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Applied Ocean Research 47 (2014) 255–269

Contents lists available at ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

onstrained optimal stochastic control of non-linear wave energyoint absorbers

.T. Sichania,∗, J.B. Chenb,c, M.M. Kramera, S.R.K. Nielsena

Department of Civil Engineering, Aalborg University, 9000 Aalborg, DenmarkSchool of Civil Engineering, Tongji University, Shanghai 200092, PR ChinaState Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, PR China

r t i c l e i n f o

rticle history:eceived 25 June 2013eceived in revised form 25 April 2014ccepted 13 June 2014

eywords:ave energy point converter

onlinear buoyancy forceonstrained optimal stochastic control

a b s t r a c t

The paper deals with the stochastic optimal control of a wave energy point absorber with strong nonlinearbuoyancy forces using the reactive force from the electric generator on the absorber as control force. Theconsidered point absorber has only one degree of freedom, heave motion, which is used to extract energy.Constrains are enforced on the control force to prevent large structural stresses in the floater at specifichot spots with the risk of inducing fatigue damage, or because the demanded control force cannot besupplied by the actuator system due to saturation. Further, constraints are enforced on the motion of thefloater to prevent it from hitting the bottom of the sea or to make unacceptable jumps out of the water.The applied control law, which is of the feedback type with feedback from the displacement, velocity,

rregular sea state and acceleration of the floater, contains two unprovided gain parameters, which are chosen so the mean(expected value) of the power outtake in the stationary state is optimized. In order to ensure accuracyof the results for each configuration of the controller Monte Carlo simulations have been carried out forvarious sea-states and the final results have been presented in the paper. The effect of nonlinear buoyancyforce – in comparison to linear buoyancy force – and constraints of the controller on the power outtake

tudie

of the device have been s

. Introduction

A wave energy converter (WEC) may be defined as a dynamicystem with one or more degrees of freedom with the intention toonvert the energy in the waves into mechanical energy stored inhe oscillating system. A point absorber is a WEC with a size that ismall compared to the dominating wave length. The power outtakes basically the conversion of this mechanical energy into electricity.he absorbers of the WEC are typically equipped with an electricower generator via a hydraulic force system. The reaction forcerom the latter influences the motion of the point absorber. Withinertain ranges the reaction forces can be specified at prescribedalues. In so-called reactive control these forces are used to controlhe motion of the WEC in a way that a maximum mechanical energys supplied to the absorbers. With a certain loss due to friction inhe hydraulic force actuators the control forces are next transferred

o the generators, where they are converted into electric energy.

Basically, the reactive control may be either of the open-loopfeed forward) or of the closed-loop (feedback) type. Open-loop

∗ Corresponding author. Tel.: +45 9940 8570; fax: +45 9814 8552.E-mail addresses: mts@civil.aau.dk (M.T. Sichani), chenjb@tongji.edu.cn

J.B. Chen), mmk@civil.aau.dk (M.M. Kramer), soren.nielsen@civil.aau.dkS.R.K. Nielsen).

141-1187/$ – see front matter © 2014 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.apor.2014.06.005

d in details and supported by numerical simulations.© 2014 Elsevier Ltd. All rights reserved.

control implies that the control effort is brought forward based onobservation (measurement) of the wave elevation. Open-loop doesnot affect the dynamics of the system, i.e. angular eigenfrequen-cies and structural damping ratios are unchanged by the control.Closed loop control is entirely based on the observed motion ofthe absorbers. Typically, this involves the displacement, veloc-ity and acceleration components, which easily can be measuredby accelerometer or laser vibrometer measurements onboard thefloating devices. A closed loop control always changes the dynamicsproperties of the system (inertia, damping or stiffness parame-ters), as specified by the poles and zeros of the frequency responsefunctions relating the wave excitation forces to the displacementresponses of the absorber system.

Latching, independently proposed in [1,2], is probably the sim-plest and definitely the most investigated control strategy. Thecontrol is based on the observation of the hydrodynamic force, forwhich reason latching control should be classified as a open-loopcontrol strategy. Latching control requires that the hydrodynamicforce can be predicted at least a semi-wave period ahead of thepresent time. In broad-banded irregular sea-states this prediction isrelated with uncertainty, which may affect the stability of the con-

trol. Normally, merely the sea surface elevation in the vicinity of theconverter is observed. This makes observation of the wave excita-tion force components difficult, due to the non-causal dependenceon this quantity on the sea-surface elevation [3]. Further, the power

2 cean R

ozf

dfitfbeonofsIcfirif

miePpSc

cwhoiceavnottctcss

octcCiaba

tpttpg

56 M.T. Sichani et al. / Applied O

uttake from the control changes between finite time intervals withero and non-zero power production, which may cause problemsor the mechanical implementation of the method.

The simplest closed loop control law is achieved by a so-callederivative controller, where the reactive control force is speci-ed to be proportional to and opposite directed to the velocity ofhe WEC. The controller has insignificant influence on the eigen-requency of the absorber, for which reason the controller onlyecomes optimal for frequencies in the auto-spectrum of the wavexcitation force in the vicinity of the undamped eigenfrequencyf the absorber. By augmenting the controller with a force compo-ent proportional to either the displacement (proportional control)r the acceleration (acceleration control) a broader spectrum ofrequencies can be absorbed. Proportional control will change thetiffness of the absorber, and acceleration control changes the mass.n both cases the eigenfrequency of the absorber can be changed to aertain extent. Finally, so-called integral control can be introducedor which the control force component appears as a convolutionntegral of the absorber velocity with respect to a given impulseesponse function. It turns out that integral control needs to bentroduced, if perfect phase locking between the wave excitationorce and the velocity of the absorber is attempted at all frequencies.

The idea of extracting energy from the waves is very old andany WEC devices have been proposed in the past [4,5]. This has

nitiated commercial WEC projects using devices such as differ-nt buoy concepts, Oscillating-Water-Column (OWC) plants, theelamis [7], overtopping WEC types like the Wave Dragon [8],oint absorber approaches used for the Wavestar device [9], or theEAREV multi-degree-of-freedom point absorber device [10]. Manyontrol strategies have been indicated and reviewed in [2,11].

