Research Article Stochastic Control of Inertial Sea Wave Energy …downloads.hindawi.com/journals/tswj/2015/980613.pdf · 2019-07-31 · Research Article Stochastic Control of Inertial

Research ArticleStochastic Control of Inertial Sea Wave Energy Converter

Mattia Raffero1 Michele Martini2 Biagio Passione1 Giuliana Mattiazzo1

Ermanno Giorcelli1 and Giovanni Bracco1

1Department of Mechanics and Aerospace Engineering Polytechnic University of Turin Corso Duca degli Abruzzi 2410129 Turin Italy2Environmental Hydraulics Institute ldquoIH Cantabriardquo CIsabel Torres 15 Santander 39011 Cantabria Spain

Correspondence should be addressed to Mattia Raffero mattiarafferopolitoit

Received 27 June 2014 Revised 19 August 2014 Accepted 21 August 2014

Academic Editor Linni Jian

Copyright copy 2015 Mattia Raffero et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The ISWEC (inertial sea wave energy converter) is presented its control problems are stated and an optimal control strategy isintroduced As the aim of the device is energy conversion themean absorbed power by ISWEC is calculated for a plane 2D irregularsea stateThe response of theWEC (wave energy converter) is driven by the sea-surface elevation which is modeled by a stationaryand homogeneous zero mean Gaussian stochastic process System equations are linearized thus simplifying the numerical modelof the device The resulting response is obtained as the output of the coupled mechanic-hydrodynamic model of the device Astochastic suboptimal controller derived from optimal control theory is defined and applied to ISWEC Results of this approachhave been compared with the ones obtained with a linear spring-damper controller highlighting the capability to obtain a highervalue of mean extracted power despite higher power peaks

1 Introduction

Oceans represent a wide distributed reservoir of energy andin them waves are by far the most conspicuous form ofenergy The global power potential represented by waves inopen oceans where energy is not dissipated due to seabedfriction or wave breaking has been estimated to be in theorder of 10 TW (1 terawatt = 1012 W) a quantity comparablewith the present world power demand [1]

For more than two centuries many devices have beenproposed for harvesting such a huge power source theearliest patent was filed in 1799 in France [2] Traditionallythe father of modern wave energy exploitation is consideredMasuda who started his studies in the 1940s and developeda navigation buoy powered by an air turbine which has beenlater commercialized [3] Since then a lot of devices have beenconceived and developed while a few of them arrived to theprecommercial stage [4]

Among these machines a considerable role is played bygyroscopic converters Gyroscopes have been widely used onships with the task of roll stabilization [5 6] The first use ofgyroscopes for wave energy extraction is due to Salter who

invented the Duck device at the University of Edinburgh inthe 1970s [7 8] the ISWEC belongs to the last generationof this kind of energy converters Many problems have stillto be solved in order to develop a reliable and economicallysustainable wave energy converter (WEC) A proof of that isgiven by the little number of survivingWEC concepts [9 10]Themain issue is the ldquoreaction problemrdquo in order to generatean action on the power take-off (PTO the component aimedat the power conversion eg the electric generator) togenerate energy a reaction is needed and has to be given byeither the seabed the water inertia or other structures [11]Moreover sea waves involve low-frequency and alternatinghigh forces making it necessary to use strong structuresand heavy conversion systems and therefore increasing thetechnology costs Other problems to be faced are relatedto corrosion of components in contact with the sea waterpossible leakage of oil (if hydraulic conversion systems areused) survivability in case of extreme events maintenanceand environmental and visual impacts [12ndash15]

In this paper one of the most important issues for thepower optimization of a WEC is faced the control problemDeveloping a good control scheme is challenging and many

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 980613 14 pageshttpdxdoiorg1011552015980613

2 The Scientific World Journal

solutions have been proposed in the recent years [16] InSection 2 the main existing WECs control algorithms aredescribed The reviewed algorithms are as follows the linearproportional-derivative (PD) controller the latching anddeclutching controller the optimal controller and finally thestochastic suboptimal controller Afterwards the ISWEC ispresented and the equations describing its working principleare discussed Given the physical characteristics of the fullscale prototype which will be installed in 2014 in real seaa performance analysis is carried out comparing the resultsobtained with the PD controller and the stochastic controlalgorithm for some representativewave conditions registeredat the installation site Moreover the effect of the maximumPTO torque constraint is analyzed in order to take intoaccount the real machine limits

2 WECs Control System Outlook

In this section a review of the existing control algorithms forwave energy converters is given so that the reader can havean overview of the state of the art in this field In most caseswhen analyzing the power extraction capabilities of a WECa one degree of freedom system is analyzed As described inSection 3 of the paper in the simplest case the hydrodynamicmodel of the device may be approximated by a 2nd orderlinear differential equation whose coefficients are frequencydependent In the following considerations such a simplemodelmay be a good reference for a reader that does not havea deep knowledge of this field

Often the first step is to develop a control strategy ableto maximize the power output under plane (2D problem)monochromatic waves Of course this means that the waveprofile is composed of a single frequency contribution andthis is not what happens in real sea Afterwards the case ofplane polychromatic wave is analyzed generating a wave timeseries based on the spectrum of a specific sea state or usingacquired wave data In the most recent studies a 3D sea stateis analyzed taking into account wave contributions comingfrom different directions

Many control strategies have been proposed with varyinglevels of complexity The main ones are described here

21 PD Controller One of the simplest ways to control aWEC is to apply on the floater an action proportional to itsvelocity This kind of controller can be called ldquoproportionalcontroller (P)rdquo and the ratio of force to velocity is thedamping coefficient In this case the power output is relatedto the square of the wave height moreover if the waveis monochromatic and its frequency matches the naturalfrequency of the device the velocity and the force are inphase and the power absorbed by theWEC is maximum [16]The natural frequency of a floating body is dependent on itsphysical features and could be varied acting on its mass forexample in order to match the incident wave frequency thusmaximizing its response amplitude Another way to obtainsuch a result without acting on the physical quantities of thedevice is to use a reactive controller This kind of controllercan also be called ldquoproportional-derivative controller (PD)rdquosince the torque acting on the floater is composed of two

contributions the first one proportional to the speed suchas in the P controller and the second one proportional tothe displacement of the body (with respect to the hydrostaticequilibrium condition)The ratio between the last force termand the displacement is the stiffness coefficient As shownin Section 41 in this case it is possible to tune the responseof the device in order to make the device resonant with theincoming wave [17] A problem often arises with this kind ofcontroller the PTO can provide an action up to a maximumvalue thus limiting the capacity of the system to adapt itselfto the incoming wave Moreover the PD controller impliesreactive power thus increasing the power losses due to theaction generated by the PTO on the floater After theseconsiderations it is clear that the floater has to be designedproperly in order to reduce the control reactive componentfor most of the incoming waves

22 LatchingDeclutching Controller The latching controltechnique has been firstly proposed for a heaving bodyindependently by Falnes and Budal [18] French [19] andGuenther et al [20] This strategy is particularly suitable forwaves longer than the WEC natural period it basically con-sists in locking the floating body when its velocity approachesthe zero value by means of a clamping mechanism and thenreleasing it at some point so that its velocity will be at itshighest point simultaneouslywith thewave force at this pointthe PTO force is set to its maximum value The action on thesystem can thus be regarded as binary that is either the bodyis locked or it is moving under maximum PTO actionmdashthusresulting in a highly nonlinear control force The declutchingcontroller is similar to the previous one but it is applied forwaves shorter than the WEC natural period [21] Differentfrom before the floater is normally free to move and whenits velocity reaches some desired value the maximum PTOforce is applied

The use of genetic algorithms indicated that if applicablethe latching and declutching control is among the best controltechniques for a wave energy converter see Nolan et al [22]A drawback of these strategies is that they need some kind ofprediction of the incoming wave force in order to actuate thedevice at the right time (autoregressive models and Kalmanfilter have been widely used in this context) however asan advantage with respect to the previously mentioned ldquoPDcontrollerrdquo any reactive power flow is eliminated from thepower take-off The result is a suboptimal control strategythat is best suitable using hydraulic power take-off systemsExperimental tests have been carried out during time includ-ing wave prediction which proved the reasonable goodnessof these control strategies especially if compared to applyinglinear damping see Budal et al [23] Hals et al [24] Falnesand Bjarte-Larsson [25] and Lopes et al [26]

