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two-body heaving WECs with latching control

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Renewable Energy 45 (2012) 31e40 Contents lists available at SciVerse ScienceDirect Renewable Energy journal homepage: www.elsevier.com/locate/renene On the annual wave energy absorption by two-body heaving WECs with latching control J.C.C. Henriques a,*, M.F.P. Lopes b, R.P.F. Gomes a, L.M.C. Gato a, A.F.O. Falcão a a IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b Wave Energy Centre, Av. Manuel da Maia, 1049-001 Lisboa, Portugal article info Article history: Received 13 October 2011 Accepted 31 January 2012 Available online 23 March 2012 Keywords: Wave energy Time domain Latching control Threshold unlatching time Two-body system Cummins equations abstract Although the latching control strategy has been recognized as an important mean of increasing the efficiency of one-body point-absorbing wave energy converters (WECs), its effectiveness in two-body floating point-absorbers has been questioned in some studies. The current work investigates the increase in annual absorbed energy achieved with a simple threshold unlatching control strategy when applied to a generic two-body heaving WEC. The WEC performance is evaluated for a set of sea-states characteristic of the wave climate off the Portuguese west coast. To achieve this computationally intensive task, a new high-order numerical method for the solution of the Cummins equations is presented and used. This approach is based on a polynomial representation of the solution, whose coefficients are computed using a continuous least-squares approximation. The code has been parallelized and computations were performed at the IST cluster. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The energy extraction by offshore WECs is, in general, obtained through the relative motion of two or more parts of the converter. The moving parts may have relative motion in translation modes, as in the Wavebob [1] and the IPS Buoy [2], or the relative rotation between parts of the converter, as the case of SEAREV [3]. Other particular cases, as devices connected directly to the seabed and bottom mounted oscillating water columns can be seen as particular cases of the first one, where the second body (the earth) has infinite mass. The two-body problem in heave is very relevant in the scope of offshore wave energy conversion. This problem has been studied by a number of authors both theoretically [4,5] and numerically [6e9]. The latching control technique, first proposed by [10] for onebody systems, consists in locking the body motion when its velocity vanishes and releasing it such that ideally the velocity becomes in phase with the excitation force during the unlatched part of the cycle. Under certain conditions, a significant increase in * Corresponding author. Tel.: þ351 218417296. E-mail addresses: joaochenriques@ist.utl.pt (J.C.C. Henriques), mlopes@wave- energy-centre.org (M.F.P. Lopes), ruigomes@ist.utl.pt (R.P.F. Gomes), luis.gato@ ist.utl.pt (L.M.C. Gato), antonio.falcao@ist.utl.pt (A.F.O. Falcão). 0960-1481/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2012.01.102 the energy extraction can be achieved, when compared with the non-controlled case. This strategy can be adapted for the relative motion of two-body systems, using the relative velocity as reference. The possibility offered by latching control of achieving larger efficiencies over a broad range of incident frequencies can significantly improve the device economics. Moreover, the efficiency gain is larger at frequencies smaller than the device resonance frequency, which may result in a substantially smaller device for the same wave energy absorption capability [11]. The practical application of the latching strategy raises several problems, mainly related to the need of fast response of mechanical and/or hydraulic components. The study of this strategy for one-body systems has been quite extensive including successful experimental investigations [12e15]. The utilization of latching control in two-body heaving systems has been justifiably questioned (see e.g. [16]) because the two bodies keep moving as one-body after latching, which makes the system less efficient as the dynamics becomes different from what was idealized in one-body latching control. However, the effect of this is dependent on the mass ratio of the two bodies. In the limiting case, when the mass of one of the bodies increases to infinity, the gain from latching control increases since the system becomes equivalent to a floater reacting against the seabed [4]. This paper examines the influence of the relative mass of the two bodies on the latching control efficiency for a given annual wave climate. 