A Method for Energy and Resource Assessment of Waves in Finite Water Depths

Article A Method for Energy and Resource Assessment of Waves in Finite Water Depths

Wanan Sheng 1,* and Hui Li 2 1 Centre for Marine and Renewable Energy (MaREI), Environmental Research Institute, University College Cork, Cork P43 C573, Ireland 2 College of Mechanical and Energy Engineering, Jimei University, Xiamen 361021, China; [email protected] * Correspondence: [email protected]; Tel.: +353-21-4864378

Academic Editor: Stephen Nash Received: 21 December 2016; Accepted: 28 March 2017; Published: 2 April 2017

Abstract: This paper presents a new method for improving the assessment of energy and resources of waves in the cases of finite water depths in which the historical and some ongoing sea wave measurements are simply given in forms of scatter diagrams or the forms of (significant) wave heights and the relevant statistical wave periods, whilst the detailed spectrum information has been discarded, thus no longer available for the purpose of analysis. As a result of such simplified wave data, the assessment for embracing the effects of water depths on wave energy and resources becomes either difficult or inaccurate. In many practical cases, the effects of water depths are simply ignored because the formulas for deep-water waves are frequently employed. This simplification may cause large energy under-estimations for the sea waves in finite water depths. To improve the wave energy assessment for such much-simplified wave data, an approximate method is proposed for approximating the effect of water depth in this research, for which the wave energy period or the calculated peak period can be taken as the reference period for implementing the approximation. The examples for both theoretical and measured spectra show that the proposed method can significantly reduce the errors on wave energy assessment due to the approximations and inclusions of the effects of finite water depths.

Keywords: wave energy; wave energy assessment; ﬁnite water depth; energy assessment method; wave energy resources

1. Introduction Wave energy resource assessments have been an important factor for wave energy developments in recent years, and the focus has been on the assessment and characterisation of wave energy resources [1–13], with the global wave energy resource assessments on the overall resources and the distributions of wave energy [3–6], and the regional and national wave energy resources on the potentials for developing wave energy [7–13]. As pointed out by Cahill et al. [7], the primary purpose of these studies is to examine the available potential energy at the locations of interest, and of the seasonal and annual trends in the resources. Another important issue is the identiﬁcation of extreme events and wave conditions, which may be of critical importance on the operation and survival of wave energy converters (WECs). Recent global efforts on standardisation of wave resource assessments [14,15] have led to the development and publication of the International Electrotechnical Commission (IEC) technical speciﬁcation [16]. An application of the IEC standard to the wave energy resources at the Irish West Coast has been published [17]. For the practical purposes and the cost of wave energy production, so far nearshore and shallow water regions have been frequently considered for deploying wave energy converters and wave farms due to their closeness to the shore and the available infrastructure for cable connection and for easy

Energies 2017, 10, 460; doi:10.3390/en10040460 www.mdpi.com/journal/energies Energies 2017, 10, 460 2 of 17 accessEnergies for the 2017 operation, 10, 460 and maintenance. Magagna et al. [18] collected the information of2 of current 17 and proposedeasy access wave for the energy operation deployments and maintenance. (see Figure Magagna1), and et it al can. [18] be collected seen that the most information of the installed of wavecurrent energy and converters proposed havewave beenenergy deployed deployments in water (see Figure depths 1), lessand thanit can 50be m,seen and that these most trendsof the can be alsoinstalled seen forwave the energy proposed converters wave have farms. been Similarly, deployed in Johanning water depths et al.less [ 19than] indicated 50 m, and these early trends that wave energycan converters be also seen will for bethe very proposed likely wave installed farms. in Similarly, the shallow Johanning to intermediate et al. [19] indicated depths typically early that at the 50 mwave contour energy in the converters open areas will forbe very wave likely energy installed production. in the shallow For such to intermediate developments, depths the typically availability of waveat the energy 50 m resources contour in in the those open regions areas for would wave be energy important production. and it For was such conﬁrmed developments, by a study the in Folleyavailability et al. [8] of that wave the energy reduction resources in the in netthose wave regions energy would resource be important from and a water it was depth confirmed = 50 by m to a a study in Folley et al. [8] that the reduction in the net wave energy resource from a water depth = 50 nearshore location of water depth = 10 m is about 10%. In this regard, the available wave resources m to a nearshore location of water depth = 10 m is about 10%. In this regard, the available wave in nearshore areas may be still good enough for promoting wave energy production. All of these resources in nearshore areas may be still good enough for promoting wave energy production. All of considerationsthese considerations have been have conﬁrmed been confirmed by the by installed the installed and proposed and proposed wave wave energy energy converters converters and and wave farmswave (see farms Figure (see1). Figure 1).

Figure 1. Wave energy deployment: water depth vs. distance from shore. Note: the size of the bubble Figure 1. Wave energy deployment: water depth vs. distance from shore. Note: the size of the bubble refers to the capacity of installed projects (full circle) or the maximum site capacity (circles). Source: refers to the capacity of installed projects (full circle) or the maximum site capacity (circles). Source: courtesy of Davide Magagna [18]. courtesy of Davide Magagna [18]. Due to the deployments of wave energy converters in relatively small water depths, it is very Dueimportant to the to deployments assess the wave of energy wave and energy wave converters energy resources in relatively more accurately small water because depths, the effects it is very importantof finite to water assess depths the wave must energy be considered. and wave In the energy cases resourcesof regular waves, more accurately the effect of because water depths the effects on of finitewave water energy depths can must be easily be considered. calculated due In to the their cases well of mathematically regular waves, defined the effect wave of periods. water In depths the on cases of the sea waves (i.e., the irregular waves), it would be straightforward to assess their energy if wave energy can be easily calculated due to their well mathematically defined wave periods. In the the detailed spectra (i.e., their spectral distributions and shapes) are known, since the spectra can be cases of the sea waves (i.e., the irregular waves), it would be straightforward to assess their energy if regarded as a sum of many narrow bandwidths, and each narrow bandwidth corresponds to a the detailedregular wave spectra with (i.e., the their given spectral wave amplitude distributions and frequency/period, and shapes) are hence known, its corresponding since the spectra wave can be regardedenergy as can a sum be accurately of many narrowcalculated bandwidths, for including and the each effect narrow of water bandwidth depth. The correspondssum of the energy to a regularof waveall with the theregular given wave wave components amplitude is the and wave frequency/period, energy for the sea hencewave/spectrum. its corresponding However, wavein many energy can bepractical accurately cases, calculated the historical for and including ongoing the sea effect wave of measurements water depth. are The frequently sum of thegiven energy in simple of all the regularforms wave of componentsscatter diagrams is the or wavesome statistical energy for parameters, the sea wave/spectrum. such as the significant However, wave in height many and practical cases,stat theistical historical periods, and while ongoing the important sea wave spectrum measurements information are (spectral frequently shape given and distribution) in simple forms for of scatterassessing diagrams the effect or some of water statistical depths parameters, on wave energy such has as been the discarded significant and wave no longer height available and statistical for analysis. As such, the effect of water depths on wave energy and resource assessments becomes periods, while the important spectrum information (spectral shape and distribution) for assessing difficult or inaccurate when only the significant wave heights and characteristic periods are available, the effect of water depths on wave energy has been discarded and no longer available for analysis. and the simple option is to use the formulas for deep-water waves by ignoring the effect of water As such,depths. the Such effect simplif of waterication depths may cause on wave large errors energy in andthe wave resource energy assessments and resource becomes assessment difficult (up or inaccurateto 14% when under only-estimation). the significant For improving wave heights the wave and energy characteristic and resource periods assessment, are available, a method and is the simpleproposed option in is this to useinvestigation, the formulas with the for inclusion deep-water of the waves effects byof finite ignoring water thedep effectths on wave of water energy depths. Suchassessment. simplification From may the examples, cause large it can errors be seen in thethatwave the proposed energy method and resource is capable assessment of improving (up the to 14% under-estimation).wave energy assessment For improving in the the cases wave of finite energy water and depths: resource for assessment, both popular a method theoretical is proposed spectra in this investigation, with the inclusion of the effects of finite water depths on wave energy assessment. From the examples, it can be seen that the proposed method is capable of improving the wave energy Energies 2017, 10, 460 3 of 17 assessment in the cases of finite water depths: for both popular theoretical spectra (Bretschneider spectrum and JONSWAP spectrum) and the measured spectra from the Irish Coast, the proposed method could reduce the errors of wave energy assessment to less than 5% for the individual sea state (from the error of more than 14% when using the deep-water formulas), and to less than 4% for wave resources (compared to the error of more than 10% using the deep-water formulas).

