Effect of Control Strategies and Power Take-Off Efficiency on the Power Capture From Sea Waves

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IEEE TRANSACTIONS ON ENERGY CONVERSION 1

Effect of Control Strategies and Power Take-OffEfficiency on the Power Capture From Sea Waves

1

2

Elisabetta Tedeschi, Member, IEEE, Matteo Carraro, Marta Molinas, Member, IEEE,

and Paolo Mattavelli, Member, IEEE

3

4

Abstract—The choice of the most suitable control strategy for5

wave energy converters (WECs) is often evaluated with reference6

to the sinusoidal assumption for incident waves. Under this hy-7

pothesis, linear techniques for the control of the extracted power,8

as passive loading and optimum control, are well known and widely9

analyzed. It can be shown, however, how their performances are10

fundamentally different when irregular waves are considered and11

the theoretical superiority of optimum control is questionable un-12

der real wave conditions. Moreover, the global optimization of13

WECs implies a rational design of the power electronics equip-14

ment. This requires the analysis of the instantaneous extracted15

power in addition to the average one. In this paper, the impact16

of irregular waves on the power extraction when using different17

control techniques is analyzed in the case of a point absorber in18

heave. It is also shown how a convenient tradeoff between high19

average power extraction and limited power electronics overrating20

can be obtained by applying simple power saturation techniques.21

Moreover, the impact of power conversion efficiency on the control22

strategy is analyzed.23

Index Terms—Control strategies, irregular waves, power take-24

off (PTO), wave energy.25

I. INTRODUCTION26

AMONG renewable energies, the exploitation of ocean27

waves is certainly one of the most recent. In spite of its28

consistent potential [1], due to high availability and stability of29

the source and low environmental impact, a single leading tech-30

nology has not yet been established and different concepts are31

being studied and tested worldwide. Among them, a promising32

solution [2]–[4] is represented by point absorbers, i.e., floating33

buoys having small dimensions compared to the wavelength of34

incident sea waves. The principle of the power extraction lies35

in creating some kind of destructive interference between the36

Manuscript received December 17, 2010; revised April 2, 2011 and June 13,2011; accepted July 5, 2011. Paper no. TEC-00495-2010.E. Tedeschi was with the Norwegian University of Science and Technology,

Trondheim 7491, Norway. She is now with Tecnalia Research and Innovation,Bilbao 48012, Spain (e-mail: [email protected]).M. Carraro is with the University of Padova, Vicenza 36100, Italy (e-mail:

[email protected]).M. Molinas is with the Norwegian University of Science and Technology,

Trondheim 7491, Norway (e-mail: [email protected]).P. Mattavelli is with the Center for Power Electronics Systems, Virginia

Polytechnic Institute and State University, Blacksburg, VI 24060 USA (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TEC.2011.2164798

