Application of the Contour Line Method for Estimating Extreme Responses in the Mooring Lines of a Two-Body Floating Wave Energy Converter

Transcription

1 Application of the Contour Line Method for Estimating Extreme Responses in the Mooring Lines of a Two-Body Floating Wave Energy Converter Made Jaya Muliawan made.muliawan@ntnu.no Zhen Gao zhen.gao@ntnu.no Torgeir Moan torgeir.moan@ntnu.no Centre for Ship and Ocean Structures (CeSOS), Norwegian University of Science and Technology, Otto Nielsens vei 10, NO-7491, Trondheim, Norway The ultimate limit state (ULS) is one of the design criteria used in the oil and gas industry in mooring system design for floating platforms. The 100 year level response in the mooring line should be applied for the ULS design check, which is ideally estimated by taking into account the dynamic mooring line tension in all sea states available at the operational site. This approach is known as a full long-term response analysis using the allsea-state approach. However, this approach is time consuming, and therefore, the contour line method is proposed for estimation of the 100 year response by primarily studying the short-term response for the most unfavorable sea states along the 100 year environmental contour line. Experience in the oil and gas industry confirmed that this method could yield good predictions if the responses at higher percentiles than the median are used. In this paper, the mooring system of a two-body wave energy converter (WEC) is considered. Because this system involves the interaction between two bodies, the estimation of the ULS level response using the all-sea-state approach may be even more time consuming. Therefore, application of the contour line method for this case will certainly be beneficial. However, its feasibility for application to a WEC case must be documented first. In the present paper, the ULS level response in the mooring tension predicted by the contour line method is compared to that estimated by taking into account all sea states. This prediction is achieved by performing coupled time domain mooring analyses using SIMO/RIFLEX for six cases with different mooring configurations and connections between two bodies. An axisymmetric Wavebob-type WEC is chosen for investigation, and the Yeu site in France is assumed as the operational site. Hydrodynamic loads including second-order forces are determined using WAMIT. Finally, the applicability of the contour line method for prediction of the ULS level mooring tension for a two-body WEC is assessed and shown to yield accurate results with the proper choice of percentile level for the extreme response. [DOI: / ] Keywords: contour line method, mooring analysis, extreme response, two-body wave energy converter Introduction As with floating structures in general, a floating wave energy converter requires station keeping during operation. At present, mooring systems are considered the most economical stationkeeping systems for floating WECs. However, experience shows that mooring system design is a fundamental issue for the survival of a floating WEC and must be critically assessed. With reference to practices in the oil and gas industry [1], one of the design criteria for mooring systems is the ultimate limit state; consideration of this criterion ensures that the individual mooring lines have adequate strength to withstand the load effects imposed by extreme environmental actions. It is recognized [2] that the response for the ULS design check should correspond to a 100 year response level that ideally should be estimated by performing a full long-term analysis, but the full long-term response analysis that considers all sea states is very time consuming. The contour line method [3] has, therefore, been proposed for estimation of the long-term extreme mooring tension. The contour line method is Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received April 3, 2012; final manuscript received March 29, 2013; published online June 6, Assoc. Editor: Lance Manuel. based on the idea that the q-probability response can be estimated by primarily studying the short-term response for the most unfavorable sea state along the q-probability environmental contour line, independent of the information about response properties. However, application of the contour line method neglects the variability in the short-term extreme value. Therefore, it is recommended to consider a higher percentile value than the median tension of 3 h distributions to artificially account for the variability of the short-term extreme value, which will depend in part on the nature of the response problem [4]. Compared to the oil and gas facilities, two-body WEC is different in the way it is operated. Two bodies in the WEC system are designed to have relative motion especially in heave mode from where the power take-off (PTO) system extracts the wave power. To optimize the wave power absorption, the PTO setting, which is actually based on a mechanical connection, is controlled as a function of available sea states. However, during extreme conditions, the survival of the WEC systems becomes more important than the expectation that the WEC absorbs power. A survival strategy such as the introduction of a locking system in the interface between the two bodies should be considered. It is to minimize the instantaneous force that is experienced by the PTO system. Both operational and survival settings should be taken Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 / Copyright VC 2013 by ASME