2. Wave Energy Assessment

2.1. Wave Equation Linear wave theory is very well developed, and the details can be found in many standard text books [20–22]. For completeness and reference, some basic equations are given. In the linear wave theory, the wave velocity potential in water depth, h, is given as

gA cosh k(z + h) φ = sin(kx − ωt), (1) ω cosh kh where A is the wave amplitude, h the water depth, k the wave number (k = 2π/λ, λ is the wave length), ω the wave circular frequency. x and z are the coordinates along the x-axis (wave travel direction) and z-axis (vertical up positive with z = 0 at the calm water surface). The wave elevation has a form as

η = A cos(kx − ωt), (2) with a dispersion relation as ω2 = gktanh(kh). (3)

2.2. Wave Energy in Deep Water For convenience of the presentation, the formulas in deep water are given ﬁrst. The deep water condition here is given when the water depth, h, is larger than half of the wave length, λ0, i.e., h ≥ λ0/2 (λ0 is the wave length without seabed effect, and hereafter the subscript “0” means the parameter in deep water), though some other deﬁnitions for deep water conditions can be also seen, for instance, h ≥ λ0/4 ([22], p. 287). However, when the deep water condition (h ≥ λ0/2) is used, the maximal error of the wave length is about 0.36%, given by following formula:

|λ − λ | err = 0 × 100%, (4) λ0 where λ is the actual wave length at a water depth h = λ0/2. Similarly, when the deep water condition (h ≥ λ0/4) is used, the maximal error in wave length is 6.67% (also for a comparison, 2.67% for h ≥ λ0/3). It will be seen in the analysis below that the deep-water condition (h ≥ λ0/4) may cause a large error in terms of wave energy assessment. In deep water conditions, the wave energy of regular waves can be expressed in a popular form of wave height and period, as 1 P = ρg2 H2T, (5) reg0 32π where H is the wave height (i.e., H = 2A). For a given spectrum, corresponding to the frequency ω with a frequency bandwidth ∆ω, the wave power is calculated from (5), as

1 2π 1 ∆ω ∆P = ρg2[8S(ω)∆ω] = ρg2S(ω) . (6) irr0 32π ω 2 ω Therefore, the total time averaged wave power for the given spectrum is calculated as Energies 2017, 10, 460 4 of 17

∞ 1 Z dω P = ρg2 S(ω) . (7) irr0 2 ω 0

For the given spectrum, the wave height is given by √ Hs = 4 m0, (8) and the wave characteristic/statistic periods can be deﬁned as: r m−1 m0 m0 Te = 2π , T01 = 2π , T02 = 2π , (9) m0 m1 m2 where Te, T01 and T02 are the energy period, the spectral mean period and the mean zero-upcrossing period, respectively, with the spectral moment being deﬁned as

Z ∞ n mn = S(ω)ω dω, (10) 0 where n = −2, −1, 0, 1, 2, . . . Using the deﬁnitions of the signiﬁcant wave height, the energy period and the spectral moments yields ∞ Z dω Te 1 2 S(ω) = m− = m = H T . (11) ω 1 2π 0 32π s e 0 Thus, the wave energy in irregular waves can be calculated as

1 P = ρg2 H2T , (12) irr0 64π s e which is same as that given by Tucker and Pitt [22]. A more popular form is given in the unit of the wave power of kW/m, as 2 Pirr0 = 0.49Hs Te. (13) It must be emphasised that this formula for the wave energy of a sea state is the only correct and universal formula for calculating wave energy in deep water. From the formula, it can be seen that the wave energy is proportional to the significant wave height squared and proportional to wave energy period. The formula has been specified in the recent international standards, the IEC technical specification (IEC 62600-101 [16]), indicating that the wave scatter diagram is given only by the significant wave height, Hs, and the energy period, Te. It is worth mentioning that some historical and ongoing wave measurement data (or the scatter diagrams) are still given either in the spectral peak period, Tp, or the mean zero upcrossing period, Tz (note: in such cases, the relation between Tp or Tz vs. Te must be specified so to make the data useful). A typical example is the wave data for a few locations at Irish Coast (the Irish Ocean Energy Expertise [23]), the available wave periods are only given in Tp or Tz or both, but Te is not an option in the available data.

2.3. Waves in Finite Water Depths From the dispersion relation (3), the wave length in a ﬁnite water depth, h, can be calculated using an iterative method (the initial wave length can be taken as the deep-water wave length, λ0) as

2πh λ = λ tanh , (14) 0 λ and the wave length in ﬁnite water depths can be seen in Figure2. The celerity in the ﬁnite water depth is (also see [21]) Energies 2017, 10, 460 5 of 17

Energies 2017, 10, 460 5 of 17 1  2kh    2kh  k C  1   C 1  0 2  sinh(2kh)  k 0  sinh(2kh)  k (15)     , Energies 2017, 10, 460 1 2kh ω 2kh k0 5 of 17 C = 1 + = C0 1 + , (15) where C0 (=ω/k0) is the celerity2 of thesinh wave(2kh in) deepk water. sinh(2kh) k 1  2kh    2kh  k Thus, the wave power in finiteC  water1 depth is calculatedC 1 as  0 where C (=ω/k ) is the celerity of the2  wavesinh(2 inkh) deep k water.0  sinh(2kh)  k (15) 0 0     , Thus, the wave power in finite water depth is calculated2kh  ask P  P    0 where C0 (=ω/k0) is the celerity of thereg wave inreg 0deep1 water.  (16)  sinh(2kh)  k Thus, the wave power in finite water depth is calculated as , 2kh k0 Preg = Preg0 1 + , (16)  sinh(2kh k) k which indicates that the wave celerity and the wave 2powerkh in0 finite water depth can be regarded as Preg  Preg0 1   (16) the celerity and wave power in deep water modified sinh( with2kh) a kcoefficient, defined as which indicates that the wave celerity and the wave power in finite, water depth can be regarded as the celeritywhich and wave indicates power that inthe deepwave watercelerity modified and the wave with2 powerkh a coefficient, ink finite water defined depth as can be regarded as C  1   0 the celerity and wave power in deep waterh modified sinh( with2kh) a coefficient,k defined as (17)   .  22khkh  k k0 Ch =C 1+1   0 . (17) h  sinhsinh( (kh2kh)  )k k (17)  2  .

