estimating the potential of ocean wave power resources
TRANSCRIPT
Ocean Engineering 38 (2011) 177–185
Contents lists available at ScienceDirect
Ocean Engineering
0029-80
doi:10.1
n Corr
E-m1 G2 R.
journal homepage: www.elsevier.com/locate/oceaneng
Estimating the potential of ocean wave power resources
Amir H. Izadparast a,1, John M. Niedzwecki b,n,2
a Zachry Department of Civil Engineering, Texas A&M University, USAb Zachry Department of Civil Engineering, Texas A&M University, CE/TTI 201, 3136 TAMU, College Station, TX 77843-3136, USA
a r t i c l e i n f o
Article history:
Received 29 March 2010
Accepted 17 October 2010Available online 5 November 2010
Keywords:
Ocean waves
Random seas
Wave power resource
Wave energy conversion
Probability distribution estimates
Narrow-band processes
Field data
18/$ - see front matter & 2010 Elsevier Ltd. A
016/j.oceaneng.2010.10.010
esponding author. Tel.: +1 979 845 7435.
ail address: [email protected] (J.M. Nie
raduate Research Assistant.
P. Gregory ’32 Professor.
a b s t r a c t
The realistic assessment of an ocean wave energy resource that can be converted to an electrical power at
various offshore sites depends upon many factors, and these include estimating the resource recognizing
the random nature of the site-specific wave field, and optimizing the power conversion from particular
wave energy conversion devices. In order to better account for the uncertainty in wave power resource
estimates, conditional probability distribution functions of wave power in a given sea-state are derived.
Theoretical expressions for the deep and shallow water limits are derived and the role of spectral width is
studied. The theoretical model estimates were compared with the statistics obtained from the wave-by-
wave analysis of JONSWAP based ocean wave time-series. It was shown that the narrow-band
approximation is appropriate when the variability due to wave period is negligible. The application of
the short-term models in evaluating the long-term wave power resource at a site was illustrated using
wave data measured off the California coast. The final example illustrates the procedure for incorporating
the local wave data and the sea-state model together with a wave energy device to obtain an estimate of
the potential wave energy that could be converted into a usable energy resource.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The process of developing technologies for the efficient extraction ofenergy from ocean waves provides a range of technical and practicalchallenges. One of the most critical challenges is the need to accuratelycharacterize the random behavior of ocean waves and to design waveenergy conversion devices that can function optimally at variousoffshore sites. A first step in the wave energy conversion studies is todevelop reliable estimates of the short- and long-term available wavepower of the wave fields of interest. This requires a thorough analysis ofthe wave power random variability and estimation of the wave powerprobability distribution. The wave power probability distributions canbe utilized in obtaining the risk associated with a wave power estimateand eventually may be utilized to obtain a more realistic estimate of thewave energy unit cost.
To assess the wave power resources at a site, the long-termmean wave power is commonly considered to be the characteristicvalue used for design purposes, see for example (Pontes, 1998;Bedard et al., 2005; Henfridsson et al., 2007; Defne et al., 2009; orIglesias et al., 2009). This long-term value can be obtained byestimating the mean wave power in a sea-state and the associatedprobabilities obtained from the sea-states scatter diagram. In thisapproach, the mean wave power in the sea-states is estimated from
ll rights reserved.
dzwecki).
its relationship with wave spectral characteristics, i.e., significantwave height, wave energy period, and wave direction. For fieldswhere the measured wave time-series are not available, the long-term wave power may be obtained from the application of therelation between the wave power with either the individualobserved wind speeds or the probability distribution of wind speeddata in the field (Bretschneider and Ertekin, 1989; Ertekin and Xu,1994). Each of these approaches characterizes the random nature ofwave power, using a single statistical variable and consequentlythe uncertainty in the process may not be adequately captured.
Recently, Myrhaug et al. (2009a, b) studied the probabilitydistribution of wave power in a sea-state, using an empirical bi-variate distribution of wave height and period for waves on theNorwegian continental shelf. In their study, the random variabletransformation was utilized in order to obtain the bi-variateprobability distributions of wave power and the associated sea-state parameters of wave height and period. In this study, amethodology similar to Myrhaug et al. (2009b) is pursued, butthe bi-variate probability distributions of wave power, wave heightand wave period are derived using the joint probability distribu-tions first obtained by (Longuet-Higgins, 1975, 1983). The role ofspectral width and the assumption of narrow bandwidth on thebehavior of the bi-variate probability distributions are examined.The dimensionless asymptotic forms of these distributions forboth the shallow and deepwater limits, commonly used in oceanengineering analyses, are developed in the investigation. Thestatistics of these models are compared with those obtainedfrom a wave-by-wave analysis of waves generated from a
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185178
JONSWAP spectrum for sea-states of varying severity. The severityof the wave conditions is investigated based upon the variation ofthe JONSWAP model peakedness parameter. The application of theshort-term distributions in the long-term wave power resourceevaluation of a site is discussed and, as an example, the calculationsare carried out for a site off the California coastline.