2 into account in the estimation of ULS level response by the full long-term analysis. However, when the contour line method is applied, the sea state condition along the contour line that gives the largest response will be considered as the design condition. In principle, the WEC will be either in the operational mode or in the survival mode. However, for the mooring system response considered in this paper, it is the extreme conditions (when the WEC is in the survival mode) that will dominate the long-term extreme response. Therefore, the feasibility of using the contour line method for prediction of the extreme mooring tension for a twobody WEC should be investigated. Moreover, according to the authors knowledge, it has not yet been reported. Therefore, the purpose of this paper is to investigate the feasibility of using the contour line method for prediction of the extreme mooring tension for a two-body WEC by comparing the extreme value obtained by taking into account all sea states in the site (the all-sea-state approach). An axisymmetric Wavebob type WEC and the Yeu site in France are chosen as the device and the operation site, respectively, for consideration in the present study. Two different mooring configurations and three different connections between the two bodies are included to provide a comparative basis. The WEC properties used in the present study are based on available information in the literature and assumptions made by the authors. Therefore, there may be differences between the WEC analyzed in this paper and that of the Wavebob project. Table 1 Main Parameters of the WEC system used in the present analysis Property Value Unit Torus Outer diameter 20 m Inner diameter 10 m Draft 2 m Height 8 m Displacement 278 m 3 Mass 278 tons Centre of mass 0 m below MWL Moment of inertia I xx & I yy 12,400 tonm 2 Moment of inertia I zz 16,500 tonm 2 Stroke length 6 m Stiffness of upper end stop spring 10 6 kn/m Stiffness of lower end stop spring 10 6 kn/m Float Diameter at water level 8 m Draft 50 m Height 66 m Displacement 4680 m 3 Mass 4680 tons Centre of mass 35 m below MWL Moment of inertia I xx & I yy 1,740,000 tonm 2 Moment of inertia I zz 1,510,000 tonm 2 Axisymmetric Wavebob-Type WEC According to the Wavebob website [5], the WEC is an axisymmetric, self-reacting point absorber that primarily operates in the heave mode. This device consists of two concentric bodies with different heave natural frequencies, which are known as Torus for the shallower body and Float for the deeper body, as illustrated in Fig. 1(a). The present analysis uses the dimensions and properties of the system referred to in Ref. [6] (see Table 1 and Fig. 1(b)). These data are obtained or estimated from available published information and may deviate from the true Wavebob concept. Operational Site and Mooring Configurations In this study, the Yeu site (046 40, N latitude, , W longitude) in France, shown in Fig. 2(a), is considered as the operational site. The distribution of significant wave heights and spectral peak periods at this location based on a full year of real sea measurements [8] is illustrated in Fig. 2(b). Two of the six different mooring configurations considered in Ref. [6] are included in the study. They are presented in Fig. 3 and Fig. 1 (a) The wavebob concept [7] and (b) dimensions of the WEC used in the present analysis Fig. 2 (a) Location of the Yeu site specified as A (google map [9]) and (b) scatter diagram (presented in percentage) of waves at Yeu / Vol. 135, AUGUST 2013 Transactions of the ASME

3 Fig. 3 Mooring configurations considered in the simulations (only line 1 shown in the figure, symmetric for other lines) Table 2 Summary of mooring configurations (LINE 1 only; symmetric for other lines) shown in Fig. 3 Mooring Configuration Description MC1 MC4 Fairlead (F) (x,y,z) , , , 7.071, 0 Anchor (A) (x,y,z) 85, 85, , 220, 100 Slope (1:m) 1:2 1:3 Distance from A to F (m) Line length (m) Line Diameter (m) EA (kn) 1,711, ,000 Submerged weight per unit length (N/m) Fig. 4 One hundred year contour line for the wave conditions at the Yeu site summarized in Table 2. Each configuration consists of four steel wire lines arranged in a symmetric manner, as shown in Fig. 3. Ultimate Limit State and Environmental Conditions Because it is a new and unique system, the design criteria of the WEC mooring system are not yet available. Therefore, established practices taken from the oil and gas industry (e.g., DNV-OS-E301 [1]) are referred to in the present paper. With reference to Ref. [1], there are three design criteria used in the design of the mooring system: a ULS, a fatigue limit state, and an accident limit