Maximizing output power of linear generators for wave energy conversion

Maximizing output power of linear generators for waveenergy conversion

Antonio de la Villa Jaén1*,†, Agustín García-Santana2 and Dan El Montoya-Andrade3

1Department of Electrical Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain2Department of Electrical Engineering, AG Ingeniería, Palmas Altas, 41012 Seville, Spain

3Department of Electrical Engineering, Venezuela Central University, Ciudad Universitaria, Caracas, Venezuela

SUMMARY

The aim of wave energy converters is to extract energy from ocean waves and turn it into electricity. Sincethe earliest developments, optimal control strategy has been considered the one that optimizes the powerextracted from the oscillating system. This technique requires energy exchanges between the oscillatingsystem and an auxiliary storage energy system, and has been shown to be potentially capable of substantiallyincreasing the amount of energy absorbed. Nevertheless, it requires very efficient devices due to the hugelosses, which occur when operating conditions are far from the natural frequency of the oscillating system.Moreover, this optimization criteria cause large heave excursions in the oscillating system and high peak-to-average power ratios of the system. In this paper, a point absorber converter that uses a linear direct-drivegenerator and power electronics converters jointly is considered.This article presents a novel method of control, which optimizes the power transferred from the generator

to the power electronic converter considering the copper losses in the electric generator. This approach ofthe optimization method allows a significant increase in the wave energy converter’s capacity of energyconversion. Applying the proposed control strategy, power exchanges between the oscillating system andthe generator can be reduced, and in consequence, system efficiency significantly increases. In addition,the oscillating system’s heave excursions and the peak-to-average power ratio decrease. The formulationof the proposed method is presented as well as numerical simulations in irregular waves. Copyright ©2013 John Wiley & Sons, Ltd.

key words: wave energy; control systems; permanent magnet linear generators; point absorbers

1. INTRODUCTION

Among the emerging electrical power generation choices from renewable sources, the energycontained in the seas and oceans is one of the most promising. A diversity of prototypes has beendeveloped over the last few decades [1]. This paper focuses on a simple heave-buoy system that pow-ers a permanent magnet linear generator (PMLG) [2].The available primary energy features are pulsing and thus, the electrical energy should be properly

adapted to be injected into the grid. One of the most interesting proposals about connecting the waveenergy converter (WEC) to the power grid is based on using linear generators and full-scale back-to-back voltage source converters jointly [3–5].Theoretically, it is possible to establish an optimal control strategy, called reactive control, that

enables the power take-off (PTO) system to capture the maximum amount of energy [6]. The reactivecontrol requires that during certain parts of the cycle, energy has to be delivered from the PTO systemto the oscillating system. In other words, the instantaneous power delivered from the waves to the PTOsystem is sometimes reversed [7], so they are considered active strategies.

*Correspondence to: Antonio de la Villa Jaén, Department of Electrical Engineering, University of Seville, Camino delos Descubrimientos s/n, 41092 Seville, Spain.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMSInt. Trans. Electr. Energ. Syst. 2014; 24:875–890Published online 7 April 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1747

This optimal strategy cannot be applied in irregular waves because they are non-causal systems. Toapproximate the optimal operation conditions, several causal strategies have been reported in previousstudies, for example, [8–11].To apply active strategies when operation conditions move away from the natural frequency of the

oscillating system, the electrical machine and especially the power converters would have to beconsiderably oversized because of the high peak-to-average power ratios of the system [12,13]. It isalso a drawback that optimality conditions can only be achieved if large displacement amplitudesare possible. The end-stop problems will impose a physical constraint, and mechanical friction losseswill also be increased [14].Note that reactive control strategies optimize the PTO power absorption, but do not consider PTO

losses. A control strategy where PTO losses are taken into account assuming that the PTO systemefficiency is constant, is proposed in [15].Direct electrical control represents an attractive option for direct-drive systems largely due to its

simplistic approach [16].Power electronic converters nowadays make it possible to develop reactive control [17]. If this control

strategy is applied to systems that include back-to-back power converters, energy could be delivered fromthe linear generator to the oscillating device during small fractions of the oscillation cycle [18]. Thismeans that the generator switches to work as a motor. In this case, it is necessary to use power conversionto enable both instantaneous power flux directions. In references [4,5], control techniques are applied inthe Archimedes wave swing device to a back-to-back voltage source converter using the dq0 frametransformation of the electrical parameters.This paper proposes a control strategy to be applied on point absorbers that activate linear generators.

The model includes the losses whereas many previous studies only are focusing on the power absorption.First, the power that the linear generator transfers to the power converter is analytically calculated. Forthis, linear generator copper losses are taken into account and are expressed as a function of the hydrody-namic and the PTO system parameters. Next, optimal conditions are calculated to maximize, not theenergy captured by the oscillating system, but the energy transferred from the linear generator to thepower converter in irregular waves. Finally, the optimality conditions obtained are used to tune a causalcontrol strategy considering an irregular wave spectrum. Results show how the application of thisapproach significantly reduces the disadvantages traditionally associated with active control strategies.