The power extracted from ocean waves by a wave energyconvertor (WEC) is a function not only of the wave conditions atthe offshore site, but also depends on the hydrodynamic character-istics of the device, the power-take-off system, the control strategy,etc. Considering all these variables explicitly in the probabilitydistribution of absorbed power would be quite complicated andwould require detailed design information for a particular WEC. Thattype of detailed information is proprietary, so in order to demonstratehow the wave information modeled in this study would be utilized, asimplified model of the WEC is presented that still captures theessence of the process. The probability distribution of the wave powerabsorbed by a slack-moored, phased controlled heaving-cylindricalbuoy in random seas is suggested and the numerical simulationsresults published in a report by Eidsmoen (1996) are utilized.
2. Mathematical background
2.1. Wave power estimates for the deep and shallow water limits
The wave power per unit crest length P of regular seas asdeveloped from linear wave theory is well-known, see for example(Dean and Dalrymple, 1991), and can be expressed as the product ofthe wave group velocity cg and the total average wave energy perunit surface area E, i.e.,
P¼ cgE ð1Þ
For the deepwater limit, indicated by the subscript d, with depthd to wave number l ratio of d=l41=2, the wave power can beexpressed as
Pd ¼rg2
32pT H2 ð2Þ
where g is the gravitational acceleration, r is the fluid density, T isthe wave period, and H is the wave height. For the shallow waterlimit, denoted by the subscript s, with d=lo1=20, the wave powercan be expressed as
Ps ¼rg3=2
8d1=2H2 ð3Þ
In this limit, the wave power varies with the square-root ofwater depth d and is independent of wave period. For irregular seaconditions, the corresponding mean deepwater wave power Pd in arandom sea-state can be estimated from the following equation,(Falnes, 2002):
Pd ¼
Z 10
rg2
4pfGZZðf Þdf ¼
rg2
4p T�1m0 ð4Þ
where the over-bar symbol denotes the mean value, GZZ(f) is thesingle-sided wave energy spectrum, f is the wave frequency in Hz, andm0 is the area under the wave energy spectrum. The variable T�1 is thewave energy period and is defined in terms of spectral momentsT�1¼m�1/m0, where the nth spectral moment is defined as
mn ¼
Z 10
f nGZZðf Þdf ð5Þ
Similarly, it can be shown that for the shallow water limit thecorresponding expression for the mean power is in the form of
Ps ¼
Z 10
rg3=2 d1=2GZZðf Þdf ¼ rg3=2 d1=2m01=2 ð6Þ
As a brief comparative illustration of the estimates obtained fromthese average power equations, consider the random seas describedby a JONSWAP wave energy spectrum. Assuming that Hs¼2.0 m,Tp¼15.0 s, and, g¼3.3, then T�1¼13.5 s. Given that the waterdensity r¼1023 kg/m3, the deepwater mean wave power calcu-lated from Eq. (4) is Pd ¼ 26:5 kW/m. For the same wave spectrumwith the same spectral characteristics measured at a water depth ofd¼4.0 m, the shallow water mean wave power is estimated asPs ¼ 15:7 kW/m. The significant difference between the deep andshallow water mean powers is caused by the difference between thegroup velocities in these water limits. A precise evaluation of thedeepwater and shallow water mean wave power requires detailedanalysis of wave transformation and energy loss mechanisms,e.g. wave breaking, on a given bathymetry.
The focus in the analyses to follow is on the dimensionless formsof wave power equations. For the deepwater limit, the dimension-less wave power pd is obtained as
pd ¼Pd
Pd
¼1
8
T
Tptz2¼ adtz
2ð7Þ
where Pd is a characteristic wave power utilizing the spectral peakperiod Tp¼1/fp, and is defined as
Pd ¼rg2
64pTpH2
s ð8Þ
and, z¼H/sZ is the dimensionless wave height, t¼ T=T is thedimensionless wave period, ad ¼ ð1=8ÞðT=TpÞ is the deepwatermodel parameter and H and T are the instantaneous waveheight and period, respectively. The sZ is the standard deviationof linear and narrow-banded surface wave elevation and isestimated from it a relation with the significant wave heightsZ¼Hs/4. The ratio of the mean wave period to the spectral peakperiod T=Tp is a function of the spectral shape and width, and itsvalue generally decreases as the spectral width increases. In case ofnarrow-banded random processes, different wave periods con-verge and consequently the mean deepwater power Pd limits to thecharacteristic value Pd and the dimensionless parameters take onthe following limit values tE1, adE1/8. Subsequently, one findsthat the deepwater dimensionless wave power equation reduces tothe form
pd ¼z2
8ð9Þ
Similarly, the dimensionless form of the shallow water wavepower can be obtained from
ps ¼Ps
Ps
¼z2
8ð10Þ
As shown in this equation, the non-dimensional wave powerps is independent of water depth. Note that both dimensionlesslimits Eqs. (9) and (10) converge to the same result, where theonly difference in the derivation is that for the shallow water limitno assumption was made regarding the ocean wave energyspectrum.
