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Ying-guang Wang. A second-order random wave model for predicting the power performances of a wave energy converter[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-020-0000-0
Citation: Ying-guang Wang. A second-order random wave model for predicting the power performances of a wave energy converter[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-020-0000-0

A second-order random wave model for predicting the power performances of a wave energy converter

doi: 10.1007/s13131-020-0000-0
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  • Corresponding author: Ying-guang wang E-mail: wyg110@sjtu.edu.cn
  • Available Online: 2021-04-01
  • In this paper the power performances of a point absorber wave energy converter (WEC) operating in a nonlinear multi-directional random sea are rigorously investigated. The absorbed power of the WEC Power–Take- Off system has been predicted by incorporating a second order random wave model into a nonlinear dynamic filter. This is a new approach, and, as the second order random wave model can be utilized to accurately simulate the nonlinear waves in an irregular sea, avoids the inaccuracies resulting from using a first order linear wave model in the simulation process. The predicted results in this paper have been systematically analyzed and compared, and the advantages of using this new approach have been convincingly substantiated.
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  • [1] Tom N M, Madhi F, Yeung R W. 2019. Power-to-load balancing for heaving asymmetric wave-energy converters with nonideal power take-off. Renewable Energy, 131: 1208–1225. doi:  10.1016/j.renene.2017.11.065
    [2] Tom N, Yu Y H, Wright A D, et al. 2018. Balancing power absorption against structural loads with viscous drag and power-takeoff efficiency considerations. IEEE Journal of Oceanic Engineering, 43(4): 1048–1067. doi:  10.1109/JOE.2017.2764393
    [3] Manuel L, Nguyen P T T, Canning J, et al. 2018. Alternative approaches to develop environmental contours from Metocean data. Journal of Ocean Engineering and Marine Energy, 4(4): 293–310. doi:  10.1007/s40722-018-0123-0
    [4] Sirnivas S, Yu Y H, Hall M, et al. Coupled mooring analyses for the WEC-SIM wave energy converter design tool. In: Proceedings of the ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering, Volume 6: Ocean Space Utilization. Busan, South Korea: ASME, 2016.
    [5] Tom N M, Yu Y H, Wright A D, et al. Balancing power absorption and fatigue loads in irregular waves for an oscillating surge wave energy converter. In: Proceedings of the ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering, Volume 6: Ocean Space Utilization. Busan, South Korea: ASME, 2016.
    [6] Fernandes M A, Fonseca N. 2013. Finite depth effects on the wave energy resource and the energy captured by a point absorber. Ocean Engineering, 67: 13–26. doi:  10.1016/j.oceaneng.2013.04.001
    [7] Lindgren G. 2015. Asymmetric waves in wave energy systems analysed by the stochastic Gauss–Lagrange wave model. Proceedings of the Estonian Academy of Sciences, 64(3): 291–296. doi:  10.3176/proc.2015.3.13
    [8] Wang Yingguang. 2019. Comparison of a Lagrangian and a Gaussian model for power output predictions in a random sea. Renewable Energy, 134: 426–435. doi:  10.1016/j.renene.2018.11.051
    [9] Wang Yingguang. 2018. A novel simulation method for predicting power outputs of wave energy converters. Applied Ocean Research, 80: 37–48. doi:  10.1016/j.apor.2018.08.011
    [10] Wang Yingguang. 2018. A novel method for predicting the power outputs of wave energy converters. Acta Mechanica Sinica, 34(4): 644–652. doi:  10.1007/s10409-018-0755-2
    [11] Wang Yingguang, Wang Lifu. 2018. Towards realistically predicting the power outputs of wave energy converters: Nonlinear simulation. Energy, 144: 120–128. doi:  10.1016/j.energy.2017.12.023
    [12] http://wec-sim.github.io/WEC-Sim/. (请补充完整本条文献内容)
    [13] Ochi M K. Ocean Waves: the Stochastic Approach[M]. Cambridge: Cambridge University Press, 1998
    [14] Wang Yingguang. 2014. Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform. Ocean Engineering, 87: 16–24. doi:  10.1016/j.oceaneng.2014.05.012
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A second-order random wave model for predicting the power performances of a wave energy converter

doi: 10.1007/s13131-020-0000-0

Abstract: In this paper the power performances of a point absorber wave energy converter (WEC) operating in a nonlinear multi-directional random sea are rigorously investigated. The absorbed power of the WEC Power–Take- Off system has been predicted by incorporating a second order random wave model into a nonlinear dynamic filter. This is a new approach, and, as the second order random wave model can be utilized to accurately simulate the nonlinear waves in an irregular sea, avoids the inaccuracies resulting from using a first order linear wave model in the simulation process. The predicted results in this paper have been systematically analyzed and compared, and the advantages of using this new approach have been convincingly substantiated.