2. POINT ABSORBERS AND DIRECT-DRIVE LINEAR GENERATORS

Point absorbers [6] can generally be described as oscillators activated by waves, whose horizontaldimensions are much smaller than the prevailing wavelength. Figure 1 shows a general scheme with dif-ferent stages of a point absorber WEC where the oscillating system is made up of a buoy, a translator ofthe linear generator and a spring. The spring is attached to the translator to act as a restoring force in wavetroughs [19]. The second stage symbolizes the PTO system, which turns the power extracted from wavesinto electrical energy. This conversion process is bidirectional when reactive control strategy is applied. Inthe next stage, power captured from waves by the PTO system is exchanged with the power electronic

Figure 1. General scheme of a direct-drive point absorber.

A. DE LA VILLA JAÉN ET AL.876

Copyright © 2013 John Wiley & Sons, Ltd. Int. Trans. Electr. Energ. Syst. 2014; 24:875–890DOI: 10.1002/etep

converter. The conversion process here can also be bidirectional when reactive control is applied. In thelast stage, the power electronic converter injects the electrical power considering the distribution gridoperation codes [20].

2.1. Hydrodynamic model

It is assumed that the device can move only in the heave motion (i.e., along the vertical axis), with allthe other degrees of freedom ideally restricted. The dynamic equation, which describes the bodymotion with a single degree of freedom (DoF), oscillating in heave is [21]:

fe tð Þ þ fpto tð Þ þ fr tð Þ þ fs tð Þ ¼ m€z tð Þ (1)

where fe is the wave excitation force, which is the sum of pressure forces on the body surface due toincident and diffracted waves; fpto is the force provided by the linear generator; fr is the wave radiationforce due to the radiated wave when the body moves;m is the mass of the oscillating system; z is the heaveexcursion and fs is the net restoring stiffness force, which is the difference between the gravitational andbuoyancy forces, and can be obtained as follows:

fs tð Þ ¼ �c z tð Þ (2)

where the stiffness coefficient cmay be written as c=rgS+ ks being r the water density, g the accelerationof gravity, S the water plane area of the floating body and ks the spring stiffness force constant. The WECmodel assumes that the hydrostatic stiffness force is linear, which implies a constant water-plane area forthe buoy. Moreover, the amplitudes of waves and oscillations are considered sufficiently small for lineartheory to be a good approximation.This paper uses the model in the frequency domain. Thus, applying the Fourier transform to Equation

(1) results

Fe oð Þ þ Fpto oð Þ þ Fr oð Þ � c

joU oð Þ ¼ mjoU oð Þ (3)

where U(o) is the Fourier transform of the velocity. The frequency-response function, also calledradiation-impedance function, can be decomposed into real and imaginary parts

Fr oð ÞU oð Þ ¼ � b oð Þ þ jomadd oð Þ½ � (4)

where b(o) is the radiation damping coefficient and madd(o) is the added mass. Both parameters are realfunctions that depend on the device geometry. Substituting Equation (4) in Equation (3), yields

Fe oð Þ þ Fpto oð Þ ¼ Zi oð ÞU oð Þ (5)

where the complex intrinsic impedance is defined by [6],

Zi oð Þ ¼ b oð Þ þ jxi oð Þ (6)

and the intrinsic reactance is

xi oð Þ ¼ o mþ madd oð Þð Þ � c

o(7)

then, the PTO force can be expressed in terms of PTO complex impedance Zpto(o) [22].

Fpto oð Þ ¼ �Zpto oð ÞU oð Þ (8)

MAXIMIZING POWER OF LINEAR GENERATORS FOR WEC 877


This way, the excitation force and the oscillating device velocity are related by

Fe oð Þ ¼ Znet oð ÞU oð Þ (9)

where the net impedance of the oscillating system is

Znet oð Þ ¼ Zi oð Þ þ Zpto oð Þ (10)

2.2. Optimum control

The energy transferred from the oscillating system to the PTO system during a time interval T is givenby [6]

Epto ¼ �Z T

0fpto tð Þu tð Þdt (11)

where u(t) is the velocity of the oscillating system and T is much larger than the peak period of thewave spectrum. Assuming that the PTO force vanishes for t< 0 and t>T, the limits of integrationin Equation (11) maybe extended to infinity, and this energy can be expressed in the frequency domainby the following expression [15]

Epto ¼ � 12p

Z þ1

�1Fpto oð ÞU� oð Þ do (12)

where the asterisk denotes the complex conjugate. Taking into account Equations (8) and (9)

Epto ¼ 12p

Z þ1

�1Zpto oð Þ jFe oð Þj2

jZnet oð Þj2 do (13)

and considering that the imaginary part of Zpto(o) is an odd function [6]

