordingto the linear wave theory hydrodynamic force actingon the floater is given by:

Fwave = Fimp + Frad + Fd, (2)

The impulsion force Fimp is given by:

Fimp = −ρgSx1, (3)

Where ρ represents the density of water, g is themagnitude of the acceleration of gravity and S isthe area of the horizontal section the floater at thelevel of undisturbed water free surface.

The radiation force can be expressed by two com-ponents that are computed in the frequency domain[5]:

Frad(ω) = −A(ω)x1(ω)−B(ω)x1(ω), (4)

where A(ω)x1 represents an added mass term, A(ω)is the added mass, B(ω)the damping radiation co-efficient and the term B(ω)x1(ω) represents energybeing transferred from the body to the water. Bothare dependent on the floater’s geometry and on itsfrequency response.

This force can be written in the time domain by:

Frad(t) = −∫ t

−∞K(t−τ)x1(τ)dτ−A1∞ x1(t). (5)

where A1 is the added mass coefficient of the floaterand K(t) represents the hydrodynamic coefficientsof the floater. These hydrodynamic coefficients arenot in the study area of this work and are here inter-preted as an input from other previous study, andwhich numeric values are set to a 6 meter radiushemispherical-shaped floater.

Difraction force

Since we are assuming linear water wave theory,the resulting diffraction force is obtained as a su-perposition of the frequency components

Fd(t) =

n∑j=1

Γ(ωj)Aωjcos(ωjt+ 2π rand()), (6)

with

Aωj= Hs

√2 ∆ωj S1

ζ (ωj), (7)

In the current work, real irregular waves are repre-sented as a superposition of regular waves assumingthe Pierson-Moskowitz spectrum.

Sζ(ω) = 262, 9H2s

ω5T 4e

exp

(− 1054

ω4T 4e

). (8)

For axisymmetric bodies oscillating in heave, theexcitation force coefficient is related to the radiationdamping coefficient by:

Γ(ω) =

√2g3%B(ω)

ω3(9)

A further detailed explanation about the calcula-tion of this force is described in the dissertation [6]

3.1.2. BallastThe ballast is the body that will hold the capturedenergy of the waves on the form of gravitationalpotential energy. When the ballast is suspended bythe floater, its dynamic equation of motion is

m2d2x2

dt2= −m2g

′ + f − FR, (10)

Where g denotes the equivalent gravitational accel-eration, which takes into consideration the buoy-ancy force that the water exerts on the body of theballast and FR represents the sum of hydrodynamicdrag and friction forces on the rods.

3.1.3. The cablesThe cables connect the ballast to the floater andhave two different functions depending on the phasein which they are in the energy cycle, this means,they have the function of raising the ballast fromxmin to xmax during the accumulation phase or ac-celerate the generator with the fall of the ballastduring the production phase.

These are subject to elastic deformation whenthey are under tension, and this deformation isgiven by [4]:

δC =f

EcAc(x1 − x2) (11)

where f is the force transmitted by the cables, ECthe Young modulus of the cable, AC its full cross-sectional area (this means, the area of each cablemultiplied by the number of cables) and (x1 − x2)at each instant the distance between the floater andthe ballast.

where f is the force transmitted by the cables, ECthe Young modulus of the cable, AC its full cross-sectional area (this means, the area of each cablemultiplied by the number of cables) and (x1 − x2)at each instant the distance between the floater andthe ballast.

The variation of the cable force with time is:

f =(EcAc − f) (x1 − x2) + θEcAcRD

(x1 − x2)(12)

3.1.4. DrumThe drum performs the function of winding and un-winding the cable around itself throughout the en-ergy cycle, as well as transmitting the cable force

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to the generator through the gearbox, thus speedingit.