FigureFigure 2. Wave 2. Wave lengths lengths in in finitefinite water depths depths and and deep deep water. water. Figure 2. Wave lengths in finite water depths and deep water. From the modification coefficient curve in non-dimensional frequency shown in Figure 3, it can FromFrombe seen thethe that, modificationmodification for a given coefficientwatercoefficient depth, curvethecurve modification inin non-dimensionalnon-dimensional coefficient, Ch, frequency frequencyis a function shownshown of wave in infrequency, FigureFigure3 3,, it it can can bebe seenseenhence, that,that, forforit is a anoted givengiven as waterwater Ch(ω) hereafter. depth,depth, thethe For modificationmodification a given spectrum coefficient,coefficient, in a finite C Cwaterhh, is a depth, function the spectr of wave waveum canfrequency, frequency, be divided into many small bandwidths Δω, and each bandwidth corresponds to a regular wave hence,hence, itit isis notednoted asasC Chh((ωω)) hereafter.hereafter. ForFor aa givengiven spectrumspectrum inin aa finitefinite waterwater depth,depth, thethe spectrumspectrum cancan component. For each regular wave component corresponding to the frequency ω, Ch(ω) can be be divided into many small bandwidths ∆ω, and each bandwidth corresponds to a regular wave be dividedcalculated into accuratelymany small for a bandwidthsgiven water depth Δω,. Asand such, each the bandwidth accurate wave corresponds energy assessment to a forregular the wave component.component.spectrum For can each be conducted regular (similar wave wave component componentmethodologies corresponding corresponding can be found in references to to the the frequency frequency [4,7,16,24]). ω ω,, CChh((ωω)) cancan be be calculatedcalculated accuratelyaccurately forfor aa givengiven waterwater depth.depth. AsAs such,such, thethe accurateaccurate wavewave energyenergy assessmentassessment forfor thethe spectrumspectrum cancan bebe conductedconducted (similar(similar methodologiesmethodologies cancanbe befound found in in references references [ 4[4,7,16,24],7,16,24]).).

Figure 3. Modification coefficient of celerity in a non-dimensional form.

Figure 3. Modification coefficient of celerity in a non-dimensional form. Figure 3. Modiﬁcation coefﬁcient of celerity in a non-dimensional form.

Energies 2017, 10, 460 6 of 17

The wave power of a regular wave component in the irregular waves can be calculated as

1 ω ∆Pirr = ρg[2S(ω)∆ω] Ch(ω). (18) 4 k0

Therefore, the total average wave power for the given spectrum is thus

∞ 1 Z dω P = ρg2 C (ω)S(ω) . (19) irr 2 h ω 0

The direct numerical integral of Equation (19) in the case of the available spectrum can be taken 0 as the accurate wave power for the spectrum, denoted as Pirr, and the error (%) of the result from the approximation is calculated as P − P0 = irr irr × error 0 100%, (20) Pirr where Pirr is the calculated wave power using the proposed method. Generally, the accurate integration of Equation (19) can only be possible when the spectral components are known (the corresponding Ch can be calculated accurately). However, in the case C of deep water, the modification√ coefficient is a constant, i.e., h = 1.0, when the non-dimensional frequency is larger than π in Figure3. Hence, it becomes obvious that to accurately assess the wave power in finite water depth, the wave spectrum must be used. Now, a method is proposed for solving the problem as follows. To more accurately assess the wave power in the cases of finite water depths, it is reasonable to assume that an actual wave spectrum may concentrate on a limit bandwidth of frequencies. For instance, for a Bretschneider spectrum (Tp = 8 s, Hs = 2.0 m), 50% of the energy is concentrated within a bandwidth of 0.305 rad/s around the peak period (see Figure4). Hence, it can be suggested that Ch in the integral in Equation (19) can be approximated with a constant, the constant modification coefficient corresponding to a specific spectral characteristic frequency, ωc, as,

∞ 1 Z dω P ≈ ρg2 C (ω )S(ω) . (21) irr 2 h c ω 0

With this assumption, the wave power (in kW/m) in irregular waves in ﬁnite water depths can be calculated as 2 Pirr = 0.49Hs TeCh(ωc), (22) where ωc can be either the spectral peak frequency, ωp (=2π/Tp), or energy frequency, ωe (=2π/Te). For comparison, the other statistic frequencies/periods may be used as well in the following analysis. For convenience in the analysis, the modiﬁcation factor in Equation (22) can be referred to one of the statistic periods; hence, the wave power estimation is given as

2 Pirr = 0.49Hs TeCh(Tc), (23) where Tc can be one of the popular statistic periods, such as Tp, Te, T01 or T02, and it will be seen that the ﬁrst two periods may be the best choice in this regard. Energies 2017, 10, 460 7 of 17 Energies 2017, 10, 460 7 of 17

ωp

Figure 4.4. 50%50% energyenergy concentrated concentrated in in the the bandwidth bandwidth of of 0.305 0.305 rad/s rad/s (Bretschneider (Bretschneider spectrum: spectrum:Hs H=s 2.0 = 2.0 m, Tm,p =Tp 8.0 = 8.0 s). s).

2.4. Theoretical and Measured Wave SpectraSpectra Sea waveswaves areare irregular, irregular, meaning meaning the the wave wave period period and and wave wave height height vary vary from from wave wave to wave, to wave, thus irregularthus irregular waves waves are often are characterisedoften characterised by some by popularsome popular and most and used most statistical used statistical parameters, parameters, such as s p e thesuch signiﬁcant as the significant wave height, waveH height,s, the characteristic H , the characteristic wave periods: wave pTeriods:p (spectral T (spectral peak period), peak period),Te (energy T (energy period), T01 (mean spectral period), and T02 (mean zero upcrossing period). In addition, the period), T01 (mean spectral period), and T02 (mean zero upcrossing period). In addition, the wave spectralwave spectral shape/type shape/type is an importantis an important factor factor in applications. in applications. The most accepted theoretical wave spectra include thethe Bretschneider spectrum for the fully developed waves, and the JONSWAP JONSWAP spectrum spectrum for for the the limited limited developed developed waves. waves. They can be expressed usingusing aa generalisedgeneralised JONSWAPJONSWAP spectrum,spectrum, asas

4 4 5 p 5  4 5ωp4 2 45ω4 p  S()  (1  0.287ln  ) p5 H s2e−  α (24) S(ω) = (1 − 0.287 ln γ)16 H e 4ω4 γ , (24) 16ω5 s ,

2     0(.07,    p with   p  2 and   . γ can take a value between 1.0–5.0 depending on the   exp  (ω−2 ω2p)  0.07, ω ≤ ωp     0.09,    p with α = exp −2 p 2 2  and σ = . γ can take a value between 1.0–5.0 depending 2ωpσ 0.09, ω > ωp Det Norske Veritas (DNV) onactual the actualwave conditions wave conditions and locations and locations (see (see Det Norske Veritas (DNV)standard standard [25]), with [25]), γ with= 3.3 γfor= the 3.3 forstandard the standard JONSWAP JONSWAP spectrum, spectrum, and γ = and 1.0 γfor= the 1.0 forBretschneider the Bretschneider spectrum. spectrum. As it can can be be seen seen from from Equation Equation (24), (24), the the theoretical theoretical spectrum spectrum is defined is deﬁned using using the spectral the spectral peak frequency/period, ωp/Tp. For measured waves, their spectra can often be spiky. Figure 5 show two peak frequency/period, ωp/Tp. For measured waves, their spectra can often be spiky. Figure5 show twomeasured measured spectra spectra for the for thedifferent different measured measured sea sea states states at atAtlantic Atlantic Marine Marine Energy Energy Test Test Site Site (AMETS), (AMETS), Belmullet, Ireland. Physically, thethe spectralspectral peakpeak period can be easily decided by simply choosing the corresponding frequency/periodfrequency/period to to the the spectral spectral peak. However, However, this may be very dependent on how the measured datadata have been recorded andand analyzed; forfor instance, the sampling length and sampling frequency. AA betterbetter choicechoice forfor deciding thethe spectral peak period would be the calculated spectral peak period recommended byby TuckerTucker andand PittPitt [[22]22] (p.(p. 41).41). As such,such, the calculated spectral peak period has similar features as those well-defined statistical periods including Te, T01, T02 (Tz) in Equation (9). similar features as those well-deﬁned statistical periods including Te, T01, T02 (Tz) in Equation (9).