The non-linear stochastic control of a single wave energy pointonverter without constraints on displacement and control forceas considered in [12]. The expected value (the mean value) of thearvested power was used as objective for optimal control. In casef a linear buoyancy force and linear wave mechanics it was shownn the paper that the optimal controller at a given time is a feedbackontroller with feedback from the present displacement and accel-ration and a non-causal feedback from all future velocities of thebsorber within the considered control horizon. In order to circum-ent the indicated non-causality a causal control law applicable foronlinear buoyancy forces was proposed by a slight modificationf the optimal controller. The basic property of the devised con-rol law is to enforce the wave excitation force into phase withhe velocity of the absorber to insure a constant power supply. Theontroller contains a single undetermined gain factor, which haso be optimized to given irregular sea-state in accordance with thehosen stochastic optimality criteria. The devised controller washown to be optimal under monochromatic wave excitation for apecific choice of the gain parameter.

Hansen and Kramer [13] considered the influence of constrainsn the control force on the mean power outtake of a Wavestar pointonverter based on a PD reactive control law. It was concluded thathe constrains significantly influence the mean power outtake, andhange the values of the optimal gain factors of the PD controller.onstraints on the control force need to be taken in consideration

n praxis in order to prevent large structural stresses in the floatert specific hot spots with the risk of inducing fatigue damage, orecause the demanded control force cannot be supplied by thectuator system due to saturation problems.

As argued by [14] the aim of the control system is to optimizehe generated electrical power rather than the harvested power. Aositive power harvest indicates a power flow from the ocean to

he generator, whereas a negative power implies a power flow inhe opposite direction. Both power flows are related to inevitableower losses. Applying a PD controller they demonstrated that theenerated electric power can be increased significantly by using

esearch 47 (2014) 255–269

the generated electric power rather than the harvested power atthe optimization of the gain parameters of the control law.

Classical optimal control has indeed been applied to waveenergy absorbers before, even with Pontryagins maximum prin-ciple involved for the case of saturation in the power take-off.However, in all cases known to the authors the canonical equa-tions (the equations for the state and co-state vector) of the relatedtwo-point boundary value problem have been solved numerically.The inherent non-causality of the control law has been handled bya state predictor (Kalman- or Luenberger filter). The idea of [12] isto provide a closed-form analytical solution to the control of a sin-gle point absorber with non-linear buoyancy in terms of the basichydrodynamic functions, which are obtained numerically by a BEManalysis. The optimal control law is a pure feed-back controller,with feed-back from the present displacement and acceleration,and all future velocities, which makes the controller non-causal.Hence, a prediction of future velocities, but not of surface eleva-tions, is required for the optimal control. Next, a closely relatedcausal controller is suggested, which of course is sub-optimal. How-ever, it is demonstrated that the suggested causal controller is closeto optimal, and superior to any PD feed-back controller.

In the present paper the problem considered in [12] is revisited.The idea here is to consider the practical limitations on the WEC.Here instead of the harvested power the generated electrical powerwill be the objective at the optimization of the control law. Further,the described control law is modified somewhat to take the indi-cated necessary constraints on the control force into consideration.Depending on the water depth the WEC may hit the sea-bottom atlarge motions with the risk to damage the floater at the impact, orit may make unacceptable jumps out of the water. To prevent theseevents constraints have been imposed on the allowable displace-ments of the WEC in the optimization of the mean of the generatedelectric power. These limitations are of considerable importancesince the thresholds of the control force cause a sudden change insystem’s momentum hence introduces transients into the motionof the device. This indeed changes behavior of the system and can-not be neglected in analysis of the devices. A similar effect will beexpected if the device reaches its ultimates on its range of motion.

The problem with saturation of the control force may be for-mulated as an outcrossing of a stochastic process from fixedboundaries. However, derivation of an analytical solution for powerouttake of such a system requires an extensive study that is beyondscope of this work. In [12] the mean harvested power was eval-uated based on covariance information for the response process,which was evaluated analytically. Due to the strong nonlinearitiesintroduced by the buoyancy force at finite displacements and theintroduced constraints this approach is no longer applicable. To cir-cumvent this difficulty, a computational approach has been takeninto account and the optimal gain values for the proposed con-troller are estimated using Monte Carlo simulations. The effect ofincluding nonlinearity of buoyancy force in the model on the meanpower outtake of the device is studied in details. Effect of thresh-old level of the control force has been extensively studied in thepaper. Proper ways of treating thresholds in simulations have beenaddressed. Extensive discussion on the effect of various parametersof the controller on power extraction of the device are provided.Finally, conclusions are supported by large number of simulationsfor various cases.

2. Equation of motion of a WEC

2.1. Hydrodynamic forces

The motion of the WEC is described relative to the (x,y, z)-coordinate system shown in Fig. 1. The figure shows acylindrical heave absorber in the static and dynamic deformed

M.T. Sichani et al. / Applied Ocean R

generator

a) b)z

xyMWL O

G

G mg + fp,0

mg + fp,0

η(t)

fb,0

fc(t) fh(t) + fb,0

v(t)

shcf(atetta

f

f

f

wpIdbtbWTb

m

At

f

fsfaf

f

A

f

kasi

f

Fig. 1. Loads on heave absorber. (a) Static equilibrium state. (b) Dynamic state.

tates. Although, the analysis will be performed for the indicatedeave absorber all results including the equation of motion andontrol laws may easily be carried over to other single-degree-of-reedom systems by slight modifications. Only two-dimensionalplane) regular or irregular waves are considered, which aressumed to propagate in the positive x-direction. The motion v(t) ofhe body in the vertical z-direction is defined relative to the staticquilibrium state, where the static buoyancy force fb,0 is balancinghe gravity force mg and a possible static pre-stressing force fromhe generator fp,0. m denotes the structural mass including ballast,nd g is the acceleration of gravity. Hence,

b,0 = mg + fp,0 (1)

b,0 is given by the Archimedes’ law

b,0 = �V(0)g (2)

here � denotes the mass density of water, and V(0) is the dis-laced water volume V(v(t)) in the static equilibrium state v(t) = 0.

n the dynamic state the WEC is excited by an additional hydro-ynamic force fh(t) at the top of the static buoyancy force, andy a an additional dynamic reactive generator force fc(t) at theop of the generator pre-stressing force. fc(t) can to certain extente prescribed, so it may be used to control the motion of theEC, for which reason fc(t) will be referred to as the control force.