However these considerations apply to devices for whichthe control force is directly applied on the floatermain degreeof freedom so that this could be locked or released at thedesired time instant The wave energy converter consideredin this paper is not suitable for the implementation of thisstrategy since in such a device it is not possible to lockreleasethe relative motion between floater and gyro at a desired timeinstant

The Scientific World Journal 3

x

y

z120573

120588

120575

Wave

(a)

y

z

y998400z998400

120575

120593 120582

PTO

Flywheel

Structure

120576

x equiv x998400

(b)

Figure 1 ISWEC geometry and coordinate systems

23 Optimal Controller Optimal control theory as describedin [27 28] has already been applied on a wave energyconverter model by Nielsen et al [29] The objective of thiscontrol strategy is tomaximize the power transfer fromwavesto the floater in a wide range of sea states

Here the idea is to make the controller compensate forthe dynamics of the floater and then damp its oscillationso that its motion is in phase with the wave excitationforce and thus the power flow is unidirectional from thewaves to the WEC In this controller an infinite timehorizon is needed thus resulting in a noncausal controllaw In order to overcome such noncausality an approx-imation is introduced The convolution integral is splitinto two parts the causal part remains as it is whilstthe noncausal part is replaced by a damping term whosevalue is obtained by means of a stochastic analysis ofthe wave-structure interaction aimed at maximizing theexpected value of the power output A more detailedexplanation of this approach can be found in Section 43of this paper after the hydrodynamic model descrip-tion

3 The ISWEC

In this section the ISWEC device is introduced After abrief description of the device the hydrodynamic model ofthe floater and the mechanical model of the gyroscope aredescribed Finally the features of the ISWEC first full scaleprototype analyzed in this paper and to be deployed inautumn 2014 are reported

31 Description of the System ISWEC (inertial sea waveenergy converter) is a device designed to exploit wave energythrough the gyroscopic effect of a flywheel [30ndash33] A lot ofstudies and experimental tests have been carried out on thisdevice proving the concept feasibility [34 35] and estimatingits annual energy production [36]

Figure 1 shows the four main components of the gyro-scopic system the floater the flywheel the gyro structureand the PTO To describe the system dynamics two referenceframes have to be introduced a hull-fixed coordinate system

119909 119910 119911 and a gyroscope structure-fixed coordinate system1199091015840 1199101015840 1199111015840 Both have their origins coincident with the centre

of gravity of the system The 119909-axis is oriented towardsthe bow and coincides with the sea wave direction Thehull rotates about the 119910-axis with the induced pitchingmotion 120575 due to the wave-floater-gyro interaction Due tothe angular momentum conservation of the flywheel thecombination of the pitch speed

120575 with the flywheel speed

about the 1199111015840-axis generates a gyroscopic torque 119879

120576around

the 1199091015840-axis which can be exploited by the PTO to generate

electrical power The device involves two main phenomenathe hull hydrodynamics and gyroscope mechanics There isa strong coupling between them due to torques and energyinteractions as shown in the following paragraphs

The main advantages of the ISWEC device with respectto its competitors are the following ones All the mechanicalcomponents of the system are enclosed in a sealed hullretained by a slack mooring line and seen from outside thesystem thus looks like a moored boat This means that directinteraction between water and moving parts is avoided thusreducing corrosion problems and maintenance Moreoverthe flywheel speed is an additional free parameter that canbe tuned to increase the device performance in a wide rangeof wave conditions

32 Dynamics of the Gyroscope From the time derivationof the flywheel angular momentum the equilibrium of thesystem is described in the gyro-frame coordinate system asin [30 37]

119879120576= 119868 120576 + (119868 minus 119869)

1205752 sin 120576 cos 120576 minus 119869

120575 cos 120576 (1)

119879120593= 119869 (

120575 sin 120576 + 120576

120575 cos 120576 + ) (2)

119879120582= 119868

120575 cos 120576 + (119869 minus 2119868) 120576

120575 sin 120576 + 119869 120576 (3)

where 119868 represents the inertia of the gyroscopic system withrespect to the 119909

1015840- and 1199101015840-axes and 119869 with respect to the 119911

1015840-axis The three torques are given to the gyroscopic systemrespectively by the PTO (1) the flywheel motor (2) andthe hull (3) The torques given by the latter two equationshave a key role in the system behavior their projection on


the 119910- and 119911-axes represents the pitch and yaw torques thatthe gyroscopic system discharges to the floater In particularfor the torque related to the pitching 119910-axis it is possible towrite

119879120575= (119869sin2120576 + 119868cos2120576)

120575

+ 119869 sin 120576 + 119869 120576 cos 120576 + 2 (119869 minus 119868)

120575 120576 sin 120576 cos 120576(4)

Linearizing the mean zero position of the PTO shaft andassuming that the pitching accelerations of the system aresmall respectively from (1) (2) and (4) one gets eventually

119879120576= 119868 120576 minus 119869

120575 (5)

119879120593= 119869 120576

120575 (6)

119879120575= 119869 120576 (7)

These simple equations are very useful for a preliminarydesign of the hull the gyroscope the PTO and the controlsystem logic to be implemented on the machine [31] Theseequations are supposed to be valid for small angles of oscilla-tion for the purpose of this study results are considered validfor PTO oscillations amplitudes up to 45 degrees The strongcoupling between the floater and the gyroscope can be shownhere The action torque 119869

120575 given by the gyroscope to the

PTO is function of the pitch speed The result of such torque(combined with the control torque 119879

120576) is the acceleration

of the PTO shaft 120576 The reaction torque 119869 120576 given by thegyroscope to the floater is function of the PTO speed 120576 andas shown in the next paragraph interacts with the floaterdynamics thus affecting the pitch motion

120575

33 Hydrodynamic Model and Full System Equation

331 Cumminsrsquo Equation For the pitch motion of a rigidfloating marine structure with zero forward speed assumingthat coupling with the other degrees of freedom is negligiblethe equation of motion in the time domain can be written inbody-fixed coordinates as

(119868119865+ 119860infin

)

120575 + int

119905

0

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 + 119870120575 = 119879119908

minus 119879119898

minus 119879120575

(8)

according to Cumminsrsquo decomposition (1962) which studiedthe hydrodynamic problem under the assumption of linearphenomena [38] This equation is valid only for small pitchoscillations in the present study oscillations up to 10 degreesin amplitude are considered physically meaningful In theexpression above 120575 represents the pitch angle 119868

119865the floater

inertia moment 119860infin

the added mass for infinite oscillationfrequency and 119870 the constant hydrostatic restoring forcedue to buoyancy and gravity and finally ℎ

119903 120575is the impulse

response function of the radiation forces The convolutiontermmodels the radiation hydrodynamic problem in an idealfluid with a linear pressure force distribution and it is oftenreferred to as ldquofluid memory effectrdquo

The terms on the right hand side of the equation representthe pitch torque due to the incoming wave 119879

119908 the pitch

torque due to the mooring forces 119879119898 and the pitch control

torque 119879120575acting on the floater Note that as anticipated in

Section 22 the control torque119879120575is generally directly given by

the PTO while in the ISWEC it is given by the gyroscope as areaction torque due to itsmotion 120576 (7)Mooring contributionwill be neglected here under the assumption that its effect onthe pitching motion of the device is small

Ogilvie converted Cumminsrsquo equation for a free-floatingbody in the frequency domain under only wave excitationforces and found out the following relationships [39]

[minus1205962(119868119865+ 119860 (120596)) + 119895120596119861 (120596) + 119870] sdot 120575

0= ℎ119908

sdot 119891119908(120596) (9)

119867119903(119895120596) = 119861 (120596) + 119895120596 [119860 (120596) minus 119860

infin] (10)

where 119860 and 119861 are respectively the frequency-dependentaddedmass and potential damping while119867

119903is the frequency

response function of the radiation On the right hand sideof the equation the wave excitation torque 119879

119908is given by

the frequency-dependent force coefficient 119891119908 representing

the torque per wave amplitude unit multiplied by the waveamplitude ℎ

119908evaluated at the center of gravity of the floater

Note that relation (9) is written in the frequency domainthus involving linear quantities and steady state conditionsmoreover such relation is valid undermonochromatic excita-tion force However relation (10) that describes the frequencyresponse function of the radiation is very useful because itwill be used in the next section for the implementation of thetime domain model