Ying-guang Wang. A second-order random wave model for predicting the power performances of a wave energy converter[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-020-0000-0
Citation: Ying-guang Wang. A second-order random wave model for predicting the power performances of a wave energy converter[J]. Acta Oceanologica Sinica. doi: 10.1007/s13131-020-0000-0
    • Ocean wave energy refers to the harnessing of the Herculean power of ocean waves. Ocean waves hold a gargantuan amount of untapped energy, some of which we can use to power at least a portion of the world’s everyday electricity. The ocean wave energy has the advantages of being highly predictable, renewable and eco-friendly. An engineering device utilized for exploiting the ocean wave power is called a wave energy converter (WEC). In order to successfully design a wave energy converter, accurately simulating the random ocean waves in the WEC dynamic analysis process is of uttermost importance. However, until present, the majority of the people in the worldwide wave energy research community have applied simple linear irregular waves in their WEC dynamic simulation processes (see e.g.: [1-5]). The linear irregular wave model has the disadvantages that it can only generate unrealistic waves with horizontal symmetries, i.e. the generated waves have statistically symmetric wave crests and troughs. This linear wave model is only suitable for approximately simulating random waves from a very mild sea state in a very deep sea. However, real world ocean waves will become statistically asymmetric (i.e. having sharper and higher crests but smoother and shallower troughs) in a harsh deep sea or at a shallow water coastal site.

      Fernandes and Fonseca [6] has pointed out that most of the proposed WECs will be installed and operated in shallow water coastal sites where the water depths are less than 90 meters. Because the ocean waves in these shallow water sites will become statistically asymmetric, the linearly simulated statistically symmetric waves obviously should not be used as the inputs in the analysis and design of most of the proposed wave energy converters.

      In order to generate shallow water random waves with statistical asymmetries, Lindgren [7] and Wang [8] applied quasi-linear wave models for simulating the movements of individual water particles. However, the random wave simulations presented in [7] and [8] assumed that wave energy is traveling in only one direction (considered the same direction as the wind). That is to say, Lindgren [7] and Wang [8] had respectively performed their stochastic wave simulations based on a uni-directional wave spectrum. Similarly, when Wang [9-10] and Wang and Wang [11] used nonlinear wave models for studying the power performances of wave energy converters, they also applied uni-directional wave spectra during their stochastic simulation of asymmetric waves.

      In the real world, however, wind-generated ocean wave energy does not necessarily propagate in the same direction as the wind; instead, the ocean wave energy usually spreads over various directions. Therefore, for an accurate description of random seas, it is necessary to clarify the spreading status of energy. The wave spectrum representing energy in a specified direction is called the directional spectrum, denoted by $S\left( {\omega ,\theta } \right)$. Obviously, the energy of a sea state characterized by the wave spectrum $S\left( {\omega ,\theta } \right)$ continuously varies as the wave frequency $\omega $ and direction $\theta $ changes. However, to the best of this author’s knowledge, in the current literature there exists no research work that has performed nonlinear wave simulations based on a multi-directional spectrum during the power performance analysis of wave energy converters.

      Motivated by the aforementioned facts, in this paper the power performances of a wave energy converter (WEC) operating in a nonlinear random sea characterized by a multi-directional spectrum $S\left( {\omega ,\theta } \right)$ will be rigorously investigated. The absorbed power of the WEC Power Take Off system will be predicted by incorporating a second order random wave model into a nonlinear dynamic filter. This will be a new approach, and, as the second order random wave model can be utilized to accurately simulate the nonlinear waves in an irregular multi-directional sea, avoids the inaccuracies resulting from using a first order linear wave model in the simulation process. The predicted results in this paper will be systematically analyzed and compared, and the advantages of using this new approach will finally be substantiated.

      The reminder of this paper will be organized as follows: In Section 2 the theories behind the second order random wave simulation based on a multi-directional spectrum will be elucidated, and the measured wave elevation data from a multi-directional coastal sea will be utilized to validate the accuracies of the second order nonlinear random wave simulation method. In Section 3 the theoretical background of the nonlinear dynamic filter of a wave energy converter will be provided. In Section 4 the calculation results of some specific calculation examples will be presented and discussed, with concluding remarks finally summarized in Section 5.