Lets consider θ as the variable which measuresthe angular position of the drum, and its positivesense is that in which the cable is wound. We alsoconsider that the cable has a diameter dC , the drumhas an initial radius R0 and that the cable is woundonto the drum around itself in one spiral. Thedrum’s equation of motion is:

ID θ = −TD − TbD − TW + fRD, (13)

Where TD is the torque transmitted by the gearboxto the drum, TbD is the braking torque of the drum,TW is the torque applied by the cble winding mech-anism (this is one actuated when there is a slack inthe cable or lack of tension, otherwise it’s value isnul), and at last fRD is the torque the cable trans-mits to the drum, which is proportional to the cableforce and the radius of the loop at that instant.

3.1.5. Gearbox or transmission boxAs already said, when the clutch is actuated thegenerator is connected to the drum through a gear-box. So, we have the generator connected to theshaft on one side of the gearbox with speed ω andtorque TG; and on the drum side the shaft has rota-tion speed θ and torque TD. The relation betweentorques is given by:

TG = NtrTD (14)

And the relation between speeds is:

ω = −Ntr θ (15)

where Ntr is the speed ratio between both partiessaid, while its value is Ntr > 1. The rotation speedshave opposite signs, given by the convention in fig-ure 4, where it considers that the angular positionof the drum θ increases when the cable is windedon the drum and decreases as it unfolds. Regardingthe direction of rotation of the generator, this has apositive velocity direction in the only way it spins,which is the same sense as the drum rotates as thecable is unwound.

Figure 4: Senses of the variables: angular position of the drumθ and generator rotating speed ω. Schematic representation ofrolling the cables on the drum

3.1.6. Electric GeneratorThe equation of motion is given by:

IGω = TG −PGω− TRG , (16)

Where TG is the torque transmitted to the gen-erator shaft by the multiplying gearbox when theclutch is activated, TRG representing the torque re-sistance by friction in the bearings of the generatorand the flywheel attached to it. This binary TRGis practically independent of the rotational speed ofthe generator and is therefore assumed as a constantvalue.

Clutch unactuated When the clutch is not ac-tuated, the torque transmitted by the gearbox tothe generator shaft vanishes, TG = 0 , and so byequation (16) the generator slows down:

IGω = −PGω− TRG < 0. (17)

Clutch actuated The production cycle beginswith a small acceleration of the ballast when it isreleased and therefore accelerates the drum untilthe drum speed of rotation, affected the speed ratioimposed by the gear box, equals the rotation speedof the generator at that instant. It is at this mo-ment that begins the main stage of the productionphase, the clutch is actuated and the movement ofthe generator will be accelerated by the fall of theballast. At this stage neither the brake nor the bal-last of the drum are actuated, so that the only forcethat is exerted upon the ballast under cable is f itgenerates binary fRD on the drum.The equation of motion already simplified is:(

IG +IDNtr

)ω = −PG

ω− TRG +

fRDNtr

(18)

3.2. Equations resolution method

The main objective of this work, as has been said,is to simulate the behavior of this wave energy har-nessing system in order to validate its correct opera-tion and the applicability of the equations describedin its modeling. Also we want to check what is thepower that it will be able to produce according tothe sea state that is subjected.

For this we used a simulation performed usingthe program Simulink / Matlab. In this two state-space were used to model the corresponding systemsof equations of each stage, that were solved usingthe Runge-Kutta method.

The system consists of equations of first and sec-ond degree strongly nonlinear, so it can not besolved by the usual methods. This leads to endeav-our to reduce the degree of quadratic equations forpairs of equations of the first degree, which is de-ducted in the equation (1):

m1x1 = Fwave − f ⇔ (19)

from here, we get:

⇔{v1 = (Fwave − f) /m1

x1 = v1(20)

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All equations are rearranged in this manner sothat on the left side stays the first derivatives of thevariables to be calculated and the right side remainsthe known variables, calculated by this method inthe previous time-step. It is this procedure that willbe done to all systems of equations corresponding toeach stage of the energy cycle to simulate and willallow to use of the computational method describedabove.