2.2. Probability distributions of wave height and period in a sea-state
The joint distribution of the wave height and wave period for agiven sea-state S, as developed by (Longuet-Higgins, 1975, 1983),can be expressed as
f ðz, t9SÞ ¼ 1
8uffiffiffiffiffiffi2pp LðuÞ
zt
� �2
exp �z2
81þ 1�
1
t
� �2 1
u2
!" #
0rzo1, 0oto1 ð11Þ
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185 179
where the spectral width u is expressed as
u¼m0m2
m21
�1
!1=2
ð12Þ
and for convenience, the following combination of parameters isintroduced
LðuÞ ¼ 2½1þð1þu2Þ�1=2��1 ð13Þ
Based on this distribution, the marginal probability densityfunctions of normalized wave heights and periods in a sea-state canbe expressed as
f ðz9SÞ ¼Z 1
0f ðz, t9SÞdt¼ z
4expð�z2=8ÞLðuÞF
z2u
� �ð14Þ
f ðt9SÞ ¼Z 1
0f ðz, t9SÞdz¼ LðuÞ
1
2ut21þ 1�
1
t
� �2 1
u2
!�3=2
ð15Þ
where the normal distribution function is
FðxÞ ¼1ffiffiffiffiffiffi2pp
Z x
�1
expð�u2=2Þdu ð16Þ
Assuming that waves have a narrow-banded spectrum withnE0, it can be shown that wave heights follow a Rayleighdistribution, i.e.,
fNBðz9SÞ ¼z4
expð�z2=8Þ ð17Þ
For no0.6, Ochi, 1998 has demonstrated that Eq. (14) is wellapproximated by Eq. (17).
3. Probability distribution of wave power
3.1. Characterization in the sea-state
Utilizing the random variable transformation rule along withthe relation between random variables pd, z, and tdefined in Eq. (7),the joint probability distribution of f (z, pd9 S) can be estimatedfrom
f ðz, pd9SÞ ¼ f z, t¼ pd
adz29S
!@
@pd
pd
adz2
!ð18Þ
thus,
f ðz, pd9SÞ ¼adz
4
8ffiffiffiffiffiffi2pp
up2d
LðuÞexp�z2
81þ 1�
adz2
pd
!21
u2
0@
1A
24
35 ð19Þ
Similarly, the joint probability distribution of f(z, t9 S) can betransformed to f (pd, t9 S) as
f ðpd, t9SÞ ¼p1=2
d
16uffiffiffiffiffiffi2ppðadÞ
3=2t7=2LðuÞexp
�pd
8adt1þ 1�
1
t
� �2 1
u2
!" #
ð20Þ
Knowing the joint probability distribution of wave power andwave height, the short-term probability distribution of the wavepower f (pd9S) can be obtained from the integration f ðpd9SÞ ¼R1
0 f ðz, pd9SÞ@z. In these distributions, the random variability ofboth wave height and wave period along with the linear correlationbetween these random variables is considered. Assuming thatwaves have narrow-banded energy spectrum, the marginal prob-ability distribution of pd is derived in the form of
fNBðpd9SÞ ¼ expð�pdÞ ð21Þ
which is the well-known exponential distribution that has anexpected value and standard deviation of unity, i.e., E(pd)¼spd¼1.
In this approximation, the wave period is treated as a deterministicvariable and this leads to the result that the random variability inthe wave power estimate is caused only by the variation in theocean wave heights.
Utilizing the relation between the wave power and waveheights from Eq. (10) and the probability distribution of waveheight from Eq. (14), the probability distribution of wave power inthe shallow water limit is determined to be of the form
f ðps9SÞ ¼ 1þu2
4
� �expð�psÞF
ffiffiffiffiffiffiffiffi2ps
pu
!ð22Þ
For the narrow-band approximation uE0, this reduces to theform
fNBðps9SÞ ¼ expð�psÞ ð23Þ
where again this is an exponential probability distribution similarto the deepwater limit with an expected value and standarddeviation of unity, i.e., E(ps)¼sjs¼1.