state. However, this study focuses on the extreme response in mooring lines due to extreme environmental conditions, which corresponds to the ULS criteria, to ensure that the individual mooring lines have adequate strength to withstand the load effects imposed by extreme environmental actions. It is recognized from Ref. [2] that the response for ULS design should correspond to a 100 year response level. The environmental loads for mooring line response calculations should be based on the wind and wave conditions in a 100 year return period and applied with currents over a 10 year return period [1]. However, only waves are considered as the environmental load applied in the current study, although waves from different directions should be considered in general. As shown in Fig. 2(a), the Yeu site is mostly surrounded by other landmasses, except to the west. According to this situation, the mooring configuration is set to the orientation of the incoming waves to minimize the line tension, as illustrated in Fig. 3. It is assumed in this work that the long-term variation of sea states is described by a joint distribution in which the significant wave height H s follows a Weibull distribution, and the spectral peak period T p, conditional on a given H s, follows a log-normal distribution. Therefore, the 100 year environmental contour line can be depicted as shown in Fig. 4, which represents a set of combinations of significant wave height and peak periods that have similar 100 year probabilities of exceeding. It should be noted that the joint distribution is obtained by fitting an analytical model to the data from 1 year measurement at the site. There is a statistical uncertainty in the extrapolated 100 year contour line. Mechanical Coupling of the Two-Body System The modeling of a two-body system in the present analysis is similar to the modeling method explained in detail in Ref. [6]. Therefore, the present paper describes only an overview of the method. The Wavebob-type WEC is modeled using the features available in SIMO [10]. SIMO is a computer program developed by Marintek, and its essential features for the present analysis include flexible modeling of multibody systems that can accommodate the introduction of both mechanical and hydrodynamic couplings between Float and Torus. Figure 5 illustrates the introduction of mechanical coupling using SIMO s features to model the two-body WEC system, i.e., docking cone and end stop features. In the Wavebob-type WEC operation, the power take-off system extracts the wave power based on the relative motion between the two bodies. In the present study, the PTO system is modeled in a simplified manner as ideal linear damper and spring terms that couple the motion of the two bodies. Reference [6] has presented how those PTO damping and spring parameters affect the power absorption of the WEC under different wave conditions. Moreover, it has been estimated that the Wavebob-type WEC absorbs approximately 220 kw of wave power during a full year of service at the Yeu site when a constant Bpto ¼ 8000 kn s/m was applied in the simulation. This result is in good agreement with the power production ranges estimated by Babarit et al. [11]. Therefore, a constant Bpto ¼ 8000 kn s/m is used as the PTO setting during operational mode of the WEC in the present model. During extreme conditions, the survival of the WEC systems becomes more important than the expectation that the WEC absorbs power. In the present study, the introduction of a locking system in the interface between the two bodies is considered as survival mode to minimize the instantaneous force that is otherwise experienced by the PTO system. Case Studies Six simulation cases, as summarized in Table 3, are considered to obtain the goal of the present study. These cases differ in mooring configuration and in connection between the two bodies. Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 /

4 Fig. 5 Introduction of (a) mechanical coupling to allow the two bodies to move together in sway, surge, roll, and pitch yet move freely in heave and yaw; and (b) end stops to limit the relative heave motion Table 3 Cases considered in the present study Mooring configuration MC1 MC4 Connection between two bodies Locked at all sea states (fully one-body system) MC1-LA MC4-LA Locked when H s > ¼ 6 m else, Bpto ¼ 8000 kns/m MC1-L6 MC4-L6 Bpto ¼ 8000 kns/m (fully two-body system) MC1-UA MC4-UA MC1 and MC4, as shown in Fig. 3, represent two different mooring configurations in terms of the location on the body at which the fairlead is attached as well as the slope, length, and stiffness of the mooring lines. To include the operational and survival modes of the WEC, three different connections between the two bodies are considered. They are classified as follows: LA connection condition: when the two bodies are fixed to each other. It is achieved by applying a very high damping term during simulation in the all sea states. Therefore, the two bodies will behave as a body system. Although one may say that this is not a realistic situation in term of power production, this