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Modelling of a wave energy converter array with non‐linear power take‐off using a mixed time‐domain/frequency‐domain method

2021; Institution of Engineering and Technology; Volume: 15; Issue: 14 Linguagem: Inglês

10.1049/rpg2.12231

1752-1424

Yanji Wei, Alva Bechlenberg, Bayu Jayawardhana, Antonis I. Vakis,

Electromagnetic Simulation and Numerical Methods

IET Renewable Power GenerationVolume 15, Issue 14 p. 3220-3231 ORIGINAL RESEARCH PAPEROpen Access Modelling of a wave energy converter array with non-linear power take-off using a mixed time-domain/frequency-domain method Y. Wei, Y. Wei orcid.org/0000-0002-4280-4918 Ship Hydrodynamics, Aktis Hydraulics BV, Hanzelaan 351, Zwolle, 8017 JM The NetherlandsSearch for more papers by this authorA. Bechlenberg, A. Bechlenberg orcid.org/0000-0002-0564-8717 Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The NetherlandsSearch for more papers by this authorB. Jayawardhana, B. Jayawardhana orcid.org/0000-0003-0987-0347 Discrete Technology and Production Automation, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The NetherlandsSearch for more papers by this authorA. I. Vakis, Corresponding Author A. I. Vakis a.vakis@rug.nl orcid.org/0000-0001-8652-251X Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The Netherlands Correspondence A. I. Vakis, Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, Groningen 9747AG, The Netherlands. Email: a.vakis@rug.nlSearch for more papers by this author Y. Wei, Y. Wei orcid.org/0000-0002-4280-4918 Ship Hydrodynamics, Aktis Hydraulics BV, Hanzelaan 351, Zwolle, 8017 JM The NetherlandsSearch for more papers by this authorA. Bechlenberg, A. Bechlenberg orcid.org/0000-0002-0564-8717 Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The NetherlandsSearch for more papers by this authorB. Jayawardhana, B. Jayawardhana orcid.org/0000-0003-0987-0347 Discrete Technology and Production Automation, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The NetherlandsSearch for more papers by this authorA. I. Vakis, Corresponding Author A. I. Vakis a.vakis@rug.nl orcid.org/0000-0001-8652-251X Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Groningen, The Netherlands Correspondence A. I. Vakis, Computational Mechanical and Materials Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, Groningen 9747AG, The Netherlands. Email: a.vakis@rug.nlSearch for more papers by this author First published: 21 June 2021 https://doi.org/10.1049/rpg2.12231Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract A mixed time-domain/frequency-domain method is proposed for modelling dense wave energy converter (WEC) arrays with non-linear power take-off (PTO). The model is based on a harmonic balance method which describes the system response in the frequency domain, while evaluating the non-linear PTO force and solving the system equations of motion in the time domain. The non-linear PTO force is computed with Lagrange multipliers. In order to apply the proposed method for WEC array responses in real sea states, the time series is split into time windows and the simulation is carried out individually per window. The method is demonstrated by investigating the dynamics of the Ocean Grazer WEC array (OG-WEC) with an adaptable piston pumping system. The key parameters thought to possibly influence model accuracy, including the number of harmonic components, the length of the time window and overlay, are discussed. It is shown that the proposed model can significantly reduce the computational cost with an acceptable accuracy penalty. 1 INTRODUCTION Ocean wave energy is a sustainable and abundant energy source but extracting energy from ocean waves has not become a commercially viable technology yet. The costs, survivability, and power quality are still major obstacles. Wave energy converter (WEC) arrays may partly overcome the current obstacles, as it can potentially increase the overall energy production, reduce the operation and maintenance costs and smoothen the power output. WEC arrays, in particular dense ones, are prone to exhibiting complex and non-linear behaviours due to strong wave interactions between the buoys and coupling effects with the non-linear power take-off (PTO) system. Development of an accurate and computationally cost-effective numerical model for array configuration optimization and WEC array production prediction is therefore of great value. It is common to use non-linear PTOs (e.g. hydraulic