32 J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 2. Description of the problem The present paper is focused on the performance of a generic two-body heaving WEC with latching control. The device consists of a floating hemisphere (body 1) rigidly connected to a deeply submerged body (body 2), see Fig. 1. The two bodies are only allowed to move in heave and they are connected through a power take-off system (PTO) modelled as a linear damper. It is assumed that the distance from the submerged body to the free surface is large enough for the excitation and radiation forces to be negligible. Furthermore, the hydrodynamic interaction between the floater and the submerged body is neglected. The latching strategy used here consists in locking the bodies when their relative velocity vanishes and releasing them at a specified time (threshold unlatching) after the sign of the floater excitation force (assumed known) multiplied by its vertical coordinate becomes negative. This strategy is based on the results presented in [2] and [15]. The parameters to be studied are the ratio between the floater mass and the submerged body inertial mass (body mass plus the added mass of surrounding fluid), the PTO damping coefficient and the threshold unlatching time. In each investigation, the ratio between the floater mass and the mass plus added mass of the submerged body is kept fixed, while the other parameters are allowed to vary in order to find how to improve the annual energy absorption of the device. In this work, it is investigated how to optimize twoheaving-body point absorbers with latching control. A criterion is established for the ratio between the floater mass and the mass plus added mass of the submerged body if this type of control is to be effective. The WEC response is computed for a set of 14 sea-states characterizing an Atlantic location off the western coast of Portugal. Time series of 120 min were synthesized for each sea state, considering a Pierson-Moskowitz spectrum. The energy absorbed in each sea state is then weighted according to the sea state probability of occurrence in order to estimate the annual averaged power output. Fig. 1. Generic two-body wave energy converter oscillating in heave. The optimization of one-body WEC with latching control regarding a specific wave climate has been presented in [17]. In the current work, contour plots for the dimensionless annual averaged power output are shown for varying threshold unlatching time and the PTO damping. Although it requires more computational time, more information about the system behaviour can be retrieved from these contour plots than from just the results from the optimum operating point calculation. Since latching implies a non-linear process, the equations of motion must be solved in the time domain. In this paper, a new highorder numerical method for the solution of the Cummins equations [18] is presented. This approach is based on a continuous polynomial representation of the solution whose coefficients are computed using a continuous least-squares approximation. A six-order polynomial was used for the computation of the bodies’ positions. The convolution integral is calculated using the continuous polynomial approximation of the velocity as computed by the least-squares method. This constitutes an important advantage of the proposed method in comparison to the usual high-order Runge-Kutta methods, where the convolution integral is computed with the trapezoidal rule which limits the overall accuracy of the solution to second-order. In the present approach, the convolution integral is computed using a Gauss-Legendre quadrature rule [19] which results in a product of a matrix by a vector of instantaneous velocities. The code has been successfully parallelized using the OpenMPI library [20] to largely reduce the computational time. For the computation of the annual energy absorption, each spectrum is assigned to one processor. Optimal speed-up was achieved since the amount of information transferred between processors is very small. The computations were performed at the IST cluster. 