Energies 2017, 10, 460 8 of 17 Energies 2017, 10, 460 8 of 17

Figure 5.5. Spectra from the measuredmeasured waveswaves (AMETS,(AMETS, Belmullet,Belmullet, waterwater depthdepth == 50 m).

Following Tucker and Pitt [22], the calculated peak period is calculated using the relevant Following Tucker and Pitt [22], the calculated peak period is calculated using the relevant spectral spectral moments, as moments, as m−2m1 = π mm21 TpcT 22π 2 . (25) pc m2 (25) 00 . This formula works generally well for both the Breschneider and the standard JONSWAP spectrum.This formulaHowever, works it is also generally found that well this for formula both the gives Breschneider a small yet a and consistent the standard over-estimation JONSWAP of thespectrum. peak periods However, by 2–3%. it is also It is found therefore that suggested this formula that gives a slightly a small modified yet a consistent formula can over be-estimation used as of the peak periods by 2–3%. It is therefore suggested that a slightly modified formula can be used as m−2m1 Tpc = 2π mm/1.025. (26) T  2π m212 /1.025 pc m02 (26) 0 . Formula (26) gives much better approximations for the peak period for the theoretical spectra. However,Formula in the (26) practical gives much applications, better approximations especially when for the the wave peak spectrum period for has the double theoretical peaks, spectra. such as thoseHowever, waves in combining the practical the applications, swells and wind-generated especially when waves, the wave the formula spectrum will has over-estimate double peaks, the peaksuch periodas those very waves much. combining In that case, the swells a limit and may wind be applied-generated as, waves, the formula will over-estimate the peak period very much. In that case, a limit may be applied as,  m−2m1 m−2m1  2π 2 /1.025, f or 2π 2 /1.025 ≤ 1.25Te  m2 mm 1 m 2 mm 1 Tpc = 2π 0 /1.025, for 2π 0 /1.025 1.25 T (27)  22m−2m1 e  1.25mmTe, f or 2π 2 /1.025 > 1.25Te  00m0 Tpc   (27) mm 1.25T , for 2π 21/1.025 1.25 T Note that the coefficient 1.25 inee the second equation in2 Equation (27) was obtained using the wave  m0 measurements in 2010 in AMETS, Ireland, and the total number of the measurements. is 13,189 for two data buoysNote that in the the year coefficient of 2010. 1.25 in the second equation in Equation (27) was obtained using the wave measurements in 2010 in AMETS, City, Ireland, and the total number of the measurements is 3. Results and Analysis 13,189 for two data buoys in the year of 2010. In this section, the assessment of the wave energy in finite water depths will be conducted and3. Results compared. and Analysis The accurate assessment of wave energy in a finite water depth is the one using the availableIn this section, spectra the per assessment Equation of (19), the which wave energy is taken in as finite the referencewater depths in the will comparison be conducted (in and the figurescompared. as “Ref”), The accurate while the assessment approximations of wave using energy different in aC h finiteare identified water depth as “Mod_Tp”, is the one “Mod_Te”, using the “Mod_T01”available spectra and “Mod_T02”per Equation in (19), the figures,which isdenoting taken as the reference modifications in theC comparisonh are calculated (in the using figures the peak period T , energy period T , mean spectral period T and mean zero upcrossing period, T , as “Ref”), whilep the approximationse using different Ch 01are identified as “Mod_Tp”, “Mod_Te”,02 respectively. In addition, the uncorrected wave energy is plotted as “DeepWater”, which is the case “Mod_T01” and “Mod_T02” in the figures, denoting the modifications Ch are calculated using the without including the effect of water depth on wave energy assessment. peak period Tp, energy period Te, mean spectral period T01 and mean zero upcrossing period, T02, respectively. In addition, the uncorrected wave energy is plotted as “DeepWater”, which is the case 3.1. Bretschneider Spectra (Hs = 2 m for All Cases) without including the effect of water depth on wave energy assessment. Figure6 shows the comparison of the wave energy calculations using different correction methods. It3.1. can Bretschneider be seen that Spectra for the (H smalls = 2 wave m for periodsAll Cases) (T e < 7.0 s), the water depth of 50 m would be considered

Energies 2017, 10, 460 9 of 17

Figure 6 shows the comparison of the wave energy calculations using different correction methods. It can be seen that for the small wave periods (Te < 7.0 s), the water depth of 50 m would be considered as deep water, thus all corrections are all very small. For the long waves, the deep water condition generally under-estimate the wave energy assessment by up to 13% (see Figure 7). The correction with the peak period slightly underestimates the wave power for the short waves (similar to other corrections), and then over-estimate the wave power for the wave energy periods between 8–17 s, with the maximal error less than 5%. For other correction methods including the one with the energy period tend to under-estimate the power in short waves and then to over-estimate the wave power in the long waves. The maximal errors are 6% for “Mod_Te”, 8% for “Mod_T01” and 9% for “Mod_T02”.

Energies 2017, 10, 460 9 of 17

Figure 6 shows the comparison of the wave energy calculations using different correction Energies 2017, 10, 460 9 of 17 methods. It can be seen that for the small wave periods (Te < 7.0 s), the water depth of 50 m would be considered as deep water, thus all corrections are all very small. For the long waves, the deep water ascondition deep water, generally thus allunder corrections-estimate are the all wave very energy small. Forassessment the long by waves, up to the 13% deep (see water Figure condition 7). The generallycorrection under-estimatewith the peak period the wave slightly energy underestimates assessment bytheup wave to 13%power (see for Figure the short7). The waves correction (similar withto other the corrections), peak period and slightly then underestimates over-estimate the the wave wave power power for for the the wave short energy waves periods (similar between to other corrections),8–17 s, with the and maximal then over-estimate error less than the wave 5%. For power other for correction the wave methods energy periods including between the one 8–17 with s, with the theenergy maximal period error tend less to thanunder 5%.-estimate For other the correction power in methodsshort waves including and then the oneto over with-estimate the energy the period wave tendpowerFigure to in under-estimate the 6. Modificationslong waves. the ofThe power the maximalwave in shortenergy errors waves assessments are and 6% then usingfor “Mod_Te”, to different over-estimate characteristic 8% for the “Mod_T01” wave periods power (h = 50 and in m). the 9% long for waves.“Mod_T02”. The maximal errors are 6% for “Mod_Te”, 8% for “Mod_T01” and 9% for “Mod_T02”. In the case of water depth of 25 m, it can be seen from Figures 8 and 9 that the deep water condition gives under-estimations for short waves (Te < 15 s) with a maximal error of 13.5%. For the correction using Tp, it slightly over-estimates the wave power in the short waves (Te < 12 s), and then under-estimates the wave power, with a maximal error about 5% (slightly larger errors in the very long waves for Te > 16 s). For the correction using energy period, it under-estimates the wave power for short waves (Te < 8 s), and then over-estimates the power almost constantly larger than 5% when Te > 10 s. For the correction with T01 and T02, the corrected wave powers are similar to the case using the energy period, but with larger errors (maximal errors are 11% and 16%, respectively). Figure 10 shows the comparisons of the errors in different water depths (50 m and 25 m). The general trends are the same; for instance, the maximal errors are similar but occur at different wave periods. The error curves shift to longer periods in a deeper water condition. If we plot these using the non- * g dimensional wave energy periods, Te , defined as T *  T , the differences for the different water e e h depthsFigure disappear 6. ModificationsModiﬁcations (see Figure of of the the 11). wave wave energy energy a assessmentsssessments using using different different characteristic characteristic periods periods (h( h= =50 50 m). m).