PTO) in WEC design. Cargo et al. [1] presented a study of implementable active tuning methods for WECs with hydraulic PTOs, and they found that over-simplification of the PTO in simulations was inadequate for WEC design studies. Penalba and Ringwood [2] presented a high-fidelity wave-to-wire model for WECs with various non-linear PTOs and including different conversion stages, which can be used for WEC design and optimization. Hansen et al. [3] developed a Simulink-based time domain model to investigate an array with 20 devices combined with hydraulic PTOs, but the computational times were prohibitive for real time operation. These studies have shown the significance of assessing the non-ideal efficiency of the PTO and taking into account the non-linear effects of the PTO for the WEC (array) performance evaluation. The development of accurate and highly efficient numerical models is of great value for such purposes. Conventional frequency-domain models have been widely used for WEC array studies [4, 5]. Accompanied by a boundary element method, they can account for arbitrary geometries of WECs and wave interactions with low computational effort. Despite their computational advantages, one main drawback with this approach is its limitation in dealing with non-linear effects. On the other hand, time-domain models incorporating, for example, the Cummins' equation with a convolution operator, can incorporate such non-linearities, albeit with much higher computational costs. Alternatively, the state-space representation can approximate the representations of the convolutions, which can dramatically increase computational speeds [6]. The state-space approach has been widely employed for single WECs and small size WEC array studies [7]. However, this approach is rarely applied to large size, dense WEC arrays because of the difficulty of dealing with cross-coupled terms [8]. The more advanced computational fluid dynamics (CFD)-based models can describe the comprehensive flow details of waves and WEC array interactions. However, their applications are still limited to cases with a small number of WECs subjected to regular waves [9] as the computational cost is extremely high or even unacceptable. The Harmonic Balance Method (HBM) is a popular method to approximate the frequency response of non-linear systems, as it is a computationally efficient method to obtain steady-state responses for non-linear dynamics problems and is relevant for WEC systems; examples include non-linear electrical circuits [10, 11], non-linear mass-spring-damper systems [12-15] and computational fluid dynamics problems [16-18]. Recently, an HMB-based frequency-domain representation of WECs subject to non-linearities was proposed by Bacelli and Ringwood [19] and Merigaud and Ringwood [20]. In these papers, the displacement and the non-linear force were approximated by a truncated Fourier series. Thus, instead of the time consuming convolution integral in time-domain modelling, the HBM solves a set of equations of motion in the frequency domain. The HBM was extended to WEC real-time control applications [21] due to its competitive computational performance. In their work, an explicit Jacobian was applied in order to speed up the convergence. The paper [22] demonstrated that the HBM can be used to model large dense WEC arrays with non-linear pumping forces. They applied a numerical Jacobian in the computation due to occurrence of discontinuity in the pumping force, hence the vector field is not differentiable everywhere. Although the classical HBM is simple in its principle, it becomes computationally inefficient when a large number of harmonics is required, for example for irregular wave simulations [23]. This is due to the large amount of numerical integration operations required when computing the harmonic coefficients of a non-linear force. The present work extends the classical HBM model and propose a mixed frequency-domain/time-domain method with a windowing technique for a large dense WEC array with non-linear PTO. The appealing feature of the proposed method is that it can describe the non-linear dynamics of the complex WEC array in random sea states while significantly reducing the computational cost. In this model, the real sea state is split into time windows with uniform length; thus, the state of each time window can be described with a finite number of harmonic components. A mixed frequency-domain/time-domain method was applied for each time window. The non-linear term is calculated with a time-marching procedure in the time domain, using the Lagrange multipliers method. The OG-WEC device was used to demonstrate the validity of the proposed approach. This device consists of a large array of single floaters, each connected to a PTO utilizing adaptable piston pumps that introduces a strong non-linearity into the system. The paper is organized as follows. The system equations of motion of the OG-WEC and the numerical solution are described in Section 2. The model validation and the discussion about the key parameters are presented in Section 3. Finally, conclusions are drawn in Section 4. 