hen, assuming linear structural dynamics the equation of motionecomes

v(t) = fh(t) − fc(t) (3)

ssuming linear wave theory, fh(t) may be written as a superposi-ion of the following contributions

h(t) = fb(t) + fr(t) + fe(t) (4)

b(t) denotes the increment of the buoyancy force during quasi-tatic motions from the static equilibrium state, fr(t) is the radiationorce, and fe(t) is the wave excitation force caused by the wavection, when the absorber is fixed in the static equilibrium state.

b(t) is given as

b(t) = �(V(v(t)) − V(0))g = −r(v(t)) (5)

ssuming small vertical motions, fb(t) may be linearized as

b(t) = �V ′(0)gv(t) = −kv(t), k = −�V ′(0)g = �gA (6)

is the hydrostatic stiffness coefficient, and A is the water planerea at the static equilibrium state. k may also include an elastictiffness contribution from a mooring system. fr(t) may be written

n terms of the following differential-integro relation [15,4]

r(t) = −mhv(t) −∫ t

−∞hr v(t − �) v(�) d� (7)

esearch 47 (2014) 255–269 257

mh is the added water mass at infinite frequency, and hr v(t) is theimpulse response function for the radiation force caused by theabsorber velocity v(�). Being caused by the velocity of the absorberthe impulse response function hr v(t) in the convolution integral (7)must be causal i.e. hr v(t) = 0, t < 0. The Frequency Response Func-tion (FRF) for the radiation force and the wave excitation forceare shown in Fig. 2a. The wave excitation force fe(t) may be givenin terms of the following convolution integral of the sea-surfaceelevation �(t) [4]

fe(t) =∫ ∞

−∞he�(t − �)�(�) d� (8)

�(t) refers to the sea-surface elevation observed at a sufficient dis-tant position, so no disturbances from radiation waves are present.The impulse response function he�(t) is not causal [3]. This is so,because the fe(t) is not caused by �(t). Rather, fe(t) and �(t) are bothresponse quantities to same hidden physical cause, e.g. by distantwind shear forces on the sea-surface or by a wave generator in awave tank. The related frequency response function becomes

He�(ω) =∫ ∞

−∞e−iωthe�(t) dt (9)

The hydrodynamic parameters and functions k, mh, Hr v(ω), andHe�(ω) can be calculated by linear potential theory. In the presentpaper the programme WAMIT [16] based on the boundary elementmethod (BEM), has been used. The magnitude and phase of the fre-quency response function of the wave excitation force are shownin Fig. 2b.

Since the real parts are symmetric functions of ω, and the imag-inary parts are skew-symmetric functions of ω, results have onlybeen indicated for positive angular frequencies. Insertion of (4), (5)and (7) in (3) provides the following integro-differential equationof motion for the absorber

(m + mh)v(t) + r(v(t))

+∫ t

0

hr v(t − �) v(�) d� = fe(t) − fc(t) t > 0

v(0) = v0, , v(0) = v0

⎫⎪⎬⎪⎭ (10)

where v0 and v0 indicate given initial conditions.

2.2. Constraints on motion and control force

The absorber is not allowed to jump out of the water. Neither,is it allowed to hit the bottom of the sea. Hence, allowable motionsof the absorber are constrained as follows

vmin ≤ v(t) ≤ vmax (11)

In reality the values of vmax and vmin are decided based on phys-ical limits in mooring and/or PTO-stroke lengths (see Fig. 3a). hdenotes the water depth. Here as a simple yet physically meaning-ful assumption vmax is considered the distance from the bottom ofthe floater to the sea surface, and −vmin = h − vmax is the distancefrom the bottom of the floater to the sea bottom,

The control force fc(t) may induce critical stresses �j(t) = bjfc(t)j = 1, 2 at hot spots on the floater (see Fig. 3b). bj denote stressintensity factors, which may be calculated numerically by the finiteelement method. The indicated stresses may cause yielding orfatigue damage, for which reason limits on their allowable vari-ation must be specified. Additionally, the control force can only bespecified within certain limits due to saturation in the hydraulic

actuator system. In combination, the indicated limitations implythe following constraints on the control force

fc,min ≤ fc(t) ≤ fc,max (12)

258 M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269

Fig. 2. Frequency response functions: (a) real and imaginary parts of the radiation force Hr v(ω), (b) amplitude and phase of the wave excitation He�(ω).

a) b)

Statical reference state

v(t)

hvmax

−vmin

Fc

σ 1σ 2

F t absoo

T

f

F

f

w

H

2

f

wtw

f

Tttd

ig. 3. Constraint on allowable displacement and control force of a Wavestar poinn allowable stresses in hot spots or due to saturation of control force.

Let fc,0(t) denote the control force, when no constraint is active.hen, the constrained control force is given as

c(t) =

⎧⎪⎨⎪⎩

fc,max, fc,0(t)≥fc,max

fc,0(t), fc,min < fc,0(t) < fc,max

fc,min, fc,0(t) ≤ fc,min

(13)

ormally, (13) may be written as

c(t) = H(fc,0(t) − fc,max)fc,max + H(fc,min − fc,0(t))fc,min

+ (1 − H(fc,0(t) − fc,max))(1 − H(fc,min − fc,0(t)))fc,0(t) (14)

here H(·) denotes the Heaviside’s unit step function

(x) ={

1, x≥0

0, x < 0(15)

.3. Unconstrained control force

The unconstrained control force is taken as the causal controller

c(t) = −(m + mh)v(t) − r(v(t)) + 2cc v(t) − fr,0(t)