3. Two-body WEC modelling 3.1. Governing equations Consider the two-body heaving WEC represented in Fig. 1. Let y and x be the coordinates for positions of the heaving floater and submerged body, with y ¼ x ¼ 0 at equilibrium position and both increasing upwards. The heaving floater is subjected to a hydrodynamic force, Fy, and a force transmitted through the PTO to the submerged body, FPTO. During latching, a braking force Fbrake is applied to keep both bodies fixed to each other (zero relative velocity). The equations of motion for the two-body system are m1 y€ ¼ ðm2  Fy t; y; þ A2Þ€x y_ ; ¼  y€ÀFþPTOFPÀTy_O; Àx_yÁ_ ;Àx_ ÁFþbraFkberÀatk;eyÀ_ ;t;x_yÁ_ ;; x_ Á; (1) where mi is the mass of body i (i ¼ 1,2). Since body 2 is assumed to be deeply submerged, it does not radiate waves and its added mass A2 is independent of the frequency of motion. The hydrodynamic force Fy is the vertical component of the force due to water pressure on the floater wetted surface (Fy ¼ 0 for a motionless floater at y ¼ 0 in calm water). The force Fy depends on known quantities. According to the linear wave theory, if the amplitudes of the wave and of the floater motion are small, then the hydrodynamic force can be decomposed as    Fy t; y; y_ ; y€ ¼ FhðyÞ þ Fr t; y_ ; y€ þ FdðtÞ; (2) where Fh is the hydrostatic restoring force, assumed to be zero when y ¼ 0, Fr is the force exerted by fluid on the body as a result of its oscillation in the absence of incident waves and Fd is the vertical force produced by the incident waves on the floater fixed at the y ¼ 0 position. J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 33 In the linearized hydrostatic modelling, it is Fh ¼ À9gSy; (3) where 9 is the water density, g is gravity acceleration and S is the floater’s waterplane area. The coefficients of the radiation force are computed in the time domain as  Fr t; y_ ; y€ ¼ ÀFc À t; Á y_ À AN 1 y€ðtÞ; (4) where Fc À t; y_ Á ¼ Zt Kðt À sÞy_ ðsÞds; (5) ÀN aandddeAdN 1ma¼ssulc/ imoNefAfi1ciðeunÞt, where A1ðuÞ is for the floater. the frequency dependent Since we are assuming linear water wave theory, the resulting diffraction force is obtained as a superposition of the frequency components Xn À Á À Á FdðtÞ ¼ G uj Auj cos ujt þ 2prandðÞ ; (6) j¼1 with Auj ¼ qffi2ffiffiDffiffiffiffiuffiffiffijffiSffiffizffiffiÀffiffiffiuffiffiffijffiÁffiffi; (7) Duj ¼ ð1 þ 6randðÞÞDu; (8) uj ¼ ujÀ1 þ 1 2 ÀDuj þ DujÀ1Á; j ¼ 2; .; n; (9) where u1 ¼ 0:1 rad/s, Du ¼ 3:0=n rad/s, 6 ¼ 0:2 and randðÞ is a random number generator between 0 and 1. All the spectra SzðuÞ calculations were performed assuming n ¼ 300 spectrum components. For axisymmetric bodies oscillating in heave, the excitation force coefficient is related to the radiation damping coefficient by [21] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðuÞ ¼ 29gu3B3ðuÞ: (10) In the current work, real irregular waves are represented as a superposition of regular waves assuming the Pierson-Moskowitz [22] spectrum ! SzðuÞ ¼ 262:9 Hs2 u5Te4 exp 1054 À u4Te4 : (11) a b Fig. 2. Diffraction force using n ¼ 300 spectrum components and (a) equally spaced frequencies, Duj ¼ Du, (b) non-equally spaced frequencies, Eq. (8). In case (a), the diffraction force is almost periodic repeating approximately each 660 s. 34 J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 Numerical results obtained for equally spaced frequencies and 300 spectrum components show an almost periodic diffraction force pattern approximately each 660 s, see Fig. 2a. A large increase in the probability of getting non-periodic series for longer time periods was obtained in the current approach by using the same number of spectrum components with non-equally spaced frequencies, see Fig. 2b. The PTO is modelled in the present work as a linear damper with coefficient CPTO, together with a spring of stiffness k ¼ 9gS=10 introduced to avoid the vertical drift of the submerged body ÀÁ ÀÁ FPTO y_ ; x_ ¼ CPTO y_ À x_ þ kðy À xÞ: (12) The