Mooring Pattern Optimization using Genetic Algorithms

Transcription

1 6th World Congresses of Structural and Multdscplnary Optmzaton Ro de Janero, 30 May - 03 June 005, Brazl Moorng Pattern Optmzaton usng Genetc Algorthms Alonso J. Juvnao Carbono, Ivan F. M. Menezes Luz Fernando Martha, Department of Cvl Engneerng and Computer Graphcs Technology Group (Tecgraf) Pontfcal Catholc Unversty of Ro de Janero (PUC-Ro), Ro de Janero - RJ, Brazl [email protected] [email protected] [email protected]. Abstract Ths paper presents the development of a Genetc Algorthm (GA) to solve the problem of the moorng pattern of floatng unts used n ol explotaton operatons. The dstrbuton of moorng lnes s one of the factors that drectly nfluence the dsplacements (offsets) suffered by floatng unts when subjected to envronmental condtons such as wnds, waves and currents. Thus, the GA seeks an optmum dstrbuton of the moorng lnes whose fnal goal s to mnmze the unts dsplacements. The computaton of the floatng unt s statc equlbrum poston s accomplshed by applyng the catenary equlbrum equaton to each moorng lne n order to obtan the out-of-balance forces on the unt, and by usng an teratve process to compute the fnal unt equlbrum poston. The objectve functon conssts of the sum of the squares of the floatng unt s dsplacements for each set of envronmental condtons. The developed GA as well as a representatve example and some conclusons are also presented.. Keywords Moorng Pattern, Structural Optmzaton, Genetc Algorthm 3. Introducton Wth the ncreasng demand for ol, ol companes have been forced to explot new felds n deep waters. Currently, n Brazl, Petrobras extracts about 7% of ts producton n shore, 9% n shallow waters, and 64% n deep waters (over a thousand meters deep). Due to the hgh cost of ol explotaton operatons, the development of technologes capable of ncreasng effcency and reducng costs s crucal. In ths context the use of floatng unts becomes more frequent. The postonng of the floatng unts durng ol explotaton operatons s done usng moorng lnes, whch are flexble structures usually made of steel wre, steel chan and/or synthetc cables. The dstrbuton of the moorng lnes s one of the factors that drectly nfluence the dsplacements suffered by the floatng unts when subjected to envronmental condtons (e.g. wnds, waves and currents). Therefore, the determnaton of an optmum moorng pattern results n an optmzaton problem whose fnal goal s to mnmze the dsplacements of floatng unts. As an optmzaton strategy, Genetc Algorthms (GAs) seem to be approprate to solve ths problem. They belong to the feld of Evolutonary Computaton and consst of heurstc combnatoral search technques that are based on the concepts of Darwn s evoluton theory [] and genetcs. When searchng for the global optmum soluton for complex problems, especally those wth many local mnmums, tradtonal optmzaton methods fal to effcently provde relable results. Accordng to Andre et. al. [] the standard bnary-encoded GA could consttute an nterestng alternatve to perform the global mnmzaton of a functon (objectve functon) of several contnuous varables. Ths motvated much research to employ GAs as effcent tools for solvng contnuous optmzaton problems. The frst GAs were developed n the 960s. In 975, Holland s poneerng book, Adaptaton n Natural and Artfcal Systems [3], whch had been orgnally conceved for the study of adaptve search n Artfcal Intellgence, formally establshed GAs as vald search algorthms []. GAs dffer from the most common mathematcal programmng technques n several aspects, such as: they use a populaton of ndvduals or solutons nstead of a sngle desgn pont; they work on a codfcaton of the possble solutons nstead of the solutons themselves; they use probablstc transton rules nstead of determnstc operators; they can handle, wth mnor modfcatons, contnuous, dscrete or mxed optmzaton problems; and they do not requre further nformaton (such as the gradent of the objectve functon) [4]. One of the dsadvantages of GAs s ther hgh computatonal cost, due to the large number of evaluatons of the objectve functon necessary to acheve numercal convergence. In the present work, the steady-state GA [] has been mplemented, whch performs the substtuton of only one or two ndvduals per generaton [5]. The basc operators used n ths algorthm are mutaton and crossover. The remanng sectons of ths paper are organzed as follows: frst, an ntroducton to the desgn of moorng systems as well as the mathematcal formulaton of the optmzaton problem are dscussed; second, a bref lterature revew about GAs and a detaled dscusson of the proposed GA are presented; next, a representatve example s shown n order to demonstrate ts effcency and robustness; and fnally some conclusons and suggestons for future works are made. 4. Moorng Systems Desgn There are several factors to be consdered when desgnng a moorng system, such as the type of anchorage, the strength of the moorng lnes, the poston of the anchors, the dfferent sets of envronmental condtons the unt wll be subjected to, the seabed s shape and the tme the floatng unt wll reman anchored. Moorng lnes are usually composed of dfferent types of materals, each wth dfferent physcal and mechancal propertes. Buoys and/or clump weghts may be attached to them n order to deal wth the restrcton mposed ether by other moorng systems or by the seabed s layout.