= − (m + mh)v(t) − r(v(t)) + 2cc v(t) −∫ t

−∞hr v(t − �) v(�) d�

(16)

here fr,0(t) = −fr(t) − mhv(t) =∫ t

−∞ hr v(t − �) v(�) d�, c.f. (7). Then,he control law enforces the velocity of the absorber v(t) into phaseith fe(t) as given by

e(t) = 2cc v(t) (17)

his will work in the stationary state, where the influence fromhe initial value response has been dissipated. However, each timehe absorber leaves a displacement or control force constraint, theisplacement and velocity of the absorber at the time where the

rber, (Wave star energy). (a) Constraint on allowable displacement. (b) Constraint

said constraint is released, will introduce a transient motion in thesucceeding unconstrained state, which is unaffected by the controllaw (16). The transients are merely damped eigenvibrations at thesystem’s damped natural frequency. A proportional controller canbe tuned to absorb maximum mean power under monochromaticwave excitation. For this reason it is believed that a proportionalcontrol force component kcv(t) may be used to absorb the transientmotions. In any case such a term needs to be introduced at practicalimplementations of the controller in order to prevent a drift of theabsorber. The displacement constraints (11) are taken into consid-eration by adding large control buoyancy stiffness to the nonlinearbuoyancy force fb(t) = −r(v(t)) in the vicinity of the displacementconstraints as indicated by the functions −r1(v(t)) and −r2(v(t)). Theresulting control buoyancy force −rc(v) has been shown in Fig. 5.Then, the unconstrained control force attains the form

fc,0(t) = −(m + mh)v(t) − rc(v(t)) + kcv(t) + 2cc v(t)

−∫ t

0

hr v(t − �)v(�) d� (18)

The unprovided control gain parameters kc and cc are determined,so the constrained control force (14) absorbs maximum meanpower. With reference to Fig. 4 the motion of the absorber in theinterval [t1, t4] is described, where an upper control force con-straint is active during the sub-interval [t1, t2], followed by anun-constraint motion during the interval ]t2, t3[, and ending witha constrained motion in the interval [t3, t4], where a lower controlforce constraint is prescribed. During the interval ]t1, t2] the motionof the absorber is defined by the following integro-differentialequation of motion, cf. (10)∫ t ⎫⎪⎪

(m + mh)v(t) + r(v(t)) +

t1

hr v(t − �) v(�) d�

= fe(t) − fc,max, t > t1

v(t1) = v1, v(t1) = v1

⎬⎪⎪⎭ (19)

M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269 259

fc(t)

t

fc,max

fc,min

t1 t2 t3 t4

fc,0(t)

fc,0(t)fc,0(t)

Fig. 4. Constrained control force.

−rc(v)

−r1(v)

−r2(v)

εv

vmin vmax

ε

Fm

waadtif

w

�

tvtd

Tfotv

Ph(t)

tPh = E Ph

−

P+h (t)

ig. 5. Control buoyancy force with additional control stiffness close to the displace-ent constrains.

here v1 and v1 denote the displacement and velocity of thebsorber at the instant of time t1, where the constraint becomesctive. At the time t2 the constraint becomes inactive, where theisplacement is v2 = v(t2) and the velocity v2 = v(t2). Then, withhe unconstrained control law (18) the displacement and veloc-ty response during the interval ]t2, t3[ is obtained by solving theollowing differential equations

�r(v(t)) + kcv(t) + 2cc v(t) = fe(t)

(∂�r(v(t))

∂v+ kc

)v(t) + 2cc v(t) = fe(t)

⎫⎪⎬⎪⎭ , t > t2

v(t2) = v2, v(t2) = v2

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(20)

here

r(v(t)) = r(v(t)) − rc(v(t)) (21)

Obviously, �r(v(t)) = 0 for v ∈ [vmin + ε, vmax − ε]. At the time3 the lower constraint becomes active, where the displacement is3 = v(t3) and the velocity v3 = v(t3). Then, during the interval ]t3,4] the motion of the absorber is defined by the following integro-ifferential equation of motion, cf. (10)

(m + mh)v(t) + r(v(t)) +∫ t

t1

hr v(t − �) v(�) d�

= fe(t) − fc,min, t > t3

v(t3) = v3, v(t3) = v3

⎫⎪⎪⎬⎪⎪⎭ (22)

hroughout the time integration the unconstrained control force

c,0(t) as given by (18) is updated. The activation and deactivationf the constraints is decided based on the sign and magnitude ofhis quantity. In case the absorber hits the bottom despite the pre-entive measure imposed via the unconstrained control law, the

Ph (t)

Fig. 6. Positive and negative loops of harvested power.

absorber displacement should be set to v(t) = vmin during the timeintegration as long as v(t) < 0.

Fig. 6 shows the instantaneous harvested power Ph(t) = fc(t) v(t),and P+

h(t) and P−

h(t) describes positive and negative side loops. At

positive power harvest the power is transferred from the oceanto the generator, whereas at negative power harvest the power istransferred in the opposite direction. The mean harvested power ofthe device using ergodic sampling can be calculated as

Ph(T) = 1T

∫ T

0

fc(t) v(t)dt (23)

The mean (expected) value of the harvested power for theunconstrained control force (18) reads

Ph(t) = −E{(m + mh)V(t)V(t)} − E{r(V(t)V(t))}

−∫ t

−∞hr v(t − �)E{V(�)V(t)}d� + 2ccE{V(t)V(t)}

+ kcE{V(t)V(t)} (24)

The first two terms vanish since E{v(t) v(t)} = E{v(t)}E{ v(t)} = 0and E{r(v(t)V(t))} = E{r(v(t))}E{V(t)} = 0 assuming independencyof the two terms V(t) and r(v(t)). Next due to the independence ofdisplacement and its time derivative we have E{V(t)V(t)} = 0. Dueto the stationarity Ph = Ph(t) becomes independent of t as followsby introducing the change of variables u = t − � in Eq. (24)

Ph = 2cc�2V

−∫ ∞

0

hr v(u)VV (u)du (25)

insertion of (18) into (10) results in the following control law

2cc v(t) + kcv(t) = fe(t) (26)

which can be used to obtain the auto covariance function of thevelocity in this case

SVV (ω) = 1

4c2c + (k2

c /ω2)SFeFe (ω) (27)

VV (�) =∫ ∞

−∞eiω� SFeFe (ω)

4c2c + (k2

c /ω2)dω (28)

using (27) and (28) and in (25) results in

Ph = 2cc�2V

−∫ ∞

0

∫ ∞

−∞eiω� hr v(�)

SFeFe (ω)

4c2c + (k2

c /ω2)dω d� (29)

since hr v(�) = 0 , � < 0 hence

Ph =∫ ∞

−∞(2cc − Ch(ω))

SFeFe (ω)

4c2 + k2c

dω (30)

c ω2

where Ch(ω) = Re(Hr v(ω)) is the hydrodynamic radiation dampingcoefficient. (30) shows the harvested power of the system withunconstrained controller.