    • The fluid region is described by using the 3D Cartesian coordinates (x, y, z), with x the longitudinal coordinate, y the transverse coordinate, and z the vertical coordinate (positive upwards). Time is denoted by t. The location of the free surface is at z = η(x, y, t) at a specific time of t.

      For real fluid flows in the sea surface, it is reasonable to neglect the effects of viscosity. Meanwhile, sea water has a low compressibility, and therefore normally sea water flows are incompressible. Considering the aforementioned facts, it is reasonably accurate to assume that the fluids on the sea surface are ideal (i.e. incompressible and inviscid). Furthermore, rotation of a fluid particle can be caused only by a torque applied by shear forces on the sides of the particle. Since shear forces are absent in an ideal fluid, the flow of ideal fluids is essentially irrotational. Then, the velocity potential $\Phi \left( {x,y,z,t} \right)$ exists. If the water depth d at the sea bottom is constant, then for constant water depth d the velocity potential $\Phi \left( {x,y,z,t} \right)$ and the free surface elevation η(x, y, t) can be determined by solving the following boundary value problem:

      In this study the following expansion is utilized to solve the system (1)–(4):

      In equation (5) ε is a small parameter which is typically proportional to the wave steepness. For an irregular sea state characterized by a specific wave spectrum ${S_{\eta \eta }}\left( \omega \right)$ in which $\omega $ denotes the angular frequency, it can be shown that a first order linear solution of the system (1)–(4) can be expressed as follows:

      In the above two equations $N$ is usually chosen to be a sufficiently large positive integer. Meanwhile in the above two equations, i stands for the imaginary unit, ${c_n}$ denotes the sinusoidal wave component amplitude and ${\varepsilon _n}$ is the phase angle uniformly distributed in the interval [0, 2${\text π} $]. Furthermore, in the above two equations ${\omega _n}$ denotes the radian frequency (${\omega _n} = 2{\text π} n/T$, T is the time interval) and ${k_n}$ denotes the wave number. ${\omega _n}$ and ${k_n}$ are related through the following linear dispersion relation:

      In equation (8) $d$ and $g$ are respectively the water depth and the gravitational acceleration.

      Equation (7) describes an ideal linear irregular sea model and this model has been derived under the assumption that the wave heights are small compared to the wave length. Then the surface elevation resulting from irregular ocean waves can be approximated as a superposition of multiple harmonic waves with different amplitudes and phases. Furthermore, in this ideal linear irregular sea model the random phase angles ${\varepsilon _n}$ are assumed to be uniformly distributed between 0 and 2π. However, for modelling shallow water nonlinear waves, the above linear irregular sea model should be corrected by including second order terms as follows:

      The terms $P({\omega _n},{\omega _m})$, ${r_m}_n$ and ${q_m}_n$ in equations (9) and (10) are called quadratic transfer functions, and there expressions are given as follows:

      In equation (11) the Kroenecker delta (${\delta _{ - n,m}}$=1 if n+m = 0, zero otherwise) is introduced to avoid a singular $P({\omega _n},{\omega _m})$. The wave surface elevations for the second order nonlinear waves can finally be obtained by combining equation (7) and equation (10) as follows:

      The proposed second order random wave simulation method starts with taking a multi-directional spectrum $S\left( {\omega ,\theta } \right)$ and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum $S\left( \omega \right)$ is then utilized to generate a nonlinear wave elevation time series by applying equations (8) and (14) at a specific site of the sea. Finally it should be pointed out that the coefficients ${c_n}$(or ${c_m}$) in equation (14) are determined by using a given spectrum $S\left( \omega \right)$ as follows:

      where $\Delta \omega = {\omega _C}/N$ and ${\omega _C}$ is the upper cut off frequency beyond which the wave spectrum $S\left( \omega \right)$ may be assumed to be zero for either mathematical or physical reasons.

      As we know, a wave spectrum is actually obtained from the observed time series of water elevation. The transformation from water elevation to wave spectrum is based on the assumption that irregular waves can be treated as a combination of linear waves with different amplitudes, different frequencies and different phases. It would be better to give an explanation for how to reasonably consider the nonlinearity of waves in the inverse transformation (i. e. the transformation from the wave spectrum to the nonlinear water elevation). In the following we will raise a specific calculation example of this inverse transformation:

      We first show as an example to use equation (7) to simulate the linear Gaussian wave time histories of a sea state with a JONSWAP spectrum with a significant wave height ${H_S}$= 6 m, a spectral peak period ${T_p}$=8 s, a peakedness factor $\gamma $=1 and corresponding to a water depth of 10 m.