incident wave and the oscillation of the buoy. This requires an 37

active control of the device that needs to be tuned according to 38

the sea state. Thus, the control of the motion is one of the cru- 39

cial aspects to improve the power performance of wave energy 40

converters (WECs), andmany scientific contributions have been 41

devoted to this topic [5]–[12]. The most of them focus on linear 42

control techniques that are quite easy to implement and have 43

been systematically studied in order to maximize the power ex- 44

traction. It is worth noting that a large number of these studies 45

are developed under the ideal assumption of sinusoidal incident 46

waves [6]–[9]. It will be shown in the following, however, that 47

this can lead to misleading conclusions when real waves are 48

considered. On the other hand, some studies on practical con- 49

trol techniques to improve power performance have also been 50

proposed, but being limited to few very specific applications, 51

without necessarily claim to generality [10]–[12]. 52

Moreover, when dealing with WEC performance analysis, 53

the attention is usually focused mainly on mechanical and hy- 54

drodynamic aspects, while the issues related to the sizing and 55

efficiency of electric and electronic equipment are often over- 56

simplified, except for a few specific cases [13]–[15]. The goal of 57

this paper is at first to underline the impact of irregular waves on 58

the performance of linear control techniques. Following, some 59

considerations on the relationship between the choice of the con- 60

trol strategy and the rating of power electronics equipment are 61

presented. Consequently, the usefulness of simple power satura- 62

tion techniques [15] in order to maximize the average extracted 63

power, while avoiding excessive over ratings of the converter, 64

is analyzed and exemplified through computer simulations. The 65

attention is then focused on the effects of a nonideal power con- 66

version efficiency, and its influence on the choice of the best 67

control strategy is also shown. Finally, a complete model of the 68

system, including the electrical machine and power electronics 69

interface, has been developed in MATLAB/Simulink; hence, 70

the proposed control solutions have been tested, confirming the 71

validity of the previous analyses. 72

II. MODEL OF THEWEC 73

A. Hydrodynamic Model 74

The following analysis is focused on a spherical point ab- 75

sorber in heave [16], i.e., a single-degree-of-freedom device 76

as the one schematically depicted in Fig. 1, which is directly 77

connected to the electric power take-off (PTO) without any in- 78

termediate hydraulic or pneumatic stage. 79

Under the assumption of plane progressivewaves propagating 80

in an infinite water depth and if small motion is assumed, linear 81

0885-8969/$26.00 © 2011 IEEE

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Proof

2 IEEE TRANSACTIONS ON ENERGY CONVERSION

Fig. 1. Simplified model of the point absorber wave energy converter.

TABLE IDATA OF THE SELECTED SIMULATION TEST CASE

theory can be applied and the hydrodynamic diffraction model82

can be used to represent the interaction between the buoy and83

the waves [17].84

1) Frequency-Domain Model: If sinusoidal incident waves85

(regular waves) are considered, the system behavior can be de-86

scribed in the frequency domain as87

−ω2(M + a(ω))⌢

S + jω B(ω)⌢

S + K⌢

S =⌢

FE +⌢

FL (1a)

where ω is the angular frequency of the incident wave, S is88

the buoy position, and “⌢. ” denotes complex quantities. M is89

the mass of the device, including also the contribution of the90

generator inertia Jm , scaled by the gear ratio, n. Moreover, a is91

the added mass at the considered frequency. Added mass takes92

into account the water mass involved in the device movement93

and depends on the radiation force caused by device oscillation.94

B(ω) is the mechanical damping, also including the radiation95

resistance b, which is frequency dependent, too; K is the hydro-96

dynamic stiffness. Finally, FE is the excitation wave force and97

FL is the force applied by the PTO. Hydrodynamic parameters98

for the reference test case are listed in Table I and refer to the99

point absorber described in [7].100

Fig. 2. Electric analogue of a point absorber wave energy converter.

2) Time-Domain Model: In order to analyze the WEC be- 101

havior in irregular waves and to cope with any nonlinearity, a 102

time-domain model [18] is required, as the following one based 103

on the Cummins equation [19]: 104

(M + a∞)s(t) +

∫ t

−∞

Hrad(t − τ)s(τ)dτ + Ks(t)

= FE (t) + FL (t). (1b)

In (1b), “.” is time derivation operation, while a∞ represents 105

the value of added mass for ω→∞, i.e., 106

a∞ = limω→∞

a(ω).

Moreover, Hrad is the radiation impulse response function 107

(RIRF). 108

B. Electric Analogue of the WEC Under Sinusoidal Condition 109

The system described by (1a) corresponds to a mass–spring– 110

damper system and, in order to gain a better understanding of its 111

behavior, it can be useful to introduce its electric analogue (see 112

Fig. 2), which is valid as long as regular waves are assumed. 113

The excitation force FE corresponds to the sinusoidal voltage 114

e and the buoy velocity s to the current i. Moreover, the device 115

mass M and added mass a are represented by the inductance L, 116

the spring stiffness K by the inverse of the capacitance C and, 117

finally, the mechanical damping B by the resistance R. Thus, the 118

buoy, as a whole, is represented by the impedance Z. The force 119

applied by the PTO corresponds to the load voltage uL , whose 120

value is related to the control parameters represented by load 121

resistance RL and load reactance XL , as will be better explained 122

in Section III. 123

C. Model of the WEC for Irregular Waves Tests 124

The time-domain model of the WEC has been built from (1b) 125

and is reported in Fig. 3. The excitation force FE coming from 126

the waves is considered as a system input and its derivation is 127

described in detail in the following section. The control force 128

applied by the PTO FL is a system input as well. 129

The point absorber considered in this paper is the same as 130

that analyzed in [7]. Its hydrodynamic coefficients, damping 131

b(ω) and addedmass a(ω) have been obtained using a boundary- 132

element code (ANSYSAQWA),working on the geometrical and 133

physical properties of the system. For simple standard shapes 134

IEEE

Proof

TEDESCHI et al.: EFFECT OF CONTROL STRATEGIES AND POWER TAKE-OFF EFFICIENCY ON THE POWER CAPTURE FROM SEA WAVES 3

Fig. 3. Time-domain model of the wave energy converter used in irregularwaves analyses.

