A novel method for predicting the power outputs of wave energy converters

Yingguang Wang

Abstract This paper focuses on realistically predicting the power outputs of wave energy converters operating in shallow water nonlinear waves.A heaving two-body point absorber is utilized as a specific calculation example,and the generated power of the point absorber has been predicted by using a novel method (a nonlinear simulation method) that incorporates a second order random wave model into a nonlinear dynamic filter.It is demonstrated that the second order random wave model in this article can be utilized to generate irregular waves with realistic crest–trough asymmetries,and consequently,more accurate generated power can be predicted by subsequently solving the nonlinear dynamic filter equation with the nonlinearly simulated second order waves as inputs.The research findings demonstrate that the novel nonlinear simulation method in this article can be utilized as a robust tool for ocean engineers in their design,analysis and optimization of wave energy converters.

Keywords Wave energy converters·Nonlinear simulation ·Nonlinear dynamic filter

1 Introduction

Wave energy is a kind of green and renewable energy,and the global wave power resources potential is tremendous(Gunn and Stock-Williams[1]).Recently the interest in exploiting the vast global wave energy resources has been growing rapidly,and different kinds of wave energy converters have been or are being developed in various parts of the world.For the robust design,analysis and optimization of a wave energy converter,realistically describing the ocean wave environment in which the converter will operate is of vital importance.However,in the current literature,the performance analysis of a wave energy converter is almost always carried out by assuming that the converter is operating in linear random waves(seee.g.,the publications:Cargoet al.[2],Eriksson et al.[3],Fan et al.[4],Fernandes and Fonseca[5],Gomes et al.[6],Herber and Allison[7],Sánchez et al.[8],Stelzer and Joshi[9],Wang and Isberg[10],Yu et al.[11]).However,the linear random wave model can only produce ideal waves with crest–trough symmetries,and this model is only suitable to approximately describe irregular waves in very deep sea or from a very moderate sea state.In the real world,however,when deep water waves become too steep,or as the water depth decreases, the waves will become statistically asymmetric,i.e.,the wave crests will become higher and sharper and the wave troughs will become shallower and flatter.This is called the crest–trough asymmetry in the ocean engineering literature.

As stated in Ref.[5],most of the proposed wave energy converters will be installed in shallow sea areas with water depths less than 90 m.Because sea waves in shallow water areas are a nonlinear random process, the linear random wave model obviously cannot be used as an input in the design,analysis and optimization of most of the today’s wave energy converters.

In order to generate shallow water nonlinear irregular waves,Lindgren[12]used a quasi-linear wave model for the movements of individual water particles.In Ref.[12]the power output performance of a wave energy converter in a linear sea,as well as in a quasi-linear sea has been investigated,and a conclusion has been drawn that the use of a linear wave model tends mostly to overestimate the generator power compared to what can be produced by a quasi-linear wave model. However, because the wave model used in Ref. [12] is only quasi-linear, it obviously cannot produce “truly” nonlinear waves. Most importantly, in the study of Lindgren [12] a simplest formulation of a linear converter (a mass-springdamper filter) with hydrostatic excitation only has been used, disregarding any hydrodynamical forces. Obviously this is not a correct model. Therefore, the prediction results in Ref.[12]cannot be deemed reliable.As pointed out in the conclusion of the paper of Lindgren[12]:“A nonlinear filter,taking also hydrodynamical forces into account,may give different results.”At this point it should also be mentioned that in the existing literature there are many studies dealing with the simulation and prediction of nonlinear irregular waves(see,e.g.,Forristall[13],Toffoli et al.[14],Wang and Xia[15,16],Wang[17,18],Dias et al.[19]).However,except Lindgren[12],none of the existing publications has investigated the power output performance of a wave energy converter operating in nonlinear irregular waves.

Motivated by this, in this paper, the power outputs of a specific wave energy converter,i.e.,a heaving two-body point absorber in a realistic second order nonlinear random sea will be rigorously predicted by a novel nonlinear simulation method.A nonlinear dynamic filter,taking also hydrodynamic forces into account will be utilized to model the wave energy converter.Meanwhile,the power outputs of the wave energy converter in an ideal linear random sea will also be predicted for comparison purpose.The calculation results in this paper will be systematically analyzed and compared,and some valuable conclusions will finally be pointed out.

