Short-Term Wave Forecasting for Real-Time Control of Wave Energy Converters
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 approach 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 different locations shows how the relatively simple linear autoregressive 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.
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ABSTRACT: With the recent sharp increases in the price of oil, issues of security of supply, and pressure to honor greenhouse gas emission limits (e.g., the Kyoto protocol), much attention has turned to renewable energy sources to fulfill future increasing energy needs. Wind energy, now a mature technology, has had considerable proliferation, with other sources, such as biomass, solar, and tidal, enjoying somewhat less deployment. Waves provide previously untapped energy potential, and wave energy has been shown to have some favorable variability properties (a perennial issue with many renewables, especially wind), especially when combined with wind energy [1].IEEE control systems 01/2014; 34(5):30-55. · 2.37 Impact Factor -
Conference Paper: Fuzzy logic based reactive controller for heaving wave energy converters
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ABSTRACT: This paper presents a new control strategy based on fuzzy logic for maximizing the absorbed energy of heaving wave energy converters (WEC). The fuzzy logic controller is responsible of minimizing the discrepancy between the optimal reference buoy velocity and the actual buoy velocity, hence maximizing the captured power for irregular sea environment. The proposed controller performed well compared to a conventional fixed reactive control strategy.Renewable Energies for Developing Countries (REDEC), 2012 International Conference on; 01/2012 - SourceAvailable from: John Ringwood[Show abstract] [Hide abstract]
ABSTRACT: Energy-maximizing controllers for wave energy devices are normally based on linear hydrodynamic device models. Such models ignore nonlinear effects which typically manifest themselves for large device motion (typical in this application) and may also include other modeling errors. The effectiveness of a controller is, in general, determined by the match between the model the controller is based on and the actual system dynamics. This match becomes especially critical when the controller is highly tuned to the system. In this paper, we present a methodology for reducing this sensitivity to modeling errors and nonlinear effects by the use of a hierarchical robust controller, which shows small sensitivity to modeling errors, but allows good energy maximization to be recovered through a passivity-based control approach.IEEE Transactions on Sustainable Energy 01/2014; 5(3):958-966. · 3.84 Impact Factor
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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 WECs 101
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
andwas 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
atinstantcanbe 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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