3.3. Central controls

In practice, monitoring of installations will be donethrough a small number of mechanisms to decidehow this will behave, and these controls are the ON/ OFF type.

Ballast Brake (BB) - This is what will catch theballast rods to prevent it from losing vertical eleva-tion.

Drum brake (DB) - Will block the drum when itwants the cable exert strain on the ballast.

Mechanism Winding Cape (CWM) - Is coupledto the drum to allow rotate it and roll the cable.

Clutch (CLU) - This is the mechanism that allowsto couple the movements of rotation of the drumand gear box to the generator Electrical.

Power Generator (PG) - This is the electricalpower that will be requested the generator to pro-vide, controlled by the power controller of PI type.

3.4. Stages of plant operation

A logic that controls the processes of the plant op-eration was defined. We know that the energy cy-cle is divided into two phases: Accumulation andPower Production, and each of these is divided intostages, which are characterized by a different sys-tem of equations. In Figure 5 described the flowdiagram which symbolizes the logical sequence be-tween stages and phases, as well as the necessaryconditions for there to be an exchange of stage.

4. Power controller of the plant

The objective of power control is to keep the samerotational speed of the generator, in order to ob-tain a power output with constant characteristics,namely, always with the same electrical frequency.However, the generator speed is not constant, it isimmediately set to the average rotating speed of thegenerator a reference value (omega-ref) 1500 [rpm]which intends to keep. As it is not possible to main-tain a zero error in instantaneous power value, aPI (Proportional-Integral) controller that controlsthe electrical power to be drawn from the generatorbased on the error of the generator speed is used,and so it is intended to stabilize the speed value in arange of values considered acceptable, for example:

1400 rpm 6 ω 6 1600rpm, (21)

Figure 5: Scheme of the stages of system control

in order to control the variation of all the otherparameters of the plant. For example, in the simu-lation corresponding to the referenced figure it wasachieved by keeping the relative error of the angularvelocity of less than 7 % over the period of 12,000seconds, which is the same period used in all othersimulations.

Calculating the Plant efficiencyThe maximum power that can be extracted by a

floater in heave for a Pierson-Moskowitz spectrumfor deep water is given by:

Pmax = 149.5H2sT

3e . (22)

And for regular waves of height H and period T isgiven by:

Pmax = ErcgLm. (23)

where Er is the energy per unit of horizontal area,cg the rate at which energy spreads, or the groupvelocity and LM the maximum width of capture forabsorbing spot.

Er =ρgH2

8, (24)

cg =gT

4π, (25)

Lm = g

(T

2π

)2

. (26)

Computing the ratio between the power generatorand the maximum power that could be extractedfrom this spectrum

η =PGPmax

, (27)

we define η as representing the performance or effi-ciency of the plant, or in other words, the fraction ofthe maximum power that the plant is producing.

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5. Results Discussion

5.1. Regular waves case

For the case with regular waves, two types of sim-ulations were made: one varying the wave period(T ), and the other one varying de wave height (H).The values for the average power of the generator,maximum power of the wave, and efficiency of theplant for each simulation are described in table 1.

Table 1: Simulations for regular waves.

n H[m] T [s] PG[kW ] Pmax[kW ] η[%]

1 2,0 6 76,2 210,7 36,16Variation of T 2 2,0 8 311,0 499,3 62,27

3 2,0 10 547,4 975,3 56,134 1,5 8 174,4 280,9 62,085 2,0 8 311,0 499,3 62,27

Variation of H 6 2,5 8 442,9 780,2 56,767 3,0 8 558,7 1123,5 52,408 3,5 8 762,2 1529,2 49,84

5.1.1. Wave period variation (T )After the analysis of the results obtained, it is pos-sible to see that the main effect of the increasing ofthe power of the wave on the system is decreasingthe period of each energy cycle, or the time thatit takes for the ballast to go from its lowest heightto the highest, coming back to the lowest. This isseen when analyzing a 1000 seconds interval: thenumber of cycles completed is 11, 21 and 32 forthe wave periods of 6, 8 and 10 seconds, respec-tively. Therefore, there is less time between eachenergy production cycle, which results on the gen-erator being accelerated more frequently, producingmore energy.