In order to estimate the wave power absorbed by the WECdevice, i.e., absorbed power, the concept of device capture width isneeded. The device capture width is a fundamental parametercommonly used to evaluate the performance of a three-dimen-sional isolated body. For a given frequency, the device capturewidth is defined as the ratio of the total mean power absorbed bythe body Pa to the mean power per unit crest length P of theincident wave train; see e.g. (Thomas, 2008)
l¼Pa
Pð24Þ
The device capture width is a function of, e.g. the seastate, thehydrodynamic characteristics of the device, power-take-off mechan-ism, and control system. Knowing the analytical relation of the capturewidth and the sea-state parameters, wave height and period, or itscorrelation with wave power, one can derive the theoretical probabilitydistribution of an absorbed power. This however requires a fairlycomprehensive and detailed simulation of the device in differentrandom sea-states. In order to simplify the problem, here, it is assumedthat absorbed power has an exponential distribution. This approxima-tion is consistent with the idea of modeling waves as narrow-bandedrandom processes. Additionally, it is assumed that the mean absorbedpower in the sea-state Pa is known from numerical or experimentalmodel simulations. Considering these assumptions, the probabilitydistribution of non-dimensional absorbed power pa ¼ Pa=Pa, with Pa
being the instantaneous absorbed wave power in a given sea-state,simplifies to the exponential distribution as
f ðpa9SÞ ¼ expð�paÞ ð25Þ
The result is based upon the stated assumptions, and in practiceshould be validated with the appropriate data.
3.2. Long-term probability distributions
A commonly used long-term estimate of the wave powerresources at a specific site is the long-term, e.g. yearly, seasonal,monthly, and the averaged mean wave power. This long-termestimate is obtained from the summation taken over the entirepossible seastates of mean wave power in each seastate multipliedby the long-term averaged probability of occurrence of thecorresponding seastate. Following a similar procedure, the long-term probability distribution of a random variable X can beestimated using the equation
f ðXÞ ¼X
i
f ðX9SiÞFrðSiÞ ð26Þ
The random variable of interest X represents either wave powerP or absorbed power Pa. In Eq. (26), f (X9Si) is the conditional
ζ
2
4
6
8
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185180
probability density function of the random variable given asea-state Si with the probability of occurrence of Fr (Si). Theconditional probability density functions were presented in theprevious section and the sea-state probabilities are generallyestimated based upon the scatter diagram of the significantwave height and wave peak period. In this case, the summationis taken over the entire sea-states with non-zero probability andthe statistics represent the long-term averaged values. In Eq. (26),the dimensional form of the conditional probability distributionwas utilized and the corresponding dimensional form is obtainedusing the following equation:
f ðXi9SiÞ ¼1
Xi
f xi ¼X
Xi
9Si
� �ð27Þ
where Xi is the characteristic value of the random variable X in agiven sea-state Si. For random variables that have exponentialconditional distributions, i.e., pd and ps of narrow-banded waves,and pa, f (X) becomes a hyper-exponential distribution with the firsttwo moments of the form, (Stewart, 2009)
EðXÞ ¼X
i
XiFrðSiÞ ð28Þ
EðX2Þ ¼ 2X
i
ðXiÞ2FrðSiÞ ð29Þ
τ0 0.5 1 1.5 2
0
ζ
pd
0 0.5 1 1.50
1
2
3
4
5
6
2
2.5
4. Illustrative examples
4.1. Characterization of the sea-state
In order to more easily study the effects of spectral width on theprobability distribution of wave power in a sea-state, a triangularenergy spectrum of the form
GZZðf Þ ¼
H2s
8Df
f�fmin
fp�fminfminr f r fp
H2s
8Df
fmax�f
fmax�fpfpo f r fmax
8>>>><>>>>:
9>>>>=>>>>;
ð30Þ
is introduced. In Eq. (29), fp, fmin, and fmax are peak, minimum, andmaximum frequencies, respectively, and Df ¼ fmax�fmin is thefrequency range of interest. The parameters are schematicallyintroduced in Fig. 1. One should be noted that the triangularspectrum is only introduced for illustrative purposes, and in thesections to follow the JONSWAP spectrum is used.
First consider the situation where the waves are propagating indeepwater and have an energy spectrum with finite width of u¼0.6,which corresponds to the following frequencies fmin ¼ 0:01Hz,fp¼0.1 Hz, fmax ¼ 1Hz, and significant wave height Hs¼4.0 m. InFig. 2, subplots (a), (b) and (c), the contours of constant probabilitydensities of wave height and period f (z, t9S), Eq. (11), of wavepower and wave height f (z, pd9S), Eq. (19), and of wave power and
Gηη
( f )
f fmaxfmin fp
Hs2 / (8� f )
Fig. 1. Schematic representation of the triangular wave spectrum.