2 The theories behind the non linear simulation method

2.1 The theories behind the second order nonlinear random wave model

We describe the fluid region 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).We also denote the time by t.The location of the free surface is at z=η(x,y,t)at a specific time of t.Assuming that the fluid is ideal,incompressible and inviscid,and the fluid motion is irrotational,then the velocity potential Φ(x,y,z,t)exists.If the water depth d at the sea bottom is constant,then for constant water depth d the velocity potentialΦ (x,y,z,t)and the free surface elevation η(x,y,t)can be determined by solving the following boundary value problem(see,e.g.,Marthinsen[20])

In this paper we try to solve the system Eqs.(1)–(4)by utilizing the following expansion(see,e.g.,Marthinsen[20])

In this equation ε is a small parameter that is typically proportional to the wave steepness.For a random sea state characterized by a specific wave spectrum Sηη(ω)in which ω denotes the radian frequency,it can be shown that a first order linear solution of the system Eqs.(1)–(4)can be expressed as follows(see,e.g.,Marthinsen[20])

as N tends to infinity.In the above two equations,cndenotes the random complex valued amplitude for each elementary sinusoidal wave,ωndenotes the angular frequency,kndenotes the wavenumber,εndenotes the random phase angle and g is the gravitational acceleration.

However,for modelling shallow water nonlinear waves,the above linear random sea model should be corrected by including second order terms as follows(see,e.g.,Langley[21]and Hasselmann[22])

The terms P(ωn,ωm),rmnand qmnin Eqs.(9)and(10)are called quadratic transfer functions,and their expressions are given as follows

The wave surface elevations for the second order nonlinear waves can finally be obtained by combining Eqs.(7)and(9)as follows

For the numerical implementation of Eq.(13),we consider η(1)in Eq.(13)at a specific reference location(say x=0).For a total time period T of computation,η(1)in Eq.(13)can be written as

Assuming that there are npts computed points in the time period T and the regular spacing between points is Δt=T/npts,η(1)in Eq.(13)can then be written as

In the above equation Xnare complex Fourier coefficients.The lower half of these are expressed as

The upper half of these are expressed as:

Because we have the relations Δt·Δω =2π/(npts)and ωn=n·Δω,Eq.(15)can be evaluated at t=tjto give that

In Fig.1 the red curve shows an example to use Eq.(18)to calculate a linear wave time series of a sea state with a P-M spectrum with a significant wave height of Hs=3 m and a spectral peak period of 6 s.The water depth is 49.5 m.This wave time series contains 200 elevation points simulated with a regular spacing Δt=0.2s.

The termη(2)in Eq.(13)can be generated similarly as the first order one and can be expressed as follows

In Fig.1 the green curve shows an example to use Eqs.(18)and(19)to calculate a nonlinear wave time series(η(1)(t)+η(2)(t))of a sea state with a P-M spectrum with a significant wave height of Hs=3 m and a spectral peak period of 6 s.The water depth is 49.5 m.This wave time series contains 200 elevation points simulated with a regular spacing Δt=0.2s.We can notice that in comparison with the linear waves,the crests of the nonlinear waves have become higher and sharper,and the troughs have become flatter and shallower.

2.2 The nonlinear dynamic filter for the wave energy converter

The study by Lindgren[12]used a quasi-linear wave model when investigating the power output performance of a wave energy converter.However,in that study a simplest formulation of a linear converter(a mass-spring-damper filter)with hydrostatic excitation only was used,disregarding any hydrodynamical forces.The formulation of the linear filter specified in Ref.[12]is shown in Eq.(20)with L and Z denoting surface elevation and system elevation relative to the surface:

In Eq.(20),m is the dry mass of the system and mais the added mass.? denotes the total damping coefficient and k denotes the spring constant.The parameter c depends on the geometry and size of the wave energy converter.The term c(L?Z)on the right side of Eq.(20)represents the hydrostatic excitation force.Because the equation does not include any term representing the hydrodynamic force,it obviously is not a correct model.Therefore,the calculation results in Ref.[12]can hardly be deemed reliable.In order to obtain more accurate calculation results,in this article we will utilize a nonlinear dynamic filter also taking hydrodynamic forces into account to model a wave energy converter operating in a realistic,nonlinear random sea.