Epto ¼ 12p

Z þ1

�1rpto oð Þ jFe oð Þj2


It is well known that this energy is maximum when the intrinsic reactance is canceled and the PTOresistance equals the intrinsic resistance [6]. Thus, the so-called optimum complex-conjugate controlmay be expressed by the following condition:

Zoppto oð Þ ¼ Z�

i oð Þ (15)

The following equivalent expressions can also be used considering the real and imaginary parts ofthe PTO impedance

roppto oð Þ ¼ b oð Þ (16)

xoppto oð Þ ¼ �o mþ madd oð Þ½ � þ c

o(17)

2.3. Causal control strategies

When a complex-conjugate control strategy is implemented, optimal generator force depends onfuture values of the buoy’s velocity. This means that optimal generator force cannot be implementedadequately in practice. For this reason, some alternative suboptimal control strategies have been



proposed using causal functions [7,11]. In the latter section, two of them are described and used forthe simulations.

2.3.1. Approximate complex-conjugate control. Approximate complex-conjugate (ACC) controlstrategy is based on a linear second-order model that considers constant PTO coefficients. In this case,the PTO impedance can be expressed as

eZpto oð Þ ¼ erpto þ j oempto �ekptoo

" #(18)

where erpto , empto and ekpto are respectively the damping, mass and spring PTO constant coefficientsassociated with the ACC control. The impedance matching expressed by Equations (16) and (17) isverified for a properly chosen frequency ok which depends on the sea state [11,22], so

erpto ¼ b okð Þ (19)

and

expto ¼ okempto �ekptook

¼ �ok mþ madd okð Þ½ � þ c

ok(20)

Using the peak frequency of the wave spectrum is a common approach [23]. Condition of Equation

(20) involves a single DoF when selecting the parameter values empto andekpto. Several control strategiesare analyzed in [22] depending on the coefficients considered in the PTO impedance. The damping-spring strategy has been the one considered in this paper, so the PTO coefficient values are limited to:

empto ¼ 0 (21)

ekpto ¼ o2k mþ madd okð Þ½ � � c (22)

2.3.2. Resistive loading control. On the other hand, the passive strategy called resistive loading control(RLC) does not require reactive power exchange between the PTO system and the oscillating system.Thus, the electrical machine never has to work as a motor. This strategy is stated in the followingexpression:

Fpto oð Þ ¼ �r�ptoU oð Þ (23)

and the PTO resistance for a chosen frequency ok is [11]

r�pto ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 okð Þ þ x2i okð Þ

q(24)

The �rpto value given by Equation (24) optimizes the power transferred to the PTO system applyingRLC strategy in regular waves.Figures 2 and 3 show the results obtained when the causal control strategies described in this section

are applied to a semi-submerged sphere of radius 2.5m, considering a regular wave period of 4 secondsand a wave height of 1m. Time series of normalized excitation force (the maximum value of theexcitation force is 4.4 � 104N), heave position and velocity signals for this strategy are shown.Figure 2 shows the performance of the ACC strategy when the system is tuned to the wave

frequency. Note that the wave excitation force and velocity are in phase when this control strategyis applied, as described in [6].Figure 3 shows the performance of the passive control. Note that the amplitude of the oscillations is

smaller than in the previous case. Furthermore, the excitation force and velocity are not in phase.



2.4. Electrical system model

The electrical system is composed of a linear generator, a generator side power electronic converter, adirect current (DC) link and a grid side power electronic converter as is shown in Figure 1. Next, themodels used for the linear generator and the power electronics are described.

2.4.1. Linear generator. In this paper, a generic direct-drive three-phase synchronous PMLG is used[1,19,24]. A coordinate transformation for the PMLG between the abc frame of reference and the dq0frame of reference is considered, so the model of the PMLG in dq0 reference frame is used.The dq0 components regarding the linear generator model used in this paper can be expressed as

follows [25]:

vd tð Þ ¼ Rsid tð Þ þ Lsdid tð Þdt

� om tð ÞLsiq tð Þ (25)

vq tð Þ ¼ Rsiq tð Þ þ Lsdiq tð Þdt

þ om tð ÞLsid tð Þ þ om tð Þc (26)

v0 tð Þ ¼ Rsi0 tð Þ þ Lsdi0 tð Þdt

(27)

where om is the angular velocity of the stator variables

om tð Þ ¼ pu tð Þtp

(28)

54 56 58 60 62 64−1.5

−1

−0.5

0

0.5

1

1.5

t (s)

Heave position (m)Velocity (m/s)Normalized excit. force

Figure 3. Time series examples for the passive control strategy.