wave period f (pd, t9S), Eq. (20), are, respectively, illustrated. Notethat the contours in each of the sub-plots correspond to the valueslisted in the figure title starting with the first value and propagatinginward. It is observed that the joint distribution f (z, t9S) isasymmetric with regard to t and this is also observed in the jointdistribution f (pd, t9S). The bi-variate probability distribution of f (z,pd9S) clearly shows the quadratic relation between pd and z. In asimilar fashion, the characterizations for narrow-banded sea-states
τ
pd
0 0.5 1 1.50
0.5
1
1.5
Fig. 2. Contours of constant probability densities in a seastate with u¼0.6 (a) f (z,
t9S)¼0.05, 0.10, 0.20, 0.3, and 0.35, (b) f (z, pd9S)¼0.10, 0.30, 0.5, 0.7, and 0.9, and
(c) f (pd, t9S)¼0.10, 0.30, 0.5, 0.7, and 0.9.
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185 181
are presented in Fig. 3, again with three subplots to aid in thecomparison of the various distribution functions. The waves in thissea-state are generated using the triangular energy spectrum, Eq.(30), with fmin¼0.08 Hz, fp¼0.1 Hz, fmax ¼ 0:13Hz, and the samesignificant wave height Hs¼4.0 m. This results in a much smallerspectral width of u¼0.1. It is observed that as the width of the wavespectrum decreases, the distributions f (z, t9S) and f (pd, t9S) becomemore symmetric about the mean period t¼1. Further, f (z, pd9S)
τ
ζ
0 0.5 1 1.5 20
2
4
6
8
ξ
pd
0 1 2 3 40
1
2
3
4
5
6
τ
0 1 2 3 4 50
0.5
1
1.5
2
2.5
pd
Fig. 3. Contours of constant probability densities in a seastate with u¼0.1
(a) f (z, t9S)¼0.05, 0.10, 0.20, 0.3, and 0.4, (b) f (z, pd9S)¼0.10, 0.30, 0.5, 0.7, and
0.9, and (c) f (pd, t9S)¼0.10, 0.30, 0.5, 0.7, and 0.9.
becomes narrower around the quadratic curve and possesses alonger tail.
Estimates of the probability density functions (PDF) and cumu-lative distribution functions (CDF) of wave power obtained fromnumerical integration of Eq. (19) for sea-states with differentspectral widths are presented in Fig. 4a and b, respectively. Alsoshown in these figures are the distributions obtained using thenarrow-band approximation, i.e., Eq. (21). The distributions clearlyindicate that the deepwater wave power formulation is quitesensitive to the spectral width parameter. One source contributingto the significant difference between the distributions is the ratio ofT=Tp that is approximated with a value of unity in the narrow-banded formulation. The results presented in Fig. 4 suggest that thenarrow-band approximation is reasonable when uo0.40. Forcomparison, the shallow water PDF and CDF distributions arepresented in Fig. 5a and b, respectively. As can be observed, thevariation of the probability distributions with wave spectral widthis very nearly negligible, when compared to the narrow-bandapproximation for ps40.5. This is most likely a consequence of thefact that the shallow wave power formulation is not a function ofwave period.
4.2. Wave-by-wave analysis
The empirical probability distribution functions of wave powerfor a sea-state can be obtained from directly analyzing time-seriesrecords developed from simulations or measured in the laboratory
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
pd
f (p d
| S
)
Narrow-Band
ν = 0.1
ν = 0.3
ν = 0.4
ν = 0.6
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
pd
Narrow-Band
ν = 0.1
ν = 0.3
ν = 0.4
ν = 0.6
f (p d
| S
)
Fig. 4. Variation of deepwater wave power distribution in respect to the wave
spectrum width: (a) probability density functions and (b) cumulative distribution
functions.
0 1 2 3 4 50
0.5
1
1.5f (
p s |
S)
Narrow-Band
ν = 0.1
ν = 0.3
ν = 0.4
ν = 0.6
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
ps
ps
Narrow-Band
ν = 0.1
ν = 0.3
ν = 0.4
ν = 0.6
f (p s
| S
)
Fig. 5. Variation of shallow water wave power distribution in respect to the wave
spectrum width: (a) probability density functions and (b) cumulative distribution
functions.
Table 1Characterization of the simulated sea-states and the associated wave power
resource.
