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IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 1, NO. 2, JULY 201099

Short-Term Wave Forecasting for Real-Time Control

of Wave Energy Converters

Francesco Fusco and John V. Ringwood

Abstract—Real-time control of wave energy converters requires

knowledge of future incident wave elevation in order to approach

optimal efficiency of wave energy extraction. We present an ap-

proach where the wave elevation is treated as a time series and it

is predicted only from its past history. A comparison of a range of

forecasting methodologies on real wave observations from two dif-

ferent locations shows how the relatively simple linear autoregres-

sive model, which implicitly models the cyclical behavior of waves,

can offer very accurate predictions of swell waves for up to two

wave periods into the future.

Index Terms—Time series, wave energy, wave forecasting.

I. INTRODUCTION

T

bodies or on oscillating pressure distributions within fixed or

moving chambers. Oscillators generally have pronounced reso-

nances, which enable efficient power absorption only over a re-

strictedrangeoffrequencies.Inorder,however,tocopewiththe

variations of wave spectra, a control system can be designed to

alter the oscillator dynamics such that the efficient energy con-

version occurs over a wide range of wave conditions [1].

The control approach, in the early stages of wave energy con-

version, consisted of frequency domain relationships regulating

the dynamics of the system to be tuned for maximum energy

absorption at different peak frequencies corresponding to dif-

ferentincomingwavespectra[1],[2].Althoughbeinganadvan-

tageous approach for real sea spectra, frequency domain tech-

niques do not generally allow real-time control on a wave-by-

wavebasis,whichcansignificantlyraisethedeviceproductivity

and, therefore, its economical viability. Real-time optimal con-

trol can be directly derived from the aforementioned optimal

frequency relationships [1], [2]. The main difficulties arise from

the fact that the transformation into the time domain results in

noncausal transfer functions, so that the conditions for optimal

powerabsorptioncanberealizedonlyiffuturemotionofthede-

vice, or of the future incident wave profile, are known [1]–[3].

Theproblemofshort-termwaveprediction,forsomeseconds

into the future, of the actual wave elevation profile at a specific

point oftheseasurface,is central, therefore,to themoregeneral

HE energy conversion in most wave energy converters

(WECs) is based either on relative oscillation between

Manuscript received December 17, 2009; revised February 22, 2010;

accepted March 29, 2010. Date of publication April 19, 2010; date of current

version June 23, 2010. This work was supported by the Irish Research Council

of Science, Engineering, and Technologies (IRCSET) under the Embark

Initiative.

The authors are with the Department of Electronic Engineering, National

University of Ireland Maynooth, Maynooth, Ireland (e-mail: francesco@eeng.

nuim.ie; fusco@eeng.nuim.ie).

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/TSTE.2010.2047414

Fig. 1. Wave predictions are required in order to generate an optimal reference

for a generic time-domain control of a WEC.

issue of time domain control of WECs. As an example, Fig. 1

illustrates a possible digital control scheme for a generic oscil-

lating body, where the controlled variable is its oscillation ve-

locity

and the control action is performed through a con-

trolforce

.Theestimateoftheoptimalreference

computed by an algorithm that requires future values

oftheincidentwaveelevation

principle of the specific device, as well as the control strategy

adopted and the available instrumentation, the involved quanti-

ties may vary, but the logic can always be described by a frame-

work such as that proposed in Fig. 1. In particular, the quantity

to be predicted may be the wave excitation force or the oscilla-

tion velocity for an oscillating body, the air pressure inside the

chamber of an oscillating water column, and they all depend,

ultimately, on the incident wave elevation.

The main approach followed in the literature is based on a

spatialprediction ofthewaveelevation,as inFig.2(b),meaning

that the wave field at a certain location is reconstructed from

one or more observations at nearby locations [4]–[7]. The fore-

casting model, in thissituation, requiresan array of spatial mea-

surements and can become very complex, because it has to take

into account the possible multidirectionality of waves [5], the

presence of radiated and diffracted waves [8], and eventual non-

linearities in the waves propagation (refer to [9] for a more de-

tailed literature survey).