2 NUMERICAL MODEL 2.1 Motion equations of OG-WEC system The Ocean Grazer is a hybrid renewable energy device which combines adaptable WEC technology with on-site energy storage to harvest renewable energy offshore. An OG-WEC system comprises a finite number of buoys with a wind turbine in the centre and its storage system is housed in a gravity-based concrete structure. The OG-WEC is designed to react to the high variability of waves by utilising an adaptable pumping system – M P 2 P T O (multi-piston, multi-pump power take-off) system – and store potential energy in the storage system. The detailed description can be found in Wei et al. [22]. Penalba et al. [24] applied a numerical model to investigate the hydrodynamic interactions in three WEC arrays with different sizes, and suggested that arrays with about 20 devices were hydrodynamically more efficient. We applied the numerical model to an OG-WEC system with 18 buoys, arranged as depicted in Figure 1(a). All buoys have a uniform geometry (see dimension in Figure 1(b)) and are connected to the central pillar via trusses (not sketched) which restrict their motion in heave only. Each buoy is linked to a controllable transmission system which drives the hydraulic piston-type PTO system to pump the working fluid from the inner reservoir to the outside flexible bladder. The stored potential energy can be converted into electricity with hydro-turbines. In the model, we further assume that the cable linking the buoy and the pumping system is stiff enough so that we can simply describe the piston displacement as the buoy displacement adjusted via a transmission ratio. FIGURE 1Open in figure viewerPowerPoint Sketch of the OG-WEC system: (a) honeycomb arrangement of an array; (b) geometry of a cone-cylinder buoy; and (c) schematic of the power take-off system. The pillar and rigid reservoir are maintained the atmospheric pressure (ATM). Reprinted from Wei et al. [22] The motion of the buoy is governed by the following equation: M b X ̈ b = F e + F r + F h s + F c (1)where M b is the mass of the buoy, X b is the heave displacement of the buoy, and F e , F r , F h s and F c represent the excitation, radiation, hydrostatic restoring and cable forces, respectively. The motion of the piston in an adaptable piston pump can be described by: m p X ̈ b α = − F p − α F c (2)where m p is the mass of the piston, α is the transmission ratio, F p is the non-linear pumping forces which is expressed as: F p = ρ g D − h r + ρ l p z ̈ p + ρ z ̇ p 2 A c z ̇ p > 0 0 z ̇ p ≤ 0 (3)where ρ is the density of the working fluid (assumed in this work to be the same as sea water density), D is the water depth, h r is the depth of the reservoir, l p is the length of the piping between the reservoir and the bladder, z ̇ p and z ̈ p are the piston velocity and acceleration, and A c is the area of the check valve (used to adapt the piston pump to the wave excitation). The pumping force F p is very large during the upstroke but becomes zero during the downstroke. In the upstroke component of the pumping force, the first term represents the hydraulic head which saves potential energy and is the largest of the three terms; the second term represents the inertia effect which can lower the resonant frequency of the system; the third term is the kinetic energy which is considered as loss in the system. The present paper focused on the energy extracted from the waves, while the energy losses in the pumping system, electricity generation, and electricity to wire were not accounted for in this study. These losses should be investigated in further studies, for example in the context of a power matrix assessment. Combining (1) and (2), we obtain the governing equation of one OG-WEC unit: M b + m p α 2 X ̈ b = F e + F r + F h s − F p α . (4)Assuming a steady state periodic response for the WEC