Frequency-domain and stochastic model for an articulated wave power device

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1 Frequency-domain stochastic model for an articulated wave power device J. Cândido P.A.P. Justino Department of Renewable Energies, Instituto Nacional de Engenharia, Tecnologia e Inovação Estrada do Paço do umiar, 9- isboa, Portugal Abstract To have the first look into device performance, analytical numerical tools must be used. Assuming that the wave power system hydrodynamics has a linear behaviour, diffraction radiation coefficients can be computed. If the power take-off equipment may be, for the first approach, regarded as holding a linear behaviour then overall (i.e. hydrodynamic plus mechanical device performance can be studied for regular waves. In this study a frequency-domain model describes the articulated system behaviour for regular waves. For this paper a stochastic model is found for an articulated wave power device, probability density functions are defined for the relevant parameters that characterize the wave power system behaviour. For these parameters for different sea states the probability density functions are found. The articulated system is characterized by these probability density functions. Also, average values for capture width are obtained for these sea state conditions. Keywords: Wave energy, frequency-domain, stochastic model. Nomenclature A ij  = added mass hydrodynamic coefficient = complex wave elevation amplitude  n = complex rom amplitude B ij = damping hydrodynamic coefficient C i = hydrostatic restoring coefficient for body i D = damping coefficient of the power take-off equipment E {} = expected value of F D i = complex amplitude for the diffraction force on body i F = load force HG, HG = transfer function H s = significant wave height K = spring coefficient of the power take-off equipment M i = mass of body i P I = incident power for a regular wave P u = average useful power S η = spectral density T = time interval T e = wave energy period Z = load impedance g = acceleration of gravity h = water depth k = wave number t = time ϕ n = rom phase η = sea surface elevation λ c = capture width θ = wave direction angle = water specific mass ρ σ = variance ξ i = complex amplitude displacement for body i * = complex conjugate Proceedings of the 7th European Wave Tidal Energy Conference, Porto, Portugal, 7 Introduction Since the beginning of the th century man has been thinking on how to make use of wave power. So far more than about thouss of patents have been registered. To have the first look into device performance, analytical numerical tools must be used. Assuming that the wave power system hydrodynamics has a linear behaviour, diffraction radiation coefficients can be computed using, for example, WAMIT or Aquadyn. If the power take-off equipment can be, in the first approach, regarded as holding a linear behaviour, then overall (i.e. hydrodynamic plus mechanical device performance can be studied for regular waves. In this study a frequency-domain

2 model describes the behaviour for regular waves of an articulated system consisting of two main concentric bodies. Optimal mechanical damping spring coefficients are computed for some parameters (relative absolute displacements restrictions. Useful power, capture width, other variables, like relative displacements (displacement of one body for a coordinate reference system that oscillates with the same amplitude phase as the other body, i.e. a coordinate reference system fixed with respect to the second body, are computed for regular waves the mechanical coefficients mentioned above. In [] a theory for wave power absorption by two independently oscillating bodies has already been devised. Frequency analysis has been used to study devices such as Searev wave energy converter, [], also the hydrodynamic performance of arrays of devices as in []. As mentioned before software using panel methods can be used to compute the hydrodynamic coefficients of the two concentric bodies. In [] hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders were already computed. A stochastic model has already been developed for OWC power plants []. This stochastic model has been used for optimization procedures of the FOZ do Douro OWC plant, [], [7]. In this paper a stochastic model is found for the articulated wave power device, probability density functions are defined for the relevant parameters that characterize the wave power system behaviour. Assuming that the overall system behaviour is linear that the wave elevation for the irregular waves may be regarded as a stochastic process with a Gaussian probability density function, the variables that define the system behaviour, like, for example, displacements for the articulated system elements, will also have a Gaussian probability density function. For these parameters for different sea states the probability density functions (i.e. variances are found. The articulated system is characterized by these functions. Also, average values for capture width are obtained for these sea state conditions. Mathematical models The wave energy device is made of two concentric axisymmetric oscillating bodies, Fig.. The relative heave motion between bodies allows extracting power from sea waves. As usual it is assumed that the bodies have linear hydrodynamic behaviour also that the power take-off can be modelled by spring damping terms proportional to the relative displacement between bodies to the relative velocity, respectively. A frequency domain model a stochastic model are subsequently presented. Frequency domain model Applying Newton s second law, the governing equations for the wave energy device are, M ( ( ˆ ω ξ = ω A iωb ξ + ω A iωb ξ + FD Cξ ( K + iωd ( ξ ξ, (, M ( ( ˆ ω ξ = ω A iωb ξ + ω A iωb ξ + FD Cξ + ( K + iωd ( ξ ξ, ( where ω is the angular frequency, ξ i is the complex amplitude displacement for body i, M i the mass of body i, C i the hydrostatic restoring coefficient for body i, Aij B ij the added mass damping hydrodynamic coefficients, F D the complex amplitude for the diffraction i force on body i, K D the spring damping coefficients of the power take-off equipment. As given in [] for the angular frequency ω the average useful power can be given by, P ˆ ˆ u = Dω ξ ξ. ( Thus, choosing values for K D it is possible to compute for each angular frequency wave elevation amplitude the heave motions of the two concentric floating bodies, ξ ˆ ξ, as well as the average useful power, P u. The capture width, λ c, may be computed by, ˆ ˆ P Dω ξ ξ u λ c = =, ( PI ˆ kh ρg A( ω + tanh( kh sinh( kh where P I is the incident power for a regular wave with angular frequency ω complex wave elevation amplitude  ( ω. The water depth is h, k is the wave number given by the positive root of the dispersion relationship, ω = k tanh( kh, ρ is the water specific g mass g the acceleration of gravity. Stochastic model As in [] we will consider a time interval of duration T, assume that the sea surface elevation, η ( t, is a Gaussian rom variable given by, ( ˆ η t = An exp( inωt, ( n= where ω = π / T, Aˆ n = Aˆ n exp( iϕ n is a complex rom variable with ϕ n being a rom variable uniformly distributed in the interval [ ;π [ assuming that Aˆ E n = σ n { ˆ * ' } = E A n A n, for n n'.

