Multi-Objective Optimization of A Semisubmersible for Ultra-Deep Water

China Oean Engnering, Vol.22, No.1, pp.11- 20口2008 China Ocean Press, ISSN 0890- 5487Multi-Objective Optimization of A Semisubmersiblefor Utra-Deep Water'CHEN Xin-quan (陈新权)' and TAN Jia-hua (谭家华)School of Naxal Archiecture, Ooean and Cruril Enginering,Shanghai JiaoTong Uniersity, Shanghai 200030, China(Received I9 April 2007; roceived revised fom 14 August 2007; accepled 15 November 2007)ABSTRACTSemisubmersible will work well when oil exploitation goes to ultra-deep water because of its variable loed capacities,and good motion performance in extreme waves. It is considered to be a main type of platform while the water depth ismore than 300 meters. This paper establishes a multi- objective optimization model of semisubmersible for ulra- deep wa-ter, and it is solved by a muliobjective genetic algorihmr -NSGA-II. The model is applied to a practical design, andPareto results are obtained. The efectiveness of the method is verified by hydrodynamic analysis.Key words: semisubmersible; ultrar deep water; hull steel weight ; heaxe ; muli objetive optinization; Pareto; hydro-dynamics1. IntroductionAs oil exploitation is moving towards ultra-deep water semisubmersible will be a more suitabletype of plaform because of its good motion performance in waves, large deck area, variable load, mul-tifunctions, and so on. There are already 18 ultra-deepwater semisubmersibles in service. All of theseplatforms have an operational depth more than 3048 meters ( 10000 feet), and some can work at themaximal depth of 3657 meters. Now, there are 22 new generation semisbumersibles under construc-tion.In order to have better performances of semisubmersible, considerable studies have been cariedout focusing on the effect of its underwater form on hydrodynamic behavior. Many optimizations are putforward to restrain the movements of platfom. Hooft (1970, 1971) proposed a systematic method forhydrodynamic forces calculation, and studied how to achieve minimum motion of a semisubmersible inregular waves. Minkenberg and van Sluijs (1972) made comparison of the motion among semisubmer-silbes with circular, square and eliptical cross-section pontoons. They found that resonance responseof semisubmersible with square cros- section pontoons is much less than the other two. Akagi and Ito(1985) utilized a simplified method to calculate hydrodynamic cofficients, and worked out the basicdimensions of 8 semisubmersible with minimum motion in random waves by quadratic program. 0wingto the high nonlinearit of hydrodynamic response, limitatiol中国煤化工algoithm, op-* This research was supported by absorption of introduced technologJMYHC N M H Gnghai (Grant No.05CBJT-32)1 Coresponding author. E mail: chenxinquan@ sjtu. edu.cn12CHEN Xin-quan and TAN Jia-hua /China Ooean Enginering, 22(1), 2008, 11-20 .timization design can hardly be presented and methods based on repetitious trial- error exercises and lo~cal searching were widely employed to decide the principle dimensions. However, these methods aretime consuming and ineffcient.Semisubmersible construction cost is much more than before when the water depth increases andthe platform migrates to large dimensions. Performance, cost and safety all should be under considera-tion in design. Coming to the optimum position between the best performance and lowest cost is a mul-ti-bjective optimization issue. In the beginning of this paper, a numerical model of hull steel weightand hydrostatics is presented. The dynamic equation of heave motion is solved with this model. Hullsteel weight and heave amplitude in random seas are selected as design objectives, which are mini-mized with respect to principal dimensions. The most advanced multiobjective optimization algorithm- NSGA-II is adopted in the optimization of a semisubmersible for ultra-deep water, and Pareto resultsare obtained. Finally, optimal dimensions of the platfom are determined. The efctiveness of thisproposed method in initial design phase is venified by hydrodynarmic analysis.2. Construction of the SemisubmersibleConstruction of the semisubmersible has changed much since it came into being. It trends towardsimplification and finally it is mainly composed of pontoons, columns and a deck. The pontoons pro-vide sufficient buoyancy and offer enough storage space for oil， ballast water and mooring lines.Columns connect pontoons and upper deck, and provide sufcient waterplane area for platform stabili-ty. Pontoons and columns of square cross-section enable easier manufacture, more