The time domain vector-form motion equations of the wave energy converter subjected to wave loads and other loads can be expressed as

In this equation MRBis the rigid body inertia matrix,A(∞)is the constant infinite-frequency added mass matrix,x(t)is a vector of linear and angular displacements,Phsis a vector of the hydrostatic restoring forces and moments.Meanwhile,in Eq.(21)is a matrix of retardation functions.Because of the wave energy converter’s motion,waves will be generated in the free surface.These waves will persist at all subsequent times and affect the wave energy converter’s motion.This is known as the hydrodynamic memory effects,and they are captured in Eq.(21)by the convolution integral term that is a function of(τ)and the retardation functionsOn the right hand side of the equality(21),Pwave(t)denote the wave excitation loads(forces and moments).That is,the Pext(t)terms are the hydrodynamic forces and moments.Pext(t)denote the external loads(forces and moments),and Pvisc(t)denote the viscous loads(forces and moments)due to the hydrodynamic viscous effects.

Fig.2 The WEC-Sim model of the chosen heaving two-body point absorber

Up to now the wave excitation loads Pwave(t)acting on a wave energy converter are usually calculated by using linear random waves as inputs(i.e.,by inputting free surface elevation values generated by using Eq.(15)).As pointed out in Sect.1,this is not a very accurate approach.Therefore,in the present work we will also calculate the wave excitation loads Pwave(t)acting on a wave energy converter by using nonlinear random waves as inputs(i.e.,by inputting free surface elevation values generated by using Eqs.(18)–(19)).Our calculation results regarding a specific wave energy converter will be summarized and discussed in the next section.

3 The calculation example and discussions of the calculation results

3.1 The chosen wave energy converter

In this paper we will show our calculation results regarding a chosen wave energy converter,i.e.,a heaving two-body point absorber.Figure2shows the WEC-Sim model of this specific wave energy converter.Pleasenote that WEC-Sim is an open source code for simulating the performances of wave energy converters(http://wec-sim.github.io/WEC-Sim/).Figure 3 shows the dimensions of this specific wave energy converter.

The two-body point absorber is free to move in all 6 degrees of freedom in response to incident waves.Power is primarily captured in the heave direction.This two-body point absorber consists of a float and a spar/plate.The diameter of the float is 20 m,and the thickness of the float is 5 m.The mass of the float is 727.01 t.The height of the spar/plate is 38 m,and the diameter of the spar is 6 m.The mass of the spar/plate is 878.3 t.

Fig.3 The main dimensions of the chosen heaving two-body point absorber

At this point it should also be noted that the water depth is only 49.5 m at the site where the two-body point absorber is installed.Therefore,the shallow water irregular waves at this site will obviously be nonlinear(see,e.g.,the nonlinear wave theory in Ref.[23]).

3.2 The calculation results and discussion

We have calculated the power outputs of the aforementioned two-body point absorber in eight different sea states by numerical simulations.The widely used P-M wave spectrum is adopted in the process of simulation of the random sea waves.The mathematical expression of the P-M wave spectrum S(ω)is written as[24]

where ω is the wave angular frequency(rad/s),Hsis the significant wave height of a specific sea state.In the above equationωpis the spectral peak angular frequency andωp=2π/Tp.Tpis the spectral peak period of a specific sea state.

Fig.4 Power output time series under the sea state of linear waves with Hs=3m

We next show our calculated power outputs of the abovementioned two-body point absorber operating in an ideal linear random sea versus operating in a realistic nonlinear random sea.Figure 3 shows our calculation results of the two-body point absorber power output time series under the sea state of linear random waves with a P-M spectrum with Hs=3 m,Tp=6 s.Figure 4 shows our calculation results of the power output time series under the sea state of nonlinear random waves with a P-M spectrum with Hs=3 m,Tp=6 s.These calculation results have been obtained by numerically integrating the wave energy converter motion equations(Eq.(21))in WEC-Sim.However,WEC-Sim does not has the ability of generating nonlinear random waves,which are the necessary inputs for calculating the hydrodynamic wave excitation load term Pwave(t)in Eq.(21).Therefore,in our research,we have separately generated the nonlinear waves in MATLAB according to the theories in Eqs.(18)–(19).The generated nonlinear random waves time series are saved as a.mat file and imported into WEC-Sim for subsequent calculations.For comparison purpose,we have also separately generated the linear random waves in MATLAB.The generated linear random waves time series are also saved as a “.mat” file and imported into WEC-Sim for subsequent calculations.After obtaining the calculation results as shown in Figs.3 and 4,we have continued to calculate the descriptive statistics based on these two time series and our calculation results are shown in Table 1.From Table 1 we can notice that the mean value of the wave energy converter 12,000 s power output time series under the sea state of linear random waves is smaller than the mean value of the wave energy converter 12,000 s power output time series under the sea state of nonlinear random waves.This result is different from that published in Ref.[12],which states that inputting a linear wave time series will lead to the overestimation of the generated power.However,the simple linear dynamic filter model without hydrodynamical forces terms in Ref.[12]is obviously not correct and cannot predict reliable power output results.To the contrary,the nonlinear dynamic filter model with hydrodynamical forces terms in this paper is obviously more rational,and thus can predict more reliable power output values.Furthermore,because the second order randomwave model can produce realistic waves which are the necessary inputs to the nonlinear dynamic filter(i.e.,Eq.(21))of the wave energy converter,therefore,the wave excitation loads(forces and moments)Pwave(t)can be more accurately calculated with the nonlinearly simulated second order waves as inputs.Consequently,more accurate generated power can be predicted by subsequently solving Eq.(21)in this paper with the nonlinearly simulated second order waves as inputs. At this point it should also be reminded that the author of Ref.[12]also admitted that“A nonlinear filter,taking also hydrodynamical forces into account,may give different results.”