3 Following [] we find for the variance of the sea surface elevation, assuming that the sea state can be represented by a discrete power spectrum, * { } exp( ( ' { ˆ ˆ * σ η = E ηη = i n n ωt E An A ' } = n n= n' = = σ n. n= ( As it is well known, if the sea surface power spectrum is continuous the variance of the elevation is given by ση = S η ( ω dω, (7 S η ω is a spectral density defined in the range where ( ] [ ;. Thus in the limiting case when T we will get ω dω E Aˆ n = σ n Sη ( ω dω. If the power spectrum is dependent not only on the wave frequency but also on the direction of the incoming waves we get for ( M ( ˆ η t = Anm exp( inωt, ( m= n= where M is the number of considered directions for the interval [ ;π [, Aˆ E nm = σ nm { ˆ * ' ' } = E A nm A n m for n n' m m'. The variance of the elevation for a continuous power spectrum is now π ση = S η ( ω, θ dωdθ. (9 Taking into consideration that the two oscillating bodies are axisymmetric that their behaviour in the frequency domain can be described by eqs. ( (, it is possible to find transfer functions, HG ( nω HG ( nω, that relate the amplitude of the incident wave A n the displacement amplitude for body, ξ ( nω = HG ( nω An, (, ξ ( nω = HG ( nω Aˆn. ( According to ( the vertical displacements for bodies are described by, ( t = HG ( nω Aˆ exp( inω t, ξ n= n ( ( t = HG ( nω Aˆ exp( inω t. ξ n ( n= Note that just as η, also ξ ξ are Gaussian rom variables, with variances * { ξ ξ } = HG ( nω σ, σ ξ = E n ( n= * { } σ ξ = E ξ ξ ( = HG nω σ n. ( n= If the sea state can be represented by a continuous power spectrum, then we get for ξ ξ variances σ = ( (, S η ω HG ω dω ( ξ σ = ( (. S η ω HG ω dω (7 ξ It is straightforward that the variance for the velocity of bodies may be computed by, σ & = Sη ( ω ω HG ( ω dω, ( ξ σ & = Sη ( ω ω HG ( ω dω. (9 ξ Assuming that the load force, F, can be given by ( exp( inω t, ( ˆ F t = ± ( K + inωd ξ( nω ξ ( nω n= ( we get, for the variance of the load force, * σ = E{ FF } = F * * ( ( ' ( ˆ ˆ ( ˆ ˆ = + Z nω Z n ω E ξ ξ n ξ ξ n' n= n' = exp( i( n n' ωt, ( with Z ( nω = K + inωd. Since ξ ξ are given by eqs. (- E{ ( ξ ξ n ( ξ ξ n' } = for n n', we get for the variance of F σ = Z ( nω HG ( nω HG ( nω σ n. ( F n= Thus, for a continuous power spectrum we may write σ = Sη ( ω Z ( ω HG ( ω HG ( ω dω. ( F The average useful power may be written as P = DE & u ξ ( t & ξ ( t, (