displacement andlarger capacity, and now are widely used in semisubmersibles under construction. Deck is the impor-tant space for equipments and inhabilancy. Bracings either connect hulls or support deck. The numberof bracing joints is reduced in consideration of cost and damage caused by joints fatigue. Manysemisubmersibles have only two horizontal bracings with a diameter of 1 to 2 meters between thecolumns to ensure structure safety when the hull is sagging or hogging (Brown et al., 2004).Fig. 1. Sketch of semisubmersible.X63xIAll of the new generation semisubmersibles can be operated in ultra-deep water with water depthmore than 3000 meters. And they are becoming larger and中国煤化工~ 120 meters, .70~ 80 meters in width, 35 ~ 45 meters in deph.MHCNMHGThe semisubmersible designed in this paper has two pontoons and four columns (Fig. 1)，andCHEN Xin-qun and TAN Jia -hua /China Oeean Enginering, 22(1), 2008, 11-201can work in ultra-deep water. Its operational depth is 3000 meters, drill depth 9000 meters, opera-tional displacement about 50000 tones, draft 24 meters, and variable load 10000 tones.3. Hull Steel WeightA semisubmersible can be divided into three sub- systems: hull steel, machinery and equipment.Semisubmersibles with the same dimensions for different functions vary much in equipments and vari-able load. However, the difference in hull steel weight is not signifcant. In initial design phase, hullsteel weight should be minimized to reduce cost and increase variable load .Penny et al. ( 1984) divided a platform into four parts: pontoon, column, bracing and deck, thesteel weight of each part is decided by its configuration, dimension and function. Formulas for steelweight of each part have already been established respectively. For example, the pontoon' s main func-tion is to provide sufficient displacement, and its inner structure, such as the frames ’dimension anddistribution, should ensure pontoon ' s structure safety under hydrostatic load. Thus, the pontoon’ ssteel weight is a function of its surface area and draft:wp = 9.4x 10-3(SpTC)1.05;sp = 2L(B,+ D,);(1)where Top is draft; Lp, Bp, and Dp, are pontoon length, breadth and height, respectvely. Then thehull steel weight is given by .Wh = EW,(2)which is a function of principal dimensions. W; is the respective steel weight of each part.Penney also gave the weight formulas for equipment weight calculation, which have been widelyused in initial design of semisubmersible (Ma and Luo, 1993).4. Motions in Waves4.1 Heave Response in Regular WavesThe motion of the semisubmersible in waves has generally 6-degree of freedom. However, onlyheave motion in head seas is considered in this paper because of the following reasons.(1) Heave is nomally considered to be the principal motion that afctse operation. Since one endof the riser is fixed onto the semisubmersible, the platform has less flexibility against heave motionwhen compared with motions in other degrees of freedom.(2) Surge and roll can be reduced by adjusting the gravity center or regulating inertia moment af-ter principal dimensions are given.(3) Assisted by dynamic position or mooring system, the semisubmersible is usually placed withhulls parallel to wave direction to minimize motion.(4) It is assumed that the semisubmersible is symmetrical about its longitudinal and transverseaxes, which makes heave and pitch uncoupled.中国煤化工Then the dynamic equation of heave motion is given byMYHCNMHG(M+M,)Z+BZ+CZ=F(3)14CHEN Xim quan and TAN Jia-hua / China Ocean Enginering, 22(1), 2008, 11-20where M is the platform mass, Ma is hull added mass, B is damping cofficient, C is restoring coffi-cient, and F is wave excited force.The following assumptions are made in the computation.1. The hydrodynamic interactions between platform components are neglected;2. Hydrodynamic force on the hull is the sum of them on members;3. Forward velocity is zero.The origin of coordinate is the center of platformn in water plan. x axes is parallel to longitudinalsymmetrical axes, and z axes is vertical with upward direction.Linear theory is adopted in this paper. The velocity potential φ, the wave profile η, the velocityη and the acceleration 7 of water particle, and the pressure p at the water depth of z are given respec-tively as follows:φ = elsin(kx - wt);(4)η= 5ae"cos(kx - wt);(5)η = Sjve"sin(kx - wt);(6)η =- ζw2el(kx - wt);(7)驶= p5age*cos(kx - wt). .