In the case of water depth of 25 m, it can be seen from Figures 8 and 9 that the deep water condition gives under-estimations for short waves (Te < 15 s) with a maximal error of 13.5%. For the correction using Tp, it slightly over-estimates the wave power in the short waves (Te < 12 s), and then under-estimates the wave power, with a maximal error about 5% (slightly larger errors in the very long waves for Te > 16 s). For the correction using energy period, it under-estimates the wave power for short waves (Te < 8 s), and then over-estimates the power almost constantly larger than 5% when Te > 10 s. For the correction with T01 and T02, the corrected wave powers are similar to the case using the energy period, but with larger errors (maximal errors are 11% and 16%, respectively). Figure 10 shows the comparisons of the errors in different water depths (50 m and 25 m). The general trends are the same; for instance, the maximal errors are similar but occur at different wave periods. The error curves shift to longer periods in a deeper water condition. If we plot these using the non- * dimensional wave energy periods, Te , defined as * g , the differences for the different water Te  Te h depths disappearFigure 7. Errors (see Figure for deep-waterdeep 11).-water formulas and modiﬁcationsmodifications using different periodsperiods ((hh == 50 m).m).

In the case of water depth of 25 m, it can be seen from Figures8 and9 that the deep water condition gives under-estimations for short waves (Te < 15 s) with a maximal error of 13.5%. For the correction using Tp, it slightly over-estimates the wave power in the short waves (Te < 12 s), and then under-estimates the wave power, with a maximal error about 5% (slightly larger errors in the very long waves for Te > 16 s). For the correction using energy period, it under-estimates the wave power for short waves (Te < 8 s), and then over-estimates the power almost constantly larger than 5% when Te > 10 s. For the correction with T01 and T02, the corrected wave powers are similar to the case using the energy period, but with larger errors (maximal errors are 11% and 16%, respectively). Figure 10 shows the comparisons of the errors in different water depths (50 m and 25 m). The general trends are the same; for instance, the maximal errors are similar but occur at different wave periods. The error curves shift to longer periods in a deeper water condition. If we plot these using the non-dimensional q * ∗ g wave energy periods, Te , deﬁned as Te = Te h , the differences for the different water depths disappearFigure (see 7. Figure Errors 11 for). deep-water formulas and modifications using different periods (h = 50 m).

Energies 2017, 10, 460 10 of 17 Energies 2017,, 10,, 460 10 of 17

Figure 8. ModificationsModiﬁcations of of the the wave wave energy energy assessments assessments using using differen differenttt characteristiccharacteristic characteristic periodsperiods periods ((h( =h 25= 25 m) m)...

Figure 9. Errors for deep deep-water--water formulas and modiﬁcationsmodifications usingusing differentdifferent periodsperiods ((hh == 25 m).m)..

Figure 10. Errors at different water depths (50(50 mm vs.vs. 25 m).

Energies 2017, 10, 460 11 of 17 Energies 2017, 10, 460 11 of 17

FigureFigure 11. 11.Errors Errors for for different different water water depth in in non non-dimensional-dimensional wave wave periods. periods.

Figure 11. Errors for different water depth in non-dimensional wave periods. 3.2. Standard JONSWAPFigure 11. Spectra Errors for (H differents = 2 m for water All Cases) depth in non-dimensional wave periods. 3.2. Standard JONSWAP Spectra (Hs = 2 m for All Cases) 3.2. StandardIn the cases JONSWAP of the standardSpectra (H JONSWAPs = 2 m for All spectra, Cases) it can be seen from Figures 12–15 that similar In3.2.results the Standard casescan be JONSWAP of obtained the standard toSpectra the cases (H JONSWAPs = of2 mBretschneider for All spectra, Cases) spectra, it can be but seen with from the fo Figuresllowing 12differences:–15 that similar(i) In the cases of the standard JONSWAP spectra, it can be seen from Figures 12–15 that similar resultsthe can deep be water obtained condition to the gives cases worse of Bretschneider under-estimations spectra, of the but wave with power the following(maximal error differences: is 14.5%); (i) the resultsIn canthe casesbe obtained of the standardto the cases JONSWAP of Bretschneider spectra, spectra,it can be but seen with from the Figures following 12– 15differences: that similar (i) deep(ii) water the corrections condition using gives peak worse period under-estimations and energy period of give the wavebetter powerassessments, (maximal and the error maximal is 14.5%); resultsthe deep can water be obtained condition to gives the cases worse of under Bretschneider-estimations spectra, of the but wave with power the fo(maximalllowing errordifferences: is 14.5%); (i) (ii) theerrors corrections are smaller using (about peak 3%), period even in and the shallower energy period water condition give better (water assessments, depth 25 m). and the maximal the(ii) deepthe corrections water condition using givespeak worseperiod under and energy-estimations period of give the wavebetter power assessments, (maximal and error the ismaximal 14.5%); errors(ii)errors are the smaller are corrections smaller (about (about using 3%), 3%),peak even eve period inn in the theand shallower shallower energy period water give condition condition better (water assessments, (water depth depth 25and m). 25 the m). maximal errors are smaller (about 3%), even in the shallower water condition (water depth 25 m).

Figure 12. Modifications of the wave energy assessments using different characteristic periods (h = 50 m). Figure 12. Modifications of the wave energy assessments using different characteristic periods (h = 50 Figure 12. Modiﬁcations of the wave energy assessments using different characteristic periods Figurem). 12. Modifications of the wave energy assessments using different characteristic periods (h = 50 (h = 50m) m)..

Figure 13. Errors for deep-water formulas and modifications using different periods (h = 50 m).

Figure 13. Errors for deep-water formulas and modifications using different periods (h = 50 m). FigureFigure 13. Errors13. Errors for for deep-water deep-water formulas formulas andand modiﬁcationsmodifications using using different different periods periods (h = (5h0 =m) 50. m).

Energies 2017, 10, 460 12 of 17 Energies 2017,, 10,, 460 12 of 17

Figure 14. 14. ModificationsModiﬁcations of ofthe the wave wave energy energy assessments assessments using using different different characteristic characteristic periods periods (h = 25 (m)h =. 25 m).

Figure 15. Errors for deep-waterdeep-water formulas and modiﬁcationsmodifications usingusing differentdifferent periodsperiods ((hh == 2525 m).m).

3.3. Spectra from Measured Waves (AMETS)(AMETS) InIn this section, the comparison will be made for for the the spectra spectra from from the the measured measured waves waves from from seas. seas. InIn getting getting the the correct correct wave wave conditions conditions for for comparison, comparison, the measured the measured waves waves are binned/grouped are binned/grouped using using the given significant wave height (Hr = 0.5 m, 1.0 m, …, representing the significant wave height theusing given the given significant significant wave wave height height (Hr = (H 0.5r = m,0.5 1.0m, 1.0 m, m,... …,, representing representing the the significant significant wavewave height bins: 0–0.5 m, 0.5–1.0 m, …) and the given reference energy periods (Tr + 0.25 s, with Tr = 6 s, 6.5 s, bins: 0–0.50–0.5 m, 0.5–1.00.5–1.0 m,m, ...…) )and and the the given given reference reference energy energy periods periods ( (TTrr ++ 0.25 0.25 s, s, with with Trr == 6 6 s, s, 6.5 6.5 s, 7 s, …, 15 s) for the bins (bin size: 0.5 s for Te). All the binned spectra are averaged to represent the 7 s, ...…, ,15 15 s) s) for for the the bins bins (bin (bin size: size: 0.5 0.5 s s for Te).). All All the the binned binned spectra spectra are are averaged to represent the respectiverespective wavewave spectraspectra forfor the given wavewave conditions. Thus, the calculated wave significantsignificant heights and energyenergy periodsperiods may may be be slightly slightly different different from from the the given given values, values, but theybut they may may be very be closevery close to these to giventhese given values. values. Figures Figures 16 and 16 17 and show 17 show the binned the binned average average spectra spect forra two for different two different sea states sea states (for a comparison,(for a comparison, the corresponding the corresponding Bretschneider Bretschneider and JONSWAP and JONSWAP spectra are spectra alsoplotted). are also In plotted). Figure 16 In, theFigure spectrum 16, the correspondsspectrum corresponds to the waves to the of awaves short of energy a short period, energy and period, it can and be it seen can thatbe seen the that spectra the arespectra closer are to closer the Bretschneider to the Bretschneider spectra spectra (and their (and spectral their spectral bandwidths bandwidths are closer). are closer). In the In cases the cases of a of a long wave period of Te = 13.99 s, the recorded waves have double peaks, including both the locally longof a long wave wave period period of T ofe = T 13.99 = 13.99 s, thes, the recorded recorded waves waves have have double double peaks, peaks, including including bothboth thethe locally wind-generatedwind-generated waveswaves (at(at aa higherhigher frequency)frequency) andand thethe swellsswells (at(at aa lowerlower frequency)frequency) (see(see FigureFigure 17 17).).