Fig. 4. RIRF of the considered point absorber.

as the spherical one, those coefficients could be as well found135

tabulated in an adimensional form [20].136

The radiation force FR has been determined as described137

in the following. If we consider the system represented by the138

transfer function −Hrad (ω), whose input is the buoy velocity s139

and the output is the radiation force FR , it can be shown that the140

impulse response of such system −Hrad (t) corresponds to the141

inverse Fourier transform of −Hrad (ω), which in turn is related142

to the hydrodynamics coefficients as follows [17]:143

Hrad(t) = F−1 [Hrad(ω)] = F−1 [b(ω) + jω a(ω) − jω a∞].(2)

Thus, for the considered system, the RIRF showed in Fig. 4 was144

derived by an inverse Fourier transform F−1 , starting from the145

hydrodynamic coefficients. Once the RIRF is available, there146

are several identification methods that can be used to identify147

the system, i.e., in the form of transfer function or of state-space148

model. In our case, the state-space model of the radiation force149

has been derived and its order conveniently reduced by proper150

MATLAB functions to finally obtain a fourth-order state-space151

model of the radiation force (see Fig. 3).152

Fig. 5. (a) Energy spectrum according to Bretschneider model. (b) Exampleof an excitation force profile in irregular waves.

D. Irregular Incident Waves 153

In order to evaluate the real performance of aWEC, the analy- 154

sis of its behavior when subject to irregular waves is mandatory. 155

It is especially meaningful to develop some time-domain stud- 156

ies related to the power extraction. The stochastic character of 157

real waves can be well understood if the superposition of in- 158

finite sinusoidal waves of infinitesimal height, random phase, 159

and different frequencies is considered. Several different mod- 160

els have been developed to provide a suitable description of a 161

real sea. Most of them use few fundamental parameters that 162

are considered representative of a specific sea state to build 163

up the energy spectrum of the sea itself. In the following, the 164

Bretschneider model [21], [22] will be adopted, where the en- 165

ergy spectrum [see Fig. 5(a)] can be obtained as a function of 166

the selected wave period and significant wave height of the sea 167

state. From the energy spectrum, the time-domain expression 168

of incident waves is found. The force coefficients are derived 169

from the corresponding damping coefficients by applying the 170

Haskind relations [17]. Then, the time-domain expression of 171

the excitation force that the buoy receives from the sea can be 172

found [23]; an example of excitation force profile is shown in 173

Fig. 5(b). 174

It is worth noting that, for the purpose of control strategies 175

comparison, the energy period, also known as “true average 176

period,” Te = 6 s and the significant wave height Hs = 1.41m 177

that are used to build the energy spectrum for the following 178

analyses, are selected according to a criterion of “equal energy 179

period, Te” (3a) and “equal energy transport, EN” (3b) between 180

the cases of sinusoidal and irregular waves [22]: 181

Te = T (3a)

ENsin =ρg

2A2

ENirr =ρg

16H2

s

ENsin = ENirr → Hs = 2√

2A. (3b)

IEEE

Proof


In (3a) and (3b), ρ is the water density and g the gravitational182

constant.183

E. Model of the PTO184

The PTO is composed by an electrical machine, and con-185

trolled by a bidirectional switching converter. Field-oriented186

control (FOC) is used to control the torque (i.e., the force) ap-187

plied by the PTO. One of the goals of the following power188

performance analysis is to provide information about the sizing189

of the generator and the power converter.190

III. CONTROL OF THEWEC191

A. Linear Control Strategies192

The simplest and most widespread control strategy for heav-193

ing buoys is passive loading, where the force exerted by the PTO194

is proportional to the buoy velocity. Acting this way, only the195

amplitude of the buoy motion can be controlled. On the other196

hand, both the amplitude and the phase of the motion need to197

be controlled in order to extract the absolute maximum average198

power. This is realized by implementing the so-called optimum199

(reactive) control, where the applied force has a component that200

is proportional to the buoy acceleration (reactive component) in201

addition to the one proportional to the velocity (damping com-202

ponent). It is worthwhile to clarify that this “optimum control”203

is typically used in wave energy applications to denote the case204

when, under the assumption of sinusoidal incident wave, the205

load reactance XL cancels the reactance impedance given by206

inductance L and capacitance C shown in Fig. 2, so that the207

impedance seen by voltage generator E is purely resistive. In208

this case, the theoretical condition of resonance between the209

motion of the device and incident waves is achieved; however,210

a bidirectional power flow between the buoy and the PTO must211

be allowed.212

B. Sinusoidal Incident Waves213

The power performance of point absorbers is well known if214

sinusoidal incident waves are considered [17] and it is here re-215

called with reference to the most consolidated control strategies216

for comparison purposes.217

Passive loading application corresponds to have a zero control218

reactive component, i.e., XL = 0.219

To obtain the maximum possible extraction of average power220

under such condition, the resistive component must be221

RL =

√

R2 +

(

ω L −1

ω C

)2

. (4)

On the other hand, application of optimum control implies222

XL = −

(

ω L −1

ω C

)

(5a)

RL = R (5b)

leading to the resonance condition and, thus, to the absolute223

maximum of the average power extraction.224

Fig. 6. (a) Incident wave, (b) instantaneous and average extracted power ap-plying passive loading, and (c) optimum control.