(8)P=-ρtSince the pontoon is large- scale structure, the vertical wave excited forces including radiationforce, diffraction force and Froude Kriloff force (F-K force) are calculated by diffraction theory. Theadded mass and damping coefficient of members are hard to calculate and are determined by model testpreviously. In consideration of pontoon being slender body, some papers adopted strip theory used in .ship hydrodynamics to calculate the hydrodynamic coefficients. It is found that the computed data areconsistent with the experimental results. In this paper strip theory and Frank close fit method (Frank,1967) are applied to calculate the added mass a and damping ceffcient b of the pontoon section(Fig.2), and the diffraction forces FD on members are obtained by relative motion analysis.6.0x10*p盈冒4.0x10p身4.0x0)号3.6x10*号亩2.0x10*包3.2x100.5 1.01.5一2.0 2.5‘.5 1.0一 1.5 2.0 2.5Wave frequency (nd/s)Wave frequency (rad/s)Fig. 2. Added mass and damping cofficient of pontoon section.Then,M。=2]ac| adl,B = 2| bdl;(9)中国煤化工Fo= 2()axdl+jb. indl) =-2a5YHCNMHGkCHEN Xin-quan and TAN Jia hua / China Oean Enginering, 22(1), 2008, 11-20 .152sin( )- 2bζwe"-●一sin( wt),(10)where L is pontoon length, Zm is mean z coordinate of pontoon section.The F-K force acting on per-unit length of pontoon can be calculated by integrating pressure overthe section boundary.dFp-K= Jpods.(11)And the F-K force can be expressed as:Fp-K= 2]dFp-K-OFp-Kkle__= 2|dFp-K - 4pgSae*.S●cos 2 cos(wt),(12)where△Fp- K is the correction factor since there is no pressure on the area where pontoon and columnconnected, Zu is z coordinate of pontoon’s upper surface, le is longitudinal distance between columns,and S is section area of column.There are no vertical wave excited forces on column. The forces on bracings are calculated byMorison formula due to their small scale.Thereafter heave response of semisubmersible in regular waves can be obtained by solving the dy-namic equatin. The result turms to be heave response amplitude operator (RAO) by putting 5a=1.4.2 Response in Irregular WavesThough sea waves are iregular, there are still some rules making the inegular process be a sta-tionary random one. Wave spectrum analysis is then introduced into dynamic response analysis of ma-rine and ocean engineering." Semisubmersible' s operation gets worse with increasing water depth,where natural frequency of the platform has the great chance to be similar with the peak frequency ofwave energy. Therefore, the traditional design wave method can not satisfy the requirement, and spec-tral method should be taken into the design.The ISSC wave spectrum is used in this paper (Fig. 3)12 rFg.3. ISSC wave spectrum.心4中国煤化工MHCNM H G-田(rad/s)16CHEN Xin- quan and TAN Jia-hua /China OCcan Enginering, 22(1), 2008, 11-20s(w)= Aexp(- 昌)，(13)2。where A=173 and B='are decided by significant wave height h and wave period T.Then the variance of heave motion amplitude in iregular waves is given byc2=。R40| S(w)dw. .(14)5. Optimization ModelBoth performance and cost factors are considered in building an optimization model. While di-mensions and draft of semisubmersible are selected as design variables X= (x|, x2,.. ,x), minimumhull steel weight and minimum variance of heave motion amplitude are the two objectives. The less thehull steel weighs, the less the platform costs and the more the platform loads. Smaller variance ofheave motion amplitude means natural frequency of the platform is away from the peak frequency ofwave energy and the platform has a better performance in seas.Constraint conditions of variables are defined based on the attainable infonmation of semisub-mersibles and design criterion, which make displacement, initial metacentric height, deck area, andair gap meet design requirements. In this paper, DNV offshore standard is used, which requires thatinitial metacentric height be more than 1 meter and air gap be positive in waves with a 1/100 annualprobability of exeedance.All above, the optimization model is defined as follows:min(Wn) = SW; = s&j(x});min(σ) = g(x;);s.l. Cmin≤C≤Cmax;xmin≤勾≤xmax.(15)NSGA-II (nondominated sorting genetic algorithm II) (Deb et al., 2002) is used in this paper,which need not compute the gradient of the obhjective function and can find the global optimum solutionscompared with traditional numerical optimization method. It is the most advanced multi-objective ge-netic algorithm at present.6. Results and AnalysisFig. 