braking force is applied only for latching and is modelled as À Fbrake t; y_ ; x_ Á ¼ CbrakeðtÞÀy_ À x_ Á: (13) Realistically, in the case of latching, a braking mechanism is introduced with a non-zero time response. The brake damping coefficient is defined as a cubic function of time to ensure a continuous transition between zero and the maximum value CbrakeðtÞ ¼ (  Cmax 3~t2 À  2~t3 ; Cmax ; 0 ~t ~t>1 1; (14) with ~t ¼ ðt À tbÞ=tmax, where tmax is target braking time and t À tb is the elapsed time since the braking command instant. The equations of motion for the two-body system in the time domain are then written as À m1 þ AN 1 ðm2 Áy€ þ þ A2 CT Þ€x ðþt ÞCÀTy_ ðÀt ÞÀx_x_Á þ À y9_ ÁgSþy þ kðy kðx À À yÞ xÞ ¼ ¼ 0; FsðtÞ; with (15) Fig. 3. Stages of the threshold unlatching algorithm written in Cþþ. The code is executed once at the end of each time step. Ds. y_ ¼ y_ 0 þ Xn iaisiÀ1; i¼2 (21) FsðtÞ ¼ FdðtÞ À FcðtÞ; (16) CT ðtÞ ¼ CPTO þ CbrakeðtÞ: (17) The system of differential equations is subjected to the initial conditions yð0Þ ¼ y0, y_ ð0Þ ¼ y_ 0, xð0Þ ¼ x0 and x_ ð0Þ ¼ x_ 0. 3.2. Numerical solution of the equations of motion The numerical solution of the motion equations are based on the polynomial approximation presented in [23], adopting a more generic continuous least-squares method for the computation of the the polynomials coefficients instead of the the semi-analytical approach of [23]. The time step will be denoted by Ds. From this point onwards we will refer to a relative time s starting at the beginning of the current absolute time step, tn, such that s ¼ t À tn: (18) We approximate the coordinates of the floater and the submerged body by an n-th order polynomial Xn y ¼ y0 þ y_ 0s þ aisi; i¼2 (19) Xn x ¼ x0 þ x_ 0s þ bisi: i¼2 Consequently (20) Xn y€ ¼ iði À 1ÞaisiÀ2; i¼2 (22) and similar equations for x. The use of the relative time allows us to impose the initial conditions yð0Þ ¼ y0, y_ ð0Þ ¼ y_ 0, xð0Þ ¼ x0 and x_ ð0Þ ¼ x_ 0 directly in the polynomial approximations (19) and (20). Additionally, we approximate FsðtÞ and CT ðtÞ also by interpolating polynomials of degree n at n þ 1 equally spaced points, fsðsmÞ ¼ Fsðtn þ smÞ; (23) Table 1 Characteristics of the 14 sea states used in the calculation of the average annual power absorption. n Hs [m] Te [s] 4 [%] 1 1.10 2 1.18 3 1.23 4 1.88 5 1.96 6 2.07 7 2.14 8 3.06 9 3.18 10 3.29 11 4.75 12 4.91 13 6.99 14 8.17 5.49 6.50 7.75 6.33 7.97 9.75 11.58 8.03 9.93 11.80 9.84 12.03 11.69 13.91 7.04 12.35 8.17 11.57 20.66 8.61 0.59 9.41 10.07 2.57 4.72 2.81 1.01 0.39 J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 35 a b c d e f Fig. 4. Dimensionless annual averaged power absorption P*, regarding the wave climate of Table 1, as function of threshold unlatching time and PTO damping coefficient for: (a) L ¼ 1, (b) L ¼ 2, (c) L ¼ 5, (d) L ¼ 10 and (e) L ¼ N. In case (f), it is L ¼ 5, and a single sea state is used (Pierson-Moskowitz spectrum Hs ¼ 2:8 m and Te ¼ 8:14 s) with the same annual energy content as the considered wave climate. 36 J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 cT ðsmÞ ¼ CT ðtn þ smÞ; where Xn fsðsÞ ¼ fisi; i¼0 (24) (25) Xn~ cT ðsÞ ¼ cisi; i¼0 (26) with sm ¼ mDs=n, m ¼ 0; .; n and n~sn in the general case. Details of computation of (25) are given in appendix Appendix A. The polynomial coefficients ai and bi can be computed, using the continuous least-squares method, by replacing (19) to (26) in the residual (see appendix Appendix B for details) ZDsh i r12ðsÞ þ r22ðsÞ ds; 0 (27) with r1ðsÞ r2ðsÞ ¼ ¼ À m1 þ AN 1 Áy€ ðm2 þ A2Þ€x þ þ À CT ðtÞ À y_ À CT ðtÞ x_ À x_ Áþ y_ Á þ 9gSy þ kðy kðx À yÞ: À xÞ À FsðtÞ; (28) The resulting system of equations must be solved iteratively until convergence since the velocity need to be known when computing the convolution integral. The following stopping criterion was used Cepffiffiffiyffi < 10À16; (29) where y ¼ 1 Ds ZDsh yðnÞ À yðnÀ1Þi2ds; 0 Ce ¼ m9S1þþkA=N 1g ; (30) (31) The different stages of the latching control algorithm are executed at the end of each time step, see Fig. 3. The function ZeroCross() detects if the relative velocity of the two-bodies vanishes in the time interval s˛½0; 2:5DsŠ using a parabolic approximation of the velocity within the current Ds. This allows the prediction of the