system, X b , F e , F r , F h s and F p are approximated as a truncated Fourier series: X b ( t ) = X b , 0 + ∑ n = 1 N h Re X ̂ b , n e i n ω t (5) F e ( t ) = ∑ n = 1 N h Re A ̂ w , n F ̂ e , n e i n ω t (6) F r ( t ) = ∑ n = 1 N h Re n 2 ω 2 A r , n − i n ω B r , n X ̂ b , n e i n ω t (7) F h s ( t ) = − K h s X b , 0 + ∑ n = 1 N h Re − K h s X ̂ b , n e i n ω t (8) F p ( t ) = F p , 0 + ∑ n = 1 N h Re F ̂ p , n e i n ω t (9)where N h being the number of harmonic components, ω is the frequency step for the signal with period T p = 2 π ω , X ̂ b , n corresponds to the nth harmonic complex amplitude of the buoy displacement, F ̂ e , n A r , n and B r , n are the excitation force, added mass and radiation damping coefficients , and F ̂ p , n is the nth harmonic pumping force coefficient. For a dense WEC array, interactions between WECs are considered via cross-coupling hydrodynamic coefficients obtained by NEMOH. Substituting Equations (5)–(9) into Equation (4), the equation can be further transformed to the frequency domain with a set of linearised equations over each harmonic. Sorting out the terms with the same frequency, the n-th harmonic motion equation for the ith WEC, is expressed as: − n 2 ω 2 M b , n ( i ) + m p ( i ) α 2 + A r , n ( i , i ) + i n ω B r , n ( i , i ) + C c ( i ) + K h s , n ( i ) + K c ( i ) ( 1 ) ( ∑ 1 1 ) 1 ( 1 ) X ̂ b , n ( i ) + ∑ j = 1 , j ≠ i N b − n 2 ω 2 A r , n ( i , j ) + i n ω B r , n ( i , j ) X ̂ b , n ( j ) − α i i n ω C c ( i ) + K c ( i ) z ̂ p , n ( i ) + F ̂ p , n ( i ) α = F ̂ e , n ( i ) , (10)The matrix form of the motion equations of the WEC array can be written as: Z r X ∼ b + F ∼ p α = F ∼ e (11)where Z r is the condensed dynamic stiffness assembled from (10). The vector X ∼ b contains the unknown Fourier coefficients of displacement, X ∼ b = X b , 0 ( 1 ) , Re X ̂ b , 1 ( 1 ) , … , X ̂ b , N h ( 1 ) , Im X ̂ b , 1 ( 1 ) , … , X ̂ b , N h ( 1 ) , … , X b , 0 ( N b ) , Re X ̂ b , 1 ( N b ) , … , X ̂ b , N h ( N b ) , Im X ̂ b , 1 ( N b ) , … , X ̂ b , N h ( N b ) T ∈ R ( 2 N h + 1 ) N b (12)with N b being the number of buoys. Solving (11) with unknown X ∼ b is not straightforward as F ∼ p is undetermined. 2.2 A mixed frequency-domain/time-domain model According to (3), the pumping force is a piston state-dependent variable, which cannot be explicitly obtained in the frequency domain. Alternatively, it is more convenient to calculate the pumping force using a time-marching procedure. The procedure of the A mixed frequency-domain/time-domain (MFT) approach is structured as follows: a. With the input of the time series of wave elevation per time window (either waves pre-generated with a defined wave spectrum or real time measurements), the pre-processing is carried out to transfer the wave elevation into the frequency domain; then, the wave excitation force is obtained in frequency domain. The buoy displacement is initialized as zero. b. The excitation force is transferred into the time domain, and the time series of the pumping force is predicted with the Lagrange multiplier method (see Section. 2.3), using time domain buoy displacement as input. c. The system equations of motion are solved in MFT form, cf. (15). d. An iterative procedure is applied to each time window to obtain converged results. The iteration terminates once the system residual is smaller than a given tolerance. Otherwise, the buoy displacement is updated, the procedure returns to step 2 and the iteration continues. For temporally periodic oscillations, the Fourier coefficients and the time variation can be transferred back and forth via a discrete Fourier transform (DFT) and its inverse (iDFT), for example: X ∼ b = T X ¯ b and X ¯ b = T inv X ∼ b , (13)where T is the DFT matrix, and T inv is the iDFT matrix, and X ¯ b is the vector of the buoy displacements at all the selected time steps. However, the present paper deals with a WEC array in real sea states where, due to the random nature of ocean waves, the incident wave series is non-periodic; this makes the situation more complicated. Ekici and Hall [17] suggested to determine the Fourier coefficients with non-square matrices to replace T inv in (13), where T inv is a block diagonal matrix with each block written as T inv ( i ) = 1 cos ω t 1 ⋯ cos N h ω t 1 sin ω t 1 ⋯ sin N h ω t 1 1 cos ω t 2 ⋯ cos N h ω t 2 sin ω t 2 ⋯ sin N h ω t 2 1 cos ω t 3 ⋯ cos N h ω t 3 sin ω t 3 ⋯ sin N h ω t 3 ⋮ ⋮ ⋮ 1 cos ω t N t ⋯ cos N h ω t N t sin ω t N t ⋯ sin N h ω t N t (14)where