      Fig. 1 shows our simulation of the linear part wave elevation time series ${\eta ^{\left( 1 \right)}}(x,t)$ which contains 150 wave elevation points (with an equal time distant of 0.1 s between two successive points). A wave elevation point has two coordinates (The first coordinate is time. The second coordinate is the value of the wave elevation above the mean water level.). The computation is based on equation (7) and the aforementioned JONSWAP spectrum.

      Figure 1.  The simulation of the linear part $ {\eta ^{\left( 1 \right)}}(x,t) $ which contains 100 wave elevation points.

      Fig. 2 shows our simulation of the second order correction part wave elevation time series ${\eta ^{\left( 2 \right)}}(x,t)$ which contains 150 wave elevation points (with an equal time distant of 0.1 s between two successive points). The computation is based on equation (10) and the aforementioned JONSWAP spectrum. This is precisely an example for how to reasonably consider the nonlinear correction part of waves in the inverse transformation (i. e. the transformation from the wave spectrum to the nonlinear correction part water elevation).

      Figure 2.  The simulation of the second order correction part ${\eta ^{\left( 2 \right)}}(x,t)$ which contains 100 wave elevation points.

      In Fig. 3 the blue curve shows our simulation of the entire $\eta (x,t)$ (the linear part ${\eta ^{\left( 1 \right)}}(x,t)$ plus the second order correction part ${\eta ^{\left( 2 \right)}}(x,t)$) wave elevation time series which contains 150 wave elevation points (with an equal time distant of 0.1 s between two successive points). The computation is based on equation (14) and the aforementioned JONSWAP spectrum. In Fig. 3, the red curve shows our simulation of the linear part wave elevation time series ${\eta ^{\left( 1 \right)}}(x,t)$ which contains 150 wave elevation points based on the aforementioned JONSWAP spectrum. In Fig. 3 the green curve shows our simulation of the second order correction part wave elevation time series ${\eta ^{\left( 2 \right)}}(x,t)$ which contains 150 wave elevation points based on the aforementioned JONSWAP spectrum. We can see that the wave crests of the entire nonlinear waves $\eta (x,t)$ have become steeper and higher. This is precisely an example for how to reasonably consider the nonlinearity of the entire waves in the inverse transformation (i. e. the transformation from the wave spectrum to the entire nonlinear waves).

      Figure 3.  The simulation of the blue curve entire nonlinear waves $\eta (x,t)$, the red curve linear waves ${\eta ^{\left( 1 \right)}}(x,t)$ and the green curve second order correction waves ${\eta ^{\left( 2 \right)}}(x,t)$ respectively of 150 points wave elevation time series.

    • In this sub-section the accuracy of the proposed second order random wave simulation method will be validated by a calculation example. Specifically, the proposed second order random wave simulation method will be applied for calculating the wave crest amplitude exceedance probabilities of a sea state with a multi-directional wave spectrum based on the measured surface elevation data at the coast of Yura. The measured surface elevation data at the coast of Yura were obtained at a location 3 km off Yura fishing harbor facing the Sea of Japan. The observations were carried out during the period from 11:10 a. m. to 14:08 p. m. on November 24, 1987 by the Ship Research Institute, Ministry of Transport of Japan. Temporal sea surface elevations were measured with ultrasonic-type wave gages installed at three points in 42 m water depth. The sampling time interval during the measurement was 1 s. Based on these measured Yura coast surface elevation data, a multi-directional wave spectrum was estimated by using the Maximum Likelihood Method and is shown in Fig. 4. Fig. 5 shows a part of the measured wave elevation time series by the mid-point wave gage at this site.

      Figure 4.  A multi-directional wave spectrum from the measured data at the coast of Yura.

      Figure 5.  A part of the measured wave elevation time series at the coast of Yura.

      In the field of ocean engineering, based on a specific wave spectrum, an empirical (or theoretical) model or numerical simulation methods can be used to calculate the exceedance probabilities of the wave crest amplitudes (which is directly related to the occurrence probability of the extreme waves). In the following, taking the multi-directional wave spectrum in Fig. 4 as a calculation example, we will test our proposed second order random wave simulation method to calculate the exceedance probabilities of the wave crest amplitudes, and compare its accuracy with those of a theoretical model or a linear simulation method.