2 cean R

2

f

wpc(

P

Tdort

P

Pt

3

ptGdpv

S

w

Tqgbtebs

�

�

wd

gravity G is placed at a distance a below MWL in the static referentialstate, and the water depth is h. The relevant data of the absorberand the wave excitation have been indicated in Table 1.

hD2

O

D

G

a

z

x

H

y

vmax

−vmin

staticequilibrium

state

MWL

60 M.T. Sichani et al. / Applied O

.4. Power outtake of point absorber

The indicated power transfers are related with inevitable lossesor which reason the following relations apply [14]

P+e (t) = �+P+

h(t)

P−h

(t) = �−P−e (t)

}(31)

here P+e (t) and P−

e (t) indicate the obtained and lost electricalower of the device and �+ ∈ [0, 1] and �− ∈ [0, 1] denote the effi-iency coefficients related to the respective power flows. Based on31) the instantaneous electrical power, Pe(t), can be calculated as

e(t) = P+e (t) + P−

e (t) = �+P+h

(t) + 1�− P−

h(t)

=(

�+H(fc(t) v(t)) + 1�− (1 − H(fc(t) v(t)))

)fc(t) v(t) (32)

he instantaneous harvested power is a random variable, whichepends on the gain parameter cc, i.e. P(t) = P(t ; cc). The idea is toptimize the expected value (the mean value) of this quantity withespect to cc and kc in the stationary state as t→ ∞, correspondingo

e,opt = supcc,kc

limt→∞

E[Pe(t; cc, kc)] (33)

e,opt indicates the optimal mean electric harvested power underhe constraints (11) and (12).

. Stochastic wave load model

The irregular plane waves are assumed to propagate in theositive x-direction. The surface elevation at the position x at theime t is described by the homogeneous and stationary zero meanaussian process {�(x, t), (x, t) ∈ R2}. The double sided auto-spectralensity function of the sea-surface elevation process at a givenosition is defined by the following slightly modified double-sidedersion of the JONSWAP spectrum [6]

��(ω) = H2s

ωp�˛

(|ω|ωp

)−5

exp

(−5

4

(ω

ωp

)−4)

(34)

here

= exp

(−1

2

(|ω| − ωp

�f ωp

)2)

Tp = 2�

ωp

� = 3.3

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(35)

p is the peak period, ωp = 2�/Tp is the related angular peak fre-uency, �f is a band-width parameter and g is the acceleration ofravity. As seen, the band-width parameter �f has been assumed toe frequency independent in contrast to the original formulation ofhe spectrum [6]. However the modification poses negligible differ-nce to the original power spectrum. Notice that the abscissa haseen normalized with respect to the angular peak frequency ωp. Theurface elevation field admits the following spectral representation

(x, t) =J∑

j=1

√2�j cos(ωjt − kj x − ˚j), ωj = (j − 1)�ω (36)

j =√

2S��(ωj)�ω (37)

here J is the number of harmonic components in the spectralecomposition, �j denotes the standard deviation of harmonic

esearch 47 (2014) 255–269

components with angular frequencies in the interval ]ωj, ωj + �ω],and ˚j are mutually independent identical distributed randomvariables, uniformly distributed in [0, 2�]. kj is the wave numberof the jth harmonic wave component. This is related to the angularfrequency ωj by the well-known dispersion relation of lineargravity waves [17]

kjh tanh(kjh) =ω2

jh

g(38)

The phases ˚j are referred to as the basic variables of the model.The number J of these variables is typically 102–104. The wave exci-tation force can be calculated from its frequency response function(9) which may be written on the form

He�(ω) = |He�(ω)|e−i ϕ(ω) (39)

where |He�(ω)| denotes the modulus, and the negative argumentϕ(ω) indicates the phase lag of the wave excitation force relative tothe surface elevation, c.f. Fig. 2b, given as

tan ϕ(ω) = − Im(He�(ω))Re(He�(ω))

(40)

(39) is assumed to indicate the transfer function for the surfaceelevation at the position x = 0 of the absorber. Then, combining (34)and (39) provides the following spectral representations for thewave excitation force

fe(t) =J∑

j=1

√2 fj cos(ωjt − ˚j − ϕ(ωj)) (41)

fj = |He�(ω)|�j =√

2|He�(ωj)|2S��(ωj)�ω (42)

4. Numerical modeling of the WEC

The theory is illustrated with the heave absorber shown in Fig. 7,consisting of a cylindrical volume with a diameter D. The bottomconsists of a hemisphere with the same diameter as the cylinder. Tostabilize the absorber the bottom is filled with ballast. The center of

Fig. 7. Geometry of heave absorber.

M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269 261

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30−2

−1

0

1

2

3

4

5

6x 10

4

WAMIT

State−Space

Real

yranigamI

t [s]

h rv(t)

a) b)

Fig. 8. State space model representation of the radiation force: (a) impulse response function, (b) poles of the state space model.

Table 1Heave absorber and wave excitation parameters.

a 2.00 m �f 0.1 NH 7.00 m m 1.84×106 kgD 14.00 m mh 0.44×106 kgh 30.00 m �+ 0.75vmax 14.00 m �− 0.75

rW

tRitsm

d

−

Tsaietdd

ef

-15 -10 -5 0 5 10

-15

-10

-5

0

5

10

−rc(v)/k

]m[

v [m]

11

vmin vmax

vmin −16.00 m Tp 7.45 sε 0.25 m c0 1.0×105 Ns/mHs 3.00 m

In order to implement the radiation force (7) a state space rep-esentation has been calibrated to the hydrodynamic results of

AMIT. Then (7) can be written in the following state space form

x(t) = Ax(t) + B v(t)

fr(t) + mhv(t) = Cx(t) + D v(t)(43)

The Prony method [18] with order 14 has been used to calibratehe state space model for the radiation force in (43). The Impulseesponse Function (IRF) of the calibrated radiation force is shown

n Fig. 8a. As seen in this figure, calibrated IRF fits perfectly to thearget IRF obtained by WAMIT. The poles of the calibrated discretetate space model together with the unit circle, e.g. the stabilityargin, are shown in Fig. 8b.The nonlinear buoyancy force with artificial stiffness close to the

isplacement constraints included reads

c(v)k

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0.5016.00 + v

, v ∈] − 16.00, −15.75]

2, v ∈] − 15.75, −2.00]

−v, v ∈] − 2.00, 7.00]

−v + (v − 7)3

147, v ∈]7.00, 13.75]

− 2.914514.00 − v

, v ∈]13.75, 14.00[

(44)

he buoyancy force has been normalized with respect to the hydro-tatic stiffness coefficient k as given by (6), and v as well as fb(v)/kre in m. Notice, the results for v ∈]7.00, 13.75[ are merely approx-mations, obtained by a cubic polynomial interpolation fitting thexact ordinates and slopes at the endpoints of the interval. As seen,he artificial stiffness at the end of the allowable interval for theisplacement are assumed to increase inverse proportional to the

istance from the constraint c.f. Fig. 9.