The solution proposed in this paper is the prediction of the

wave elevation based only on its past history at the same point

of the sea surface, as in Fig. 2(a). Such an alternative approach

certainly introduces significant advantages with respect to the

spatial prediction, in terms of complexity of the models (mul-

tidirectionality and radiation do not need to be considered un-

less the device itself requires it, e.g., directional WECs) and on

the amount of instrumentation required (measurements at only

one point are required). Some preliminary work, following this

purelytimeseriesapproach,maybefoundin[1]and[10],where

autoregressive (AR) models are proposed for prediction of fu-

ture oscillation velocity of a generic device, and in [11], where

is

.Dependingontheoperating

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100 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 1, NO. 2, JULY 2010

Fig. 2. Two main approaches to wave forecasting. (a) Prediction based only

on local single-point measurements. (b) Prediction based on reconstruction of

wave field from array of distant measurements.

ARandhybridKautz/ARmodelsareutilizedtopredictthewave

elevationtimeseries.Bothofthemonlypresentresultswithdata

numerically generated from standard wave spectra.

This paper presents a variety of forecasting models and vali-

dates them against real observations, following the preliminary

results presented in [12] and [13]. The real data available is first

analyzed in Section II. The forecasting models are then pre-

sented in Section III and a methodology to evaluate the con-

fidence intervals of the predictions is outlined in Section IV.

Results are finally discussed in Section V.

II. AVAILABLE DATA

The data utilized for this study comes from two locations:

1) The Irish Marine Institute provided real observations from

a data buoy located in Galway Bay, on the West Coast of

Ireland,atapproximately53 13 N,9 18 W(waterdepth

nearly 20 m). Data consist of 20-min records sets for each

hour, collected at a sampling frequency of 16.08 rad/s

(2.56 Hz). The location is sheltered from the Atlantic

Ocean so that the wave height magnitude, in general, is

relatively small.

2) Wave elevation time series are also available from the At-

lantic Ocean at the Pico Island, in the Azores Archipelago,

atapproximately38 33 N,28 34 W(waterdeptharound

40 m). The Pico data are collected in the form of two con-

tiguous 30-min record sets for each hour, with a sampling

frequency of 8.04 rad/s (1.28 Hz) [14]–[16]

The climate at the Galway Bay site is dominated by relatively

low energy sea states (significant wave height less than 2 m)

which, most of the time, have a broad spectral distribution with

no clear energy peak, due to the superposition of low-frequency

swell(s) and high-frequency wind waves of similar energy con-

tent. Wave systems off the coast of Pico, on the other hand, usu-

ally have a more defined low-frequency peak (around 0.7 rad/s)

and their significant wave height ranges, usually, from 1 to 5 m.

A more detailed analysis of the wave data at the two locations

can be found in [17].

The wave spectra of three significant data sets at the two lo-

cations, shown in Fig. 3, will be utilized to test the forecasting

models, in Section V. In particular, one wide-banded and one

narrow-banded sea state from each of the two sites is consid-

ered. Then, a situation where wind waves predominate is se-

lected from the Galway Bay data and a very high-energy wave

system, where the sea bottom slightly affects the wave sym-

metry (this was analyzed through higher order spectral analysis

Fig. 3. Wave spectra of sample data set for the two locations: (a) Galway Bay;

(b) Pico Island.

andskewnessandkurtosisindices[13]),ischosenfromthePico

Island data.

One main characteristic that emerges from the wave data

analysis, which is well known, is that a significant portion of

the wave energy is usually concentrated at low frequencies. In

contrast, high frequencies dominate the spectrum only for very

low energy sea states, which are of diminished interest for wave

energy conversion. It is also known that low-frequency waves

(the swell) are more regular and less affected by nonlinearities

(which can be verified by means of the Bispectrum [13], [17],

[18]). It is reasonable, therefore, to low-pass filter the wave

elevation and focus the prediction only on the low frequencies,

which, intuitively, might improve the accuracy of the forecasts

and the length of the forecasting horizon.