The sinusoidal reference wave used in the following analysis 225

as a design wave has amplitude A = 0.5m and period T = 6 s 226

[see Fig. 6(a)]. 227

IV. ANALYSIS OF POWER PERFORMANCEWITH DIFFERENT 228

CONTROL STRATEGIES 229

A. Impact of Control Strategies on the Converter Rating 230

In order to rationally choose the WEC control strategy, the 231

rating of the electric and power electronics equipment must be 232

taken into account. For such purpose, one interesting parameter 233

is the instantaneous extracted power or equivalently the ratio k 234

between peak and average power. 235

By simple analytical considerations, it can be shown that un- 236

der sinusoidal conditions (T = 6 s and A = 0.5m), if passive 237

loading is applied, the peak power always doubles the average 238

one. In the presented case, the average power that can be ex- 239

tracted is of 23.73 kW [see Fig. 6(b)]. On the other hand, if 240

optimum control is adopted under sinusoidal conditions, k is 241

largely varying depending on the specific device and incoming 242

waves: in the considered example, the adoption of optimum con- 243

trol leads to an average power of 56.22 kW and peak to average 244

power ratio is equal to 4.54 [see Fig. 6(c)]. Thus, under ideal 245

conditions, optimum control enables the extraction of a higher 246

average power compared to passive loading at the expense of 247

significantly higher power electronics rating. 248

It is then interesting to verify if the same trend applies in the 249

case of irregular incident waves. 250

If the system is completely linear, the power performance 251

under irregular waves can be theoretically predicted, once that 252

the energy spectrum of the incident waves is known. With ref- 253

erence to the Bretschneider spectrum of Fig. 5(a), the total av- 254

erage power extraction is derived as the sum of the extractable 255

power at the different frequency components. For the single fre- 256

quency, the system corresponds to the ideal case described in 257

IEEE

Proof


Section II-A1. In the considered case, the control coefficientsRL258

and XL /(2pfp ) are selected as the optimal ones for a sinusoidal259

wave whose frequency fp corresponds to the peak of the energy260

spectrum for both passive loading and optimum control [see261

(4a) and (4b)] and they are then kept constant at all frequencies.262

Under such conditions, the average extracted power is ex-263

pected to be 18.71 kW when applying passive loading and264

28.94 kW when applying optimum control.265

B. Power Saturation Effect266

The analysis of the instantaneous trend of the power extracted267

in irregular waves shows that it is extremely fluctuating with268

sporadic very high peaks. If the conflicting need to maximize269

the average extracted power, while limiting excessive overrating270

of electronic equipment is also taken into account, it is useful271

to consider the effect of saturation on the instantaneous power272

on the average power extraction. Such saturation is obtained273

by conveniently reducing the PTO applied force, so that the274

stated power limit is never exceeded. This effect is represented275

in a very simplified way, since more practically a torque lim-276

itation (possibly including some time-dependant overloading277

conditions) and a power converter current limitation are usually278

applied. All these provisions are here aggregated in the PTO279

power saturation.280

C. Efficiency of the Power Conversion281

When trying to quantify the useful extractable power from282

a WEC, another important consideration regards the efficiency283

of the whole power transfer. It must be considered that elec-284

tric machines and power converters have an efficiency that is285

affected by several factors, such as the loading conditions, the286

rotational speed, etc.; although a precise evaluation of the effect287

of efficiency is rather complex, the simplified efficiency curve288

of Fig. 7(a) will be used as a function of the load factor, i.e.,289

the ratio between the extracted power and the nominal (rated)290

one. As expected, the efficiency is strongly decreased when the291

extracted power is a small fraction of the nominal one. This has292

a different effect depending on the selected control strategy.293

Two different aspects must be taken into account, which are294

explained in the following, mainly with reference to the ideal295

sinusoidal case. The first and most important one deals with the296

fact that, in order to ensure the same average power extraction,297

optimum control implies much higher peak power values, thus298

experiencing higher losses due to nonideal efficiency. This also299

relates to the fact that if optimum control is selected, the power300

flow is bidirectional, while in the case of passive loading it is301

unidirectional. This implies that in the case of reversible power302

flow, the reduction of the extracted power needs to be taken303

into account in both directions [see Fig. 7(c)], thus being a304

disadvantage for optimum control both in regular and irregular305

waves.306

Moreover, under certain conditions, the application of opti-307

mum control can lead the system to work in a “low-load region”308

for a longer part of the period, thus contributing to disadvantage309

optimum control over passive loading. It is better to point out,310

however, that this second effect strongly depends on the specific311

Fig. 7. (a) Efficiency of electric and electronic devices and (b) its effects onpower extraction with passive loading and (c) optimum control (c).