332 Modeling of the Radiation Forces The numerical com-putation of the convolution term in (8) may be quite time-consuming and not well suited for the design and analysis ofthe wave energy converter control system Perez and Fossensuggested a smart way for overcoming this problem [40]Based on (10) it is possible to pursue a parametric frequencydomain identification of the impulse response function Theobjective is to find an appropriate order transfer functionwhich satisfies the criteria of minimum approximation errorstability and passivityThe frequency-dependent added mass119860 and potential damping 119861 can be found for a chosen set offrequencies by means of any commercial code based on theimplementation of the panel method under the assumptionof potential flow By means of the toolbox developed byPerez and Fossen it has been possible to identify the transferfunction related to the pitching degree of freedom of thestructure under investigation [41]

Figure 2 shows that it has been possible to find a transferfunction able to describe the radiation frequency responsefunction of the floater with a sufficient approximation inthe typical frequency range of the studied sea The obtainedtransfer function is stable and responds to the requiredpassivity criteria

333 Modeling of the Wave Excitation Forces With theassumption that the wave elevation process is a homogeneousand stationary zero-mean Gaussian process the sea state is


95

100

105

110

115

120

125

130

Frequency (rads)

Convolution model-pitch DoF

Data4th order fit

10minus1 100 101

Am

plitu

dehr(j120596

)

(a)

Frequency (rads)

Convolution model-Pitch DoF

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Data4th order fit

10minus1 100 101

Phas

ehr(j120596

) (de

g)

(b)

Figure 2 Frequency domain identification for the ISWEC floater 4th order transfer function magnitude and phase

given by the one-sidedwave spectrum 119878lowast

120578120578(120596) Given the RAO

(response amplitude operator) of the system 119867119890120578(120596) that

describes the amplitude and phase of the force acting on thefloater with respect to a unit amplitudemonochromatic waveit is possible to calculate the time history of the wave forcesacting on the structure as a finite sum of harmonic excitationforces

119879119908(119905) =

119872

sum

119898=1

119879119898cos (120596

119898119905 + 120601119898

+ 120579119898) (11)

where

119879119898

= radic2

10038161003816100381610038161003816119867119890120578

(120596119898)

10038161003816100381610038161003816

2

119878lowast

120578120578(120596119898) Δ120596

120601119898

= ang [119867119890120578

(120596119898)]

(12)

The angle 120579119898

between the harmonics components of thespectrum can either be chosen as random phase or can beguided by a groupiness factor [42] or in case of wave dataacquisitions may be the phase angle given by the fast Fouriertransform (FFT) analysis of the time series The approachabove described is referred to as linear stochastic wave loadmodel [43]

34 Features of the Tested ISWEC Device The first full scaleprototype of the ISWEC device will be installed in 2014 offPantelleria Island (Sicily Italy) [36∘5010158400010158401015840N 11∘5510158403910158401015840E] (seeFigure 3 and Table 1)

Table 1 Features of the full scale ISWEC prototype

Property Value UnitsHull width 8 mHull length 15 mHull natural period 55 sGyro mass 20 tonGyro diameter 3 m

For the Pantelleria site a wave gauge has measured thesea wave elevation for the whole 2010 Among the acquireddata a set of nine 20-minute-long waves has been chosen asrepresentative of the site as shown in Table 2

The reported data are the result of a spectral analysis ofthe acquired time series where119867

1198980is thewave spectral height

and 119879119890is the wave energy period

4 Optimal Control of a PitchingWave Energy Converter

In this section the control problem of a generic pitchingdevice is introduced Starting from the floater hydrodynamicequation the maximum extractable power is obtained forboth monochromatic and irregular wave The suboptimalcausal control algorithm is then introduced and the optimaldamping factor is obtained by means of a stochastic analysis


Float

Bearing housing

Gyroscope

BedframePTO (rotational)

Float pitch(wave induced) One DoF

platform

Wave direction

Figure 3 ISWEC layout concept

Table 2 Spectral properties of the chosen set of waves

Wave ID Wave spectralheight

Wave energyperiod

Wave powerdensity

[mdash] [m] [s] [kWm]

1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125

of the wave resource In the next section the causal subopti-mal control is applied on the ISWEC and its performances arecompared with the linear reactive control

41 Optimal Control under Monochromatic Wave As statedin the Introduction the PTO control force for the linearreactive controller is composed of two parts an elasticcontribution and a damping one It can be written as follows

119879120575= minus119896120575 minus 119887

120575 (13)

Adding (13) in (9) the dynamic equation of the controlledsystem in the frequency domain eventually becomes

[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)

Given the incident wave frequency 120596 the maximum poweroutput is achieved by setting the proper 119887 119896 parameters

that can be obtained applying the maximum power transfertheorem (Jacobirsquos Theorem 1840)

119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)

In such conditions the system is resonant with the incomingwave so the force and the speed are in phase and the powerextracted by the oscillator is

119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)

This result has been obtained with regular monochromaticwave to show how it is possible to maximize the powerextraction by tuning the control parameters In case ofirregular waves the optimal parameters for the linear reactivecontroller can be found using an optimization algorithm[44]

42 Optimal Control of a Pitching Wave Energy ConverterAn analytical approach may be followed to find an optimalcontrol force law which ensures the floating device to absorbthe maximum mechanical energy from a given irregular seastate This is usually called a deterministic optimal controlproblem and can be solved following basically two pathsthe Hamilton-Jacobi-Bellman method [27] and Pontryaginrsquosprinciple [28] based on a variational approach In thisanalysis the latter method has been used which is widelydiscussed and explained in [29 45] Given (8) that describesthe dynamic of the floater and assuming that all the state


variables are deterministic quantities the control force thatmaximizes the mean absorbed power is found to be

119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)

As it can be noted such a control law is noncausal dependingon the future values of the velocity

120575 Inserting the equationabove into the equation of motion of the system (8) one canget the following

int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)

Equation (18) is also known to be a Fredholm integralequation Fourier transforming it one gets the followingrelationship between the wave excitation force and the pitch-ing velocity at a general excitation frequency 120596 for optimalcontrol

119879119908(120596) = 2119861 (120596)

120575 (120596) (19)

where 119861(120596) represents the potential damping of the systemTherefore the optimal control lawhas as a direct consequencethe fact that the wave excitation force is in phase with thefloater pitching velocity for all their harmonic componentswhich is consistent with the hypothesis of maximum powertransfer to the system

43 Suboptimal Causal Feedback Control of a Pitching WaveEnergy Converter As previously stated the control law pro-posed in the previous paragraph is noncausal and cannot beimplemented on a real machine unless the future evolutionof the system is known or predicted with a sufficient levelof accuracy At this point two possible approaches may befollowed

(i) The noncausal control law is used together withsome prediction algorithmof the future incidentwaveforce as stochastic autoregressive models [46] neuralnetworks [47] or digital filters

(ii) The optimal control law is approximated by a closelyrelated causal process and the algorithm becomesthen suboptimalThis method does not need to knowthe wave elevation in order to be used

In this analysis the second approach will be followed sincethe quality of the prediction algorithms is not considered highenough to control the ISWEC with the desired accuracy Theanticausal part of the convolution term in (17) is replaced as

int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)

The new causal-control force becomes then

119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)

The reason for using this control force is evident whenreplacing it by the equation of motion of system (8) whereit follows that

119879119908(119905) = 2119887

119888

120575 (119905) (22)

This equation is similar to that one obtained for the noncausaloptimal controller with the difference that the dampingcoefficient is constant with respect to the frequency Its valuehas to be determined by means of some optimality criterionfor the mean absorbed mechanical power under given seastate conditions For linear stiffness and monochromaticwaves it is easy to find that [29]

119887119888= 119861 (120596) (23)

Instead for the case of irregular waves the calculation ofthe damping factor 119887

119888can be related to a stochastic dynamic

response analysis of the wave energy converterIt can be argued that the impulse response function of the

causal optimal stochastic controller is different everywherein the time domain from that of the optimal stochasticcontroller However as pointed out and demonstrated in[29] ldquothe causal controller absorbs almost as much power asthe optimal controller for all parameter values defining theautospectral density functionrdquo which gives confidence androbustness to the investigated methodology