4 shows the detailed analysis steps.The optimization of a semisubmersible is a high nonlinear process and Pareto results are obtainedafter hours of computation. The results are shown in Fig. 5. Fig. 5 shows that semnisubmersibles ofdifferent Wh and σ are achieved by changing dimensions, and σ decreases ereatlv while Wh increases.However, σ decreases lttle after Wh reaches a certain val中国煤化工reases with lit-tle improvement. Three groups of solutions, A, B and C:fYHCNMHGtheirRAOsareshown in Fig. 6. Natural frequencies of all three cases are far away from the peak frequency of waveCHEN Xin-quon and TAN Jia-hua /China Oeean Enginering, 22(1), 2008, 11-20 .energy. Difference of heave of platformns B and C is insignificant while the hull steel weight of C is20% more than that of B. σ of platform B decreases by 20% while Wh increases by 17. 8% whencompared with platform A. Since hull steel weight can be cut down by decreasing column height anddesk area while air-gap and general arrangement still meeting design requirements, case B is selected.Number of generation .Set variables，boundPopulation sizeC Set NSGA II parameters」Crossover andfCompute hull seel weight}mutation coefficientsInitialize generationSolve theCompute hydrostaticdynamicvaluesequationOptimizationCompute hydrodynamicmodelcofficientsNSGA II，___Compute 0N" SatisfyobjectivesTY- NLast generation>YEnd, givePareto resultsFg.4. Flow chart of solution.0.48r -A20.460.448置0.424息0.40'...B0.38 L10000 11000 12000 13 000 14 00015 0000L12 d0W,田(rad/s)F'g. 5. Pareto resuls.Fig. 6. Heave RAO.Table 1Three groups of solution(勾are principal dimensions, see Fig. 1)xx4xsx6x10draftWA 100.000 15.884| 12.00013.53216.23821.300 69.39723.500103470.4847_B| 107.381 17.783| 8.50014.54617.455 .26.838| 75.126| 24.500121940. 3975C| 118.517 16.373 8.50014.30117.16126.838 85.000 24.500146550. 3845RAOs of motions in six degree of fredom solved by中国煤化工Fig. 7, fromwhich it is found that natural periods of motion in six degree:MYHCNMH Gr wave. Vis-cosity plays a major role in computation at natural period while it is not considered in potential theory8CHEN Xi-quan and TAN jia-hua / China 0cean Enginering, 22(1), 2008, 11-20and heave RAO value at natural period is much larger than its actual value (Kim and Chou, 1973)，which was verifed by Clauss (1978). Clauss also gave the heave RAO correction function at naturalperiod for a ring pontoon semisubmersible based on model test while considering nonlinear damping:Table 2Table 2 Dataof BWaterplane( m2)1016Displacement(t)49999Deck area( m?)5586Height of gravity( m)18Air- gap( m)5.3Heave natural period(s)22Hull stel weight(t)121943p0.009 [2昌0.006号0.0038.12162024812162024Wave period (S)Wave period(s)(a) Heave(b) Pitch1.2x1060.99.0x107 一0.66.0x1070.3豆3.0x10-74 8 1216 2024812.162024Wave period (s)1.2x10>3 「(C) Roll3.60x10s t(d) Surge8.0x1042.40x105 t4.0x10点1.20*10*216202412162024中国煤化工d()(C) SwayMHCNMHGFig. 7. Head sea RAOs.CHEN Xin-quan and TAN Jia-hua / China Oleoan Enginering, 22(1), 2008, 11- 20l9Heave RAO = (43. 4/H)4,(16)where H is wave height. The result corrected by the above function is showm in Fig. 7(a) with dashedline.Heave RAOs of a semisubmersilbe with bracings of different diameters are shown in Fig. 8.Heave RAOs are almost the same with bracing diameter varying from 0.5 meter to 2.5 meter except atnatural period where viscosity plays a major role and is calculated by Morison formula. Therefore, it isdeduced that wave loads on bracings can be neglected for their small size and the diameter of bracingneed not be a variable of optimization. However, bracing size can be decided according to the require-ment of structure strength after initial design.Fg. 8. Heave RAOs of the platform with bracingsof diferent diameters.10152025Wave period (S)7. ConclusionThis paper introduces a multi-objective optimization model of ultra-deepwater semisubmersiblewhile considering cost and performance. The optimization search is completely automatic compared toprevious optimization designs, and the principal dimensions are determined with the aid of Pareto frontcurve. The effectiveness of the method is venified by analysis on initial design of a semisubmersible.ReferencesAkagi, S. and lo, K., 1984. Opimization Design of Semisubmersible Form by Minimizing its Motion in Random Seas,Jourmal of Mechanisms，Transmisins, and Automation in Design, 106, 23 ~ 30.Brown, D. T.，Patel, M. H. et al., 2004. Waxe Induced Motions and Deck loads of Unbraced SEMI- SUB-MERSIBLES，Transactins of Royal Institution of Naval Architects.Chakrabarti, S., 2005. Handbook of Ofshore Engineering, Elsevier, Amsterdam, London.CHEN Xinquan and TAN Jjiahua, 2006. Principal Dimensions Optimization of Semisubmersible for Ulura-Deepwater Basedon Genetic Algorithm, China Ofshore Platform, 6: 24 ~ 27. (in Chinese)Class, G. F., 1978. 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