Even if the power increases when the wave period,increases, the same can?t be said of the efficiency ofthe plant, defined in equation 27, which has its max-imum for the wave period of T = 8 seconds. Thismight have to do with various factors, but what isimportant to note is that the value of 8 seconds isthe closest one to the resonance value of the floater,allowing for the extraction of a greater amount ofenergy from the wave.

5.2. Wave height variation (H)

Simulations were conducted for 5 different waveheights, in which it was verified that the wave heightis related to the regular wave power, depending onit by the relation H2, as can be verified by the equa-tions 23 to 26.

It was also seen from the results of table 1, thatthe wave type that has a better efficiency in theusage of the wave energy is the one with H = 2, 0m,which is a median value compared with the othervalues tested, which makes the energy productionpossible for a wider value of wave powers.

The evolution of some of the variables studiedfor regular waves can be seen with greater detail on

figure 6.

Figure 6: Detail of the evolution of the angular velocity andgenerator power, and positions and velocities of the floater andballast, for a regular wave with T = 8 s and H = 2 m. (n=2)

5.3. Wave behaviour representative of thesea state in a location at the sea offLeixoes

A wave behaviour represents a sea state through atime span in a certain location. That behaviour canbe described by a spectrum of irregular waves, witha certain power (Pmax) and a probability of happen-ing of (φi). In this section, it will be presented theresults for the simulations of the plant behaviourwith a group of spectrums of irregular waves repre-sentative of a wave behaviour for a location at thesea off Leixoes, which can be seen in table 2.

Table 2: Wave behaviour at a location off Leixoes, during ayear [5].

Spectra Series 2n Hs[m] Te [s] φi[%] Pmax [kW] PG [kW] ηi[%] ηiφi ηiPG[kW ]1 1,10 5,49 7,04 29,92 1,18 6,50 12,35 57,23 1,23 7,75 8,17 105,3 18,8 17,8 1,46 1,534 1,88 6,33 11,57 134,0 36,9 27,5 3,19 4,275 1,96 7,97 20,66 290,8 122,6 42,2 8,17 25,336 2,07 9,75 8,61 593,7 191,3 32,2 2,77 16,477 2,14 11,58 0,59 1.063,2 206,9 19,5 0,11 1,228 3,06 8,03 9,41 724,8 313,6 43,3 4,07 29,519 3,18 9,93 10,07 1.480,3 413,2 27,9 2,81 41,6110 3,29 11,80 2,57 2.658,811 4,75 9,84 4,72 3.213,812 4,91 12,03 2,81 6.274,813 6,99 11,69 1,01 11.669,114 8,17 13,91 0,39 26.857,6

Total 23,13 % 119,96 kW

Two series of simulations were made, in which theaverage generator power (PG) and the efficiency (ηi)of each spectrum are described.

The anual average efficiency for the plant canbe calculated with the plant efficiency for a certainspectrum and its ocurrence probability:

ηannual =∑i

ηiφi. (28)

Similarly, the average annual power is given by:

PG annual average =∑i

PGφi. (29)

The values are presented in the last line of table 2and they are the best indicators of performance of

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the plant, as they consider the probability of occur-rence of each spectrum, which will say which valuesof power and efficiency are expected to happen forthe plant through the period of a year, if the plantwas placed at the sea of Leixoes.

The presented analysis was made for two differentconfigurations of the plant, that were based on thevariation of some of its dimensions and identified as”Series 1” and ”Series 2”.