Seastate no. Hs Tp c t T�1 T T=T�1 Pd
(m) (s) (s) (s) (kW/m)
1 4.0 10.0 1.0 0.42 8.57 7.73 0.90 67.62
2 4.0 10.0 3.3 0.38 9.03 8.35 0.92 71.26
3 4.0 10.0 7.0 0.35 9.31 8.78 0.94 73.46
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185182
or the field. Basically, the distributions are developed from a zero-crossing analysis, which in turn allows a wave-by-wave estimate ofthe average wave power in each wave cycle (Smith and Venugopal,2006). More specifically, the wave periods Ti and associated waveheights Hi are obtained from the zero-crossing analysis of thesurface elevation time-series. Next, for each pair of Hi and Ti thewave power Pi is estimated using deepwater limit Eq. (2) or Eq. (3)for the shallow water limit, and consequently the wave powersamples are developed. The samples are then normalized with thecharacteristic values defined in Eq. (6) or Eq. (8) depending uponthe limit selected and then these empirical distributions of thewave power are compared with the appropriate theoretical dis-tributions. For illustrative purposes, the wave surface elevationtime-series were generated for the deep water limit using aJONSWAP wave energy spectrum model for different values ofthe peakedness parameter, specifically g¼1.0, 3.3, 7.0, while thesignificant wave height Hs¼4.0 m, and peak period Tp¼10.0 s wereheld constant. In order to reduce the sample size effects, for eachsea-state 60 h of surface wave elevation (about 26,000 waves) weregenerated with the sample rate of 5 Hz employing uniformlydistributed random phase in the range of [0, 2p].
The JONSWAP model parameters, the spectral width, estimatesof average and energy period, the ratio T=T�1, and estimates of themean deepwater power Pd, obtained using Eq. (4), are presented inTable 1. The ratio T=T�1 depends on the wave spectral shape andwidth and is expected to converge to a value of unity for an ideal
narrow-banded process. To verify this statement, wave conditionswere simulated using the JONSWAP spectrum model with asignificant wave height range of 0.5rHsr5.0 m, a peak periodrange 3rTpr20s, and spectral peakedness values ranging1rgr7. For these parameter ranges, it was determined that theratio T=T�1 varied in the range 0.88–0.95. As shown in Table 1, theratio of T=T�1 varies in the range 0.90–0.94 for the sea-statesspecified. The mean wave power values given in Table 1 indicatethat waves with narrower energy spectrum have higher averagepower as the energy is focused around the peak period and thesmall wave heights are less probable in a narrow-band process ascompared to a process with the wider spectrum. These results can
be contrasted with the single value of Pd ¼ 78:89 kW/m, obtainedfor each sea-state using Eq. (8), which yields an even higherestimate. The value of Pd represents the mean wave power in anideal narrow-banded spectrum with zero width.
Estimates of the corresponding theoretical model (Eq. (21)) andnarrow-band approximation (Eq. (26)) of the probability distribu-tion functions for wave power pd are presented and comparedwith the histogram of the data in Fig. 6. In this figure, the histogrambin width was estimated using the formula presented by (Wand,1997), i.e.,
h0 ¼24p1=2
n
� �1=3
s ð31Þ
where n is the sample size and s¼min ss, IQR=1:349� �
, ss is thesample standard deviation, and IQR is the inter-quartile rangedefined as the difference between the third and first quartiles. Theexpected values and standard deviations of the wave powercorresponding to the data in Fig. 6 are presented in Table 2. Thehistograms in Fig. 6 clearly show the exponential distribution ofwave power, which is consistent with the theoretical models. Thedistributions shown in Fig. 6 and the results presented in Table 2indicate a reasonably good agreement between the theoreticalmodel and the statistics obtained from the wave-by-wave analysis.The narrow-band approximation, on the other hand, constantlyoverestimates the statistics of wave power, while once again theperformance of the narrow-band model is found to be reasonablefor the second and third sea-states, where the spectral width is lessthan 0.4.
4.3. Long-term analysis of wave power
Since the final objective of these analyses is to address the useof site specific data, the long-term wave power distribution ofwave data measured at the Harvest Point, California was selectedfor evaluation. The wave data were measured using a Datawelldirectional buoy located at 341 27.240 N 1201 46.830 W (CDIP 071)operated by Coastal Data Information Program (CDIP) of ScrippsInstitution of Oceanography (http://cdip.ucsd.edu). The 549 mwater depth at the buoy location is assumed to be adequate tosatisfy the deepwater condition d/l41/2. The scatter diagram ofthe significant wave height Hs and peak period Tp for the waves at
0 1 2 3 4 50
0.5
1
1.5
pd
f (p d
| S
)Histogram
0 1 2 3 4 50
0.5
1
1.5 Histogram
0 1 2 3 4 50
0.5
1
1.5 Histogram
f (pd | S)
fNB
(pd | S)
f (p d
| S
)f (
p d |
S)
pd
pd
f (pd | S)
fNB
(pd | S)
f (pd | S)
fNB
(pd | S)
Fig. 6. Probability density functions of deepwater wave power in seastates with
JONSWAP energy spectrum and peakedness factor of (a)g¼1.0, (b) g¼3.3, and
(c) g¼7.0.
Table 2Statistics of non-dimensional wave power in the simulated seastates.