Energies 2017, 10, 460 13 of 17 Energies 2017,, 10,, 460 13 of 17

Figure 16. MeasuredMeasured spectrum (average) and the the corresponding corresponding Bretschneider Bretschneider and JONSWAP JONSWAP spectra e s pc ((Tee = 6.09 6.09 s, s, HHss == 1.91 1.91 m) m),, TTpcpc isis calculated calculated using using Equation Equation (27). (27).

Figure 17. MeasuredMeasured spectrum (average) and the the corresponding corresponding Bret Bretschneiderschneider and JONSWAP JONSWAP spectra ((Tee = 13.99 13.99 s, s, HHss == 2.04 2.04 m), m), TTpcpc isis calculated calculated using using Equation Equation (27). (27).

Figures 18 and 19 show the corrections and the corresponding errors for the wave power Figures 18 and 19 show the corrections and the corresponding errors for the wave power assessment in the water depth of 50 m. From the figures, it can be seen that the deep-water formulas assessment in the water depth of 50 m. From the ﬁgures, it can be seen that the deep-water formulas under-estimate the wave power constantly, with a maximal error of 14%. The correction using the under-estimate the wave power constantly, with a maximal error of 14%. The correction using the peak period over-estimates the wave power for almost all the wave periods, with a maximal error peak period over-estimates the wave power for almost all the wave periods, with a maximal error less less than 5%, whilst the correction using energy period under-estimates the wave power for the than 5%, whilst the correction using energy period under-estimates the wave power for the waves of waves of Te < 11 s and then over-estimates the wave power for the long waves. The maximal error Te < 11 s and then over-estimates the wave power for the long waves. The maximal error can be larger, can be larger, about 10% at the longest wave period Te = 15 s. When using T01 and T02 as the reference about 10% at the longest wave period Te = 15 s. When using T01 and T02 as the reference periods, the periods, the corrections for wave energy assessment in the finite water depths give larger errors than corrections for wave energy assessment in the ﬁnite water depths give larger errors than those using those using peak and energy periods. peak and energy periods.

Energies 2017, 10, 460 14 of 17 Energies 2017, 10, 460 14 of 17

Energies 2017, 10, 460 14 of 17

FigureFigure 18. 18.Modiﬁcations Modifications of of the the wave wave energy energy assessments assessments using different using characteristic different characteristicperiods (h = 50 m) periods. (h = 50Figure m). 18. Modifications of the wave energy assessments using different characteristic periods (h = 50 m).

Figure 19. Errors for deep-water formulas and modifications using different periods (h = 50 m).

Figure 19. Errors for deep-water formulas and modifications using different periods (h = 50 m). 4. DiscussionsFigure 19. Errors for deep-water formulas and modifications using different periods (h = 50 m). 4. Discussions 4. Discussions4.1. Peak Periods from Spectra 4.1. PeakSpectral Periods peak from period/frequency Spectra is a very important parameter when the theoretical spectrum is 4.1. Peakdefined Periods (see from Equation Spectra (24)), together with the possible requirement of the spectral peak Spectral peak period/frequency is a very important parameter when the theoretical spectrum is period/frequency for wave energy assessment as shown in the previous sections, a reliable, consistent Spectraldefined (see peak Equation period/frequency (24)), together is a very with important the possible parameter requirement when of the the theoretical spectral spectrum peak and representative spectral peak period is very desirable. As shown in the examples of the spectra is definedperiod/frequency (see Equation for wave (24)), energy together assessment with as shown the possiblein the previous requirement sections, a ofreliable, the spectralconsistent peak from the measured sea waves in Figures 5 and 17, it is easy to pick the physical spectral peak period/frequencyand representative for wave spectral energy peak assessmentperiod is very as desirable. shown in As the shown previous in the sections, examples a reliable,of the spectra consistent periods/frequencies from the spectral curves, but these peak spectral periods may be very subjective and representativefrom the measured spectral sea waves peak period in Figures isvery 5 and desirable. 17, it is easy As shown to pick in the the physical examples spectral of the peak spectra due to the fact that it may be very dependent on the length of the measured data and the sampling periods/frequencies from the spectral curves, but these peak spectral periods may be very subjective fromrate, the as measured well as the sea method waves for in processing Figures5 theand data. 17, Generally it is easy speaking, to pick thethere physical is no well spectral-accepted peak due to the fact that it may be very dependent on the length of the measured data and the sampling periods/frequenciesdefinition for the fromspectral the peak spectral period, curves, unlike but other these wave peak statistic spectral periods, periods such mayas Te, be T01 very and subjectiveT02 in rate, as well as the method for processing the data. Generally speaking, there is no well-accepted due toEquation the fact (9). that For it solving may be this very difficulty, dependent Tucker on theand length Pitt [22] of proposed the measured a so-called data calculated and the sampling spectral rate, definition for the spectral peak period, unlike other wave statistic periods, such as Te, T01 and T02 in as wellpeak as period the method Tpc, Equation for processing (25), and theit is data.foundGenerally that the definition speaking, works there very is nowell well-accepted with the theoretical definition Equation (9). For solving this difficulty, Tucker and Pitt [22] proposed a so-called calculated spectral for thespectra, spectral but peak with period,a small and unlike consistent other wave over- statisticestimation. periods, The formula such as hasT ,beenT andrefinedT inin Equation Equation (9). peak period Tpc, Equation (25), and it is found that the definition works verye well01 with the02 theoretical (26) for a better and more consistent calculation of the spectral peak period. For solvingspectra, this but difficulty,with a small Tucker and consistent and Pitt [over22] proposed-estimation. a so-calledThe formula calculated has been spectral refined in peak Equation period Tpc, Table 1 gives a comparison of the calculated spectral peak periods for the theoretical wave Equation(26) for (25), a better and itand is foundmore consistent that the definitioncalculation works of the spectral very well peak with period. the theoretical spectra, but with spectra, including the well-developed sea state (γ = 1.0), the standard JONSWAP spectrum (γ = 3.3) a small andTable consistent 1 gives over-estimation.a comparison of the The calculated formula hasspectral been peak refined periods in Equation for the theoretical (26) for a betterwave and and the more peaked wave spectrum (γ = 5.0). It can be seen that the refined spectral peak period morespectra, consistent including calculation the well of-developed the spectral sea peak state period.(γ = 1.0), the standard JONSWAP spectrum (γ = 3.3) method (given by Equation (26)) predicts the spectral peak periods with much improved accuracy: and the more peaked wave spectrum (γ = 5.0). It can be seen that the refined spectral peak period Tablethe errors1 gives are aless comparison than 0.5%, of whilst the calculated the Tucker spectral and Pitt’s peak formula periods (i.e., for Equation the theoretical (25)) consist waveently spectra, method (given by Equation (26)) predicts the spectral peak periods with much improved accuracy: includingover-predicts the well-developed the spectral peak sea period state (byγ =about 1.0), 3%. the standard JONSWAP spectrum (γ = 3.3) and the the errors are less than 0.5%, whilst the Tucker and Pitt’s formula (i.e., Equation (25)) consistently more peaked wave spectrum (γ = 5.0). It can be seen that the refined spectral peak period method over-predicts the spectral peak period by about 3%. (given by Equation (26)) predicts the spectral peak periods with much improved accuracy: the errors are less than 0.5%, whilst the Tucker and Pitt’s formula (i.e., Equation (25)) consistently over-predicts the spectral peak period by about 3%. Energies 2017, 10, 460 15 of 17 Energies 2017, 10, 460 15 of 17

TableTable 1. Spectral1. Spectral peak peak period period calculationcalculation for for the the theoretical theoretical spectra. spectra.