Before proceeding further, we present the effect of the nonlin-ar stiffness in the model. Hence simulations have been performedor the linear case where the stiffness of the model is according

Fig. 9. Nonlinear buoyancy component of the control force with additional stiffnessclose to the displacement constraints.

to (6). Fig. 10 shows that considering the buoyancy force as anonlinear function of the displacement, changes the behavior ofthe responses. As seen in Fig. 10a when the device immerses in thewater the stiffness which is due to the buoyancy force is constant i.e.the constant negative buoyancy force within the time range t ∈ [60,140] s. This has an effect on the velocity of the WEC such that ignor-ing the nonlinearity in the buoyancy force will cause amplificationon the velocity of the WEC, c.f. Fig. 10c. This affects the harvested aswell as the electrical harvested power of the device. On the otherhand since in the nonlinear case the buoyancy force is not increas-ing proportional to the displacement when the device is immersedin water, c.f. Fig. 9, the controller needs to use more power topush the system into resonance hence the harvested power of thenonlinear model may decrease compared to the linear model.

Fig. 11 shows samples of the sea surface elevation, control forceand displacement of the nonlinear WEC with and without con-troller. The figure emphasizes the effect of the controller on theresponses of the system. In these simulations optimal value of thecc for the unconstrained controller has been used e.g. c0. This canbe calculated by invoking (30) and setting kc = 0 which results in∫ ∞

Ch(ω)SFeFe (ω) dω∫ ∞

c0 = maxcc

P(cc, t) = −∞∫ ∞−∞

SFeFe (ω) dω= 1

�2FeFe 0

hr v(�)FeFe (�)d� (45)

Fig. 12 shows the effect of the constraints – imposed on thecontrol force – on system responses in comparison with the

262 M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269

0 20 40 60 80 100 120 140 160 180 200−1

−0.5

0

0.5

1x 10

7

0 20 40 60 80 100 120 140 160 180 200−10

−5

0

5

10

0 20 40 60 80 100 120 140 160 180 200−10

−5

0

5

10

a)

b)

c)

t [s]

t [s]

t [s]r cv(t )

]N[

v (t

]m[)

v (tm[)/]s

F roller;d

umf

4

ipctc

Fd

ig. 10. Responses of the linear and nonlinear wave energy converter without contisplacement, (c) velocity.

nconstrained control force. It is seen that system responses,ainly in the regions where the constraints are activated, deviate

rom the system responses of the unconstrained controller.

.1. Effect of the controller parameters on the power outtake

The effect of the damping on the harvested power of the devices shown in Fig. 13. Fig. 13a represents convergence of the harvested

ower of the device using ergodic sampling for the unconstrainedontroller. The solid curve in this figure shows the convergence ofhe harvested power outtake of the device for the damping valuec = 1 ×105 [Ns/m] while the dashed curve shows the convergence

0 20 40 60 80 10−4

−2

0

2

4

0 20 40 60 80 10−1

0

1x 10

7

0 20 40 60 80 10−10

0

10

a)

b)

c)

t

t

t

f c( t

]N[)

v(t

]m[)

η(t

]m[)

ig. 11. Responses of the nonlinear wave energy converter; : with controlleisplacement.

: linear system, : nonlinear system; (a) buoyancy force, (b)

for the damping value cc = 4 ×105 [Ns/m]. It is seen that the deviceembedding controller with higher damping value, e.g. the dashedcurve, harvests less power than the device with less damping. Thissuggests that for the device with unconstraint controller increasingdamping of the controller decreases the harvested power of theWEC.

Fig. 13b shows the harvested power of the same device but forthe case that the controller has a constraint on its applied force

6

equal to fc,max = 2 ×10 [N]. The solid curve and the dashed curve inthis figure have similar definitions to their counterparts in Fig. 13a.However unlike the previous case, here the total harvested powerof the WEC – after 200 s – increases with damping of the controller

0 120 140 160 180 200

0 120 140 160 180 200

0 120 140 160 180 200[s]

[s]

[s]

r, : without controller; (a) sea surface elevation, (b) control force, (c)

M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269 263

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

0 20 40 60 80 100 120 140 160 180 200−1

0

1x 10

7

0 20 40 60 80 100 120 140 160 180 200−10

0

10

a)

b)

c)

t [S]

t [S]

t [S]

f c(t

]N[)

v(t

]m [)

η(t

]m[)

F ×106 [s

icct

tttoPfpo

east

Fcc

controller of the WEC both affect the harvested power but in dif-

ig. 12. Effect of the constraints on the controller on the nonlinear model, fc,max = 2

ea surface elevation, (b) control force, (c) displacement.

.e. the dashed curve converges to a higher value than the solidurve after 200 s. This is in contrast to the previous case where theontroller did not have any constraints on the force it can apply tohe device.

These results suggest that the harvested power is not only func-ion of the damping of the controller but also the saturation level ofhe controller. However such an analysis cannot determine the rela-ionship between the damping of the controller, cc, saturation levelf the control force fc,max and the harvested power of the deviceh(t). Hence a more detailed study of this effect needs to be per-ormed to determine in more detail the relationship between thesearameters. This analysis has been performed in the following partf the paper.

On the other hand clearly the stiffness of the controller will influ-

nce the harvested power as well. Therefore a similar analysis fornalyzing the effect of this term is also taken into account. Fig. 14ahows the harvested power of the unconstrained controller andwo different value of the stiffness kc. The damping term of the

20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5x 10

5

20 40 60 80 100 120 140 160 180 200−2

−1

0

1

2x 10

5

a)

b)

t [s]

t [s]

Ph(t

]W[)

Ph( t

]W[)

ig. 13. Convergence of the harvested power by ergodic sampling; (a) unconstrainedontroller, (b) constrained controller, fc,max = 2 ×105 [N]; —: cc = 1 ×105 [Ns/m], - - -:c = 4 ×105 [Ns/m].