      Fig. 6 shows our calculated wave crest amplitudes exceedance probabilities based on the multi-directional spectrum shown in Fig. 4 and the measured Yura coast wave data. The continuous green curve in Fig. 6 represents the calculation results of the wave crest amplitudes exceedance probabilities directly obtained from the measured Yura coast wave data that contains 10700 wave elevation points. This continuous green curve based on the measured Yura coast wave data are used as the benchmark against which the accuracy of the results from the various numerical simulation methods and from an existing theoretical wave crest amplitudes model is checked.

      Figure 6.  Comparison between the wave crest amplitudes exceedance probabilities from the linear simulation, from the Rayleigh distribution, from the nonlinear simulation and from the wave crest amplitudes exceedance probabilities results from the measured Yura coast wave data.

      The continuous red curve in Fig. 6 represents the results of the wave crest amplitudes exceedance probabilities obtained from using the theoretical Rayleigh distribution model expressed as follows:

      In Eq. (16) $h$ represents the wave crest amplitude and ${H_S}$ represents the significant wave height. We can clearly see that the wave crest amplitudes exceedance probabilities obtained from using the theoretical Rayleigh distribution model deviate a lot from the corresponding benchmark results obtained directly from the measured Yura coast wave data. This is not a surprise because the theoretical Rayleigh distribution is an ideal linear model and the measured Yura coast wave elevation series are obviously nonlinear. The continuous black curve in Fig. 6 represents the results of the wave crest amplitude exceedance probabilities obtained from using the linear simulation method based on the multi-directional spectrum shown in Fig. 4. The linear simulation process started with taking the multi-directional spectrum shown in Fig. 4 and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum was then used to generate a wave elevation time series of 800000 points by applying equations (7–8). Next the wave crest amplitudes time series were extracted from these 800000 wave elevation points. Finally, the wave crests amplitudes exceedance probabilities were obtained by statistical and mathematical processing the extracted wave crest amplitudes time series. Obviously, the wave crest amplitudes exceedance probabilities obtained from using the linear simulation method also deviate substantially from the corresponding benchmark results obtained directly from the measured Yura coast wave data.

      In order to more accurately calculate the wave crest amplitudes exceedance probabilities we had tried to use our proposed second order random wave simulation method. In Fig. 6 a pink * represents the result of the wave crest amplitudes exceedance probability obtained from using our proposed second order random wave simulation method based on the multi-directional spectrum shown in Fig. 4. Our proposed second order random wave simulation process started with taking the multi-directional spectrum shown in Fig. 4 and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum was then utilized to generate a nonlinear wave elevation time series of 800000 points by applying equations (8) and (14). Next the wave crest amplitudes time series were extracted from these 800000 wave elevation points. Finally, the wave crests amplitudes exceedance probabilities were obtained by statistical and mathematical processing the extracted wave crest amplitudes time series. Please note that in the figure legend the phrase “From nonlinear simulation” exactly means “From second order random wave simulation”. We can find that the wave crest amplitudes exceedance probabilities obtained from using our proposed second order random wave simulation method fit quite well with the corresponding benchmark results obtained directly from the measured Yura coast wave data. The accuracy of our proposed second order random wave simulation method is therefore convincingly validated.

    • In the practice ocean engineering, the power performances prediction of a wave energy converter (WEC) is usually carried out by inputting simulated ocean waves in a nonlinear dynamic filter and performing subsequent time domain simulations. The mathematical equations of the WEC nonlinear dynamic filter are written as follows:

      In the above equation ${{\bf{M}}_{RB}}$ is the WEC rigid body inertia matrix and ${\bf{x}}\left( t \right)$ are the WEC positions. Meanwhile, in the above equation ${\bf{A}}\left( \infty \right)$ is the WEC infinite-frequency added mass matrix and ${\bf{\bar K}}\left( t \right)$ are the kernel functions. ${{\bf{P}}_{wave}}\left( t \right)$, ${{\bf{P}}_{ext}}\left( t \right)$, ${{\bf{P}}_{visc}}\left( t \right)$ and $ - {{\bf{P}}_{hs}}$ are the wave excitation, external, hydrodynamic viscous and hydrostatic loads (forces and moments) respectively.