Tp Tp T T γ γ TTp(Given) (Given) p err0err0 (%) (%) p err1 (%)err1 (%) p viavia Equation Equation (25) (25) via Equationvia Equation (26) (26) 55 5.130 2.600 2.600 5.005 5.005 0.098 0.098 88 8.214 2.679 2.679 8.014 8.014 0.174 0.174 1.0 1.0 1010 10.269 2.692 2.692 10.019 10.019 0.188 0.188 15 15.405 2.703 15.03 0.198 15 15.405 2.703 15.03 0.198 5 5.146 2.929 5.021 0.418 5 5.146 2.929 5.021 0.418 8 8.239 2.986 8.038 0.474 3.3 8 8.239 2.986 8.038 0.474 3.3 10 10.300 2.996 10.048 0.484 1510 10.300 15.450 2.996 3.003 10.048 15.074 0.484 0.491 515 15.450 5.137 3.003 2.731 15.074 5.011 0.491 0.226 8 8.222 2.78 8.022 0.273 5.0 5 5.137 2.731 5.011 0.226 10 10.279 2.788 10.028 0.281 8 8.222 2.78 8.022 0.273 5.0 15 15.419 2.794 15.043 0.287 10 10.279 2.788 10.028 0.281 15 15.419 2.794 15.043 0.287 4.2. Wave Resource Assess in Finite Water Depth: Belmullet 50 m Water Depth In4.2. the Wave following Resource Assess analysis, in Finite the Water spectra Depth: from Belmullet the sea 50 wavem Water measurements Depth are same as those in Cahill et al.In [the7]. following The total analysis, number ofthe wave spectra measurements from the sea wave is 13,189 measurements from two dataare same buoys as inthose the yearin of 2010Cahill in the et Atlantic al. [7]. The Marine total number Energy of Test wave Site measurements (AMETS) in is the13,189 50 from m water two data depth buoys (Belmullet, in the year Ireland). of The overall2010 in availabilitythe Atlantic Marine of the measuredEnergy Test data Site is(AMETS) about 75% in the (the 50 detailedm water availabilitydepth (Belmullet, of the Ireland). data can be seen inThe [7 overall]). These availability measured of the spectra measured are used data is for about the following75% (the detailed analysis availability and comparison. of the data can be Fromseen in the [7]). available These measured measured spectra wave are used spectra, for the the following wave energy analysis and and resource comparison. assessment can be From the available measured wave spectra, the wave energy and resource assessment can be accurately calculated using the direct numerical integration Equation (19). In addition, using these accurately calculated using the direct numerical integration Equation (19). In addition, using these available wave spectra, a wave scatter diagram can be easily generated (Figure 20 showing the available wave spectra, a wave scatter diagram can be easily generated (Figure 20 showing the occurrenceoccurrence of the of the waves waves in thein the year year in in 2010), 2010), andand based on on which, which, the the wave wave energy energy and andresource resource in a in a waterwater depth depth of 50 of m 50 (a m ﬁnite(a finite water water depth) depth) will will bebe assessed by by including including the the effect effect of the of water the water depth. depth.

Te (s) <5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 14.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 13.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 13.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.01 0.00 0.00 0.00 12.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 12.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 11.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 11.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.01 0.00 0.00 0.00 0.00 0.00 10.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 10.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 9.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.01 0.00 0.00 0.02 0.02 0.00 0.00 0.00 9.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.00 0.01 0.01 0.02 0.04 0.00 0.01 0.00 8.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.00 0.02 0.02 0.01 0.01 0.00 0.01 0.01 0.00 8.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.02 0.03 0.05 0.01 0.01 0.00 0.00 0.00 0.00 0.00 7.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.05 0.05 0.02 0.00 0.01 0.02 0.02 0.00 0.00 7.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.02 0.04 0.09 0.09 0.06 0.02 0.01 0.01 0.03 0.00 0.00 0.00 6.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.04 0.09 0.13 0.11 0.09 0.01 0.02 0.02 0.01 0.00 0.00 0.00 0.00 6.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.07 0.14 0.14 0.12 0.03 0.02 0.02 0.00 0.00 0.00 0.00 0.00 0.00 5.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.07 0.14 0.24 0.21 0.14 0.11 0.03 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.11 0.28 0.34 0.35 0.31 0.22 0.14 0.05 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4.5 0.00 0.00 0.00 0.00 0.00 0.01 0.08 0.12 0.52 0.60 0.72 0.47 0.27 0.20 0.12 0.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4.0 0.00 0.00 0.00 0.00 0.00 0.04 0.16 0.50 0.83 0.95 0.86 0.55 0.52 0.37 0.10 0.03 0.00 0.05 0.01 0.01 0.00 0.00 0.00 0.00 3.5 0.00 0.00 0.00 0.00 0.07 0.11 0.52 0.89 1.49 2.06 1.57 1.05 0.68 0.51 0.34 0.10 0.08 0.05 0.00 0.00 0.00 0.00 0.00 0.00 3.0 0.00 0.00 0.00 0.09 0.21 0.74 1.15 2.14 2.61 2.58 1.58 1.14 0.90 0.56 0.39 0.26 0.09 0.10 0.02 0.01 0.00 0.00 0.00 0.01 2.5 0.00 0.01 0.02 0.17 0.57 1.71 2.03 2.15 2.42 1.87 1.53 1.18 0.68 0.36 0.21 0.14 0.15 0.13 0.09 0.03 0.02 0.02 0.02 0.02 2.0 0.00 0.11 0.67 1.08 1.74 1.93 2.81 3.43 3.71 2.68 1.79 1.15 0.49 0.20 0.17 0.10 0.03 0.02 0.00 0.01 0.02 0.00 0.01 0.01 1.5 0.02 0.46 1.34 0.92 1.50 2.18 2.38 3.12 2.25 2.27 1.74 1.02 0.55 0.20 0.03 0.05 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.0 0.09 0.14 0.55 0.30 0.71 0.95 0.79 0.49 0.33 0.40 0.14 0.06 0.03 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Significant wave height, Hs (m) Hs height, wave Significant 0.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

FigureFigure 20. Scatter 20. Scatter diagram diagram of of the the waves waves in inAMETS AMETS in 2010 2010 (available (available data data points points 13,189 13,189 for two for wave two wave buoys with 75% availability, detailed description can be found in [7]). buoys with 75% availability, detailed description can be found in [7]).