T inv ( i ) ∈ R N t × ( 2 N h + 1 ) , t 1 – t N t are equally spaced time steps per time window, and N t > 1.5 ( 2 N h + 1 ) was recommended to ensure that the aperiodic equivalents of DFT and iDFT are well-conditioned. Then, T is replaced with the Moore–Penrose inverse of T inv in (13). By using (13), (11) is equivalent to f X ∼ b = F ¯ e − T Z r X ∼ b + F ¯ p α . (15)Note that (15) becomes an overdetermined system when N t > 2 N h + 1 , which can be solved by using the non-linear least squares method in the Matlab optimization toolbox. The numerical Jacobian matrix is calculated with the same approach as presented in Wei et al. [22] in order to improve the computational efficiency. It is worth remarking that (15) mixes the frequency-domain displacement and the time-domain pumping force. The pumping force is obtained directly by Lagrange multipliers without any smoothing. Thus, no additional treatment is required to deal with the sums and differences of frequencies generated by the non-linear force in the conventional harmonic balance method. This is the major advantage of this approach compared to solving the system in the frequency domain with (11). 2.3 Lagrange multipliers Nacivet et al. [25] introduced a 'dynamic Lagrangian' algorithm for non-linear contact problems. The pumping force is computed based on the dynamic state of the piston, that is sticking, upstroke or downstroke, which is similar to the contact problem. Hence, we applied a similar algorithm in the present study. At each iteration, the pumping force is updated with Lagrange multiplier λ (corresponding to pumping forces), which is formulated as a penalization of the equations of motion in time domain: λ ¯ = F ¯ e − T Z r X ∼ b + ε X ¯ b − X ¯ b , r (16)where ε is a penalty coefficient that can be chosen arbitrarily as positive but influences the convergence speed: a good choice is to set this to half of the hydrostatic force of the pump, ε = 0.5 ρ g ( D − h r ) ; X ¯ b , r is a new vector of relative displacements which satisfies the updated dynamic state of the piston. In order to calculate λ ¯ at each iteration, (16) is separated into two parts, and rewritten as λ ¯ = F ¯ e − T Z r X ∼ b + ε X ¯ b ︸ λ ¯ I − ε X ¯ b , r ︸ λ ¯ I I (17)where λ ¯ I is determined by the displacement obtained from the non-linear equations (15), and λ ¯ I I is the corrective pumping force vector which is computed with a prediction-correction procedure. At each time step increment, it is firstly assumed that the piston is in a sticking state, that is the piston does not move and λ ¯ I I k = λ ¯ I I k − 1 . The predicted pumping force vector at time step k is λ ¯ p r e k = λ ¯ I k − λ ¯ I I k − 1 . (18)with λ ¯ I I 0 = 0 . The corrected pumping force vector λ ¯ c o r k ∈ R N b is determined by enforcing the following rules per buoy: 1. If λ ¯ p r e k , ( i ) > F ¯ p , u p k , ( i ) , the piston moves upstroke, λ ¯ c o r k , ( i ) = F ¯ p , u p k , ( i ) . (19) 2. If λ ¯ p r e k , ( i ) < F ¯ p , d o w n k , ( i ) , the piston moves downstroke, λ ¯ c o r k , ( i ) = F ¯ p , d o w n k , ( i ) . (20) 3. When F ¯ p , d o w n k , ( i ) ≤ λ ¯ p r e k , ( i ) ≤ F ¯ p , u p k , ( i ) , the piston sticks and does not move, λ ¯ c o r k , ( i ) = λ ¯ p r e k , ( i ) . (21) Here the sticking state of the piston is determined by the force rather than piston velocity as the sticking phenomenon cannot be properly captured with the reconstruction of the time series of piston velocity from frequency-domain inversion. 2.4 Time window and overlay Theoretically, the HBM can be straightforwardly used for WEC arrays oscillating in real sea states. Some researchers [15, 17] successfully demonstrated the applicability of the HBM for oscillating systems with multiple excitation forces. However, the challenge is that a large number of harmonic components is required to obtain a satisfactory description of the time series of signal data in the frequency domain, which would geometrically increase the computational cost and render the HBM as under-performing compared to a time integration approach. To overcome this difficulty, we split the time series of the incident wave signal into a series of time windows. For each time window, satisfactory accuracy can be achieved with a finite number of harmonic components and the simulation can be effectively carried out. Since each time window is independent within the HBM framework, the computation can be simply accelerated by using a parallel implementation. This is