In the following we explain why there is a discrepancy between the linear wave model and the second order nonlinear wave model energy output.The absorbed power PPTOis calculated by using the following equation

In this equation CPTOis the damping of the PTO(power take off)system of the wave energy converter.In the mentioned calculation example CPTO=1,200,000(N·s/m),i.e.,CPTOis a constant.relis the relative velocity between the two bodies of the wave energy converter in the aforementioned calculation example.We have performed specific calculations and have found thatrelwhen inputting nonlinearly simulated waves is often different from therelwhen inputting linearly simulated waves. This explains the discrepancy between the linear wave model and the second order non-linear wave model energy output.

In our research we have also calculated the wave energy converter power output time series under seven other sea states with a P-M wave spectrum with the following parameters respectively:(Hs=4 m,Tp=7 s);(Hs=5 m,Tp=8 s);(Hs=6 m,Tp=9 s);(Hs=7 m,Tp=10 s);(Hs=8 m,Tp= 11 s);(Hs= 9 m,Tp= 12 s);(Hs=10 m,Tp=13 s);(Hs=11 m,Tp=15 s)and(Hs=12 m,Tp=16 s).After obtaining these calculation results,we have continued to calculate the descriptive statistics based on the obtained time series and our calculation results are summarized in Table 1.

Analyzing the data in Table 1,we can find that inputting nonlinear random waves to the wave energy converter almost always leads to larger predicted mean power output values.In the case of the sea state with a significant wave height value of 12 m,the predicted mean power output value with the nonlinear waves as inputs is 6.12%larger than that with the linear waves as inputs.

These calculation results contradict the findings in Ref.[12]that the use of a linear wave model tends to mostly overestimate the generator power.However,as explained before,our prediction results ought to be deemed more reliable.Meanwhile,if we study the calculation results in Table1 more carefully,we can find that in all cases the standard deviation values of the wave energy converter power out-put time series under the sea states of linear random waves are smaller than the corresponding standard deviation values under these a states of nonlinear random waves.These calculation results indicate that in all the cases using a nonlinear random wave model will predict power output values that spread out over a wider range of values(Fig.5).

Fig.5 Power output time series under the sea state of nonlinear waves with Hs=3m

4 Concluding rem arks

In this paper we have realistically predicted the power outputs of a two-body point absorber wave energy converter operating in shallow water nonlinear waves.The generated power of the wave energy converter has been predicted by using a new method(a nonlinear simulation method)that incorporates a second order random wave model into a nonlinear dynamic filter.The nonlinear dynamic filter also takes the hydrodynamical forces into account.The nonlinear irregular waves time series have been generated separately in MATLAB based on the second order random wave model and have been used as the necessary inputs for calculating the hydrodynamic wave excitation load terms in the nonlinear dynamic filter.After obtaining the power output time series,we have continued to calculate the descriptive statistics based on the time series. As explained in the main part of this paper,the nonlinear dynamic filter is more sophisticated and more rational in comparison with a simple linear filter and can be utilized to obtain more reliable power output results.Meanwhile,it is shown that the nonlinear random wave model in this paper can be utilized to generate irregular waves with more realistic crest–trough asymmetries than a linear wave model can do,and when used in combination with the nonlinear dynamic filter will produce more accurate power output predictions.The research findings in this paper demonstrate that the new nonlinear simulation method in this article can be utilized as a robust tool for ocean engineers in their design,analysis and optimization of wave energy converters.

AcknowledgementsThe work was supported by the State Key Laboratory of Ocean Engineering of China(Grant GKZD010038).Special thanks are due to the two anonymous reviewers whose valuable comments have led to the improved quality of this paper.