Two approximation methods are compared here: the approximation using the deep-water formula and the proposed method in this study using the water depth modiﬁcation factor Ch (the results Energies 2017, 10, 460 16 of 17

0 are both compared to the accurate wave energy resource, Pirr). From Table2, it can be seen that the deep-water formula gives an under-estimation of 10.18% for the annual mean wave power. The correction method based on the spectral peak periods gives 3.86% over-estimation, and based on the energy periods gives 1.47% under-estimation. It seems that the correction with the energy period gives a better wave resource assessment, but it is caused by the under-estimation for short waves and over-estimation for long waves (Figure 19) for this particular water depth and wave condition, thus they cancel each other in the overall wave energy assessment so to give a better result, while for this particular water depth, the correction using the calculated peak period gives more consistent over-estimation of the wave energy for all waves (less than 5%).

Table 2. Wave power assessment for AMETS.

Water Depth (m) Method Annual Mean Power (kW/m) Error (%) Direct calculation, Equation (19) 37.41 - Deep water, Equation (13) 33.60 −10.18 50 Correction with ωp, Equation (22) 38.76 3.86 Correction with ωe, Equation (22) 36.86 −1.47

5. Conclusions In the paper, the effect of water depths on wave energy assessment has been studied, and a method has been proposed for improving the wave energy assessment in ﬁnite water depth for the cases of only the simple scatter diagram available. From the analyses and comparisons, the proposed method can improve the assessment of the wave energy and resource, and following conclusions can be drawn:

1. For the cases of ﬁnite water depths, using the deep water formulas may under-estimate the wave power by up to 14.5% in irregular waves of interest for wave energy production. 2. For the wave measurement data in simple forms of scatter diagrams, the proposed method in this investigation can improve the wave energy assessment. For both the theoretical spectra (Bretschneider and JONSWAP) and the measured spectra, the proposed method can improve the accuracy of the wave energy assessment, reducing the maximal error from about 14.5% to less than 5%. 3. The proposed method using either the calculated peak period or the energy period can signiﬁcantly improve the assessment of the annual mean wave power. For the AMETS data, the deep water formulas give an error more than 10%, whilst with the proposed method, the error can be reduced to less than 4% or even less. 4. The calculated spectral peak period for the measured waves can be reliably calculated using Equation (27) for those wind-generated waves.

Acknowledgments: The first author would like to thank Science Foundation Ireland (SFI) Centre for Marine and Renewable Energy Research (MaREI) (Grant No. 12/RC/2302), Environmental Research Institute, University College Cork, Ireland for providing funding support. The second author would like to thank the National Natural Science Foundation of China (Grant No. 51409118) for the finance support the research work. The authors would like to thank Brendan Cahill (Sustainable Energy Authority Ireland, SEAI) for providing the wave spectra for the analysis in this study. Author Contributions: Wanan Sheng planned the main research, developed the methodology, carried out the main work and wrote the manuscript; and Hui Li conducted part of the calculation and analysis and reviewed the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Pontes, M.T.; Athanassoulis, G.A.; Barstow, S.; Cavaleri, L.; Holmes, B.; Mollison, D.; Oliveira-Pires, H. An Atlas of the Wave-Energy Resource in Europe. J. Offshore Mech. Arct. Eng. 1996, 118, 307–309. [CrossRef] Energies 2017, 10, 460 17 of 17

2. Stopa, J.E.; Cheung, K.F.; Chen, Y.L. Assessment of wave energy resources in Hawaii. Renew. Energy 2011, 36, 554–567. [CrossRef] 3. Arinaga, R.A.; Cheung, K.F. Atlas of global wave energy from 10 years of reanalysis and hindcast data. Renew. Energy 2012, 39, 49–64. [CrossRef] 4. Cornett, A. A Global Wave Energy Resource Assessment. In Proceedings of the International Offshore and Polar Engineering Conference, Vancouver, BC, Canada, 6–11 July 2008. 5. Mork, G.; Barstow, S.; Kabuth, A.; Pontes, M.T. Assessing the global wave energy potential. In Proceedings of the 29th International Conference on Ocean, Offshore Mechanics and Arctic Engineering (OMAE2010), Shanghai, China, 6–11 June 2010. 6. Gunn, K.; Stock-Williams, C. Quantifying the global wave power resource. Renew. Energy 2010, 44, 296–304. [CrossRef] 7. Cahill, B.G.; Lewis, A. Wave energy resource characteristion of the Altantic marine energy test site. Int. J. Mar. Energy 2013, 1, 3–15. [CrossRef] 8. Folley, M.; Whittaker, T. Analysis of the nearshore wave energy resource. Renew. Energy 2009, 34, 1709–1715. [CrossRef] 9. Lenee-Bluhm, P.; Paasch, R.; Ozkan-Haller, H.T. Characterizing the wave energy resource of the US Paciﬁc Northwest. Renew. Energy 2011, 36, 2106–2119. [CrossRef] 10. Hughes, M.G.; Heap, A.D. National-scale wave energy resource assessment for Australia. Renew. Energy 2010, 35, 1783–1791. [CrossRef] 11. Reikard, G.; Robertson, B.; Buckham, B.; Bidlot, J.R.; Hiles, C. Simulating and forecasting ocean wave energy in western Canada. Ocean Eng. 2015, 103, 223–236. [CrossRef] 12. Iglesias, G.; Carballo, R. Wave energy and nearshore hot spots: The case of the SE Bay of Biscay. Renew. Energy 2010, 35, 490–500. [CrossRef] 13. Iglesias, G.; López, M.; Carballo, R.; Castro, A.; Fraguela, J.A.; Frigaard, P. Wave energy potential in Galicia (NW Spain). Renew. Energy 2009, 34, 2323–2333. [CrossRef] 14. Cornett, A.; Toupin, M.; Baker, S.; Piche, S.; Nistor, I. Appraisal of IEC Standards for Wave and Tidal Energy Resource Assessment. In Proceedings of the International Conference on Ocean Energy, Halifax, NS, Canada, 4–6 November 2014. 15. Folley, M.; Cornett, A.; Holmes, B.; Lenee-Bluhm, P.; Liria, P. Standardising resource assessment for wave energy converters. In Proceedings of the International Conference on Ocean Energy, Dublin, Ireland, 17–19 October 2012. 16. Marine Energy-Wave, Tidal and Other Water Current Converters—Part 101: Wave Energy Resource Assessment and Characterization; IEC TS 62600-101; International Electrotechnical Commission: Geneva, Switzerland, 2015. 17. Ramos, V.; Ringwood, J.V. Exploring the utility and effectiveness of the IEC (International Electrotechnical Commission) wave energy resource assessment and characterisation standard: A case study. Energy 2016, 107, 668–682. [CrossRef] 18. Magagna, D.; Uihleih, A. Ocean energy development in Europe: Current status and future perspectives. Int. J. Mar. Energy 2015, 11, 84–104. [CrossRef] 19. Johanning, L.; Smith, G.H.; Wolfram, J. Measurements of static and dynamic mooring line damping and their importance for ﬂoating WEC devices. Ocean Eng. 2007, 34, 1918–1934. [CrossRef] 20. Holthuijsen, L.H. Waves in Oceanic and Coastal Waters; Cambridge University Press: Cambridge, UK, 2007. 21. Newman, J.N. Marine Hydrodynamics; The MIT Press: Cambridge, MA, USA, 1977. 22. Tucker, M.J.; Pitt, E.G. Waves in Ocean Engineering; Elsevier Science Ltd.: Oxford, UK, 2001. 23. MI. Ocean Energy Expertise (Ireland). Available online: http://www.oceanenergyireland.ie/Observation/ DownloadWave (accessed on 25 July 2016). 24. Iglesias, G.; Carballo, R. Wave farm impact: The role of farm-to-coast distance. Renew. Energy 2014, 69, 375–385. [CrossRef] 25. DNV. Environmental Conditions and Environmental Loads. Available online: https://rules.dnvgl.com/ docs/pdf/DNV/codes/docs/2010-10/RP-C205.pdf (accessed on 29 March 2017).

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).