an appealing feature that can be applied to the development of real time control strategies. The main drawback of using this scheme is that discontinuities may occur when reconstructing the time series based on the results from each time window. One distinctive feature of hydrodynamic systems is the fluid memory effect. Due to the independence between each time window, the fluid memory effect from the previous time window cannot be transferred to the following one. In order to compensate for the loss of accuracy due to this discontinuity, an overlay between the time windows is included, and only the results in the middle of the time window are used for further analyses such as computing the predicted power output of the OG-WEC system. The energy generated per window E w is calculated by trapezoidal numerical integration of: E w = ∑ i = 1 N b ∫ t b + 0.5 t o t b + t w − 0.5 t o F p ( i ) ( t ) z ̇ p ( i ) ( t ) d t , (22)where t w is the length of time window, t b is the beginning time of the time window, and t o is the length of overlay. From the description above, there are several parameters that influence the simulation results and computational cost, including the length of the time window and overlay and the number of harmonic components. We investigate the influence of these parameters in the following section of this paper. 3 RESULTS AND DISCUSSION 3.1 Comparison with the time domain model The proposed MFT model is validated by comparing its results to the results obtained from our previously developed time domain (TD) model [26]. The TD model is based on the open source package WEC-Sim, augmented with the in-house developed adaptable piston pump model. The hydrodynamic coefficients were obtained from NEMOH [27], the incident waves were generated based on the JONSWAP spectrum with 1001 wave frequencies, and the configuration of the WEC array was chosen to correspond to previous research in Wei et al. [22]. In the MFT model, the input time series of incident waves was replicated from the TD model using the same pseudo-random phase, and the implemented OG-WEC parameters were chosen to be equivalent to those applied to the TD model, in order to enable the comparison of input and output values of the MFT and TD methods. Figure 2 depicts the wave elevation ( η) and the excitation force ( F e ) over the total simulation time for one floater in the OG-WEC array (floater 14 as enumerated in Figure 1). The chosen sea state is a JONSWAP spectrum with a significant wave height of 2m and a peak period of 7s. Figure 2 represents the computation of the MFT model considering a time window length of 19s, an overlap of 7.6s (40% of the time window length) and 9 harmonic components. It can be seen from the first plot in Figure 2 that the input wave elevation is the same in both models. In the second plot in Figure 2, the excitation force of each time window is described with a different colour to show the functionality of the overlap; at the point in which the excitation force of one time window matches the excitation force of the previous one, they connect to create the new starting point for the latter time window's calculation. This ensures a smooth and more accurate representation of the system dynamics. The vertical lines show the time window delimitation within the calculation. The last plot in Figure 2 gives a more detailed representation of the time windows 4 to 7 that show high excitation force in the plot above, and highlights the overlapping technique; the remaining time windows of the simulation show similar accuracy. Large discrepancy in the excitation force can be observed between the MFT and TD models in the first time window which is due to the differences in the initialization. In the TD model, the computation starts in a resting position with a time ramp to avoid strong transient flows at the beginning of the simulation, while, in the MFT model, the simulation is performed in a steady-state manner. The excitation in the following time window shows good agreement with that in the TD model, indicating that the given number of harmonic components is sufficient to approximate the real sea state. FIGURE 2Open in figure viewerPowerPoint Comparison of the time series of wave elevation (top) and wave excitation (middle) by the MFT and TD models; section from time window 4-7 of wave excitation (bottom) The floater's response is captured through the buoy displacement ( X b ), pumping force ( F p ) and pumping or predicted extracted p