case could be considered as an equivalent system for the purpose of mooring design. L6 connection condition: assumes that the system has a survival mode for facing the extreme condition. The WEC will behave as a two-body system connected with Bpto ¼ 8000 kn s/m to absorb the wave power if the H s is lower than 6 m. If the H s is higher than 6 m, the two bodies will be locked with very high damping and will thus behave as a one-body system. This case is supported by simulations results performed in Ref. [6] that show that the WEC will not produce a significant amount of power if the H s is higher than 6 m. UA connection condition: presents the WEC as a two-body system connected with Bpto ¼ 8000 kn s/m in all available sea states to gain more power from the waves. It should be noted that it is not realistic to have a WEC in operation during extreme conditions, and therefore, this case is simply used for comparison of mooring design. Mooring Analysis and Extreme Response General. In the present study, a water depth of 100 m is used, and it is assumed that wave conditions are represented by a Jonswap wave spectrum with a peakedness parameter of c ¼ 3.3. Hydrodynamic properties, including added mass, potential damping, and first and second order wave forces on the WEC bodies for each simulation case are calculated in the frequency domain using HydroD [12] based on WAMIT [13] as the core approach for the hydrodynamic analysis. These hydrodynamic properties are applied in the coupled mooring analysis using time domain simulation with SIMO/RIFLEX [14] that includes the stiffness, inertia, and viscous forces on the lines due to its motion and fluid interaction. Viscous forces on the WEC body are also included during the analysis following the Morison model. Hydrodynamic interactions between two bodies are not taken into account for all simulation cases. Figure 6 shows the panel model that was developed for the two-body hydrodynamic analysis in the present study. Full Long-Term All-Sea-State Approach. In the present study, the all-sea-state approach is applied to estimate the longterm distribution of 3 h maximums of the line tension X 3h, which can be written as in Eq. (1). ð ð F X3h ðxþ ¼ ðxjh; tþf Hs;T p ðh; tþdtdh (1) where: f Hs;T p h F X3hjH s;t p t ðh; tþn is the joint long-term distribution of the sea-state variation (H s and T p ) represented by the scatter diagram in Fig. 2(b). As stated in the section on environmental conditions, this scatter diagram has been fitted to an analytical joint distribution to cover the 100-year sea states. It is assumed that the long-term variation of sea states is described by a joint distribution in which the significant wave height H s follows a Weibull distribution, and the spectral peak period T p, conditional on a given H s, follows a log-normal distribution. ðxjh; tþ is the short-term distribution of the 3 h maximum line tension X 3h in the given sea state. In the present F X3hjH s;t p / Vol. 135, AUGUST 2013 Transactions of the ASME

5 Fig. 6 Panel model for the present analysis analysis, for each sea state, the short-term distribution of the maximum line tension is estimated by performing a coupled time domain mooring analysis in 12 different 1 h wave realizations. Subsequently, the distribution of 1 h maximum line tensions can be plotted by fitting the simulation results to a Gumbel distribution, as illustrated in Fig. 7(a) for case MC1- AL at given values H s ¼ 8.5 m and T p ¼ 12 s. Then, the fitted distribution is used to estimate the 3 h maximum tension distributions that are presented in Fig. 7(b), assuming that the 3 h period contains three independent 1 h extremes. Therefore, the line tension with an annual probability of being exceeded (q) can be estimated by solving Eq. (2) below, where N ¼ 2920 is the total number of 3 h events in 1 year and q ¼ 0.01 corresponds to the 100 year response. The extreme response value that is predicted by Eq. (2) will correspond to the most probable largest response. 1 F X3h ðx q Þ¼ q N (2) The tail parts of the long-term distributions of the 3 h maximum line tension for each case considered in the present study are presented in Fig. 8. The estimated line tensions that correspond to several return periods are summarized in Table 4. Contour Line Prediction. The contour line method is based on the idea that we can estimate the q-probability response by primarily studying the short-term response for the most unfavorable sea state along the q-probability environmental contour line, without having to conduct a full long-term analysis. The theory underlying the method has been reported by previous papers, e.g., Refs. [3,4,15,16]. Therefore, here only the major steps of the method are mentioned [2]: The q-probability contour for H s and T p must be available. Independent of the information about response properties, the IFORM technique described in Ref. [3] may be used to establish the q-probability environmental contour line from the joint environmental distribution by solving Eq. (3). ðð q N ¼ f Hs;T p ðh; tþdtdh (3) In the present study, the ULS response level that corresponds to the 100 year level or q ¼ 10 2 is considered. The 100 year environmental contour line at the Yeu site is shown in Fig. 4 The worst sea state along the q-probability contour for the system must be determined. Several sea states are listed in Fig. 7 (a) A 1 h extreme tension fitted to a Gumbel distribution on Gumbel probability paper, and (b) probability density function for both 1 h and 3 h extreme tensions for the MC1-LA case with H s m and T p 5 12 s Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 /