Figure 2. Time series examples for the approximate complex-conjugate strategy.



and tp is the pole pitch of the PMLG, Rs is the stator resistance, Ls is the stator inductance and c is theflux linkage of the stator windings due to the flux produced by the permanent magnets. Moreover nd,nq, n0, id, iq and i0 are the voltage induced and the stator winding current components in the dq0reference frame. Considering that there is no star-point connection, the sum of the phase voltages iszero and accordingly homopolar component n0 is null.The power at the generator input may be written as [4]

ppto tð Þ ¼ 32ptp

cu tð Þiq tð Þ (29)

Considering that the power produced is negative when the PTO force and the velocity have the samesign, the PTO power may be calculated by [6]

ppto tð Þ ¼ �fpto tð Þu tð Þ (30)

and substituting Equations (29) in (30), yields

fpto tð Þ ¼ � 3pc2tp

iq tð Þ (31)

Note that parameters tp and c only depend on the generator features, thus the PTO force can bedirectly controlled by iq.

2.4.2. Power electronics. The power electronic converter is a bridge rectifier/inverter based on insu-lated gate bipolar transistor devices controlled by pulse-width modulation technique. It is usual touse uncoupled control for the generator side and the grid side power electronic converter. In this sense,generator side power converter control is responsible for reducing copper losses (id = 0) and keepinggenerator reaction force in the value set by the control strategy (iq ¼ irefq ). On the other hand, the gridside power converter control has to keep the DC link voltage and the grid side electrical parameters inacceptable values.The assessment of WEC control strategies is the aim of this paper. Thus, the generator side power

converter control will be the only one considered, assuming the DC link as a DC voltage source. Theaverage model applied by Giroux [26] has been considered in this paper.The average model is based on the energy conservation principle where the instantaneous power

must be the same on the DC side and the alternating current side of the inverter [26].

vdcidc ¼ vaia þ vbib þ vcic (32)

where na,nb,nc,ia,ib,ic are the phase voltages and phase currents supplied by the linear generator to thepower converter. Moreover, ndc and idc are the DC link voltage and current, respectively.

3. INCLUDING COPPER LOSSES IN CONTROL STRATEGY

The proposed control strategy aims to maximize the power transferred to the power converter by thelinear generator.In this section, this power is first formulated as a function of the hydrodynamic and the PTO param-

eters in irregular waves, and then, the conditions that maximize this power are established.The linear generator copper losses can be expressed as follows [27]:

Eloss ¼Z þ1

�1Rs i2a tð Þ þ i2b tð Þ þ i2c tð Þ� �

dt (33)



These losses can be expressed as a function of the dq0 current components [28],

Eloss ¼ Rs

Z þ1

�1

32

i2d tð Þ þ i2q tð Þ þ 2i20 tð Þh i

dt (34)

Because the PMLG is symmetric, the variables of the zero sequence are zero [4].

i0 tð Þ ¼ 0 (35)

Moreover, the d-axis component of the stator current is normally controlled to zero to reduce thepower loss in the stator of the PMLG [4].

id tð Þ ¼ 0 (36)

Substituting Equations (35) and (36) in Equation (34), the following expression is obtained

Eloss ¼ 32Rs

Z þ1

�1i2q tð Þ dt (37)

Using Equation (31) in the expression of Equation (37), generator losses can be expressed as follows

Eloss ¼ dZ þ1

�1f 2pto tð Þ dt (38)

where the auxiliary variable d is

d ¼ 2t2p3p2c2 Rs (39)

Applying Parseval’s theorem in Equation (38), the following expression in frequency domain isobtained

Eloss ¼ d2p

Z þ1

�1Fpto oð Þ 2 do

�� (40)

Using Equations (8) and (9) in Equation (40),

Eloss ¼ d2p

Z þ1

�1jZpto oð Þj2 jFe oð Þj2


The energy transferred from the linear generator to the power converter Ec can be expressed as abalance between the average energy extracted by the PTO system and the linear generator copperlosses. This way, the following expression is obtained by using Equations (14) and (41)

Ec ¼ Epto � Eloss ¼ 12p

Z þ1

�1Fe oð Þj j2 rpto oð Þ � djZpto oð Þj2


This expression yields the average energy delivered to the converter considering the excitationforce, the hydrodynamic characteristics of the oscillating device, the main features of the linear gener-ator and the PTO control parameters.Next, PTO control parameters rpto(o) and xpto(o), which maximize this energy are calculated

analytically. The excitation force does not depend on the control strategy. Thus, the parameter valuesof the control strategy that optimize the expression (Equation (42)) can be obtained by optimizing thefollowing function:



Jc ¼ rpto oð Þ � djZpto oð Þj2jZnet oð Þj2 (43)

Taking into account optimal conditions,

@Jc@rpto oð Þ ¼ 0 ;

@Jc@xpto oð Þ ¼ 0 (44)

The following expressions are obtained through a simplification process when Equation (44) isapplied to Equation (43),

1� 2drpto oð Þ� �Znet oð Þ 2 � 2 rpto oð Þ � d Zpto oð Þ 2

�� rpto oð Þ þ b oð Þ� � ¼ 0�� (45)

dxpto oð Þ Znet oð Þ 2 þ rpto oð Þ � d Zpto oð Þ 2�� xpto oð Þ þ xi oð Þ� � ¼ 0

�� (46)

working with previous expressions, the following linear relation between rpto(o) and xpto(o) can beobtained:

1þ 2db oð Þ½ �xpto oð Þ � 2dxi oð Þrpto oð Þ þ xi oð Þ ¼ 0 (47)

substituting Equation (47) in condition of Equation (45) or Equation (46), and factoring yields

rpto oð Þ þ b oð Þ� �rpto oð Þ � b oð Þ þ 2d b2 oð Þ þ x2i oð Þ� �

g oð Þ� �

¼ 0 (48)

where

g oð Þ ¼ 4d2 b2 oð Þ þ x2i oð Þ� �þ 4db oð Þ þ 1 (49)

Equation (48) has two solutions. On one hand, rpto (o) =�(b)o, and the corresponding valuexpto(o) =� xi(o) that results from the linear expression of Equation (47). Function of Equation(43) becomes a singularity when this solution is considered. On the other hand, the second solutionthat corresponds to the operating point of maximum delivered power is

roppto oð Þ ¼ b oð Þ þ 2d b2 oð Þ þ x2i oð Þ� �g oð Þ (50)

and using the linear expression of Equation (47), yields

xoppto oð Þ ¼ � xi oð Þg oð Þ (51)

Control references assigned to the PTO parameters when generator losses are taken into account inthe control strategy, are obtained from Equations (50) and (51). These control references maximize thepower transferred from the PTO system to the power converter. These functions are a generalization ofthose used by the optimum complex-conjugate control of Equations (16) and (17), which optimizes thepower delivered to the PTO system without considering the losses.Note that the reference values used by the optimum complex-conjugate control, which are given by

Equations (16) and (17), can also be deduced through the proposed strategy by keeping the generatorresistance equal to zero and this way, d = 0 in Equation (39) and g(o) = 1 in Equation (49).In a similar way to the steps performed in Section 2.3.1 regarding the ACC control where the reference

functions of Equations (16) and (17) are implemented by a control strategy that uses the parametersdefined in Equations (19), (21) and (22); next, these parameters are obtained for the proposed control(PC) strategy.



The PTO force can be calculated by a linear second-order equation that includes the PTO coefficients

associated with the PC strategy r̂pto, m̂pto and k̂pto. Thus, a causal control strategy based on the obtainedconditions could be achieved by matching up the new expressions of Equations (50) and (51) with aproperly chosen frequency ok. This way, the following expressions are obtained

r̂pto ¼b okð Þ þ 2d b2 okð Þ þ x2i okð Þ� �

g okð Þ (52)

and

x̂pto ¼ okm̂pto�k̂ptook

¼ �ok mþ madd okð Þ½ � þ cok

g okð Þ (53)

The single DoF available when matching m̂pto and k̂pto parameters, allows different strategies to beimplemented. The damping-spring strategy has been the one considered in this paper, so the PTO co-efficient values are limited to:

m̂pto ¼ 0 (54)

k̂pto ¼ o2k mþ madd okð Þ½ � � c

g okð Þ (55)

The new parameters m̂pto and k̂pto are related to the ones obtained by the equivalent ACC controlstrategy, see expressions of Equations (21) and (22), by

m̂pto ¼ empto

g okð Þ k̂pto ¼ekptog okð Þ (56)

Note how the new parameters proposed in Equations (52) and (55) match up with those obtained bythe ACC control of Equations (19) and (22) when generator losses are not considered in the optimiza-tion process (d= 0 and g(ok) = 1). Figure 4 shows an overview diagram that includes the sequentialsteps of the simulation.

4. RESULTS AND DISCUSSION

In this section, the following control strategies are compared in irregular waves: causal PC strategy,ACC control and RLC strategy. To assess the influence of losses, the PC strategy is compared witha simple PTO model, defined only by its damping coefficient (RLC strategy), where energy only flowsfrom the oscillating system into the PTO system, never in the reverse direction. On the other hand, thePC strategy is compared with a reactive control strategy where the PTO system needs a bidirectionalpower-flow capability, so that it can put energy into the oscillating system at certain points in the cycle.

Figure 4. Block diagram of the system simulated model.



Moreover, the ACC control is chosen as comparative strategy because simulations are performed inirregular waves, and therefore, it is necessary to implement a causal suboptimal strategy. Furthermore,it is a well-known strategy, commonly applied in WECs.The hydrodynamic coefficients come from reference [29] and correspond to a 2.5m radius heaving

spherical buoy. The WAMIT software package has been used to calculate the hydrodynamic values. Thespring constant considered is 6.2 kN/m and the natural period value of this oscillating system is 2.8seconds. The buoy was placed on the sea bed at 25m depth.The main parameters of the linear generator model implemented correspond to the Lysekil project

[30] and are presented in Table I.To test the different strategies in irregular waves a time-domain simulation model has been

implemented in MATLAB/SIMULINK, which allows modeling realistic PTO control law. The methodsare applied for a single DoF heaving buoy considering a Pierson–Moskowitz spectrum. The significantheights considered are 1.4, 2.2 and 3m. Radiation and excitation forces have been obtained byapproximation using transfer functions of the buoy frequency response.To perform time-domain simulations on more realistic conditions, a great number of numerical

simulations have been carried out in random seas for different values of the peak period between2.5 and 8 seconds, repeating the same simulations for different random phase arrangements to obtaina mean estimation for each strategy.