Parameter variation of the two analyzedseries

Initially, for the first result series, it was desiredthat the heights of oscillation of the ballast werecomprised in an interval from -30 m to -5 m, butthese were found to be very close to the heightsof oscillation of the floater, which resulted in colli-sions for the most energetic states. On the secondseries of results the values were changed to preventthis, increasing the distance between the two com-ponents. The reason for trying to keep the ballastmovements closer to the floater is related to the factthat it is desired that the depth of the installationlocation is as smaller as possible, to decrease instal-lation costs. If the system created can only workon deep waters, it is necessary to place the plantfar away from the coast, as only there deep waterswill be found, increasing the cost of transmission ofelectricity to land with electric wires in the seabed,which might compromise the project economically.

Friction effect on the efficiencyA friction term was considered in the equation

of the generator, which corresponds to the internalbearing resistance. This resistance was character-ized in the simulations by a resistive binary (TRG )that opposed the direction of rotation. Therefore,there is a minimum limit of energy in the spectrumdue to that resistive term, for which the plant canproduce energy. This means that when the poweravailable on the sea state is lower than a certainvalue, the energy dissipated by friction is higherthat the energy that can be captured by the plant,leading to the stoppage of the generator.

Mass relation between ballast and floaterThe same way, after the first simulations, it was

verified that the relation value ∆ between the bal-last and the floater masses has a very relevant im-pact on the capacity for extracting energy of theplant. It was initially assumed a mass relation∆ = 0, 5, which revealed to be a very high value, asfor the lower energy spectrums it was not possible toattain enough force on the cable in order to over-come the ballast mass. The ballast brakes wouldnever be deactivated, then, as the ballast wouldn?tmove upwards. The value chosen for the mass re-lation was ∆ = 0, 3, as it already allowed for the

production of energy in less energetic spectrums,and also an acceptable efficiency for less energeticspectrum.

Numerical analysisAnalyzing the results obtained, in table 2 it can

be seen that, as opposed to what would be expected,the spectrum that most contributed for the increas-ing of the annual average power was not the one thatshowed a higher efficiency (n=8) but the a more en-ergetic spectrum with a similar occurrence probabil-ity (n=9). For n=8, it is possible to get more energyfrom the wave in a more efficient way, being higherthe percentage of available energy captured. How-ever, in absolute values, and taking into accountthe occurrence probability, the energy captured byspectrum n=9 is the highest of all the spectrums.With this, it can be concluded that it is importantnot only to optimize the plant dimensions takinginto account the efficiency increasing, but also tomaximize the value of the product of the producedpower and the occurrence probability.

Graphical analysis of the resultsIt is presented the graphic for spectrum n=8,

which was the one that obtained the best efficiency.Even if all the simulations were done for a total pe-riod of 12000 seconds, the results are presented onlyfor the last 2000 seconds so that it is possible to seethe analyzed variables with detail.

Figure 7: Evolution of the angular velocity and generatorpower, positions and velocities of the ballast and floater, for anirregular wave spectrum with Te = 8, 03 s and Hs = 3, 06 m,(Series 2, n=8).

After the analysis of figure 7, it is important tonote the irregularity of the floater and ballast move-ments, opposed to what happened with the regu-lar waves, due to the irregularity and the strengthof the incident wave diffraction, which best repre-sents what happens in nature. As for figure 7 cor-responding to spectrum n=8, the analysis is verysimilar to the one for spectrum n = 9, differing pri-marily on the range of motion of the float whichis directly connected to the power available for thespectrum, and as a consequence to the power ex-tracted. Note however that this is the spectrum

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which showed higher yield of all the analyzed spec-trums, and probably the entire configuration of theplant is more optimized for spectrums similar tothis one.

5.4. Influence of the dimension of some com-ponentes on the dynamic behaviour ofthe plant

Here a study was conducted to illustrate the im-portance of optimization of all components of thecentral regarding sizing, since it strongly influencesthe performance and the dynamic behaviour of theplant. It will only be studied the effect of the vari-ation of three characteristics of the plant: the massof the ballast, the diameter of the cables and theinertia of the generator. The simulation made witha regular wave of H = 2m and T = 8s was chosen toserve as default state, from which the variations ofthe parameters will be done. This wave was chosenfor being the regular wave for which we obtainedbetter performance of the plant, and also becausebeing a regular wave, allows us to anticipate andpredict some expected behaviours, and thus com-pare the results obtained with these.