6 Table 5 Sea states on the 100 year contour line (see Fig. 4) considered in extreme prediction using the contour line method T p (s) H s (m) is obtained using a similar approach as that performed in the all-sea-state approach by conducting coupled time domain mooring analyses in 12 different 1 h simulations using 12 different wave realizations. Then, the distribution of 1 h maximum line tensions can be plotted by fitting the simulation results to a Gumbel distribution and, subsequently, used to estimate the 3 h maximum tension distributions. The q-probability response is estimated by the a-percentile of this distribution function. By using the contour line method, we essentially neglected the short-term variability of the 3 h response extreme value. As a consequence, the corresponding 100 year responses estimated by this approximation will be nonconservative compared to the full long-term approach. To obtain an adequate estimation of the 100 year response using the contour line method, Ref. [4] reported certain corrections to artificially account for the variability of the short-term extreme value. One of these corrections selects a higher percentile as the short-term characteristic value rather than the median value. Then, the 100 year response is estimated by solving Eq. 4, where h 100 and t 100 define the worst sea state along the 100 year contour for the present problem. F X3hjH s;t p ðxjh 100 ; t 100 Þ ¼ a (4) Fig. 8 Tail parts of the long-term extreme response distributions obtained by the all-sea-state approach for the cases (a) with MC1 and (b) with MC4 Table 4 Comparison of the mooring line response levels obtained by the all-sea-state approach (the most probable largest tension for the different return periods) The target percentile is obtained by choosing the percentile that gives the same ULS tension as the all-sea-state approach, as illustrated in Fig. 9. The percentile level represented by a will depend somewhat on the nature of the response problem. For most practical problems, reasonable results are obtained for values of a in the range , as in Refs. [4,15,17]. However, a default choice of the 90th percentile has been recommended in Ref. [4]. Clearly, this choice must be validated by the all-sea-state approach results, especially when applied for the two-body WEC system. Level of the line tension (kn) Case 100 year 70 year 50 year 30 year 10 year MC1-LA 56,880 54,350 52,050 48,760 42,710 MC1-L6 56,230 53,540 51,060 47,460 40,630 MC1-UA 50,310 47,370 44,610 40,470 31,720 MC4-LA 15,010 14,640 14,290 13,760 12,630 MC4-L6 13,860 13,470 13,110 12,550 11,320 MC4-UA 12,260 11,510 10, Table 5, which are on the 100 year contour line shown in Fig. 4 and are applied in the simulation to identify the worst sea state for the system. The distribution function for the 3 h maximum response (X 3h ) for the worst sea state must be established. In the present analysis, the short-term response for each considered sea state Fig. 9 Illustration of the target percentile value that gives the same ULS tension as the all-sea-state approach result / Vol. 135, AUGUST 2013 Transactions of the ASME

7 Fig. 10 Comparisons of 100 year mooring line tension resulting from the all-sea-state approach with those predicted using the contour line method with consideration of several percentile values for cases (a) MC1-LA, (b) MC1-L6, (c) MC1- UA, (d) MC4-LA, (e) MC4-L6, and (f) MC4-UA Therefore, comparisons have been made between the 100-year level responses in the mooring line resulted from the all-sea-state approach and the responses estimated by the contour line method considering several percentile values. These results are presented in Fig. 10 and summarized in Table 6. Discussion From the comparison between the ULS level response in the mooring line resulting from both methods, as shown in Fig. 10, one can see that the expected maximum of the short-term response Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 /