4.1. Power transferred to the power converter. Figure 5 shows the average power that can be transferredto the power converter. Negative power balances in the power converter are considered null in the figure.The PC strategy generally presents the best results. ACC control is able to transfer power to the

power converter only in a few peak periods around the natural period, because generator losses in-crease considerably when the incident wave frequency moves away from the natural frequency ofthe oscillating system.

Table I. Main generator features.

Electrical characteristics Mechanical characteristics

Synchronous inductance 7.8mH Nominal velocity 0.7m/sStator winding resistance 0.45Ω Magnet dimensions 40� 6.5� 400mmArmature current rms 28.9A Stator length 1300mmRated power 10 kW Stator width 400mmElectrical efficiency 86% Piston length 1800mmPhase-to-phase voltage rms 200V Piston width 400mmFundamental frequency 7.0Hz Pole width 50mmResistive losses 1.0 kW Stator steel weight 766Kg

3 4 5 6 7 80

5

10

15

20

25

30

Wave peak period (s)

Pow

er a

t con

vert

er (

kW)

RLCACCPC

3 4 5 6 7 80

5

10

15

20

25

30


Pow

er a

t con

vert

er (

kW)

3 4 5 6 7 80

5

10

15

20

25

30


Pow

er a

t con

vert

er (

kW)

(a) (b) (c)

Figure 5. Power transferred to the power converter for each strategy, with significant heights (a) H = 1.4m,(b) H= 2.2m and (c) H= 3m.



The power delivered to the power converter when the proposed strategy is applied in high-periodranges of sea states, becomes several times higher than that delivered when applying RLC strategy,whereas the results are similar in low-period ranges of sea states.

4.2. Power absorbed by the power take-off system. Figure 6 shows the average power that can beabsorbed by the PTO system for the range of periods considered, applying each strategy.It can be seen that the maximum power extraction occurs when the ACC control is applied, because

the purpose of this strategy is to maximize the energy absorbed by the PTO system. In spite of this,power transferred to the power converter gets positive results only for a few sea states around the res-onance frequency (Figure 5).In addition, the power levels delivered to the power converter when the proposed strategy is applied

(Figure 5) can be achieved without the need for the high levels of power absorbed by the PTO systemthat take place when the ACC control strategy is applied (Figure 6).

4.3. Peak-to-average power ratio. Results associated with the peak-to-average power ratio in the PTOsystem are shown in Figure 7.The peak-to-average power ratios obtained when the ACC control is applied, significantly increase

when the peak period moves away from the natural frequency of the system, achieving inadmissiblevalues when periods are far from the natural frequency.On the other hand, note that peak-to-average power ratios are just a little higher than applying RLC

3 4 5 6 7 80

10

20

30

40

50

60

70

80


Abs

orbe

d po

wer

at P

TO

(kW

)

RLCACCPC

3 4 5 6 7 80

10

20

30

40

50

60

70

80


Abs

orbe

d po

wer

at P

TO

(kW

)

3 4 5 6 7 80

10

20

30

40

50

60

70

80


Abs

orbe

d po

wer

at P

TO

(kW

)

(a) (b) (c)

Figure 6. Power absorbed by the power take-off (PTO) for each strategy, with significant heights (a)H = 1.4m, (b) H = 2.2m and (c) H = 3m.

3 4 5 6 7 80

5

10

15

20

25

30

35


Pea

k−to

−av

erag

e po

wer

rat

io

RLCACCPC

3 4 5 6 7 80

5

10

15

20

25

30

35


Pea

k−to

−av

erag

e po

wer

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io

3 4 5 6 7 80

5

10

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35


Pea

k−to

−av

erag

e po

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io

(a) (c)(b)

Figure 7. Peak-to-average power ratio in the power take-off system for each strategy, with significantheights (a) H = 1.4m, (b) H= 2.2m and (c) H = 3m.



strategy when the proposed strategy is applied, decreasing when the peak period moves away from thenatural frequency of the system.

4.4. Amplitude and velocity of oscillation. Figure 8 and 9 show the maximum amplitude and velocityreached by the oscillating device depending on the control strategy applied.The dynamic behavior of the oscillating system when ACC strategy is applied, presents the worst

performance, reaching inadmissible values for high periods. On the other hand, RLC and PC strategiespresent admissible values.Considering high-wave periods, where wave power levels are higher, the ratio of the power

delivered to the power converter (Figure 5), to the maximum velocity or amplitude of the oscillatingsystem (Figures 8 and 9), presents higher values when the proposed strategy is applied. For example,considering amplitudes, the ratio obtained by the RLC strategy is about (2.5/1.5’ 1.7 kW/m), and theone obtained by the PC strategy is about (19/2.4’ 7.9 kW/m) when the peak period and the significantwave height considered are 8 seconds and 3m, respectively.