44 Stochastic Identification of the Damping Factor Assum-ing that the floating device is in stationary conditionsand keeping the assumption that the wave elevation canbe regarded as a stationary zero-mean Gaussian processthrough linear stochastic dynamics theory [45 48] it ispossible to derive the optimal control law for known sea stateconditions Under these assumptions the pitching velocityprocess and in turn the displacement and acceleration canbe regarded as stationary zero-mean Gaussian independentrandomprocessesMoreover due to stationary conditions thefollowing properties hold [45 49]

119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)

where the operator119864[sdot] indicates the expected value and 120581 theautocorrelation function that in case of zero-mean process isequal to the covariance function The substitution

Δ119905 = 120591 minus 119905 (28)

has been applied in order to match the notation used inthe Cummins equation (8) For the suboptimal control and


assuming that all responses processes are ergodic the meanabsorbed power becomes

119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)

Using the relations from (24) to (27) the equation abovebecomes

119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)

By means of the Wiener-Khinchin theorem which relatesthe Fourier transform of the autocorrelation function of astationary random process to its double-sided autospectraldensity function we get the following

120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)

Given the sea state the spectrum of the wave excitation forcescan be obtained with it being related to the spectrum of thewave elevation process through the wave-to-force responseamplitude operator as

119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)

Moreover the variance of the velocity process is related to thatof the wave excitation force process by

1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)

The mean absorbed power in (30) becomes then

119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890

is the autocorrelation coefficient function of thewave excitation force process and is defined as

120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)

The maximum for the absorbed power function (34) is thenfinally found for a damping value of

119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)

Therefore once the sea state and the hull hydrodynamic prop-erties are known it is possible to calculate the suboptimalunconstrained stochastic value of the damping coefficient forthe pitching motion of the system

5 Results Control Strategy for the ISWEC

The previously mentioned control law was obtained andtested for a generic pitching device (acting directly on thefloater by means of a control torque 119879

120575) in this section it will

be used to control the ISWECgyroscope in order tomaximizethe wave power conversion The ISWEC can be controlledacting through the PTO on the 120576-axis of the gyroscope bymeans of the control torque 119879

120576 Two main control strategies

are tested for the device under consideration with andwithout PTO torque saturation and the results are compared

51 Proportional Derivative (PD) Control Law for ISWEC Asimple and easy controller consists inmaking the PTObehaveas a spring-damper groupThis approach was proposed beingsimilar to the one presented in ldquoPD Controllerrdquo The PTOtorque equation can be written as

119879120576= minus119896120576120576 minus 119888120576

120576 (37)

The goal is now to maximize the PTO mean power produc-tion on a wide range of sea conditions For the chosen set ofwaves the best stiffness and damping values in terms ofmeanpower production were calculated by means of a parametricanalysis

From Figure 4 it can be noticed that the power fluxbetween the PTO and the gyroscope is bidirectional (ie thePTO sometimes acts as a motor) therefore introducing areactive power component This is why this kind of controlis also referred to as ldquoreactive controlrdquo

The goodness of the conversion efficiency in order toprovide a further comparison parameter for the same deviceusing different control laws is related here to the relativecapture width RCW calculated as

RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)

where 119882 is the floater width This term is the ratio betweenthe mean mechanical power generated by the device (whichis assumed to be equal to the electrical power ie electricalconversion efficiency equal to unity) and the power of thewave front and it represents somehow the transfer of energyfrom the wave to the floating device it should be notedhowever that its value may exceed one since the floater mayabsorb more energy than the one contained in the wave frontdue to wave-body interactions [50 51] An interesting trendis found if this indicator is plotted versus the wave energyperiod (Figure 5)

Since the ISWEC pitching undamped natural periodis approximately 55 s the waves with longer period areless suitable for power extraction with this kind of device[34]


450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104

T120576 (Nm times10)

(b)

Figure 4 Time histories of the main system variables (a) Wave excitation force versus pitch speed and (b) PTO torque and power

5 6 7 8 90

02

04

06

08

1

Wave energy period (s)

RCW

(mdash)

Figure 5 RCW for the optimized linear reactive controller

52 Suboptimal Stochastic Control of ISWEC Thesuboptimalcausal control law calculated in the previous section hasbeen implemented on the ISWEC device The objective isto control the gyroscope to ensure that a given torque 119879

120575is

discharged to the hull Once the optimal pitching torque iscalculated through (21) and (36) the PTO speed to be set forthe linear gyroscope is calculated by means of (7) as

120576set =119879120575opt

119869

=

119879120575opt

119871

(39)

where 119871 is the angular momentum of the flywheel A closed-loop speed control is implemented acting on the PTO torqueMoreover since the gyroscopic torque acting as a disturbanceon the PTO axis is known analytically from (5) it is possibleto add a feedforward torque as

119879120576119891119891

= minus119869

120575 = minus119871

120575 (40)The gyroscopic torque makes the PTO speed deviate from itstarget value and the feedforward torque is used to cancel outthis effect and help the control to work better

The resulting control system for the ISWEC device isshown in Figure 6

119896119875is the proportional gain of the closed-loop speed

controller and it has been necessary to introduce a relativelysmall stiffness 119896

119877term in order to prevent position drifting in

irregular wave conditions Notice that the hull parameters areknown since they are characteristics of the device while thesea state spectrum is given by the weather forecast and by anon-board monitoring system that will be installed for the seastate evaluation and prediction


Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP

Figure 6 Closed loop optimal speed control for the ISWEC

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)

Figure 7 Set versus feedback for gyroscope speed and pitching torque

53 Unconstrained Optimal Control It is initially consideredthat the PTO can give any torque to the shaft In theseconditions the behavior of the system is represented inFigures 7 and 8 for the representative wave number 4

As it can be seen in Figure 7 with the implemented con-trol loop the gyroscope is able to produce the pitching torquerequired by optimal control by rotating at the required speedAs already seen for the monochromatic wave in Section 41when the optimal control is implemented the wave excitationtorque and the pitching velocity of the floating device areldquoin phaserdquo that is their maxima occur at the same timeinstants (Figure 8) This is one of the first consequences ofthe implementation of the optimal control law

It is interesting that correspondingly the pitching positionof the device is reasonably in phasewith thewavemeasured atthe body centre of gravity This may be very useful in furtherdevelopment of the control algorithm of the system At thesame time the oscillations of the gyroscope are relativelysmall which ensures some grade of reliability in using thelinearized gyroscope equations The same holds for thepitching oscillations of the device Results for the other wavesare summarized in Table 4

In this section the results for the waves numbers 2 5 6 7and 8 were excluded due to high pitching floater oscillationsfor which the linear hydrodynamic model loses its validityCompared with Table 3 it can be noticed that the RCW


minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)

Figure 8 (a) Wave force versus pitching velocity and (b) wave elevation versus pitch angle

Table 3 Results obtained for the linear reactive controller main physical variables that characterize the system response to the consideredwaves

Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079

Table 4 Results for the optimal stochastic control with no constraints

Wave ID 120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131

of the optimal controlled system is higher but higher peaktorque values are registered too

54 Unconstrained Optimal Control with Constraints In areal machine the PTO undergoes some current and thustorque limitations In order to be able to apply the method-ology shown before it would be needed to recalculate

optimal control signals with respect to system constraintsfor example torque limitations This could be the object offuture investigations nonetheless it is interesting to showthe effect of imposing system constraints a posteriori to theoptimal unconstrained control signals Two different valuesfor the PTO maximum torque have been investigated andthe results are reported in Table 5 When saturations occur


Table 5 Results with saturation on PTO torque at 500 kNm


120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID

Optimal no sat PD no satOptimal Tsat

120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)

Figure 9 ISWEC RCW comparison between optimal and linearreactive controller with and without torque saturation

the gyroscope is no longer able to control the floatermotion asrequested by the optimal control algorithm This is reflectedin the fact that the wave force and the pitching velocity losetheir phasing during this transition As shown in Table 5for 500 kNm PTO saturation torque the overall effect is adecrease in the mean power extracted by the machine andthus in the RCW

It has not been possible to decrease more the PTOsaturation torque in this section since the oscillations of thegyroscope were too high for the linear model to be stillacceptable (Figure 9)

An interesting result is found since the maximum torquefor the linear reactive controllerwas about 500 kNm the samevalue has been imposed on the constrained optimal controland though the power production decreases with respect tothe optimal unconstrained case it is higher than the oneextracted with the linear reactive controller