8 Table 6 Comparison of the 100 year mooring line tension obtained by the all-sea-state approach and the contour line method 100 year tension (kn) Contour line a Case All sea-state approach Contour line (Expected max) Target percentile MC1-LA 56,880 38, % MC1-L6 56,230 38, % MC1-UA 50,310 40, % MC4-LA 15,010 10, % MC4-L6 13,860 10, % MC4-UA 12, % a To yield the all sea states max. distribution from the worst sea state considered in the contour line method always underestimates the response compared to the allsea-state approach for all cases considered in the present study. However, by taking a higher percentile than the mean value produced from the contour line, as proposed in Ref. [4], the contour line method is found to yield an adequate 100 year response prediction. The value of the percentile that gives a corresponding 100 year response varies from 80% for the two-body system (cases MC1-LA and MC4-LA) up to 98% for the one-body system (cases MC1-UA and MC4-UA). These percentile values are consistent with those reported in Refs. [4,15,17] for floating systems in the oil and gas industry. Therefore, the contour line method is considered feasible for the prediction of extreme responses in the WEC system when the response that corresponds to the specified percentile is considered. The question that remains is how to choose the correct percentile value when applying the contour line method to a new system. First, it is important to remember that the contour line method is an approximate method that is not required to give the exact extreme response. Therefore, a default percentile value of 90%, as recommended in Ref. [4], can be considered to approximate the target ULS response. However, a long-term analysis is necessary to obtain the exact ULS response. In principle, we need to have large numbers of extremes to obtain an adequate asymptotic Gumbel distribution, especially if the upper tail of the distribution (such as percentile values of 85 95%) will be of interest. Therefore, more extreme data from simulations are needed to estimate the real target percentile that gives the same ULS tension as the all-sea-state approach. Reference [16] reports the statistical uncertainty associated with the estimated percentile level when based on only 20 simulations for the worst sea state. It is shown in Fig. 11 that if the percentile levels are estimated using the distributions based on 20 extremes, the percentile levels vary from 77% to 97%. However, when they combined all 400 extremes in a Gumbel distribution, the percentile levels end up from 87% to 90% to reach the target value estimated using the all-sea-state approach in their specified cases. References [4,15,17] reported percentile levels from 85% to 95% for the other systems considered. However, from Fig. 11 taken from Ref. [16], one can see that the estimated extreme responses from the distributions based on 20 extreme samples and taking 90% percentile level vary from 10% to 30% compared to the all-sea-state result, which represents a large uncertainty band. Therefore, we need more samples of short-term extremes to generate a Gumbel distribution, particularly when we use a high percentile value such as 90% to reduce this uncertainty. However, this approach is not desirable for an approximation method because it is time consuming. Therefore, it is more desirable to refer to a lower level that has less uncertainty associated with the number of samples, such as the expected maximum. From Fig. 11, one can see that the expected maximum based on 20 extremes that correspond to a y-axis value of approximately Fig. 11 Comparison of percentile level based on different numbers of samples: (a) samples and (b) fitted Gumbel distributions. The continous-connected dots in (a) and the solid fitted line in (b) are based on 400 samples (from Ref. [16]) (the cumulative probability (F) of 0.53 is a typical value for the mean value of a Gumbel distribution) has much smaller variation, especially if the fitted distributions shown in Fig. 11(b) are considered. By taking the expected maximum of the fitted distributions, the extreme responses vary from 30% to 15% compared to the all-sea-state result, which is a much narrower band compared to the uncertainty band if the 90% percentile level is taken (based on 20 samples). Then, the user can simply multiply the expected maximum by a factor of 1.3 to reach the target extreme response when applying the contour line method. The sensitivity of the extreme responses for the six considered cases in the present study is summarized in Fig. 12, which shows that the extreme responses estimated by taking the expected maximum vary from approximately 32% to 20% compared to the all sea state results from all considered cases. This result represents the narrowest uncertainty band present in the figure and is consistent with the uncertainty band for the considered cases in Ref. [16] if the expected maximum is taken, as discussed above. These observations support the choice to use the expected maximum as a reference to predict the target extreme response with much less computational effort / Vol. 135, AUGUST 2013 Transactions of the ASME