A quantification of the possible benefits of low-pass fil-

tering on the prediction can be obtained by measuring the

predictability of the wave elevation time series. It is argued

that for predictability analysis, it is not necessary to design any

predictors; we just have to know how much information about

future signal values can be obtained from the past [19]. A sim-

pler measure of predictability than the very general approach

proposed in [19] and [20] (based on the mutual information

notion) will be adopted here, as previously proposed in [13].

It assumes that a linear relationship exists between the future

values of the wave elevation and its past values. A predictability

index

is estimated as

(1)

where

isthewaveelevation,supposedtohaveazeromean,

is the expectation operator, and

-step ahead prediction. A very efficient algorithm for the esti-

mation of

, which is utilized in this study, was proposed

in [21].

In Fig. 4, it is shown that the improvement of predictability

of the wave data detailed in Fig. 3, when only low-frequency

components are considered, is quite significant, particularly in

the case of narrow-banded sea states. The suggestion that it is

reasonable to focus the wave forecasting algorithms only on the

low-frequencywavesis,therefore,numericallyconfirmed.Note

that the choice of the cut-off frequencies

inspection of the spectral shape of each data set, such that the

is the optimal

is based on visual

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FUSCO AND RINGWOOD: SHORT-TERM WAVE FORECASTING FOR REAL-TIME CONTROL OF WECs101

Fig. 4. Predictability of wave elevation time series, low-pass filtered with different cut-off frequencies ? : (a) Galway Bay; (b) Pico Island.

amount of energy discarded is not significant (apart from the

data set

, where the wind wavesdominate the spectrum). The

cut-off frequency will ultimately represent a significant param-

eteratthedesignstageofanyspecificwavepredictionalgorithm

and it will be determined as a compromise between improve-

ment in the prediction accuracy and amount of discarded wave

components/energy.

III. FORECASTING MODELS

A. Cyclical Models

The most straightforward forecasting model is a cyclical

model, where the wave elevation

position of a number

of linear harmonic components

is expressed as a super-

(2)

Anerrorterm

plitude information for each harmonic component is contained

in the parameters

and [12], [13].

The model in (2) is completely characterized by the parame-

ters

, andbythefrequencies

through a nonlinear estimation procedure (the model is non-

linear in the frequencies) and utilized to predict the future wave

elevation. It needs, however, to be adapted to the time variations

of the wave spectrum (amplitudes, phases, and frequencies are

nonconstant), so that a first solution has been considered [13],

where the frequencies are kept constant and the model becomes

perfectly linear in the parameters

The choice of the frequencies is a crucial one. In [13], it was

pointed out how, while the range of frequencies is easy to deter-

mine, the distribution of the

lematic. A robust choice would be a constant spacing over all

the range, with the spacing as small as possible in order to give

an accurate coverage of the spectrum (appropriate constant fre-

quency intervals are also discussed in [22]). Alternative nonho-

mogeneous solutions may be considered, for example, with the

frequencies more densely distributed around the energy peaks.

Such more efficient solutions, however, would be unreliable, in

the context of a model with fixed frequencies, due to the time

hasbeenintroduced,whilethephaseandam-

.Itcouldbefittedtothedata

,.

inside this range is more prob-

variations of the energy distribution in wave spectra (which can

be verified through the Wavelet transform [17], [22]).

Once the frequencies are determined, a model for the ampli-

tude variations has to be chosen. Initially, the cyclical structural

model proposed by Harvey [23] was adopted

(3)

and

(4)

where

model the

is the sampling time, vectors

cyclical components, and it can be verified that

and . The Gaussian white disturbance

models the amplitude and phases variation.

In [12] and [13], an alternative stochastic model for the ampli-

tudes

and was proposed: the dynamic harmonic

regression (DHR) model [24]. However, the higher complexity

required (four state variables for each harmonic against two of

the Harvey’s model) was not justified by an improvement of the

modeling ability, so the DHR model it is not further considered.