6 Conclusions

Stochastic suboptimal control and linear reactive controlhave been developed tested and compared for the ISWECdevice Results were obtained with a linear dynamic model

of the system The suboptimal control maximizes the meanabsorbed power at the cost of higher power peaks and gen-erator torques if compared with an optimized linear reactivecontroller applied to the gyroscope However if torquelimitation is imposed the power production is still higherthan the one obtained with the linear reactive controllerNonetheless the optimal control theory can give an upperbound of the performance of the WEC under irregular seastate conditions and furnishes guidelines for the optimizationof other control algorithms and its parameters can be derivedanalytically given the sea state and the hull hydrodynamicproperties Further studies are needed in order to assessthe nonlinear gyroscope performance and controllabilityMoreover the hydrodynamic model has to be improvedin order to take into account nonlinear wave forces andwave-body interactions when high pitch angles are involved

Energy dissipations have to be introduced in themodel inorder to maximize the net power production of the systemComparison with experimental data will be carried out oncethe ISWEC prototype is installed and tested in real seaconditions

Nomenclature

119860 Added mass of the floater in the frequencydomain

119860infin Added mass for infinite oscillationfrequency

119861 Hydrodynamic damping coefficient in thefrequency domain

119864[] Expected value of119865119890 Wave excitation force in the frequency

domain119867119890120578 Force-to-motion response amplitudeoperator of the system

1198671198980 Wave spectral height

119867119903 Frequency response function of the

radiation119868 Inertia moment of the gyroscopic system

with respect to the 1199091015840- and 119910

1015840-axes119868119865 Inertia moment of the floater with respect

to the 119910-axis119869 Inertia moment of the gyroscopic system

with respect to the 1199111015840-axis

119870 Hydrostatic restoring force (hydrostaticstiffness)

119871 Angular momentum of the flywheel119875119886 Absorbed mean power under stochastic

optimal causal control


119875119898 Mean power extracted by the oscillator

under optimal control119875wave front Input power from the wave front119875120576 Mean extracted power by the PTO

RCW Relative capture width of the WEC119878119894119894 Double-sided spectrum of the variable

119878lowast

120578120578 One-sided wave spectrum

119879120575 Floater control torquegyro reaction

torque on 119910-axis (given to the hull)119879120575opt Optimal floater control force

119879119890 Wave energy period

119879119898 Mooring reaction torque

119879119898 Amplitude of the 119898th harmonic of the

wave excitation force119879119908 Wave excitation torque

119879120576 Torque on the 119909

1015840-axis (from the PTO tothe gyroscopic system)

119879120582 Torque on the 119910

1015840-axis (from the hull to thegyroscopic system)

119879120593 Torque on the 119911

1015840-axis (from the flywheelmotor to the gyroscopic system)

119882 Floater width119887 Damping coefficient of the optimal control119887119888 Damping coefficient of the optimal causal

control119888120576 Damping coefficient of the PD control

119891119908 Froude-Krylov forces coefficient

ℎ119903 120575 Impulse response function of the radiation

forces with respect to the pitching motionℎ119908 Wave amplitude

119896 Stiffness coefficient of the optimal control119896120576 Stiffness coefficient of the PD control

119905 Time120575 Pitch angle (rotation about the 119910-axis)120576 PTO angle (rotation about the 119909

1015840-axis)120579119898 Angle between the harmonics

components of the wave spectrum120581119909119909 Autocorrelation function

1205902 Variance

120601119898 Phase angle of 119867

119890120578

Flywheel angular velocity (rotation aboutthe 1199111015840-axis)

120596 Angular frequency

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work presented in this paper collects some resultsthat have been obtained in many research activities Suchactivities have been funded byENEAAccordo di ProgrammaMSE-ENEA sulla Ricerca di Sistema Elettrico by RegionePiemonte PROMO and by Regione Sicilia REMOTO

References

[1] N N Panicker ldquoPower resource estimate of ocean surfacewavesrdquo Ocean Engineering vol 3 no 6 pp 429ndash439 1976

[2] D Ross Power from Sea Waves Oxford University PressOxford UK 1995

[3] Y Masuda Wave-Activated Generator International Collo-quium on the Exposition of the Oceans Bordeaux France 1971

[4] E2I EPRI Assessment Offshore Wave Energy ConversionDevices Electricity Innovation Institute 2004

[5] E Sperry ldquoThe gyroscope for marine purposesrdquo Transactions ofthe Society of Naval Architects and Marine Engineers vol 18 pp143ndash154 1910

[6] E Schlick ldquoThe gyroscopic effect of flywheels on board shiprdquo inTransactions of the Royal Institution of Naval Architects vol 23pp 117ndash134 1904

[7] S H Salter ldquoWave powerrdquo Nature vol 249 no 5459 pp 720ndash724 1974

[8] S Salter ldquoRecent progress on ducksrdquo in Proceedings of the 1stSymposium on Wave Energy Utilization pp 36ndash76 ChalmersUniversity of Technology Gothenburg Sweden November1979

[9] Z Chen H Yu M Hu G Meng and C Wen ldquoA review of off-shore wave energy extraction systemrdquo Advances in MechanicalEngineering vol 2013 Article ID 623020 9 pages 2013

[10] M Faizal M R Ahmed and Y-H Lee ldquoA design outline forfloating point absorber wave energy convertersrdquo Advances inMechanical Engineering vol 2014 Article ID 846097 18 pages2014

[11] M J French ldquoOn the difficulty of inventing an economicalsea wave energy converter a personal viewrdquo Proceedings ofthe Institution of Mechanical Engineers Part M Journal ofEngineering for the Maritime Environment vol 220 no 3 pp149ndash155 2006

[12] J A Hudson D C Phillips and N J M Wilkins ldquoMaterialsaspects of wave energy convertersrdquo Journal of Materials Sciencevol 15 no 6 pp 1337ndash1363 1980

[13] S Parmeggiani J P Kofoed and E Friis-Madsen ldquoExtremeloads on the mooring lines and survivability mode for the wavedragon wave energy converterrdquo in Proceedings of the WorldRenewable Energy Congress Linkoping Sweden 2011

[14] O Langhamer K Haikonen and J Sundberg ldquoWave powermdashsustainable energy or environmentally costly A review withspecial emphasis on linear wave energy convertersrdquo Renewableand Sustainable Energy Reviews vol 14 no 4 pp 1329ndash13352010

[15] S Diaconu and E Rusu ldquoThe environmental impact of a wavedragon array operating in the Black Seardquo The Scientific WorldJournal vol 2013 Article ID 498013 20 pages 2013

[16] S H Salter J R M Taylor and N J Caldwell ldquoPower conver-sionmechanisms for wave energyrdquo Proceedings of the Institutionof Mechanical Engineers Part M Journal of Engineering for theMaritime Environment vol 216 2002

[17] J K H Shek D E Macpherson and M A Mueller ldquoControlof linear electrical generators for direct drive wave energyconversionrdquo in Proceedings of the International Conference onElectrical Machines 2006

[18] J Falnes and K Budal ldquoWave-power conversion by pointabsorbersrdquo Norwegian Maritime Research vol 6 no 4 pp 2ndash11 1978


[19] M J French ldquoA generalized view of resonant energy transferrdquoJournal of Mechanical Engineering Science vol 21 pp 299ndash3001979

[20] D A Guenther D Jones and D G Brown ldquoAn investigativestudy of a wave-energy devicerdquo Energy vol 4 no 2 pp 299ndash306 1979

[21] P A P Justino and A F D O Falcao ldquoActive relief valve foran OWC wave energy devicesrdquo in Proceedings of the EuropeanWave Energy Conference pp 295ndash300 2000

[22] G A Nolan J V Ringwood W E Leithead and S ButlerldquoOptimal damping profiles for a heaving Buoy wave energyconverterrdquo in Proceedings of the 15th International Offshore andPolar Engineering Conference 2005

[23] K Budal J Falnes T Hals L C Iversen and T OnshusldquoModel experiment with a phase controlled point absorberrdquo inProceedings of the 2nd International Symposium on Wave andTidal Energy pp 191ndash206 Cambridge UK September 1981

[24] J Hals T Bjarte-Larsson and J Falnes ldquoOptimum reactive con-trol and control by latching of awave-absorbing semisubmergedheaving sphererdquo in Proceedings of the 21st International Confer-ence on Offshore Mechanics and Arctic Engineering (OMAE rsquo02)pp 415ndash423 June 2002