The results for the default state are presented.These charts show the variations in the positionsand speeds of the float and ballast, the diffractionforce and the rotational speeds of the generator: areference and an instant speed.

Figure 8: Default state for the variations of the parameters instudy. Linear wave with T = 8s and H = 2m.

5.4.1. Effect of the variation of the mass of the bal-last

To study the effect of varying the mass of the bal-last, two comparisons were performed with the de-fault value. The values used for the mass ratio of thefloat and ballast (∆) were ∆ = 0, 3 for the defaultstate and ∆ = 0, 1 and ∆ = 0, 5 for the followingsimulations, where the mass ratio ∆ is given by:

∆ =m2

m1 +A∞1. (30)

After analysis of the variations of mass calculated,one can conclude that the change in mass of the bal-last has two major effects: its increase causes the

power accumulation cycle to take a longer time tobe concluded, as seen in the simulation for ∆ = 0, 1,each energy cycle takes 30 seconds, while for ∆ =0, 5 this cycle already takes about 90 seconds. Theother effect that it is easily visible is the variation inrotational speed of the generator that occurs dur-ing the production phase. As the accumulation cy-cle takes longer for a larger weight of the ballast, italso leads to the accumulation of more potential en-ergy (because remember that EPotencial = mg∆h),giving it a greater acceleration of the generator inthe production phase.

5.4.2. Effect of the variation of the diameter of thecables

It was studied here the influence that the variationof the diameter of the cables connecting the ballastto the floater has on their dynamic behaviour. Al-though it may not seem, from all the parametersstudied this is the one with the biggest impact onthe movement of both bodies, because it is the ca-bles that make the connection. The default valueused in the diameter of the cables was Dc = 30 mm,and therefore it is presented here the results for thediameters of Dc = 60 mm and of Dc = 10 mm. Itwas tried to vary considerably the numerical valueof each of the simulations against the default valueso that it was quite obvious and visible the effectthese changes may generate.

The conclusion after analyzing the results is thatthe diameter of the cable, which along with thenumber of cables is what defines the total numberof cables, is an element that contributed greatly tothe rigidity of the link which is given by:

Keq = EcAc, (31)

That said, it is noticed that for Dc = 60 mm thereis an extremely rigid connection and due to the ca-ble being a type of bond that only transmits ten-sile forces, it happens that whenever the cable isstretched an extremely high force is generated forextremely short time interval which in the limitcould be described as almost a shock. Thus, wehave very high and intermittent forces in the cable,which suggest that in this case there would be alarge vibration in the handle and a small capacityto absorb impacts.

On the other hand the connection cable of Dc =10mm would be much more elastic, yielding muchlower forces in comparison with the previous case,but also leading to a greater deformation of the ca-ble, which may preclude the generation of energy inthe case of low energy sea states in which the floaterhas very small oscillations.

5.4.3. Effect of the inertia of the generatorThe latter parameter studied was the inertia of thegenerator, which has an effect similar to that de-

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scribed on the weight of the ballast. As the mass ofthe ballast stores the gravitational potential energyduring lifting, the inertia of the generator stores thepotential kinetic energy of rotation when the angu-lar acceleration of the generator happens.

Knowing this, it was compared the amplitude ofthe variation of the angular velocity of the gener-ator through each energetic cycle, for a total pe-riod of 12000 seconds. The values of inertia usedwere IG = 20.000 [Kg m2] for the default state, ofIG = 40.000 [Kg m2] e de IG = 5.000 [Kg m2]. An-alyzing the results of the study with the highestvalue of inertia, the variation of the angular veloc-ity through an energetic cycle was on the order of25 rpm, while the same variation for an inertia ofIG = 5.000 [Kg m2] was on the order of 175 rpm.