zero relative velocity as will be needed in a real application. The average annual energy conversion of the two-body WEC was computed for an Atlantic location off the western coast of Portugal. A set of 14 sea-states and 120 min time series were considered, see Table 1 (for details see [26]). Each sea state is characterized by the significant wave height, Hs, and the energy period, Te. To impose a smooth start, the diffraction force (Eq. (6)) is multiplied by a cubic function SðtÞ, SðtÞ ¼ 3 t2 ts2 À 2 t3 ts3 ; (32) during the first ts ¼ 5Te, such that Sð0Þ ¼ 0 and SðtsÞ ¼ 1, resulting in the initial conditions y0 ¼ y_ 0 ¼ 0. The function SðtÞ has zero slope at t ¼ 0 and t ¼ ts. Contour plots of the dimensionless annual averaged power absorption as a function of the threshold unlatching time and the PTO damping coefficient are presented in Fig. 4 for different values of the ratio between the floater mass and the mass plus added mass of the submerged body. These contour plots allow a detailed insight into the system behaviour which can not be highlighted from the single result of an optimization procedure. Fig. 4 presents the dimensionless annual averaged power absorption. P* ¼ Psys ; Pmax; heave (33) where Psys ¼ CPTO t2 À t1 Zt2Ày_ À x_ Á2dt; t1 (34) Pmax; heave ¼ g39 4 ZN uÀ3SzðuÞdu ¼ 149:5Hs2Te3; ÀN for five mass ratios L ¼ 1, 2, 5, 10 and N, where (35) and ðnÞ denotes the nth iteration. An identical criterion is used for the variable x. In each iteration, the symmetric linearized system of equations is solved using the Cholesky decomposition LDLT [24]. 3 3 4. Results The WEC floater is a 5 m radius hemisphere connected through the PTO to the submerged body. The hydrodynamic radiation coefficients of the hemisphere where computed by the WAMIT code [25], using 160 equally spaced frequencies between 0.2 and 2.5 rad/s and also at the limiting values of 0 and N. The maximum brake damping coefficient and braking time were set to Cmax ¼ 5  108 Ns/m and tmax ¼ 0:2 s, respectively, see Eq. (14). A sixth degree polynomial was used to compute the floater and submerged body coordinates for each constant time step Ds ¼ 0:1 s. This value was considered to be the largest time step to be used with the latching control algorithm. Numerical stability and comparable accuracy were observed for much larger time steps without latching control. Fig. 5. Dimensionless annual averaged power absorption P* for a passive PTO regarding the wave climate of Table 1. The plotted P* is a function of PTO damping coefficient and inertia ratio, L. J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 37 L ¼ m2 þ A2: m1 (36) The power conversion is computed over a time period of 120 min, starting at t1 ¼ ts þ 100 s and finishing at t2 ¼ t1 þ 7200 s. The annual averaged power absorption (Eq. (34)) is computed analytically using eqs. (19) and (20). The maximum conversion power in heave, Pmax; heave, given by Eq. (35), is obtained from the definition of the Pierson-Moskowitz spectrum for deep water. The computational programme has been written in Cþþ and uses the MPI library for the parallel implementation. A library was created to perform analytically all the polynomial operations. Each contour plot of Fig. 4 has 50  21 grid points and takes 2.4 days to compute, corresponding each grid point calculation to about 3.3 min. These calculations were done at the IST cluster using 14 PowerPC G5 CPUs at 2 GHz. More recent CPUs such as the Intel Core i7 870 at 2.93 GHz would require only 2.1 min for each annual simulation. The code parallelization allowed to compute the results about 14 times faster than the corresponding serial version. Results plotted in Fig. 4 show that the maximum annual averaged wave power absorption for cases (a) and (b) is significantly smaller than for cases (d) and (e), evidencing the large influence of the mass ratio between the two bodies for the latching control performance. The difference between the maximum annual averaged power absorption is small in the last two cases. For case (c) the maximum annual averaged power is less sensitive to the PTO damping coefficient. This might be an interesting effect, since a larger PTO damping coefficient reduces the relative motion amplitude of the bodies without penalizing the performance, which might