From (3) and (4), the following state space form is derived:

(5)

where

(6)

(7)

(8)

(9)

ThestatespaceformissuitedtotheapplicationoftheKalman

filter for recursive online adaptation. The initialization is pro-

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102 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 1, NO. 2, JULY 2010

vided by regular least squares estimates on a number of batch

observations and thenthe Kalman filter is appliedonline, once a

covariance matrix for the state and output disturbances is avail-

able. When the estimate of the model’s parameters

available at any instant , the -step-ahead prediction

is obtained through the free evolution of the model in (5) as

is

(10)

B. Sinusoidal Extrapolation With the Extended Kalman Filter

Astepfurtherfromthecyclicalmodelswithfixedfrequencies

would be to consider variable frequencies which are updated

online, along with the amplitudes and phases, on the basis of

the free surface wave elevation measurements.

We can, therefore, attempt to model the wave elevation as a

single cyclical component of the type in (4), but with a time-

varying frequency

(11)

with

(12)

where

the wave elevation, and a model for the variability of

been introduced, assuming a simple random walk driven by the

additional white noise

. The model in (12), of course, is

nonlinear in

and an explicit linear state space structure

cannot be formulated.

As a consequence, a linear recursive estimator cannot be di-

rectly applied. It is possible, however, to utilize an extension

of the Kalman filter to nonlinear models, namely the extended

Kalman filter (EKF), assuming that the discrete time step

sufficiently small to permit the prediction equations to be ap-

proximated by a linearized form, based on the truncation of the

Taylor expansion of (12) at the first order [25]. The estimate of

thestatevectorateachtimestep is,therefore,givenbylinearre-

cursive equations, while the prediction

is obtained from the free evolution of the nonlinear model in

(12).

The extension to a model with

is not straightforward. It was found that a superposition of

models of the type expressed in (12) does not offer any actual

advantage[17].Suchanextensionisunderstudy,atthemoment,

and the current study is focused on a single frequency, so that

the model can be considered as a sinusoidal extrapolation.

, , and are random disturbances,is

has

is

variable frequencies

C. AR Models

The wave elevation

on a number

is assumed to be linearly dependent

of its past values, through the parameters

(13)

where a disturbance term

of the parameters at instant ,

is assumed to be Gaussian and white, the best prediction of the

futurewaveelevation

(13) as

has been included. If an estimate

, is computed and the noise

atinstant canbe derivedfrom

(14)

where, obviously,

information already acquired, no need of prediction).

The general shape of the prediction function

completely determined by the poles

In the case of

(when

jugate poles,

and , [26]

if(i.e.,

is

of the AR model in (13).

is even) couples of complex-con-

(15)

where the coefficients

,

complex-conjugate poles is implicitly a cyclical model, where

the frequencies are related to the phase

amplitudes and phases of the harmonic components are related

to the last

measurements of each time instant , so that they

adapt to the observations.

The AR coefficients

are estimated from a number

batch observations through the minimization of a multistep

ahead cost functional, referred to as long-range predictive

identification (LPRI) [27]

depend on the last

. Thus, an AR model with only

observations

of each pole and the

of

(16)

where

is to be optimized. The function

standard algorithm for nonlinear least squares problems, the

Gauss–Newton algorithm, initialized with the estimates from

regular least squares [27].

Note that, from (15), the implicit frequencies are related to

the poles of the AR model, so if the AR parameters are kept

constant, the frequencies will be constant as well. An adaptivity

mechanism based on the LPRI function (16) could be imple-

mented, as proposed in [27]. However, as will also be shown in

the results of Section V, a static AR model maintains its accu-

racy for a long time after being estimated (more than 2 hours),

in spite of spectral variations (for more details refer to [17]).

Adaptive AR models, therefore, are not a priority and are not

considered in the present work.

is the forecasting horizon over which the AR model

is minimized with a

D. Neural Networks

Inspite ofthenonlinear modelingcapability,neuralnetworks

have the great disadvantage of offering a model completely en-

closed in a black box, where inherent characteristics cannot be

analyzed by inspection or analytical calculation. So, whereas in

thecyclicalandARmodelsananalysisoftheestimatedparame-

ters and frequencies and their variations in an adaptive structure

can provide indications about the real process behavior and its

main characteristics, this would not be possible with neural net-

works.