5. CONCLUSIONS

In this paper, generator copper losses have been taken into account in the control strategy of permanentmagnet linear generators for ocean wave energy conversion. The proposed strategy maximizes thepower transferred from the linear generator to the power converter to improve the overall efficiencyof the power generation system.

3 4 5 6 7 80

1

2

3

4

5

6

7

8

9

10

Max

imum

vel

ocity

(m

/s)

RLCACCPC

3 4 5 6 7 80

1

2

3

4

5

6

7

8

9

10

Max

imum

vel

ocity

(m

/s)

3 4 5 6 7 80

1

2

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4

5

6

7

8

9

10

Max

imum

vel

ocity

(m

/s)

(b) Wave peak period (s)(a) Wave peak period (s) (c) Wave peak period (s)

Figure 9. Maximum velocity of the oscillating system for each strategy, with significant heights (a)H = 1.4m, (b) H = 2.2m and (c) H = 3m.

3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

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Max

imum

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plitu

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m)

RLCACCPC

3 4 5 6 7 80

0.5

1

1.5

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2.5

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3.5

4

4.5

5

Max

imum

am

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m)

3 4 5 6 7 80

0.5

1

1.5

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2.5

3

3.5

4

4.5

5

Max

imum

am

plitu

de (

m)

(a) Wave peak period (s) (c) Wave peak period (s)(b) Wave peak period (s)

Figure 8. Maximum amplitude of the oscillating system for each strategy, with significant heights (a)H = 1.4m, (b) H = 2.2m and (c) H = 3m.



Optimum conditions that maximize this power have been calculated and a causal strategy based on asecond-order linear PTO system has been developed from them.The performance of the WEC when the proposed control strategy is applied, has been compared

with the one obtained when the approximate complex-conjugate control and resistive loading controlare applied.It has been shown that WECs working under the proposed control strategy, achieve generally higher

conversion efficiencies because generator losses are significantly reduced and are able to perform in awider bandwidth in irregular waves. In addition, peak-to-average power ratios decrease, so the powerconverter size and consequently, the investment decrease. These improvements increase as WECoperating point moves away from the natural frequency of the oscillating system.Additionally, mechanical advantages are also achieved in the design of the WEC as a result of the

reduction of the maximum velocities and heave excursions in moving parts when the proposed controlstrategy is implemented.

6. LIST OF ABBREVIATIONS AND SYMBOLS

6.1. Symbols

fe(t), Fe(o) wave excitation forcefpto(t), Fpto(o) power take-off forcefr(t), Fr(o) wave radiation forcefs(t) net restoring stiffness forceg acceleration of gravity½ water densitym mass of the oscillating systemS water plane areaks spring stiffness force constantH incident wave heightT incident wave periodiq(t) cuadrature component of generator currentid(t) direct component of generator currenti0(t) zero sequence component of generator currentvq(t) cuadrature component of generator voltagevd(t) direct component of generator voltagev0(t) zero sequence component of generator voltageia, ib, ic phase currentsva, vb, vc phase voltagesvdc, idc DC link voltage and currentmadd added massb(o) radiation damping coefficiento angular frequencyppto(t) power at the generator inputEpto energy at the generator inputEc energy transferred to the power converterEloss linear generator copper-lossesLs stator inductanceRs stator resistanceÃ flux linkage of the stator winding¿p pole pitch of the PMLGom(t) angular velocity of the stator variablesz(t) excursion in heaveu(t), U(o) oscillating system velocityxi(o) intrinsic reactance



rpto(o) real part of Zptoxpto(o) imaginary part of ZptoZi(o) intrinsic impedanceZpto(o) load impedanceZnet(o) net impedance

6.2. Abbreviations

PMLG permanent magnet linear generatorWEC wave energy converterPTO power take-offDoF degree of freedomACC approximate complex-conjugateRLC resistive loading controlPC proposed controlDC direct currentrms root mean square

ACKNOWLEDGEMENTS

The authors are grateful for the support provided by Focus-Abengoa Foundation and the Scientific and HumanisticDevelopment Council of the Central University of Venezuela (CDCH-UCV). This work is part of the ENE2010-18867 project financed by the Spanish Ministry of Science and Education.

REFERENCES

1. Cruz J. Ocean wave energy. Current Status and Future Perspectives. Springer-Verlag Press: Berlin, 2008.2. Leijon M, Bernhoff H, Agren O, et al. Multiphysics simulation of wave energy to electric energy conversion by

permanent magnet linear generator. IEEE Transactions on Energy Conversion 2005; 20(1):219–224.3. Brooking PRM, Mueller MA. Power conditioning of the output from a linear vernier hybrid permanent magnet

generator for use in direct drive wave energy converters. IEE Proceedings C, Generation, Transmission andDistribution 2005; 152(5):673–681.