N]; : unconstrained controller, : constrained controller; (a)

controller has been set to the fixed value cc = 1 ×105 [Ns/m]. Thestiffness of the controller has been chosen proportional to the stiff-ness of the device in the linear range e.g. k = �gA. For this analysis inthe first case the stiffness term is set to kc = 0.025k e.g. solid line, andin the second case to kc = 0.5k e.g. dashed line. It is seen that increas-ing the stiffness for the device having an unconstrained controllerdecreases the harvested power.

In the next step a constraint of fc,max = 2 ×105 [N] is imposed onthe control force for the same two cases of stiffness as done in forthe unconstrained controller and its results are shown in Fig. 14b.It is seen that the harvested power of the WEC with constrainedcontroller also decreases when increasing the stiffness of the con-troller. This suggests that the stiffness and damping terms in the

ferent manners. Based on these observations an extended analysisis carried out in the following section to uncover the relationshipbetween these parameters.

20 40 60 80 100 120 140 160 180 200−2

0

2

4

6x 10

5

20 40 60 80 100 120 140 160 180 200−2

−1

0

1x 10

5

a)

b)

t [s]

t [s]

Ph(t

]W[)

Ph( t

]W[)

Fig. 14. Convergence of the harvested power by ergodic sampling; (a) uncon-strained controller, (b) constrained controller, fc,max = 2 ×105 [N]; —: kc = 0.025 k, - - -:kc = 0.5 k.

264 M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269

er of t

4

lwghsdtiTfic

Fig. 15. Harvested pow

.2. Harvested power outtake

The dependency of the harvested power of the WEC with non-inear buoyancy force is obtained using Monte Carlo simulation –

ith ergodic sampling for convergence. In order to ensure conver-ence of results, time series of 800 s have been simulated and meanarvested power for each sample has been obtained using ergodicampling. For each sample of the sea surface elevation and fc,max, 50ifferent values of cc were considered e.g. x-axis in Fig. 15. In ordero decrease the uncertainty of the estimations the same analysiss performed for 20 random sea states i.e. Monte Carlo simulation.

he estimated mean value for all these sea states has served as thenal estimation of the mean harvested power and plotted as oneurve in Fig. 15. This analysis is repeated for 6 different values of

Fig. 16. Harvested power of the

he linear system, kc = 0.

fc,max, shown in the caption of the Fig. 15, plotted together with theunconstrained controller results.

Fig. 16 shows the mean harvested power of the WEC withnonlinear buoyancy force versus controller’s damping value cc.Comparison of Figs. 15 and 16 shows the effect of including nonlin-earities in the buoyancy force of the WEC on the mean harvestedpower of the device. It is worth noting that using Monte Carlo sim-ulation with more than 20 samples for some representative casesdid not make noticeable improvement to the results shown in thepaper however the computation cost increases considerably (sim-ulating the results for either linear or nonlinear cases on an Intel(R)

Core(TM) i7 CPU with 8 GB memory needs 142 min time).

In the first sight it is noted that for the unconstrained controllerthe harvested power of both the linear and nonlinear WECs have

nonlinear system, kc = 0.

M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269 265

r of th

vpittsptwpcb

tI

Fig. 17. Harvested powe

ery similar trends. Indeed only in this special case the harvestedower of the two devices coincide closely. This can be explained by

nvoking Eq. (26) that governs the motion of the WEC for the con-roller force given in (18). It is noted that (26) has no dependency onhe damping, stiffness and mass of the WEC. Indeed for the uncon-trained controller the system’s behavior is governed merely by thearameters of controller namely cc and kc, c.f. (26). Hence no mat-er the buoyancy force is considered linear or nonlinear, the deviceill experience the same motion determined by the controller’sarameters. Clearly the same conclusions can be made for the spe-ial case of controller with only damping term that may be obtained

y setting kc = 0 [N/m] in (18) and (26).

Nevertheless when constraints are imposed on the controllerhe harvested power of the two models deviate to some extent.ndeed setting constraints on the applicable control force has

Fig. 18. Harvested power of the

e linear system, cc = c0.

more effect on the harvested power of the nonlinear WEC com-pared to the linear WEC. This is evident by noting that i.e. thedashed curve, for fc,max = 1 ×106 [N], has decreased more signif-icantly in Fig. 16 than Fig. 15 compared to the solid curves.Moreover it is seen that constraints on the maximum applicablecontrol load increase the optimal damping value of the controllercc. This is evident by noting that peak of the curves with con-strained controllers move more and more toward right as theconstrains on the control force becomes more and more strict, c.f.Fig. 15 and 16.

The mean harvested power of the linear and nonlinear WECs

with respect to the stiffness terms of the controller are analyzed ina similar way to damping and shown in Figs. 17 and 18 respectively.The figures show that the harvested power of both the linear andnonlinear WEC models decrease as the gain of the stiffness term of

nonlinear system, cc = c0.

266 M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269

powe

tsgcidm

ufsfoqw

Fig. 19. Electrical harvested

he controller kc increases. This suggests that the least amount oftiffness may be specified to the controller since additional stiffnessenerally decreases the power outtake. Comparing Figs. 17 and 18onfirms that the mean harvested power of the nonlinear models more affected by the constraints on the controller force i.e. theistance between solid and dashed lines for the nonlinear case isore accentuated.The harvested power of both models however shows an

nmarked peak for high threshold level of the controller i.e.c,max ≥ 0.4 × 106 [N]. This peak happens when the controller addsmall amount of stiffness to the device i.e. kc < 0.1k. The reason

or this can be the stiffness term changes the natural frequencyf the system, it has the ability to make the system’s natural fre-uency close to the frequency components of the coming waveshich carry most of the wave energy e.g. the peak of the power

Fig. 20. Electrical harvested power o

r of the linear system, kc = 0.

spectrum of the sea surface elevation S��(ω). In such a case clearlythe absorbed amount of electrical power will be maximized byadding some stiffness (could be negative) to the device throughthe controller. However the value needed for this depends on thenatural frequency of the original WEC device hence the device con-figurations.