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FUSCO AND RINGWOOD: SHORT-TERM WAVE FORECASTING FOR REAL-TIME CONTROL OF WECs103

For the problem under study, a nonlinear relationship is cre-

ated through a multilayer perceptron [28]

(17)

Only structures with one linear output neuron and two hidden

layers, consisting of a number of nonlinear neurons, varying

between three and seven each, were considered. Several orders

ofregression

werealsoconsidered.Themodelistrainedusing

the Levenberg–Marquard algorithm [29] on a set of batch data

and utilized for multistep-ahead prediction.

The structurein(17) is, ofcourse,not theonly possibilityand

many others could be considered. For example, a priori knowl-

edge about the process (harmonic nature) may be considered as

follows:

(18)

though such an input structure would retain some of the limita-

tions of cyclical models with fixed frequencies.

E. Other Possibilities

Other possible wave forecasting models were analyzed, but

were found unsuitable and therefore discarded (refer to [17] for

a detailed discussion).

A candidate alternative technique is Gaussian Processes

(GPs), whose basic idea is to place a Gaussian prior directly

on the space of functions [30] underlying the data, without

assuming any particular function parametrization but only

specifying the prior’s mean and covariance. In order to model

the cyclical characteristics of the sea, the harmonic frequencies

of the covariance function have to be assigned permanently

in the initial estimation of the model. This leads to the issue,

concerning the choice of frequencies, that emerged from the

discussion of cyclical models with fixed frequencies (see

Section III-A).

Also, the introduction of a moving average (MA) term in the

AR model, giving an ARMA model, has been analyzed and

it was found that no real change in the forecasting function’s

shape, with respect to AR models, can be obtained, so that no

significant improvement in the results can be expected. ARMA

models can, however, be utilized in order to obtain more par-

simonious forecasting models, but this is not the focus of the

study and it is not considered here.

Particle filters [31] may be considered for the online state

estimation of the nonlinear model (12) and can deal directly

with nonlinear models. However, while they do not assume a

Gaussian distribution of the state variable, and therefore may

give better estimation results than the EKF, they still only allow

a single frequency to be tracked, and would, therefore, be sub-

ject to the main performance limitation as the EKF.

IV. CONFIDENCE INTERVALS

The wave elevation predictions alone, as computed by any of

the models presented, do not give sufficiently complete infor-

mation about the future of the signal, as they are inevitably af-

fected byan error. It is a fundamentalneed to havean indication

about the extent of this error and about the confidence that we

can put in the forecasts computed by the prediction algorithm.

If the -step ahead prediction error is Gaussian

(19)

then the variance

ability distribution. We can assume that the error is contained

within a confidence interval, with probability , as follows:

is all we need in order to define its prob-

(20)

In (20),

that

is the value of the probability distribution such

(21)

wheretheprobabilitydensityfunction

of the forecasting error considered to be zero-mean Gaussian,

assumes the following structure:

,withthedistribution

(22)

The estimate of the variance

specific model parameters and from the statistics of the param-

eter estimation algorithm, which is not straightforward and also

could be misleading if the model is not sufficiently accurate. A

morestraightforwardalternative,however,isadopted,wherethe

estimate of the variance of the forecasting error is based purely

on the past history of the prediction errors

could be calculated from the

(23)

where

The estimate of

new observations become available [32], via

is the number of past observations available.

can also be recursively updated as soon as

(24)

V. RESULTS

Each of the data sets of Fig. 3 was split up into training and

validation sets. For the Galway Bay data, the training and vali-

dation sets consist of two consecutive data sets of 3072 samples

(20mineachatasamplingfrequencyof16.08rad/s,or2.56Hz).

In the case of the Pico Island data, because the consecutive data

sets are actually contiguous in time, training and validation sets

havebeenchosenasfourconsecutivesegmentseach(9216sam-

ples per set, meaning 2 h at a sampling frequency of 8.04 rad/s,

or 1.28 Hz).

Thepredictionaccuracyismeasuredwiththefollowinggood-

ness-of-fit index, for each forecasting horizon :

(25)

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