test case (i.e., peak over average extracted power) and it may 312

become negligible in irregular waves. In the simpler sinusoidal 313

case, this condition has been exemplified in Fig 7(b) and (c), 314

showing the behavior of passive loading and optimum control; 315

in order to highlight this phenomenon comparing the two con- 316

trol solutions, the incident wave amplitude has been adjusted to 317

ensure the same average power for passive loading and optimum 318

control in the case of unity efficiency. The optimum control per- 319

formance reported in Fig. 6(c) is assumed as a reference. Due 320

to the difference in the peak power value and in the system 321

operating conditions, optimum control application leads to an 322

average power loss of 6.79 kW (∆Pavg = −12%), while in the 323

passive loading case the average power loss is of 2.95 kW, only 324

(∆Pavg = −5%). 325

V. SIMULATION RESULTS 326

The previous analyses have been verified by MATLAB/ 327

Simulink simulations, according to the test conditions reported 328

in Table I, for both regular and irregular waves. Although tar- 329

geted to a specific test case, the analyses can be extended to 330

different system ratings and operating conditions. 331

A. Linear Controls Performance in Regular 332

and Irregular Waves 333

At first, sinusoidal ideal conditions were tested to be used as 334

a reference case and to clearly emphasize the difference in the 335

power performance when applying the different control strate- 336

gies. As already mentioned in Section IV-A, if the sinusoidal 337

reference wave is assumed, passive loading leads to extract 338

23.73 kW, corresponding to only the 42% of the power that 339

can be theoretically extracted by applying optimum control. 340

Thus, for the sinusoidal case, a clear advantage in using op- 341

timum control would be apparent. Then, in order to compare 342

the power performance of passive loading and optimum control 343

IEEE

Proof


Fig. 8. Instantaneous and average power using passive loading.

in irregular waves to the corresponding ones under ideal con-344

ditions, wave’s force time series were created, in the form of345

profiles of 1000 s each (the spectrum discretization frequency346

resolution being ∆f = 0.001Hz).347

While the data about average power extraction are the same348

irrespective of the specific force profile that is applied being349

all of them derived from the same spectrum, the information350

about instantaneous power are related to the specific test case351

since phases at different frequencies are randomly generated.352

Thus, the evaluation of the peak power or the power ratio must353

be performed on a statistical base [18], considering the results354

coming, in our case, from the simulation of 80 different force355

profiles.356

The first test was aimed at the quantification of the power357

performancewhen passive loading is used. A specific simulation358

set showed that the maximum passive loading power extraction359

is achieved when the applied damping is close to the optimal360

one [see (4)] for a perfectly sinusoidal incident wave, whose361

frequency corresponds to the peak of the considered spectrum362

of Fig. 5(a). In this case, the average power can be quantified363

into 18.38 kW (see Fig. 8).When compared to the corresponding364

sinusoidal case, it shows how, due to irregular waves, a reduction365

of almost one-fourth of the average power is to be expected.366

The same kind of test was developed while applying opti-367

mum control. Also in this case, the reactive and damping co-368

efficients are selected as the ones maximizing power absorp-369

tion under regular conditions, with a sinusoidal incident wave,370

whose frequency corresponds to the peak of the spectrum. The371

corresponding power absorption is reported in Fig. 9. The to-372

tal average extracted power is now of 28.38 kW, which is even373

more significantly reduced with respect to the corresponding374

sinusoidal case. The small difference between theoretical and375

simulated values can be due to the approximation introduced376

by the identification process used for the radiation force calcu-377

lation. Then, the statistical analysis on irregular waves shows378

Fig. 9. Instantaneous and average power using optimum control.