[25] J Falnes and T Bjarte-Larsson ldquoTheoretical and experimentalinvestigation of wave energy conversion by a phase-controlledheaving bodyrdquo Proceedings of the Institution of MechanicalEngineers Part M Journal of Engineering for the MaritimeEnvironment vol 220 no 4 pp 175ndash183 2006

[26] M F P Lopes JHals R P FGomes TMoan LMCGato andA F D O Falcao ldquoExperimental and numerical investigationof non-predictive phase-control strategies for a point-absorbingwave energy converterrdquo Ocean Engineering vol 36 no 5 pp386ndash402 2009

[27] R E Bellman Dynamic Programming Princeton UniversityPress Princeton NJ USA 1957

[28] L S PontryaginTheMathematicalTheory of Optimal ProcessesPergamon Press New York NY USA 1964

[29] S R K Nielsen Q ZhouMM Kramer B Basu and Z ZhangldquoOptimal control of nonlinear wave energy point convertersrdquoOcean Engineering vol 72 pp 176ndash187 2013

[30] M Raffero Design of a Wave Energy Convertermdasha case ofapplication ISWEC [PhD thesis] 2014

[31] G Bracco ISWEC a gyroscopic wave energy converter [PhDthesis] 2010

[32] G Bracco E Giorcelli GMattiazzoM Pastorelli and J TaylorldquoISWEC design of a prototype model with gyroscoperdquo inProceedings of the International Conference on Clean ElectricalPower (ICCEP rsquo09) pp 57ndash63 Capri Italy June 2009

[33] 2013 httpwwwwaveforenergycom[34] G Bracco E Giorcelli and GMattiazzo ldquoISWEC a gyroscopic

mechanism for wave power exploitationrdquo Mechanism andMachine Theory vol 46 no 10 pp 1411ndash1424 2011

[35] G Bracco E Giorcelli and G Mattiazzo Experimental Testingon a One Degree of Freedom Wave Energy Converter Conceivedfor theMediterranean Sea TMM2008 Liberec CzechRepublic2008

[36] M Raffero G Mattiazzo E Giorcelli and M Sorli ldquoISWEC(inertial sea wave energy converter) modeling control andproductivity analysisrdquo in Proceedings of the 3rd IFToMM Inter-national Symposium on Robotics and Mechatronics (ISRM 13)2013

[37] B Passione Modelling and control of a wave energy converter[MS thesis] 2014

[38] W E Cummins ldquoThe impulse response function and shipmotionsrdquo David Taylor Model Basin Report 1661 Departmentof the Navy Washington DC USA 1962

[39] T Ogilvie ldquoRecent progress towards the understanding andprediction of shipmotionsrdquo inProceedings of the 6th Symposiumon Naval Hydrodynamics 1964

[40] T Perez and T I Fossen ldquoTime- vs frequency-domain iden-tification of parametric radiation force models for marinestructures at zero speedrdquo Modeling Identification and Controlvol 29 no 1 pp 1ndash19 2008

[41] T Perez and T I Fossen ldquoA Matlab tool for frequency-domainidentification of radiation-force models of ships and offshorestructuresrdquo Tech Rep 2009-020 Marine Systems Simulator2009

[42] G R Rodriguez C Guedes Soares and L Ferrer ldquoWave groupstatistics of numerically simulted mixed sea statesrdquo Journal ofOffshore Mechanics and Arctic Engineering vol 122 no 4 pp282ndash288 2000

[43] W F Liu M T Sichani S R K Nielsen Y B Peng JB chen and J Li ldquoSimulation of the stochastic wave loadsusing a physical modeling approachrdquoKey EngineeringMaterialsJournal vol 569-570 pp 571ndash578 2013

[44] J Falnes Ocean Waves and Oscillating Systems Linear Interac-tions Including Wave-Energy Extraction Cambridge UniversityPress 2002

[45] S Nielsen Linear Stochastic Dynamics vol 3 of StructuralDynamics Aalborg University 2007

[46] F Fusco and J V Ringwood ldquoShort-term wave forecasting forreal-time control of wave energy convertersrdquo IEEE Transactionson Sustainable Energy vol 1 no 2 pp 99ndash106 2010

[47] A A E Price and A R Wallace ldquoNon-linear methods for nextwave estimationrdquo in Proceedings of the 7th European Wave andTidal Energy Conference Porto Portugal 2007

[48] J Li Stochastic Dynamics of Structures Wiley New York NYUSA 2009

[49] K Shin and J Hammond Fundamentals of Signal Processing forSound and Vibration Engineers John Wiley amp Sons New YorkNY USA 2008

[50] P A Lynn Electricity from Wave and Tide An Introduction toMarine Energy John Wiley amp Sons 2013

[51] K Budar and J Falnes ldquoA resonant point absorber of ocean-wave powerrdquo Nature vol 256 no 5517 pp 478ndash479 1975

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FuelsJournal of


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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom


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EnergyJournal of


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Journal ofEngineeringVolume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


solutions have been proposed in the recent years [16] InSection 2 the main existing WECs control algorithms aredescribed The reviewed algorithms are as follows the linearproportional-derivative (PD) controller the latching anddeclutching controller the optimal controller and finally thestochastic suboptimal controller Afterwards the ISWEC ispresented and the equations describing its working principleare discussed Given the physical characteristics of the fullscale prototype which will be installed in 2014 in real seaa performance analysis is carried out comparing the resultsobtained with the PD controller and the stochastic controlalgorithm for some representativewave conditions registeredat the installation site Moreover the effect of the maximumPTO torque constraint is analyzed in order to take intoaccount the real machine limits

2 WECs Control System Outlook

In this section a review of the existing control algorithms forwave energy converters is given so that the reader can havean overview of the state of the art in this field In most caseswhen analyzing the power extraction capabilities of a WECa one degree of freedom system is analyzed As described inSection 3 of the paper in the simplest case the hydrodynamicmodel of the device may be approximated by a 2nd orderlinear differential equation whose coefficients are frequencydependent In the following considerations such a simplemodelmay be a good reference for a reader that does not havea deep knowledge of this field

Often the first step is to develop a control strategy ableto maximize the power output under plane (2D problem)monochromatic waves Of course this means that the waveprofile is composed of a single frequency contribution andthis is not what happens in real sea Afterwards the case ofplane polychromatic wave is analyzed generating a wave timeseries based on the spectrum of a specific sea state or usingacquired wave data In the most recent studies a 3D sea stateis analyzed taking into account wave contributions comingfrom different directions

Many control strategies have been proposed with varyinglevels of complexity The main ones are described here

21 PD Controller One of the simplest ways to control aWEC is to apply on the floater an action proportional to itsvelocity This kind of controller can be called ldquoproportionalcontroller (P)rdquo and the ratio of force to velocity is thedamping coefficient In this case the power output is relatedto the square of the wave height moreover if the waveis monochromatic and its frequency matches the naturalfrequency of the device the velocity and the force are inphase and the power absorbed by theWEC is maximum [16]The natural frequency of a floating body is dependent on itsphysical features and could be varied acting on its mass forexample in order to match the incident wave frequency thusmaximizing its response amplitude Another way to obtainsuch a result without acting on the physical quantities of thedevice is to use a reactive controller This kind of controllercan also be called ldquoproportional-derivative controller (PD)rdquosince the torque acting on the floater is composed of two

contributions the first one proportional to the speed suchas in the P controller and the second one proportional tothe displacement of the body (with respect to the hydrostaticequilibrium condition)The ratio between the last force termand the displacement is the stiffness coefficient As shownin Section 41 in this case it is possible to tune the responseof the device in order to make the device resonant with theincoming wave [17] A problem often arises with this kind ofcontroller the PTO can provide an action up to a maximumvalue thus limiting the capacity of the system to adapt itselfto the incoming wave Moreover the PD controller impliesreactive power thus increasing the power losses due to theaction generated by the PTO on the floater After theseconsiderations it is clear that the floater has to be designedproperly in order to reduce the control reactive componentfor most of the incoming waves