9 Fig. 12 Differences between the responses obtained by the contour line method (X CL ) considering several percentile levels and the 100-year response from the all-sea-state approach (X ALL ) From another perspective, we can say that if the expected maximum tension during the design sea state (e.g., 100 year sea state for design of mooring of permanent facilities) is directly taken as the characteristic line load, as specified in the Norwegian Maritime Directorate s Regulation [18], then the mooring system will not have a similar safety level as the permanently installed facilities. From the results of the all-sea-state approach presented in Table. 4, one can see that the expected maximum tension during the 100 year sea state corresponds only to a 30 year response level and even lower than a 10 year level for some cases. As a result, if the NORSOK standard [2] is referenced, this tension cannot be used as a characteristic load for the anchor design. Simulation results in the present study also show that the application of a locking system between the two bodies during the extreme sea state increases the mooring system loads. When two bodies are connected with very high damping (as applied in simulation cases MC1-AL, MC1-L6, MC4-AL, and MC4-L6), the extreme line tension becomes approximately 15 20% higher than that of the two-body system (cases MC1-UA and MC4-UA). As a consequence, larger mooring lines would be necessary. This result is not desirable in a WEC project where the cost of the mooring system is one of the major capital costs of the project. The reason for this higher tension is that the locked body will experience higher wave forces compared to the two-body system. Figure 13 illustrates how the heave response amplitude operator (RAO) changes when different PTO damping (Bpto) values are introduced to connect the two bodies. Fig. 13 Heave RAOs of the WEC for different connections between two bodies Conclusions Coupled time domain mooring analyses have been performed by including the first and second order wave forces and considering six different cases for investigation of the feasibility of using the contour line method to predict the extreme mooring tension for a ULS check of a two-body axisymmetric Wavebob-type WEC. The contour line method is compared with the full longterm approach by taking into account all sea states. The considered cases vary in the mooring configuration and in the connection between the two bodies. In the present study, the dimensions and parameters of the device have been estimated from available open sources. Based on the present study, the following conclusions can be made: The contour line method is feasible for the prediction of the ULS response level (100 year) in the mooring line of a twobody floating WEC if one of the following approaches is considered: (1) The expected maximum tension is taken and then multiplied by a factor of 1.3. (2) The tension from the appropriate percentile level (85 95%) is considered. These empirical corrections are consistent with those applications of offshore oil and gas platforms. Considering that the contour line method is an approximate method, the suggested corrections are believed to be relevant for others systems where the extreme responses governed mostly by the waveinduced responses. The first approach is found to have a narrower uncertainty band compared to the second one if a limited number of extremes are considered. Therefore, less computational effort is required. The present study confirmed that the first approach could predict the ULS response for the considered cases in the present study and for the considered cases in the study reported in Ref. [16]. The introduction of a locking strategy between two bodies increases the ULS level response in the mooring line; therefore, the mooring system should be designed carefully if such survival mode for the specific two-body system is considered. There are challenges remaining that need to be addressed regarding the analysis of a two-body WEC with mooring with regard to optimization of the economic value of WECs. The challenges include, but are not limited to, mooring analysis of the WEC farm, fatigue damage estimation, and cost optimization. However, these issues may be a part of a future study. Acknowledgment The first author gratefully acknowledges the support of the EU- Wavetrain2 project and the Research Council of Norway (RCN) for funding. We thank the RCN for their financial support through the Centre for Ship and Ocean Structures. Valuable advice was obtained for the present study during discussions with Professor Sverre Haver; this contribution is highly acknowledged. References [1] Det Norske Veritas (DNV), 2006, Position Mooring, Offshore Standard DNV-OS301. [2] NORSOK, 2007, Actions and Action Effects, NORSOK Standard N-003. [3] Winterstein, S. R., Ude, T. C., Cornell, C. A., Bjerager, P., and Haver, S., 1993, Environmental Parameters for the Extreme Response: Inverse FORM With Omission Factors, Proceedings of the ICOSSAR-1993, Innsbruck, Austria, pp [4] Haver, S., Sagli, G., and Gran, T. M., 1998, Long Term Response Analysis of Fixed and Floating Structures, Proceedings of the 1998 International OTRC Symposium, Houston, TX, pp [5] Wavebob, 2013, Wavebob: Blue Energy, Retrieved November 30, 2010, [6] Muliawan, M. J., Gao, Z., Moan, T., and Babarit, A., 2013, Analysis of a Two- Body Floating Wave Energy Converter With the Particular Focus on the Effects of Power Take-Off and Mooring Systems on Energy Capture, J. Offshore Mech. Arct. Eng. (accepted). Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 /

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