be very important for hydraulic PTOs. The optimum threshold unlatching time for cases (d) and (e) show no significant differences. Cases (c) and (f) compare the averaged power absorption considering (i) an annual wave climate and (ii) a seastate Hs ¼ 2:8 m, Te ¼ 8:14 s and the Pierson-Moskowitz spectrum, with the same average energy as the annual average energy of the wave climate considered given in Table 1. The results show that, although the maximum annual averaged power absorption is similar in the two cases, the PTO dynamics is quite different. Therefore, the specific wave climate should be taken into account in the detailed design of this type of WEC. For reference, Fig. 5 shows the computed annual averaged power absorption for a system without latching. Fig. 6 shows the two-body system response considering a threshold unlatching time of 0.5 s, L ¼ 5, and two PTO damping coefficients, CPTO ¼ 280 kNs/m and CPTO ¼ 980 kNs/m. The same time series was used as given by the Pierson-Moskowitz spectrum of Fig. 4 (f). As seen in Fig. 4 (c), there are two operating points located in same contour line of dimensionless annual averaged power absorption, P* ¼ 0:33. This figure highlights not only the system response under the same threshold unlatching control but also the influence of the damping on the motion amplitude. Fig. 7 presents more clearly the effect of the mass ratio, L, and PTO damping coefficient, CPTO, for a fixed value of the threshold unlatching time t ¼ 0:5 s. The results show that while larger mass ratios lead to higher power absorption at the optimal PTO damping coefficient, for mass ratios L between 4 and 5 the power absorption is only 10e20% lower for optimal damping. However, for these ratios not only the cost of the system is smaller, but the optimum absorption can be achieved for a much wider range of PTO damping values. This results on the one hand in less restrictions on the PTO design and on the other hand on a better control of the forces, displacements and velocities occurring in a specific converter. a b Fig. 6. Response of the system with L ¼ 5 and threshold unlatching time ¼ 0.5 s, for two different values of the damping coefficient (a) CPTO ¼ 280 kNs/m and (b) CPTO ¼ 980 kNs/ m. The plotted values are: the position of body 1, y, the relative displacement, yr ¼ y À x, the relative velocity, y_ r ¼ y_ À x_ , and the diffraction force, Fdif . 38 J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 initiative. The second author was supported by Ph.D. grant SFRH/ BD/29275/2006 and the third one by Ph.D. grant SFRH/BD/35295/ 2007 (MIT Portugal Programme). Fig. 7. Dimensionless annual averaged power absorption P*, regarding the wave climate of Table 1, as a function of the PTO damping coefficient and the mass ratio L, for a fixed threshold unlatching time of t ¼ 0:5 s. 5. Conclusions This paper is focused on the performance of a two-body heaving wave energy converter with a threshold unlatching time control considering a wave climate data off the western Portuguese coast. A generic two-body WEC was considered consisting of a floating hemisphere connected to a deeply submerged body. The two bodies are only allowed to move in heave and they are connected through a PTO modelled as a linear damper. The studied parameters were the relation between the floater mass and the mass plus added mass of the submerged body, the PTO damping coefficient and the threshold unlatching time. The floater dimensions were assumed fixed. The system was latched using an additional brake damper. The numerical solution of the Cummins equations was performed using a new high-order numerical method. The method is based on a polynomial representation of the solution whose coefficients are computed using a least-squares approximation. This approach results in a continuous representation of the numerical solution which is not achievable with other common ordinary differential equations solvers such as the Runge-Kutta method. The code has been parallelized and executed at the IST cluster. In the present calculations a sixth degree polynomial was used to compute the floater and submerged body positions. Five inertia ratios were considered, L ¼ 1, 2, 5, 10 and N. Results show, as expected, that annual averaged power absorption increases with L (up to about 5). The optimum threshold unlatching time is similar for all cases with L ! 