that, when passive loading is applied, the peak to average power 379

value is in the range 7.7–17.1 and a power peak of 314 kW 380

can be reached. In the case of optimum control, k is between 381

25.2 and 58.3 and the peak power absorption (over the 80 tested 382

cases) exceeds 1.6MW. 383

Thus, the irregular waves study clearly confirms the need for 384

a consistent overrating of the electric and electronic devices. 385

B. Test on Power Saturation Effect 386

The second part of the analysis is focused on the effects of 387

power saturation, representing the maximum power that can be 388

handled by the electronic equipment and it is performed with 389

reference to a specific force profile. 390

Eight different saturation levels are chosen for both passive 391

loading and optimum control and analyzed with respect to both 392

irregular waves and corresponding sinusoidal waves. Results 393

are reported in Table II. It can be observed that in irregular 394

waves, a saturation of themaximum instantaneous power of 62% 395

(from Psat = 147.04 kW to Psat = 55.14 kW) leads to a loss of 396

average power only of 3% for passive loading, while in the case 397

of optimum control, a corresponding reduction of 67% in the 398

maximum power (from Psat = 1102.8 kW to Psat = 367.6 kW) 399

results in a 19.5% loss of average power, thus meaning that 400

a significant reduction in the power electronics rating can be 401

achieved with limited average power drop. With reference to 402

the optimum control, it is also worth noting that if an even 403

smaller saturation of 43% (from Psat = 255.2 kW to Psat = 404

147.04 kW) is applied under sinusoidal conditions, it leads to 405

an average power loss of 63%, thus meaning that the beneficial 406

effect of a power saturation is specifically related to irregular 407

waves and could be not appreciated if only sinusoidal waves 408

are considered. From the data of passive loading applied in 409

irregular waves, it can be also noted how the adoption of a 410

nonlinear control can lead to the absorption of a higher average 411

power than the corresponding (unconstrained) linear technique, 412

under certain conditions, as already reported in the literature [7]. 413

IEEE

Proof


TABLE IIEFFECTS OF INSTANTANEOUS POWER SATURATION

Fig. 10. Average power extraction as a function of the power saturation levelin presence of unity (dashed line) and nonunity (continuous line) efficiency ofthe power conversion, for passive loading and optimum control.

Finally, it is interesting to note the fact that if the same limit for414

peak power is assumed for both passive loading and optimum415

control (e.g., Psat = 73.52 kW), the implementation of passive416

loadingleads to significantly better results than with optimum417

control.418

It is worth noting that the application of saturation tech-419

niques leads to a decreased control over velocity and position420

of the buoy. In the considered case with a saturation threshold421

of 110 kW, the buoy excursion always remains less than +/−422

1m, which is well below the +/−3-m acceptable level for the423

specific device [7]. The adoption of saturation in extreme condi-424

tions is, however, still an open issue and its impact on the buoy425

excursion and generator speed must be carefully evaluated.426

C. Test on Nonideal Efficiency Effect427

The goal of the following set of simulations is to evaluate the428

effect of a nonideal efficiency of electric and electronic devices429

[see Fig. 7(a)] on the power conversion and its specific impact430

on the different control strategies.431

In Fig. 10, the average power extraction as a function of the432

power saturation level is reported for both ideal and nonideal433

Fig. 11. Average power extracted with nonunity efficiency as a function ofthe reactive component, evaluated as a fraction of the value used in optimumcontrol. Resistive component is the one used in passive loading.

conversion efficiency, in the case of passive loading and opti- 434

mum control. 435

The first important thing to underline is that irrespective of 436

the control technique that is applied, the effect of nonideal con- 437

version efficiency is of determining a point of maximum in the 438

average power extraction, corresponding to a specific power 439

saturation level. 440

This means that even if a higher rate of the power electron- 441

ics equipment was allowed, no gain would follow in the final 442

power extraction. The reason is that when the PTO has a con- 443

sistent overrating due to irregular waves, it works mainly in 444

low-load conditions, and consequently, the whole power con- 445

version becomes more and more inefficient. 446

It was already mentioned with reference to Table II that in 447

case of very low saturation level, passive loading leads to higher 448

average power extraction than optimum control. When con- 449

sidering also the effect of nonunity efficiency, the conclusion 450

is even more decisive. When comparing the nonideal curves 451

of Fig. 10 for passive loading and optimum control, it can be 452

clearly seen that the maximum power extraction obtained from 453

passive loading (Pavg = 16.7 kW) is more than the maximum 454

power extraction that can be reached by optimum control even 455

IEEE

Proof


Fig. 12. Integrated simulation model, including hydrodynamic and electric model of the WEC.