Model Sea-state
1 2 3
E(pd) r(pd) E(pd) r(pd) E(pd) r(pd)
Wave–wave 0.83 0.82 0.89 0.87 0.94 0.93
Theoretical model 0.91 0.92 0.96 0.97 0.98 0.98
Narrow-band approximation 1.00 1.00 1.00 1.00 1.00 1.00
Table 3Wave scatter diagram at Harvest, CA.
Hs(m) Tp (s)
5 7 9 11 13 15 17 19 22
8.7 0.00 0.00 0.00 0.00 0.00 0.05 0.00 0.00 0.00
8.1 0.00 0.00 0.01 0.00 0.02 0.02 0.00 0.02 0.00
7.5 0.00 0.00 0.01 0.00 0.02 0.06 0.05 0.02 0.00
6.9 0.00 0.00 0.01 0.06 0.10 0.14 0.10 0.14 0.00
6.3 0.00 0.00 0.01 0.18 0.17 0.21 0.21 0.19 0.00
5.7 0.00 0.00 0.10 0.46 0.50 0.71 0.55 0.32 0.00
5.1 0.00 0.00 0.22 1.38 1.65 2.27 1.10 0.57 0.00
4.5 0.00 0.02 1.53 3.25 3.60 6.12 2.20 1.26 0.01
3.9 0.00 0.35 5.57 7.02 9.84 12.18 3.69 2.35 0.07
3.3 0.04 3.44 14.78 17.29 22.19 22.50 5.91 4.79 0.17
2.7 0.37 18.29 34.24 35.36 37.18 31.26 8.65 8.42 0.49
2.1 2.95 49.25 55.88 59.31 48.42 41.91 16.80 11.96 0.50
1.5 9.93 29.59 63.42 53.28 42.95 64.90 23.18 12.29 0.30
0.9 1.47 2.59 11.98 12.14 17.36 22.89 5.61 2.82 0.12
0.3 0.00 0.00 0.00 0.03 0.07 0.10 0.00 0.01 0.01
*The values are average probability in 1000 observations.
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185 183
Harvest Point measured between 12/01/1995 and 09/28/2009 ispresented in Table 3. It is assumed that sea-states are relativelynarrow-banded and consequently the wave power distribution inthe sea-states can be approximated by the narrow-banded dis-tribution, Eq. (21). Utilizing the probabilities given in Table 3 inEq. (26), the PDF and CDF estimates of long-term wave power at theHarvest Point, CA are presented in Fig. 7. As shown in this figure, thePDF starts with a sharp slope and continues with a long flatted tail.The hyper-exponential distribution in this case has an expected
value and standard deviation of 34.26 and 60.59 kW/m,respectively.
The statistics obtained here can be interpreted from a riskanalysis perspective. Consider the risk associated with a value x asthe probability of having wave power smaller than or equal to thevalue of x, i.e., RðxÞ ¼ PrðPdrxÞ ¼ FðxÞ. Regarding the CDF shown inFig. 7, the risk associated with the yearly averaged expected wavepower E(Pd)¼34.26 kW/m, commonly used as the characteristicwave power at a site, is R(E(Pd))¼0.725(72.5%). In order to reducethe risk to 0.5(50%), the wave power resource decreases to15.20 kW/m which is less than half of the expected value, andfor an accepted risk of 0.1 the wave power is as low as 1.88 kW/m.The large difference between these estimates indicates the impor-tant role that uncertainties play in evaluating the wave powerresource potential at an offshore site.
4.4. Analysis of absorbed power by a specific wave energy device
The probability distribution of absorbed power in a slack-moored, heaving-cylinder buoy with phase control studied herebased upon the numerical simulations published by Eidsmoen(1996). The wave energy device used in their numerical investiga-tion was a spar buoy type wave energy device that was 3.3 m indiameter, 5.1 m depth, with a reaction plate, 8.0 m in diameter,0.2 m thick, submerged 10 m below the mean sea surface.Their numerical analysis was performed for the sea-states atHaltenbanken (641 10.50 N, 91 10.0 E) off the Norwegian coastand it was assumed that the waves could be modeled withJONSWAP wave energy spectrum with peakedness parameterg¼3.3. The scatter diagram of the significant wave heights andzero-upcrossing periods measured in between 1947 and 1978 at
0 1 2 3 4
x 105
0
1
2
3
4
5
x 10-5f (
Pd )
Pd (W / m)
0
0.2
0.4
0.6
0.8
1
F (
Pd
)
CDF
Fig. 7. Probability distribution functions of the wave power at Harvest, CA.
0 1 2 3 4
x 105
0
1
2
3
4
5
6
7x 10-5
Pd (W / m)
0
0.2
0.4
0.6
0.8
1
F (
Pd
)
f (P
d )
CDF
0 1 2 3 4
x 104
0
1
2
3
4x 10-4
f (P a
)
F (P
a )
Pa (W )
0
0.2
0.4
0.6
0.8
1CDF
Fig. 8. Probability distribution functions of the wave power at Haltenbanken,
Norway: (a) wave power and (b) absorbed power in a slack-moored heaving buoy.