22 LatchingDeclutching Controller The latching controltechnique has been firstly proposed for a heaving bodyindependently by Falnes and Budal [18] French [19] andGuenther et al [20] This strategy is particularly suitable forwaves longer than the WEC natural period it basically con-sists in locking the floating body when its velocity approachesthe zero value by means of a clamping mechanism and thenreleasing it at some point so that its velocity will be at itshighest point simultaneouslywith thewave force at this pointthe PTO force is set to its maximum value The action on thesystem can thus be regarded as binary that is either the bodyis locked or it is moving under maximum PTO actionmdashthusresulting in a highly nonlinear control force The declutchingcontroller is similar to the previous one but it is applied forwaves shorter than the WEC natural period [21] Differentfrom before the floater is normally free to move and whenits velocity reaches some desired value the maximum PTOforce is applied

The use of genetic algorithms indicated that if applicablethe latching and declutching control is among the best controltechniques for a wave energy converter see Nolan et al [22]A drawback of these strategies is that they need some kind ofprediction of the incoming wave force in order to actuate thedevice at the right time (autoregressive models and Kalmanfilter have been widely used in this context) however asan advantage with respect to the previously mentioned ldquoPDcontrollerrdquo any reactive power flow is eliminated from thepower take-off The result is a suboptimal control strategythat is best suitable using hydraulic power take-off systemsExperimental tests have been carried out during time includ-ing wave prediction which proved the reasonable goodnessof these control strategies especially if compared to applyinglinear damping see Budal et al [23] Hals et al [24] Falnesand Bjarte-Larsson [25] and Lopes et al [26]

However these considerations apply to devices for whichthe control force is directly applied on the floatermain degreeof freedom so that this could be locked or released at thedesired time instant The wave energy converter consideredin this paper is not suitable for the implementation of thisstrategy since in such a device it is not possible to lockreleasethe relative motion between floater and gyro at a desired timeinstant


x

y

z120573

120588

120575

Wave

(a)

y

z

y998400z998400

120575

120593 120582

PTO

Flywheel

Structure

120576

x equiv x998400

(b)




3 The ISWEC








120576around





119879120576= 119868 120576 + (119868 minus 119869)

1205752 sin 120576 cos 120576 minus 119869

120575 cos 120576 (1)

119879120593= 119869 (

120575 sin 120576 + 120576

120575 cos 120576 + ) (2)

119879120582= 119868

120575 cos 120576 + (119869 minus 2119868) 120576

120575 sin 120576 + 119869 120576 (3)






119879120575= (119869sin2120576 + 119868cos2120576)

120575

+ 119869 sin 120576 + 119869 120576 cos 120576 + 2 (119869 minus 119868)

120575 120576 sin 120576 cos 120576(4)


119879120576= 119868 120576 minus 119869

120575 (5)

119879120593= 119869 120576

120575 (6)

119879120575= 119869 120576 (7)






120575



(119868119865+ 119860infin

)

120575 + int

119905

0

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 + 119870120575 = 119879119908

minus 119879119898

minus 119879120575

(8)


119865the floater






119908 the pitch






[minus1205962(119868119865+ 119860 (120596)) + 119895120596119861 (120596) + 119870] sdot 120575

0= ℎ119908

sdot 119891119908(120596) (9)

119867119903(119895120596) = 119861 (120596) + 119895120596 [119860 (120596) minus 119860

infin] (10)




119908is given by









95

100

105

110

115

120

125

130

Frequency (rads)


Data4th order fit

10minus1 100 101

Am

plitu

dehr(j120596

)

(a)

Frequency (rads)


minus80

minus60

minus40

minus20

0

20

40

60

80

100

Data4th order fit

10minus1 100 101

Phas

ehr(j120596

) (de

g)

(b)



120578120578(120596) Given the RAO



119879119908(119905) =

119872

sum

119898=1

119879119898cos (120596

119898119905 + 120601119898

+ 120579119898) (11)

where

119879119898

= radic2

10038161003816100381610038161003816119867119890120578

(120596119898)

10038161003816100381610038161003816

2

119878lowast

120578120578(120596119898) Δ120596

120601119898

= ang [119867119890120578

(120596119898)]

(12)

The angle 120579119898












Float

Bearing housing

Gyroscope



platform

Wave direction




Wave energyperiod

Wave powerdensity


1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125



119879120575= minus119896120575 minus 119887

120575 (13)


[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)



119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)


119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)





119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















x

y

z120573

120588

120575

Wave

(a)

y

z

y998400z998400

120575

120593 120582

PTO

Flywheel

Structure

120576

x equiv x998400

(b)




3 The ISWEC








120576around





119879120576= 119868 120576 + (119868 minus 119869)

1205752 sin 120576 cos 120576 minus 119869

120575 cos 120576 (1)

119879120593= 119869 (

120575 sin 120576 + 120576

120575 cos 120576 + ) (2)

119879120582= 119868

120575 cos 120576 + (119869 minus 2119868) 120576

120575 sin 120576 + 119869 120576 (3)






119879120575= (119869sin2120576 + 119868cos2120576)

120575

+ 119869 sin 120576 + 119869 120576 cos 120576 + 2 (119869 minus 119868)

120575 120576 sin 120576 cos 120576(4)


119879120576= 119868 120576 minus 119869

120575 (5)

119879120593= 119869 120576

120575 (6)

119879120575= 119869 120576 (7)






120575



(119868119865+ 119860infin

)

120575 + int

119905

0

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 + 119870120575 = 119879119908

minus 119879119898

minus 119879120575

(8)


119865the floater






119908 the pitch






[minus1205962(119868119865+ 119860 (120596)) + 119895120596119861 (120596) + 119870] sdot 120575

0= ℎ119908

sdot 119891119908(120596) (9)

119867119903(119895120596) = 119861 (120596) + 119895120596 [119860 (120596) minus 119860

infin] (10)




119908is given by









95

100

105

110

115

120

125

130

Frequency (rads)


Data4th order fit

10minus1 100 101

Am

plitu

dehr(j120596

)

(a)

Frequency (rads)


minus80

minus60

minus40

minus20

0

20

40

60

80

100

Data4th order fit

10minus1 100 101

Phas

ehr(j120596

) (de

g)

(b)



120578120578(120596) Given the RAO



119879119908(119905) =

119872

sum

119898=1

119879119898cos (120596

119898119905 + 120601119898

+ 120579119898) (11)

where

119879119898

= radic2

10038161003816100381610038161003816119867119890120578

(120596119898)

10038161003816100381610038161003816

2

119878lowast

120578120578(120596119898) Δ120596

120601119898

= ang [119867119890120578

(120596119898)]

(12)

The angle 120579119898












Float

Bearing housing

Gyroscope



platform

Wave direction




Wave energyperiod

Wave powerdensity


1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125



119879120575= minus119896120575 minus 119887

120575 (13)


[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)



119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)


119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)





119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119879120575= (119869sin2120576 + 119868cos2120576)

120575

+ 119869 sin 120576 + 119869 120576 cos 120576 + 2 (119869 minus 119868)

120575 120576 sin 120576 cos 120576(4)


119879120576= 119868 120576 minus 119869

120575 (5)

119879120593= 119869 120576

120575 (6)

119879120575= 119869 120576 (7)






120575



(119868119865+ 119860infin

)

120575 + int

119905

0

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 + 119870120575 = 119879119908

minus 119879119898

minus 119879120575

(8)


119865the floater






119908 the pitch






[minus1205962(119868119865+ 119860 (120596)) + 119895120596119861 (120596) + 119870] sdot 120575

0= ℎ119908

sdot 119891119908(120596) (9)

119867119903(119895120596) = 119861 (120596) + 119895120596 [119860 (120596) minus 119860

infin] (10)




119908is given by









95

100

105

110

115

120

125

130

Frequency (rads)


Data4th order fit

10minus1 100 101

Am

plitu

dehr(j120596

)

(a)

Frequency (rads)


minus80

minus60

minus40

minus20

0

20

40

60

80

100

Data4th order fit

10minus1 100 101

Phas

ehr(j120596

) (de

g)

(b)



120578120578(120596) Given the RAO



119879119908(119905) =

119872

sum

119898=1

119879119898cos (120596

119898119905 + 120601119898

+ 120579119898) (11)

where

119879119898

= radic2

10038161003816100381610038161003816119867119890120578

(120596119898)

10038161003816100381610038161003816

2

119878lowast

120578120578(120596119898) Δ120596

120601119898

= ang [119867119890120578

(120596119898)]

(12)