4 taking into consideration that for a sea state represented by a continuous power spectrum we get E & ξ & ξ = E (& ξ & ξ (& ξ & ξ * = ( = Sη ( ω ω HG ( ω HG ( ω dω. Note that the probability of the instantaneous useful power being less or equal than χ D is given by Prob χ x χ exp dx. χ E & ξ & ξ ( P u D = Numerical results ( To illustrate the application of the frequency-domain model mainly the stochastic model, results were obtained for the oscillating body presented in Fig.. It is assumed that body is the body with a toroidal shape (outside body body is the inside body made of two parts (one surface-piercing body a completely submerged cylinder that oscillate together. WAMIT was used to compute the hydrodynamic diffraction radiation coefficients for a set of wave frequencies in the range of.9 rad/s to.77 rad/s. The water depth is m. Assuming two scenarios for the power take-off equipment: a it can only be simulated by a damping, D, coefficient, b it can be simulated by spring, K, damping coefficients, values were computed for K D that maximize the average useful power, P u, thus the capture width, λ c. Several restrictions were also imposed on the amplitude of the heave motion of body on the amplitude of the relative heave motion between body body, ξ ξ, for an incident wave with amplitude of m. For scenario a the capture width, mechanical damping coefficient, relative amplitude displacement between bodies absolute amplitude displacements are presented in Figs. to, respectively. They are obtained for different displacement restrictions. In all the cases it is assumed that the amplitude for the relative displacement between body body cannot exceed m for an incident wave of m amplitude. It is also considered that the absolute displacement amplitude for body cannot exceed,,, 7 m. Thus the case _ means that the amplitude for the relative displacement cannot exceed m the amplitude for the absolute displacement cannot be greater than m. It can be observed from Fig. that the device has two well defined capture width peaks, one at.s another at.7s. The value for the average useful power for the second peak (at.7s is closely related to the allowed maximum amplitude heave motion for the first body, Fig.. To be able to control this amplitude, appropriate mechanical damping coefficients need to be chosen, Fig.. Note that if there are restrictions on the relative displacement between bodies on the absolute displacement of body then there will be restrictions for the absolute displacement of body, implicitly. For scenario b, Figs. 7 to present capture width, mechanical damping spring coefficients, relative amplitude displacement absolute amplitude displacements, respectively. Again, in all the cases it is assumed that the amplitude for the relative displacement between body body cannot exceed m that the absolute displacement amplitude for body cannot exceed, m. As expected, for different maximum amplitude displacements of body we get different capture widths for lower values for body maximum absolute displacement amplitude we get smaller values for capture width Fig. 7. Indeed, damping spring coefficients must be tuned out to allow for the restrictions to be achieved, Figs. 9. It may be observed from Fig. that the absolute displacement amplitude for body is bounded, as well as the relative displacement amplitude between bodies, Fig.. For the stochastic model results are shown for irregular waves. Again, the two scenarios for the power take-off equipment above explained were considered. The spring damping coefficients that maximize eq. (, thus the respective capture width, λ c, were computed. Results are obtained for the variance of ξ ξ, as well as for the variance of the load force, F. The best K D for each sea state are also shown. The variances for ξ, ξ F were obtained for sea states with H s, the sea state significant wave height, equal to m. Knowing that σ, σ σ are given by eqs. (, (7 (, ξ ξ F respectively, it is easily found that the variances are proportional to the square of H s, i.e., if we multiply H s by, the variances will be multiplied by, assuming that T e, the sea state wave energy period, is kept constant as well as K D. To represent the sea states the following frequency spectrum was adopted ([9]: - - S η ( ω =.Hs Te ω exp( Te ω. (7 The device behaviour was simulated for wave energy periods from 7 to s for a H s equal to m. As already mentioned, two different scenarios were considered for the power take-off equipment. Figs. to present the capture width, mechanical damping, spring coefficient variances for the displacements of body, load force, for both power take-off scenarios, respectively. It can be observed that the system has a better performance for sea states with smaller wave energy period, Fig.. For the range of considered wave energy periods the damping coefficients for scenario a are greater than the ones computed for scenario b, Fig.. The spring coefficients computed for scenario b may be negative or positive,