4.3. Electrical power outtake

Since the electrical power outtake of the device is the practicaloutput of the device in need of being optimized, in the following

electrical power is calculated as a function of the damping and stiff-ness of the controller. Figs. 19 and 20 show the electrical powerouttake of the linear and nonlinear models with respect to damp-ing coefficient of the controller respectively. Here the stiffness term

f the nonlinear system, kc = 0.

M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269 267

wer of

oieteomohflcouf

Fig. 21. Electrical harvested po

f the controller is set to zero, kc = 0. Comparing Figs. 15 and 19 its noted that the optimal damping value of the harvested power,.g. c0, does not maximize the electrical power outtake. Indeed ifhis value is chosen for the damping of the controller, the extractedlectrical power would be negative i.e. there is no electrical poweruttake from the device. In fact the optimal damping value foraximizing electrical power outtake is more than two times the

ptimal damping value for harvested power e.g. cc = 2.156 c0. Thisolds true for both linear and nonlinear models. Actually it is seen

rom Figs. 19 and 20 that the electrical power of the linear and non-inear models not only for the unconstrained controller but also for

ontrollers with different threshold levels are very close to eachther. Hence there seems to be not much loss of information bysing linear model instead of the nonlinear model for buoyancyorce in this regard.

Fig. 22. Electrical harvested power of th

the linear system, cc = 2.156c0.

For the final part of our analysis we set the damping of thecontroller to its optimum value for maximizing electrical powerouttake and analyze the effect of adding stiffness term to the con-troller. The results of this analysis for linear and nonlinear devicesare shown in Figs. 21 and 22 respectively. Again it is seen that in thiscase the difference between the two models is very small makingthe linear model an eligible candidate for electrical power estima-tion in the presented working range. Moreover unlike the harvestedpower, the electrical power shows a marked peak for high thresholdlevel of the controller i.e. fc,max ≥ 0.4 × 106 [N].

4.4. Performance of the controller

In order to demonstrate the efficiency of the proposed sub-optimal controller, harvested and electrical power production

e nonlinear system, cc = 2.156c0.

268 M.T. Sichani et al. / Applied Ocean Research 47 (2014) 255–269

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4x 10

5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

x 104

a)

b)

cc

cc

Ph

]W[

Pe

]W[

F powerd

oplkthfF

lhooorlp

3

It

hImpopa

5

epddp

ig. 23. Comparison of performance of different types of controllers, (a) harvested

erivative controller, : proportional-derivative controller.

f this controller is compared to a derivative controller and aroportional-derivative (PD) controller. The control forces of the twoatter cases can be expressed as fc(t) = cc v(t) and fc(t) = cc v(t) +cv(t) respectively. The stiffness coefficient of the sub-optimal andhe PD controllers are chosen kc = 0.5k. The harvested and electricalarvested power outtake of the device, e.g. Ph and Pe respectively,

or different values of the damping of the controllers are shown inig. 23.

All controllers are unconstrained meaning that there is noimit on the value of the control force e.g. controller does notave any saturation limit. In case the controller has a threshold,nce the control force hits it the device is not enforced into res-nance with the sea state any more hence the state quantitiesf the device will be affected; please see Fig. 12c. Regarding theange of the control forces, the standard deviation of the waveoad compared to those of the three unconstrained controllersresented in the paper are as follows: �fe = 3.65 × 105 [N], �f D

c=

.11 × 105 [N], �f PDc

= 3.34 × 105 [N] and �f sub−optc

= 4.15 × 105 [N].

t is seen that the control forces are in the order of magnitude ofhe wave excitation load; hence are reasonable.

The results show that the proposed sub-optimal controller canarvest more electrical power compared to the other two cases.

t is interesting that the derivative controller generally producesore power than the PD-controller. This confirms that in principal

resence of a stiffness term in the controller decreases the poweruttake of the WEC even for other types of controllers than what isroposed in this paper. However as indicated in Section 2.3 a smallmount of stiffness is inevitable in the device.

. Concluding remarks

The focus of this paper is on modeling and analysis of a wavenergy point absorber with a controller. The dynamic modeling

rocedure of the system is described in detail. The nonlinearitiesue to the nonlinear buoyancy force on the point absorber areescribed. A controller with the aim of increasing the harvestedower of the device is implemented in the model and its effect in

, (b) electrical harvested power; : sub-optimal controller, :

the motion of the device and harvested electrical power is studied indetail. The nonlinearities in this model rise from the buoyancy force,geometrical constraints on the motion of the WEC and the satura-tion load of the controller. The dependency of the harvested powerto the controller parameters has been studied in details based onMonte Carlo simulation combined with ergodic sampling.

Regarding the limitations on the saturation load of the con-troller, the lower the maximum available control force the lesspower can be extracted by the device. For controller with high sat-uration load the harvested power has a maximum with respect tothe damping of the controller cc e.g. there exists an optimal valuefor damping. However it is shown that the choice of the optimumvalue depends on the saturation load level of the controller. Theresults suggest that the optimal damping value increases as thesaturation threshold of the controller decreases and the lowestoptimal damping gain will be obtained if the controller has no limiton imposing loads on the system. Next, there exists an optimumvalue for damping coefficient of the controller which maximizesthe electrical power outtake of the WEC. However this value dif-fers from the damping coefficient that maximizes the harvestedpower outtake. Also in this case the optimal value of the dampingcoefficient of controller increases as the saturation load of the con-trol force decreases. However the change of the optimum dampingvalue with threshold of the control force is less than the harvestedpower.

On the other hand addition of stiffness term in the controllerdecreases the harvested power with both the saturated and unsat-urated controller cases. However presence of the stiffness – evenin small values – in the system is necessary since free movement ofthe device is not allowed due to the geometric constraints on themotion of the device. The stiffness term of controller has a similareffect on the electrical power outtake however a small value for itmay slightly increase the electrical power outtake of the device.

Acknowledgements

The authors gratefully acknowledge the financial support fromthe Danish Council for Strategic Research under the Programme

cean R

C0tFa

R

[

[[

[

[

[

[

M.T. Sichani et al. / Applied O

ommission on Sustainable Energy and Environment (Contract 09-67257, Structural Design of Wave Energy Devices) which madehis work possible. The support from the National Natural Scienceoundation of China (NSFC) (Grant No. 51250110538) is gratefullycknowledged by the authors.

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