The angle 120579119898












Float

Bearing housing

Gyroscope



platform

Wave direction




Wave energyperiod

Wave powerdensity


1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125



119879120575= minus119896120575 minus 119887

120575 (13)


[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)



119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)


119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)





119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















95

100

105

110

115

120

125

130

Frequency (rads)


Data4th order fit

10minus1 100 101

Am

plitu

dehr(j120596

)

(a)

Frequency (rads)


minus80

minus60

minus40

minus20

0

20

40

60

80

100

Data4th order fit

10minus1 100 101

Phas

ehr(j120596

) (de

g)

(b)



120578120578(120596) Given the RAO



119879119908(119905) =

119872

sum

119898=1

119879119898cos (120596

119898119905 + 120601119898

+ 120579119898) (11)

where

119879119898

= radic2

10038161003816100381610038161003816119867119890120578

(120596119898)

10038161003816100381610038161003816

2

119878lowast

120578120578(120596119898) Δ120596

120601119898

= ang [119867119890120578

(120596119898)]

(12)

The angle 120579119898












Float

Bearing housing

Gyroscope



platform

Wave direction




Wave energyperiod

Wave powerdensity


1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125



119879120575= minus119896120575 minus 119887

120575 (13)


[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)



119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)


119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)





119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Float

Bearing housing

Gyroscope



platform

Wave direction




Wave energyperiod

Wave powerdensity


1 118 531 3652 197 644 12253 067 738 1614 068 654 1505 136 683 6236 220 809 19187 145 777 8068 199 727 14169 069 536 125



119879120575= minus119896120575 minus 119887

120575 (13)


[minus1205962(119868119865+ 119860) + 119895120596 (119861 + 119887) + (119870 + 119896)] sdot 120575

0= ℎ119908

sdot 119891119908 (14)



119887 = 119861

119896 = (119868119865+ 119860)120596

2minus 119870

(15)


119875119898

=

1

8

1003816100381610038161003816119891119908

1003816100381610038161003816

2

119861

ℎ2

119908

(16)





119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ int

+infin

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(17)



int

infin

minusinfin

ℎ119903 120575

(|119905 minus 120591|)

120575 (120591) 119889120591 = 119879119908(119905) (18)


119879119908(120596) = 2119861 (120596)

120575 (120596) (19)






int

+infin

119905

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591 997888rarr 2119887119888

120575 (119905) (20)


119879120575opt (119905) = minus (119868

119865+ 119860infin

)

120575 (119905) minus 119870120575 (119905)

+ 2119887119888

120575 (119905) minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591)

120575 (120591) 119889120591

(21)


119879119908(119905) = 2119887

119888

120575 (119905) (22)


119887119888= 119861 (120596) (23)






119864 [

120575 (119905)] = 0 (24)

119864 [120575 (119905)

120575 (119905)] = 0 997888rarr 119864 [

120575 (119905)

120575 (119905)] = 0 (25)

119864 [

120575 (119905)

120575 (119905)] = 1205902

120575 (26)

119864 [

120575 (119905)

120575 (119905 + Δ119905)]

= 120581 120575 120575(Δ119905) 997888rarr 119864 [

120575 (119905)

120575 (120591)] = 120581 120575 120575

(120591 minus 119905) = 120581 120575 120575(119905 minus 120591)

(27)


Δ119905 = 120591 minus 119905 (28)




119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119875119886= 119864 [119879

120575(119905)

120575 (119905)]

= minus (119868119865+ 119860infin

) 119864 [

120575 (119905)

120575 (119905)] + 2119887119888119864 [

1205752(119905)]

minus 119870119864 [

120575 (119905) 120575 (119905)] minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 119864 [

120575 (119905)

120575 (120591)] 119889120591

(29)


119875119886= 21198871198881205902

120575minus int

119905

minusinfin

ℎ119903 120575

(119905 minus 120591) 120581 120575 120575(119905 minus 120591) 119889120591

= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

(30)


120581 120575 120575(120591) = int

infin

minusinfin

119890119894120596120591

119878 120575 120575(120596) 119889120596

= int

infin

minusinfin

119890119894120596120591

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

120581119865119890119865119890

(120591)

41198872

119888

(31)


119878119865119890119865119890

(120596) =

10038161003816100381610038161003816119867119890120578

(120596)

10038161003816100381610038161003816

2

119878120578120578

(120596) (32)


1205902

120575= int

infin

minusinfin

119878 120575 120575(120596) 119889120596 = int

infin

minusinfin

119878119865119890119865119890

(120596)

41198872

119888

119889120596 =

1205902

119865119890

41198872

119888

(33)


119875119886= 21198871198881205902

120575minus int

infin

0

ℎ119903 120575

(119906) 120581 120575 120575(119906) 119889119906

= 1205902

119865119890

(

1

2119887119888

minus

1

41198872

119888

int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906)

(34)

where 120588119865119890119865119890


120588119865119890119865119890

(120591) =

120581119865119890119865119890

(120591)

1205902

119865119890

(35)


119887119888= int

infin

0

ℎ119903 120575

(119906) 120588119865119890119865119890

(119906) 119889119906 (36)









119879120576= minus119896120576120576 minus 119888120576

120576 (37)




RCW =

119875120576

119875wave front=

(119879120576

120576)

0491198671198980

1198792

119890119882

(38)




450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















450 460 470 480 490 500minus5

0

5

t (s)

Wave ID 4

Tw (Nm times105)120575998400 (rpm)

(a)

450 460 470 480 490 500minus5

0

5

t (s)

P120576 (W)P120576mean (W)

times104


(b)


5 6 7 8 90

02

04

06

08

1


RCW

(mdash)



120575is


120576set =119879120575opt

119869

=

119879120575opt

119871

(39)


119879120576119891119891

= minus119869

120575 = minus119871








Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Sea state spectrum

Suboptimalcausal

stochastic control

Hull parameters

T120575setLminus1

set+ +minus minus

minus

T120576ff

(120575 )fb

L

T120576set

120576fbkRfb

kP


450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

120576998400set (rpm)120576998400fb (rpm)

(a)

t (s)450 455 460 465 470 475 480 485 490 495 500

minus4minus2

024

T120575set (Nm)T120575 (Nm)

times106Wave ID 4

(b)







minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















minus5

0

5

450 455 460 465 470 475 480 485 490 495 500t (s)

Wave ID 4

120575998400 (rpm)Tw (Nm times105)

(a)

450 455 460 465 470 475 480 485 490 495 500minus20minus10

01020

t (s)

Wave ID 4

hw (dm)120575 (∘)

(b)



Wave ID 119888120576

119896120576

120575max 119879120575max 120576max 119879

120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [kNm s] [kNm] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 1036 minus5 139 0712 203 469 0 24 00 0832 537 746 357 2002 667 1079 277 81 34 0813 104 278 75 1026 417 203 49 5 105 0364 144 455 108 1054 376 319 76 7 104 0595 537 455 158 1026 346 434 33 21 16 0426 278 278 162 1580 660 422 77 33 23 0217 104 278 137 1722 657 371 143 19 75 0298 200 278 162 1808 666 421 111 40 28 0359 746 455 77 0485 151 250 5 8 06 079



120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 284 347 220 822 805 39 205 1313 306 590 260 897 1448 13 1107 1014 295 405 218 813 836 15 548 1259 167 165 142 576 238 14 176 131







120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















120576max 119875120576max 119875

120576mean 119875120576max119875120576mean RCW

[] [∘] [MNm] [

∘] [kNm] [kW] [kW] [] []

1 195 602 553 500 1430 382 375 1293 277 753 630 500 1340 105 1279 0834 261 648 688 500 1350 135 997 1139 156 166 149 500 237 135 176 131

1 3 4 90

02

04

06

08

1

12

14

Wave ID


120576 = 500kNm PD Tsat120576 = 300kNm

RCW

(mdash)





6 Conclusions




Nomenclature




















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















119878lowast








119879120576 Torque on the 119909


119879120582 Torque on the 119910


119879120593 Torque on the 119911











1205902 Variance

120601119898 Phase angle of 119867

119890120578





Acknowledgments


References

























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of























































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Documents

Research Article Stochastic Control of Inertial Sea Wave Energy …downloads.hindawi.com/journals/tswj/2015/980613.pdf · 2019-07-31 · Research Article Stochastic Control of Inertial