      In the field of ocean engineering, the WEC hydrodynamic memory effects are captured in equation (17) by the convolution integral term that is a function of ${\dot{\bf x}}\left( \tau \right)$ and the kernel functions ${\bf{\bar K}}\left( t \right)$ which are related to the WEC radiation damping. Meanwhile in equation (17) the external loads ${{\bf{P}}_{ext}}\left( t \right)$ represent the Power-Take-Off system forces and moments.

      However, up until present, the wave excitation loads ${{\bf{P}}_{wave}}\left( t \right)$ in the WEC dynamic filter are typically calculated by using linearly simulated irregular waves as inputs. As pointed out in the Introduction part of this paper, the linear irregular wave model has the disadvantages that it can only generate unrealistic waves with horizontal symmetries. Even though some researchers [7-11] had tried to use quasi-linear wave models or nonlinear wave models for simulating irregular waves with statistical asymmetries, all these researchers unfortunately performed their stochastic wave simulations based on some unrealistic uni-directional wave spectra. Therefore, in the Introduction part of this paper we have emphasized that the extension of these research work is very much needed in order to study the power performances of a wave energy converter in a multi-directional random sea. Motivated by these facts, in the present study we will calculate the wave excitation loads ${{\bf{P}}_{wave}}\left( t \right)$ by inputting nonlinearly simulated irregular waves based on a multi-directional wave spectrum. In the next section, some calculation examples regarding the power performances of a point absorber wave energy converter operating in a multi-directional random sea will be presented.

    • Our specific calculation examples will be carried out regarding a heaving two-body point absorber wave energy converter. Fig. 7 shows this specific wave energy converter modeled in WEC-Sim [12], an open source computer software for simulating WEC performances. The dimensions of this specific wave energy converter are shown in Fig. 8. This two-body point absorber WEC consists of a float and a spar/plate. The float has a diameter of 20 m, a thickness of 5 m and a mass of 727.01 t. The spar has a height of 38 m and a diameter of 6 m. The spar/plate has a mass of 878.3 t.

      Figure 7.  The WEC-Sim model of the chosen heaving two-body point absorber.

      Figure 8.  The main dimensions of the chosen heaving two-body point absorber.

      It should be emphasized that this two-body point absorber WEC is installed in a coastal sea area with a water depth of only 49.5 m. Obviously the random waves in this shallow water area will be nonlinear according to the nonlinear random wave theory in Ochi [13].

    • In most wave energy exploitation engineering projects that we will meet in the real world, the only information we know beforehand regarding a sea state usually will be a specific wave spectrum. In Section 2 we have already demonstrated the accuracy of our proposed second order random wave simulation method in generating irregular waves based on a measured multi-directional wave spectrum. In the following calculation examples we will utilize our proposed second order random wave simulation method for generating nonlinear irregular waves for calculating the wave excitation loads ${{\bf{P}}_{ext}}\left( t \right)$ in equation (17). We will predict the absorbed power of the aforementioned WEC Power–Take–Off system for some specific sea states characterized with multi-directional JONSWAP spectra. Fig.9 shows one of these multi-directional JONSWAP spectra with a significant wave height ${H_S}$=1 m, a spectral peak period ${T_p}$=12 s, a peakedness factor $\gamma $=1, and a cosine squared spreading function with the spreading parameter equal to 15. The mathematical expression of this multi-directional JONSWAP spectrum is as follows:

      Figure 9.  A multi-directional JONSWAP wave spectrum with $ {H_S} $=1 m, ${T_p}$=12 s, $\gamma $=1, and a cosine squared spreading function with the spreading parameter equal to 15.

      where $S\left( \omega \right)$ is a frequency spectrum and $D\left( \theta \right)$ is a spreading function.

      The mathematical expression for the frequency spectrum $S\left( \omega \right)$ is as follows ((see e.g. Wang [14])):

      where

      where $\omega $ is the wave angular frequency.

      The mathematical expression for the spreading function $D\left( \theta \right)$ in equation (18) is as follows:

      with the spreading parameter s=15.

      Fig. 10 is a corresponding polar plot of the multi-directional JONSWAP spectrum as shown in Fig. 9.

      Figure 10.  A corresponding polar plot of the multi-directional JONSWAP spectrum as shown in Fig. 9.