6. Conclusion

In the present work it was successfully achieved theimplementation and validation of a computationalprogram capable to simulate the dynamic behaviourof the new type of wave energy harnessing system.The correct working of the program was validatedwith both regular waves and irregular wave spectra,including the complete simulation of a annual waveclimate of a point off the coast of Leixoes. Finnalyit was made an analysis of the impact the sizingvariation of some critical components had over thegeneral behaviour of the plant.

The code was implemented and reviewed initiallywith a simpler system, described in the article [5].This was two body energy converter system witha power-take-out mechanism simplified as a spring-dumper connection between both bodies.

It was refined the former idea of control algo-rithm that existed over the plant. The main aspectschanged were the criteria variables for changing be-tween stages. Whilst the previous thoughts on thissubject were all focused on absolute coordinates ofthe ballast and floater (x1 and x2), this revealed tobe not robust enough, being changed to the differ-ence in length between the two bodies (x1 − x2).

From the regular wave analysis it was concludedthat despite the fact the power of a wave increaseswith both wave height (H) and period (T ), the waveparameters to which the plant has a better efficiencyare variable and depend on the plant’s geometries.For the values tested, the most efficient regular wavewas H = 2 m and T = 8 s.

Regarding the analysis for the irregular wavespectrum and wave climate of a point off the coastof Leixoes there were some important conclusionsto remember. Firstly, between the two series of ge-ometries studied the variation of the range of depthsthe ballast was oscillating around generated a con-siderable difference in their efficiency, being the se-ries which oscillated the ballast in deeper positions

the one to achieve a higher efficiency. Secondly,it was important to notice that the spectrum withthe greatest efficiency was not the one that con-tributed the most for the increase of the annual av-erage power. With this it would be a suggestionfor a future work to do a optimization of station’sparameters regarding not the increase of efficiency,but the increase of the annual average power pro-duced, as well as getting a geometry optimized tohave a broader range of wave power capture, be-cause the current geometry didn’t allow to produceanything from the very high and low power waves.

To conclude, there were several relations per-ceived between the variation of some main param-eters, as the ballast mass, cable diameter and gen-erator inertia, with the functioning, behaviour andeven stability of the power plant as a whole. Tosummarize: the increase of the ballast mass wouldincrease the amount of energy captured by each ac-cumulation cycle as well as increasing it’s time du-ration; an annalogy with this can be made for thegenerator inertia with the production phase; thesystem complete stability is very dependenton thecable total area.

References

[1] Andre, J. M. C. S. and Henriques, J. C. C.:Patent PT 104 885, A new concept of waveenergy power plant applicable to deep offshorewaters. Technical Report, Lisbon, 2009.

[2] Costa, F. L. (2011): Modelacao numerica daarfagem de um novo sistema de aproveita-mento de energia das ondas. Master’s disser-tation in Mechanical Engineering, InstitutoSuperior Tecnico, University of Lisbon.

[3] Ponte, J. A. A. (2012): Desenvolvimentode um novo sistema de aproveitamento deenergia das ondas. Master’s dissertation inMechanical Engineering, Instituto SuperiorTecnico, University of Lisbon.

[4] Andre J. M. C. S., Equacoes de modelacao deum novo sistema de aproveitamento de ener-gia das ondas, Relatrio Tcnico, IST, 2010.

[5] Henriques J. C. C., Lopes M.F.P., GomesR. P. F., Gato L. M. C., Falco A.F. O., On the annual wave energy ab-sorption by two-body heaving WEC’s withlatch control, Renewable Energy (2012),doi:10.1016/j.renene.2012.01.102

[6] Westwood, J. A. F. (2014): Desenvolvimentode um novo sistema de aproveitamento deenergia das ondas (2). Master’s dissertationin Mechanical Engineering, Instituto SuperiorTecnico, University of Lisbon.

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