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185184
this site were presented in Table 10 of Eidsmoen’s (1996)report andused in this analysis.
The probability density and cumulative probability distribu-tions of wave power at Haltenbanken, Norway, are presented inFig. 8(a). The distributions are estimated assuming that waves haverelatively narrow-banded energy spectrum and water depth is highenough to satisfy the deepwater condition. The yearly averagedestimates of the expected value and standard deviation of wave
power at this site are estimated to be 34.8 and 70.4 kW/m,respectively. The mean absorbed power for the sea-states withnon-zero probability are presented in Table 11 of Eidsmoen (1996).Based on these values, the long-term probability distribution ofabsorbed power in the WEC is estimated and the results arepresented in Fig. 8(b). From this, the yearly averaged expectedvalue and standard deviation of an absorbed power are estimatedas 4.9 and 7.3 kW, respectively. Regarding the distributions shownin Fig. 8(b), the risk associated with the expected value 4.9 kW isabout 0.69 (69%), while the absorbed powers with risks of 0.5 and0.1 are, respectively, 2.4 and 0.3 kW.
5. Summary and conclusions
The random nature of ocean waves at potential offshore sitesbeing considered for wave energy farms necessitates the develop-ment of analysis methods able to more accurately address theuncertainty associated with the evaluation of the wave powerpotential and design issues related to the optimization of waveenergy convertor designs. From a practical point of view, thisrequires accurate site specific estimates of both the short- and long-term probability distributions of the wave power available. Here, atheoretical distribution model was developed that models wavepower for a given sea-state. Introducing the linear wave theory,theoretical joint probability distribution functions for wave heightand period in a given sea-state were derived in part of the formerstudies by Longuet-Higgins (1975, 1983). Here, emphasis wasplaced on deriving approximations for both deepwater and shallowwater limits in order to better understand the engineering implica-tions. Of particular interest was the nature of the distributionfunctions, their limit forms, and the role of narrow-bandedassumptions in reducing the non-dimensional form of the deep-water and shallow water wave power distributions to the well-known exponential distribution.
A triangular spectrum model, that approximated the morecomplex wave spectrum used in practice, was introduced in orderto illustrate the effects of wave spectral width on the probabilitydistribution of wave power in the sea-state. It has been observedthat in the deepwater condition, the conditional probability con-tours of wave power in the sea-state are highly sensitive to thespectral width and consequently the narrow-band approximationmust be viewed with caution. For these examples, the narrow-bandappeared to be a reasonable approximation for wave spectra withspectral widths of uo0.4. It was also observed that the shallowwater formulation can be reasonably approximated by a narrow-band model, which was reasonable considering that the shallowwater wave power equation is not a function of wave period.A comparison of the theoretical model results and the statisticsobtained from a wave-by-wave analysis of simulated deepwaterwaves indicates a reasonable agreement with the derived distribu-tion functions.
The universal short-term probability distribution of the wavepower can be incorporated with the long-term probability dis-tribution of sea-states at an offshore site to evaluate the randomvariability in the wave power resources. A long-term estimate ofthe sea-state probabilities may be obtained from the significantwave height and peak period scatter diagram developed from long-term wave measurements. To clarify the procedure, a sample long-term analysis of wave power was illustrated utilizing the measuredwave data at a site off the California coastline. This examplehighlighted the importance of addressing uncertainty in evaluationof wave power resources.
Similarly, the sea-state dependent model of the wave energydevice can be incorporated with the local wave data to obtain anestimate of the potential wave energy that could be converted into a
A.H. Izadparast, J.M. Niedzwecki / Ocean Engineering 38 (2011) 177–185 185
usable energy resource. Here, a simplified model was used toapproximate the short-term probability distribution of the absorbedpower in a wave energy device. The model assumed that theabsorbed power had an exponential distribution and that themean absorbed power in the sea-state is known from numerical,experimental, or field studies. In a hypothetical example, the long-term probability distribution of an absorbed power was estimatedfor a slack-moored, heaving-cylinder buoy with the phase control.Data available in the open literature for a case previously simulatedat a site of the Norwegian coast was used. The long-term statisticspresented could be applied to evaluate the risk associated with anexpectation. The methodology is not limited to the examplediscussed, but requires information that is device specific and notgenerally available, but is crucial for estimating the wave powerresource and the associated economic feasibility studies.
Acknowledgements
The writers would like to gratefully acknowledge the partialfinancial support of the Texas Engineering Experiment Station andthe R.P. Gregory ’32 Chair endowment during this researchinvestigation.
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