5 depending on the wave energy period, Fig.. For scenario a the curves for the displacement variance of both bodies are similar, Fig.. However, these curves are rather different for scenario b, Fig. 7. The variance for the load force converges for both scenarios when the wave energy period increases, Fig.. As the capture width is quite different for T e equal to 7s for the considered scenarios, the variance for the load force should also be different as it is observed in Fig.. The obtained variances allow defining Gaussian probability density functions for the analyzed parameters. th European Wave Tidal Energy Conference, Glasgow,. [] J. Falnes. Ocean waves oscillating systems. Cambridge University Press. [9] Y. Goda. Rom Seas Design of Maritime Structures. University of Tokyo Press 9. Conclusions Frequency-domain stochastic models were devised for an articulated device composed by two concentric bodies. Results were obtained for regular irregular waves. The use of the stochastic model for irregular waves allows to find variances that define Gaussian probability density functions for relevant wave device parameters. It was assumed that the power take-off mechanical equipment has a linear behaviour can be modelled by spring damping coefficients. For irregular waves it was assumed that the characteristics of the power take-off system are constant for the duration of a sea state. As expected, the capture widths for the stochastic model are significantly lower than the ones obtained for the frequency-domain model. In order to have better performances for irregular waves, device control will be essential. References [] M.A. Srokosz D.V. Evans. A theory for wavepower absorption by two independently oscillating bodies. J. Fluid Mechanics, 9(:7-, 979. [] A. Clément, A. Babarit, J-C. Gilloteaux, C. Josset G. Duclos. The SEAREV wave energy converter. In Proc. th European Wave Tidal Energy Conference, Glasgow,. [] P.A.P. Justino A. Clément. Hydrodynamic performance for small arrays of submerged spheres. In Proc. th European Wave Energy Conference, Cork,. [] S. Mavrakos. Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders. Applied Ocean Research, :-97,. [] A.F. de O. Falcão R.J.A. Rodrigues. Stochastic modelling of OWC wave power plant performance. Applied Ocean Research, :9-7,. [].M.C. Gato, P.A.P. Justino A.F. de O. Falcão. Optimization of power take-off equipment for an oscillating-water column wave energy plant. th European Wave Tidal Energy Conference, Glasgow,. [7] M.T. Pontes, J. Cândido, J.C.C. Henriques P. Justino. Optimizing OWC sitting in the nearshore. In Proc. (D Mar 7 - Figure : Panel grid describing the wet surface of the concentric axisymmetric oscillating bodies in numerical evaluation. Capture Width (m T (s X Z Y _7 _ Figure : Capture width for a m maximum displacement amplitude between body body (for an incident wave of m amplitude maximum absolute displacement amplitude for body varying between m m, assuming that the power take-off equipment can only be simulated by a damping coefficient D.

6 Coefficient (Ns/m.E+.E+.E+.E+.E+.E+ _7 _ T (s Figure : Mechanical damping coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude maximum absolute displacement amplitude for body varying between m m, assuming that the power take-off equipment can only be simulated by a damping coefficient D. Relative Displacement Amplitude (m _7 _ T (s Figure : Relative displacement amplitude between body body (for an incident wave of m amplitude for maximum absolute displacement amplitude for body varying between m m, assuming that the power take-off equipment can only be simulated by a damping coefficient D. Body Displacement Amplitude (m 9 7 _7 _ T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, an absolute displacement amplitude restriction varying between m m, assuming that the power take-off equipment can only be simulated by a damping coefficient D. Body Displacement Amplitude (m 9 7 _7 _ T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, an absolute displacement amplitude restriction for body varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D.

7 Capture Width (m T (s Figure 7: Capture width for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Coefficient (Ns/m.E+.E+.E+.E+.E+ T (s Figure : Mechanical damping coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power take-off equipment can be simulated by damping spring coefficients D K. Spring Coefficient (N/m.E+.E+.E+ -.E+ -.E+ T (s Figure 9: Mechanical spring coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power take-off equipment can be simulated by damping spring coefficients D K. Relative Displacement Amplitude (m T (s Figure : Relative displacement amplitude between body body (for an incident wave of m amplitude for different maximum absolute displacement amplitudes for body of m, m m, assuming that the power take-off equipment can be simulated by damping spring coefficients D K.

8 Body Displacement Amplitude (m 9 T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, absolute displacement amplitude restrictions of m, m m, assuming that the power take-off equipment can be simulated by damping spring coefficients D K. Body Displacement Amplitude (m T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, absolute displacement amplitude restrictions for body of m, m m, assuming that the power take-off equipment can be simulated by damping spring coefficients D K. Capture Width (m.... Spring 7 9 Te(s Figure : Capture width for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can only be simulated by a damping coefficient D (black line assuming that the power take-off equipment can be simulated by both damping spring coefficients D K (red line. Coefficient (Ns/m.E+.E+.E+.E+.E+ Spring 7 9 Te(s Figure : Mechanical damping coefficient for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can only be simulated by a damping coefficient D (black line assuming that the power take-off equipment can be simulated by both damping spring coefficients D K (red line.

9 Spring Coefficient (N/m.E+.E+.E E+ -.E+ Spring Te(s Figure : Mechanical spring coefficient for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can be simulated by both damping spring coefficients D K. Displacement Variance (m Body Body 7 9 Te (s Figure 7: Variances for the displacements of body, for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can be simulated by both damping spring coefficients D K. Displacement Variance (m Body Body 7 9 Te (s Figure : Variances for the displacements of body, for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can only be simulated by a damping coefficient. oad Force Variance (N^.E+.E+.E+.E+.E+.E+.E+ Spring 7 9 Te(s Figure : Variance for load force, for wave energy periods from 7 to s H s equal to m, assuming that the power take-off equipment can only be simulated by a damping coefficient D (black line assuming that the power take-off equipment can be simulated by both damping spring coefficients D K (red line.

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