Fig. 13. Block diagram of the detailed system simulated model, including main control components.

in its most favorable condition (Pavg = 15 kW). Moreover, the456

highest power extraction in the passive loading case can be457

reached at expense of a consistently reduced power electronic458

equipment (Psat = 75 kW), compared to the optimum control459

option (having its maximum at Psat = 500 kW). Thus, in the460

considered test case, optimization of the average power extrac-461

tion can be obtained by designing electric and power electronics462

equipment whose power rating is around 100 kW and by choos-463

ing passive loading as the most convenient control strategy.464

The proven superiority of passive loading over optimum control465

when nonideal efficiency is taken into account, leads to won-466

der if an “intermediate control” solution can ensure a higher467

average power extraction than passive loading under the same468

conditions.469

A specific test was carried out by adopting a saturation level 470

of 110 kW. The resistive component of the control was fixed 471

as in the case of passive loading, while a reactive component 472

being an increasing fraction of the one prescribed by optimum 473

control was added. Average extracted power was then evalu- 474

ated as the mean value of several sets of simulations. As can 475

be seen from Fig. 11, when a reactive component being 50% 476

of the one adopted in optimum control is applied, an average 477

power of 21.8 kW can be obtained, corresponding to a 23% in- 478

crease with respect to pure passive loading. Consequently, an 479

intermediate control solution including a suitable reactive com- 480

ponent is advisable even under nonunity efficiency conditions 481

and must be carefully optimized to improve the whole power 482

performance. 483

IEEE

Proof


TABLE IIIPARAMETERS OF THE FULL-SYSTEM SIMULATION

D. Test on Nonideal Efficiency Effect484

Based on the proposed analysis, a detailed simulation model485

has been developed, integrating the wave energy source model486

and the hydrodynamicmodel of theWECpresented in Section II487

with a detailed model of the electric PTO, as can be seen from488

Fig. 12. The electrical model of the system, including the electri-489

cal drive and the grid-interface converter is shown in Fig. 13. For490

this specific example, the electric machine is an asynchronous491

generator with a nominal power of 110 kW and a rated speed492

of 1487 r/min, while other electrical parameters are reported in493

Table III.494

The machine model is implemented in the dq reference frame495

and the FOC has been adopted [25].496

Two-level voltage source inverter (VSI) has been adopted497

both for the electrical drive and for the grid-connected converter498

with fsw = 8 kHz. The second one behaves as an active front499

end (AFE), where an inner dq current control imposes the line500

currents to be in phase with the line voltages and an outer501

control regulates the dc-bus voltage. The control algorithm has502

been implemented in the dq rotating reference frame and the503

angle θ, needed for the transformation, has been generated by a504

phase-locked loop. The control used both for the grid-connected505

VSI and for the electrical drive is based on well-established506

knowledge available in the literature.Main converter parameters507

are also listed in Table III. The PI regulators of the current loops508

of both the electrical drive and the AFE are tuned to achieve509

a bandwidth of 500Hz with a phase margin of 60◦. The PI510

regulating the dc-link voltage is designed for a bandwidth of511

20Hz and a phase margin of 70◦ and a feedforward action is512

added in the dc-bus voltage loop (i.e., a term proportional to the513

generated power p(t) is added at the output of the PI_Vdc) to514

reduce dc-link voltage fluctuation.515

One example of the results obtained with the detailed model516

is reported in Fig. 14. Fig. 14(a) shows the irregular wave pro-517

file, Fig. 14(b) shows the torque applied by the electrical ma-518

chinewhen passive loading is used, and corresponding generator519

speed is shown in Fig. 14(c). It can be observed that the dc volt-520

age [see Fig. 14(d)] reflects the low-frequency oscillations of521

Fig. 14. (a) Sea waves profile, (b) generator torque, (c) generator speed,(d) dc voltage reference and dc actual voltage, and (e) current injected intothe grid in phase a.

Fig. 15. Details of the voltage and current of phase a at the grid section.

sea waves and such fluctuation is present also in the grid current 522

waveform [see Fig. 14(e)]. In this specific example, the dc-link 523

capacitor is not providing any energy smoothing, and the power 524

fluctuation generated by the PTO is directly injected into the 525

grid. 526

From Fig. 15, it can be noted that the current is injected into 527

the grid with a unity power factor, as required. 528

The detailed model confirms the analysis previously reported 529

in Fig. 10. As an example, using a power saturation equal to 530

IEEE

Proof


110 kW, the average extracted power using the detailed model531

is 16.4 kW, as marked by the square shown in Fig. 10.532

VI. CONCLUSION533

This paper has considered the effect of irregular waves in the534

control and design of point absorber WECs. The following as-535

pects have been analyzed and proved by computer simulations.536

1) The detrimental effect of irregular waves on the perfor-537

mance of both passive loading and optimum control.538

2) The usefulness of power saturation provisions to reduce539

the rating of power electronics equipment without exces-540

sive drop in the average power extraction.541

3) The impact of nonunity efficiency of the real electric power542

conversion on the selected control techniques.543

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[24] R. Krishnan, Electric Motor Drives: Modeling, Analysis, and Control. 622