5. However, the case of L ¼ 5 is the less sensitive to the PTO damping coefficient. Taking this into account, L ¼ 5 may be considered to be the best engineering option for hydraulic PTOs since it allows the use of the larger damping resulting in smaller relative motion amplitudes and velocities. Acknowledgements The authors would like to thank Prof. José Maria André for his valuable comments. This work was funded by the Portuguese Foundation for Science and Technology to IDMEC through LAETA and contracts PTDC/EME-MFE/103524/2008 and PTDC/EME-MFE/ 111763/2009. The first author was supported through Ciência 2007 Appendix A. Numerical integration of the radiation force To evaluate the convolution integral FcðsÞ at n þ 1 points, see Fig. A.1, we split it in two integrals that are computed at points sm FcðsmÞ ¼ FcþðsmÞ þ FcÀðsmÞ Zsm Z0 ¼ Kðsm À sÞy_ ðsÞds þ Kðsm À sÞy_ ðsÞds: (A.1) 0 ÀN The first convolution integral is evaluated using a Gauss- Legendre quadrature rule with p points FcþðsmÞ ¼ Zsm Kðsm À sÞy_ ðsÞds ¼ X pÀ1 Kmþ;g y_ Àsmxg Á ; 0 g¼0 (A.2) with Kmþ;g ¼ sm 2 w~ g K Àsm À 1 À xg ÁÁ ; (A.3) xg ¼ 1 2  ~xg þ  1; (A.4) where ~xg˛Š À 1; þ1½ and w~ g are the integration points and weights of the Gauss-Legendre quadrature rule. In the present work we use a cubic spline approximation of KðtÞ. To have an exact integration of FhþðsmÞ, we need   1 p ¼ integer n þ 2 2 (A.5) Gauss-Legendre integration points. The matrix Kmþ;g is computed only once. The second convolution integral can be computed up to the current time as NXÀ1 Z FcÀðsmÞ ¼ Kðsm À sÞy_ ðsÞds; l ¼ 0 Il (A.6) where Il ¼ ½^s l; ^s lþ1Š, ^s l ¼ ÀðN À lÞDs and N is the number of integration intervals such that KðNDsÞ < 10À7. As seen in Fig. A.1, for the sake of computing efficiency, the integration intervals of FcÀðsmÞ have all the same length. As a result, in the N-th interval (white bars in Fig. A.1), the non-computed values are replaced by zeros. By computing FcÀðsmÞ with the Gauss-Legendre quadrature rule with p points in each Il, we get pX NÀ1  FcÀðsmÞ ¼ KmÀ;qy_ s*q ; q¼0 (A.7) where KmÀ;q ¼ D2sw~ g*  K sm À  s*q ; h i s*q ¼ Ds xg* þ l* À N ; (A.8) (A.9) g* ¼ remainderðq=pÞand l* ¼ integerðq=pÞ. The matrix Km;q is computed only once. J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40 39 Fig. A.1. Example of the convolution integral limits in the case of a 4th -order poly- nomial approximation of FcðsÞ. The function FcðsÞ is computed at five points s0,.,s4, and divided in two parts: FcþðsmÞ and FcÀðsmÞ. Appendix B. Assembling of the system of equations Introducing expressions (19)e(26) in Eq. (27), we get r1ðsÞ ¼ Pn ð þ g1iai À g2ibiÞ À h1; i¼2 r2ðsÞ ¼ Pn ð À g2iai þ g3ibiÞ À h2: i¼2 where g1iðsÞ ¼ À m1 þ AN 1 Á g4i ðsÞ þ g2iðsÞ þ 9gSsi ; (B.1) (B.2) g2iðsÞ ¼ cT ðsÞisiÀ1 þ ksi; (B.3) g3iðsÞ ¼ ðm2 þ A2Þg4iðsÞ þ g2iðsÞ; (B.4) g4iðsÞ ¼ iði À 1ÞsiÀ2; (B.5) Xn h1ðsÞ ¼ Àh2ðsÞ þ fisi À 9gSy0 À 9gSy_ 0s; i¼0 h2ðsÞ ¼ cT ðsÞÀy_ 0 À x_ 0Á þ kðy0 À x0Þ þ kÀy_ 0 À x_ 0Ás: Differentiating in order to aj and bj results in (B.6) (B.7) v vaj ZDs  r12 þ r22 ds ¼ ZDs  2 þ r1g1j À r2g2j ds; 0 0 v vbj ZDs  r12 þ r22 ds ¼ ZDs  2 À r1g2j þ r2g3j ds; 0 0 (B.8) with j ¼ 2; .; n. The unknown coefficients aj and bj are the zeros of (B.8), giving a symmetric linear system of equations with 2  2 blocks !    G11 G12 GT12 G22 a b ¼ ha hb ; (B.9) where a ¼ ða2.anÞT ; (B.10) b ¼ ðb2.bnÞT ; (B.11) ZDs  ½G11ŠiÀ2;jÀ2 ¼ þ g1ig1j þ g2ig2j ds; 0 (B.12) ZDs  ½G12ŠiÀ2;jÀ2 ¼ À g1ig2j À g2ig3j ds; 0 (B.13) ZDs  ½G22ŠiÀ2;jÀ2 ¼ þ g2ig2j þ g3ig3j ds; 0 (B.14) ZDs ðhaÞiÀ2 ¼ ðh1g1i À h2g2iÞds; 0 (B.15) ZDs ðhbÞiÀ2 ¼ ðh2g3i À h1g2iÞds: 0 (B.16) The integrals (B.12) to (B.61) are evaluated analytically using a special purpose numerical library written in Cþþ. References [1] Mavrakos SA. Hydrodynamic coefficients in heave of two concentric surfacepiercing truncated circular cylinders. Applied Ocean Research 2005;26:84e97. [2] Falcão AFO. Phase control through load control of oscillating-body wave energy converters with hydraulic PTO system. Ocean Engineering 2008;35:358e66. [3] Clément AH, Babarit A, Gilloteaux J-C, Josset C, Duclos G. The SEAREV wave energy converter. 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