      Next we present our calculation results of the absorbed power of the aforementioned wave energy converter placed in an ideal linear sea versus in a multi-directional nonlinear random sea. Fig. 11 shows the predicted WEC absorbed power time series under the sea state of linear irregular waves based on a multi-directional JONSWAP wave spectrum with ${H_S}$=12 m, ${T_p}$=12 s, $\gamma $=1 and a cosine squared spreading function with the spreading parameter equal to 15. Fig. 12 shows the predicted WEC absorbed power time series under the sea state of nonlinear irregular waves based on the same multi-directional JONSWAP wave spectrum with ${H_S}$=12 m, ${T_p}$=12 s, $\gamma $=1 and a cosine squared spreading function with the spreading parameter equal to 15. The calculation results in Fig. 11 and Fig. 12 were obtained by solving the WEC nonlinear dynamic filter equation (17) in WEC-Sim. However, WEC-Sim does not have built-in functions for generating nonlinear irregular waves for calculating the wave excitation loads ${{\bf{P}}_{ext}}\left( t \right)$ in equation (17). Therefore, in our study externally generated nonlinear irregular waves by using our proposed second order random wave simulation method were imported into WEC-Sim. We started our nonlinear irregular wave simulation by taking the multi-directional JONSWAP wave spectrum (with ${H_S}$=12 m, ${T_p}$=12 s, $\gamma $=1) and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum was then utilized to generate a nonlinear wave elevation time series of 1200 points by applying equations (8) and (14). The generated nonlinear irregular waves time series were saved as a.mat file and then imported into WEC-Sim for calculating the wave excitation loads ${{\bf{P}}_{ext}}\left( t \right)$ in equation (17) and subsequently obtaining the WEC absorbed power time series in Fig. 12. For comparison purpose, in our study externally generated linear irregular waves by using equation (7–8) were also imported into WEC-Sim. We started our linear irregular wave simulation by taking the multi-directional JONSWAP wave spectrum (with ${H_S}$=12 m, ${T_p}$=12 s, $\gamma $=1) and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum was then utilized to generate a linear wave elevation time series of 1200 points by applying equations (7) and (8). The generated linear irregular waves time series were saved as a.mat file and then imported into WEC-Sim for calculating the wave excitation loads ${{\bf{P}}_{ext}}\left( t \right)$ in equation (17) and subsequently obtaining the WEC absorbed power time series in Fig. 11. Having obtained the prediction results as shown in Fig.11 and Fig.12, we subsequently calculated the statistical characteristic values based on the time series shown in these two figures. Our calculation results are summarized in the last row in Table 1.

      Figure 11.  WEC absorbed power time series under the sea state of linear waves with Hs=12 m

      Figure 12.  WEC absorbed power time series under the sea state of nonlinear waves with Hs=12 m

      Statistical measures
      Significant wave height (m)
      Mean value under sea state of linear waves (W)Mean value under sea state of second order nonlinear waves (W)Standard deviation value under sea state of linear waves (W)Standard deviation value under sea state of second order nonlinear waves (W)Sum value under sea state of linear waves (W)Sum value under sea state of second order nonlinear waves (W)
      1138437.04116138511.45857147967.47721148037.483743.59798E83.59991E8
      2183201.45543183729.92515219208.08886219533.048234.76141E84.77514E8
      3199926.33733199983.50274249739.80807251762.625935.19609E85.19757E8
      4274600.5756276111.71603403700.40894408900.025587.13687E87.17614E8
      5315077.7521316133.23968454553.94521454063.308638.18887E88.2163E8
      6433808.09451444136.33628726265.21959769908.283181.12747E91.15431E9
      7490131.33153503258.637776567.66295809812.45691.27385E91.30797E9
      8594029.9469621211.75487978890.289271.02946E61.54388E91.61453E9
      9796963.82665874346.209991.34261E61.53251E62.07131E92.27243E9
      10767280.02013825077.575891.17164E61.30227E61.99416E92.14438E9
      111.14921E61.2581E62.21824E62.68345E62.9868E93.2698E9
      121.0986E61.1918E62.53972E62.64326E62.85525E93.09748E9

      Table 1.  Predicted 1200s WEC absorbed power values at the sea states of (${T_p}$=16 s, $\gamma $=1, s=15)