Englewood Cliffs, NJ: Prentice-Hall, 2001. 623

Elisabetta Tedeschi (M’xx) received the Dr. Ing. de- 624

gree (Hons.) in electrical engineering and the Ph.D. 625

degree in industrial engineering from the Univer- 626

sity of Padova, Padova, Italy, in 2005 and 2009, 627

respectively. Q1628

From 2009 to 2011, she was a Postdoctoral Re- 629

searcher at the Norwegian University of Science and 630

Technology, Trondheim, Norway. In 2010–2011, she 631

was an Abel Grantee Visiting Researcher at the Uni- 632

versity of the Basque Country, Bilbao, Spain. Being 633

received a Marie Curie Fellowship in April 2011 she 634

is currently a Researcher at Tecnalia Research and Innovation, Bilbao. Her re- 635

search interests include design and control of energy conversion systems, with 636

specific focus on wave-energy applications and power-quality issues. 637

638

Matteo Carraro received the M.S. degree in au- 639

tomation engineering from the University of Padova, 640

Padova, Italy, in 2010. Since January 2011, he has 641

been working toward the Ph.D. degree in mechatron- 642

ics engineering and product innovation in the De- 643

partment of Technique andManagement of Industrial 644

Systems, University of Padova, Vicenza, Italy. 645

His main research interests include control and 646

parameter estimation techniques for electrical motor 647

drives. 648

649

IEEE

Proof


Marta Molinas (M’94) received the Diploma degree650

in electromechanical engineering from the National651

University ofAsuncion,Asuncion, Paraguay, in 1992,652

the M.Sc. degree from Ryukyu University, Ryukyu,653

Japan, in 1997, and the Dr.Eng. degree from Tokyo654

Institute of Technology, Tokyo, Japan, in 2000.Q2 655

In 1998, she was a Guest Researcher at the Univer-656

sity of Padova, Padova, Italy. From 2004 to 2007, she657

was a Postdoctoral Researcher at the Norwegian Uni-658

versity of Science and Technology (NTNU), Trond-659

heim,Norway,where she became a Professor in 2008.660

From 2008 to 2009, she was a JSPS Research Fellow at the Energy Technology661

Research Institute of Advanced Industrial Science and Technology in Tsukuba,662

Japan. Her research interests include wind/wave energy conversion systems,663

and power electronics and electrical machines in distributed energy systems.664

Dr. Molinas is an active reviewer for the IEEE TRANSACTIONS ON INDUS-665

TRIAL ELECTRONICS and IEEE TRANSACTIONS ON POWER ELECTRONICS. She666

is an AdCom Member of the IEEE Power Electronics Society667

668

Paolo Mattavelli (M’xx) received the Graduate 669

(Hons.) and Ph.D. degrees in electrical engineering 670

from the University of Padova, Padova, Italy, in 1992 671

and 1995, respectively. Q3672

From 1995 to 2001, he was a Researcher at the 673

University of Padova. In 2001, he joined as an Asso- 674

ciate Professor the University of Udine, where he was 675

leading the Power Electronics Laboratory. In 2005, he 676

joined the University of Padova in Vicenza with the 677

same duties and in 2010 Virginia Polytechnic Insti- 678

tute and State University, Blacksburg, as a Professor 679

and member of the Center of Power Electronics Systems. His major research 680

interests include analysis, modeling, and control of power converters, digital 681

control techniques for power electronic circuits, and grid-connected converters 682

for power quality and renewable energy systems. In these research fields, he 683

was leading several industrial and government projects. Q4684

Dr. Mattavelli currently (2010) serves as an Associate Editor for IEEE 685

TRANSACTIONS ON POWER ELECTRONICS, and as an Industrial Power Converter 686

Committee Technical Review Chair for the IEEE TRANSACTIONS ON INDUSTRY 687

APPLICATIONS. For terms 2003–2006 and 2006–2009, he was a member-at- 688

large of the IEEE Power Electronics Society’s Administrative Committee. He 689

received the Prize Paper Award in the IEEE TRANSACTIONS ON POWER ELEC- 690

TRONICS in 2005 and 2006 and a second place in the Prize Paper Award at the 691

IEEE Industry Application Annual Meeting in 2007. 692

693

Documents

Effect of Control Strategies and Power Take-Off Efficiency on the Power Capture From Sea Waves