4. Wu F, Zhang XP, Ju P, Sterling MJH. Modeling and control of AWS-based wave energy conversion systemintegrated into power grid. IEEE Transactions on Power Systems 2008; 23(3):1196–1204.

5. Wu F, Ping X, Ju P, Sterling H. Optimal control for AWS-based wave energy conversion system. IEEETransactions on Power Systems 2009; 24(4):1747–1755.

6. Falnes J. Ocean Waves and Oscillating Systems. Cambridge University Press: Cambridge, 2002.7. Falnes J. Optimum control of oscillation of wave-energy converters. International Journal of Offshore and Polar

Engineering 2002; 12(2):147–155.8. Yavuz H, Stallard TJ, Mccabe AP, Aggidis GA. Time series analysis-based adaptive tuning techniques for a heaving

wave energy converter in irregular seas. IMechE Proceedings Part A 2007; 221(1):77–90.9. Korde UA. Control system applications in wave energy conversion. OCEANS 2000 MTS/IEEE conference and

exhibition, September 2000, Providence, USA; 1817–1824.10. Li B, Richard C, Macpherson E. Reactive causal control of a linear generator in irregular waves for wave power

system. Ninth European Wave and Tidal Energy Conference (EWTEC), 2011; Southampton, England.11. Hals J, Falnes J, Moan T. A comparison of selected strategies for adaptive control of wave energy converters.

Journal of Offshore Mechanics and Arctic Engineering 2011; 133:(3).12. Shek JKH, Macpherson DE, Mueller MA, Xiang J. Reaction force control of a linear electrical generator for direct

drive wave energy conversion. IET Renewable Power Generation 2007; 1(1):17–24.13. García A, Montoya D, De La Villa A. Output power of linear generators under reactive control in regular waves. Ninth

International Conference on Renewable Energies and Power Quality (ICREPQ), 2011; Las Palmas de G.C., Spain.14. Skjervheim O, Sørby B, Molinas M. Wave energy conversion: all electric power take off for a direct coupled point

absorber. Second Conference and Exhibition on Ocean Energy (ICOE), 2008; Brest, France.15. Perdigão J, Sarmento A. Overall-efficiency optimisation in OWC devices. Applied Ocean research 2003; 25:157–166.16. Shek JKH, Macpherson DE, Mueller MA. Phase and amplitude control of a linear generator for wave energy

conversion. Fourth IET Conference on Power Electronics, Machines and Drives (PEMD) 2008; 66–70.17. Mueller MA. Electrical generators for direct drive wave energy converters. IEE Proceedings C, Generation,

Transmission and Distribution 2002; 149(4):446–456.



18. García A, Montoya D, De La Villa A. Control of hydrodynamic parameters of wave energy point absorbers usinglinear generators and VSC-based power converters connected to the grid. Eighth International Conference onRenewable Energies and Power Quality (ICREPQ), 2010; Granada, Spain.

19. Leijon M, Waters R, Rahm M, et al. Catch the wave to electricity. IEEE Power and Energy Magazine 2009; 7(1):50–54.

20. Das B, Pal BC. Voltage control performance of AWS connected for grid operation. IEEE Transactions on EnergyConversion 2006; 21(2):353–361.

21. Cummins WE. The impulse response function and ship motions. Schiffstechnik 1962; 9:101–109.22. Price A. New perspectives on wave energy converters. PhD thesis, The University of Edinburgh. 2009.23. Budal K, Falnes J. Optimum operation of improved wave-power converter. Marine Science Communications 1977;

3(2):133–150.24. Polinder H, Damen MEC, Gardner F. Design, modelling and test results of the AWS PM linear generator. European

Transactions on Electrical Power 2005; 15(3):245–256.25. Boldea I, Nasar S. Linear Electric Actuators and Generators. Cambridge University Press: Cambridge, 1997.26. Giroux P, Sybille G, Le-Huy H. Modeling and simulation of a distribution STATCOM using SIMULINK’s power

system blockset. IECON’01 27th IEEE Industrial Electronics Society Annual Conference, 2001; 990–994.27. Polinder H, Slootweg JG, Compter JC, Hoeijmakers MJ. Modelling a linear PM motor including magnetic

saturation. Internation Conference. on Power Electronics, Machines and Drives 2002; 632–637.28. Krause PC, Wasynczuk O, Sudhoff SD. Analysis of Electric Machinery and Drive Systems, 2nd edn. IEEE Press:

New York, 2002.29. Havelock TH. Waves due to a floating sphere making periodic heaving oscillations. Proceedings of the Royal

Society of London A 1955; 231(1184):1–7.30. Danielsson O, Eriksson M, Leijon M. Study of a longitudinal flux permanent magnet linear generator for wave

energy converters. International Journal of Energy Research 2006; 30(14):1130–1145.



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Maximizing output power of linear generators for wave energy conversion