      By carefully comparing and analyzing our calculation results we can find that the mean value of the 1 200s WEC absorbed power under the sea state of ideal linear irregular waves is smaller than the corresponding mean absorbed power value when inputting nonlinear irregular waves. In order to investigate the influences of choosing different significant wave height values on the power performances of wave energy converters, we subsequently calculated the aforementioned WEC absorbed power values under eleven other sea states characterized with a multi-directional JONSWAP wave spectrum with the following parameters respectively: (${H_S}$=1 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=2 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=3 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=4 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=5 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=6 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=7 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=8 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=9 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=10 m, ${T_p}$=12 s, $\gamma $=1, s=15); (${H_S}$=11 m, ${T_p}$=12 s, $\gamma $=1, s=15). Our calculation results were also summarized in Table 1. By carefully comparing and analyzing these calculation results we can find that in all these eleven sea states the mean value of the 1200s WEC absorbed power under the sea state of ideal linear irregular waves is always smaller than the corresponding mean absorbed power value when inputting nonlinear irregular waves.

      In order to further investigate the influences of using different random seed numbers for generating irregular waves on the power performances of wave energy converters, we subsequently calculated the aforementioned WEC absorbed power values when inputting eleven different irregular wave elevation time series simulated based on the same multi-directional JONSWAP wave spectrum (${H_S}$=12 m, ${T_p}$=12 s, $\gamma $=1, s=15) using different random seed numbers. Our calculation results were summarized in Table 2. By carefully comparing and analyzing these calculation results we can find that in all these twelve sea states the mean value of the 1200s WEC absorbed power under the sea state of ideal linear irregular waves is always smaller than the corresponding mean absorbed power value when inputting nonlinear irregular waves. In one specific case (No. 11) the predicted WEC absorbed power value with the nonlinear irregular waves as inputs is 15.8% larger than that predicted with the linear irregular waves as inputs. All the aforementioned calculation results highlight the vital importance of using the nonlinear irregular waves simulated based on a multi-directional wave spectrum when studying the power performances of wave energy converters.

      Statistical measures
      Significant wave height (m)
      Mean value under sea state of linear waves (W)Mean value under sea state of second order nonlinear waves (W)Standard deviation value under sea state of linear waves (W)Standard deviation value under sea state of second order nonlinear waves (W)Sum value under sea state of linear waves (W)Sum value under sea state of second order nonlinear waves (W)
      121.18485E61.38075E62.01713E62.76839E63.07941E93.58857E9
      121.1227E61.22233E62.04328E62.41229E62.91791E93.17685E9
      121.15923E61.27871E62.2291E62.63901E63.01285E93.32336E9
      121.47223E61.5835E62.69001E63.28096E63.82632E94.1155E9
      121.18207E61.31431E62.10716E62.70589E63.07219E93.41588E9
      121.0641E61.12036E61.74052E61.80526E62.7656E92.91183E9
      121.32898E61.44481E62.55403E62.8129E63.45403E93.75506E9
      121.12296E61.31056E61.97489E62.46568E62.91857E93.40615E9
      121.4066E61.57163E62.67732E63.01967E63.65575E94.08466E9
      121.10365E61.18066E61.7609E62.01755E62.86838E93.06853E9
      121.27946E61.4813E62.27087E62.72554E63.32531E93.8499E9
      121.04726E61.13799E61.62224E61.92767E62.72184E92.95764E9

      Table 2.  Predicted 1200s WEC absorbed power values at the sea states of (${H_S}$=12 m, ${T_p}$=16 s, $\gamma $=1, s=15)

    • In the present study the power performances of a two body point absorber wave energy converter (WEC) operating in a nonlinear multi-directional random sea have been rigorously investigated. The absorbed power of the WEC Power –Take- Off system has been predicted by incorporating a second order random wave model into a nonlinear dynamic filter. This is a new approach that is uniquely proposed to ocean wave energy research community. It has been demonstrated in this paper that the second order random wave model can be utilized to accurately simulate nonlinear irregular waves in a multi-directional sea. This will help us to avoid the inaccuracies resulting from using a first order linear wave model in the WEC simulation process. The predicted results in this paper have been systematically analyzed and compared, and the advantages of using our proposed new approach have been convincingly substantiated. The research findings in this paper highlight the vital importance of using the nonlinear irregular waves simulated based on a multi-directional wave spectrum when studying the power performances of wave energy converters. In the future if enough funding is secured, the proposed second order random wave simulation method will also be compared with the related WEC model physical test data in order to fully verify the accuracy of the model.

    • Conflict of Interest: The authors declare that they have no conflict of interest.

      Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

    • This work is supported by the National Natural Science Foundation of China (Grant No. 51979165).

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