2 As n most engneerng problems, cost vs. effectveness s certanly an ssue n the moorng system desgn process. Selectng materals, defnng the envronmental condtons to be mposed on the floatng unt and any other choces wll nfluence drectly both cost and effectveness. A typcal example of a catenary moorng system s llustrated n Fgure. Fgure. Catenary moorng system. 5. Formulaton of the Optmzaton Problem The optmum moorng pattern can be expressed as an unconstraned contnuous optmzaton problem as follows: m mnmze: ( α ) = [ x ( α) + y ( α) ] () subjected to: = m = α mn α α max, =... n () where (α) s the resultng floatng-unt dsplacement (whch can be decomposed nto the components x (α) and y (α)) for a gven set of envronmental condtons; α = (α, α,, α n ) s a vector holdng the desgn varables (.e. the azmuth of each moorng lne); n s the number of ndependent desgn varables; m s the number of sets of envronmental condtons; and the nequaltes shown n Eq. () are the sde constrants. Equatons () and () represent a typcal unconstraned optmzaton problem. Fgure shows a floatng unt anchored by 8 moorng lnes and ther dstrbuton (α ) to be consdered n the optmzaton problem. North α α F α α 7 Y α Floatng Unt 4 α 4 3 α 3 α 5 Z X Fgure. Representaton of a moorng system wth 8 lnes.

3 Ths work proposes an algorthm to mnmze the dsplacements (offsets) suffered by the floatng unt when subjected to envronmental condtons. Therefore, t s necessary to fnd an optmum moorng-lne dstrbuton n order to reduce the relatve dsplacements x and y, as shown n Fgure 3. Y x y θ Floatng Unt Z X Fgure 3. Dsplacements suffered by a floatng unt when subjected to external loads. 6. Genetc Algorthms Prelmnary deas about GAs date back to the 960s. Consderable research has been conducted n ths area. The basc concepts about GAs, used n ths work, can be found n the book by Holland [3], whch s based on the process of natural evoluton. Holland s nterests were not lmted to optmzaton problems; he was also nterested n the study of complex adaptable systems, whether bologcal (such as the mmunty system) or not (such as the global economy). GAs have proven to be a very versatle tool for the soluton of optmzaton problems. They result n a very robust search mechansm for complex and dscontnuous desgn spaces, whch are very dffcult or even mpossble to search wth tradtonal calculus-based methods. In general, the term genetc algorthm refers to any populaton-based model that uses several operators (e.g. selecton, crossover and mutaton) to evolve [6]. In a basc GA, each ndvdual could be a real or bnary-valued strng, whch s usually referred to as chromosome (genotype). The prncples on whch the GAs are based are smple: accordng to Darwn s theory, the selecton prncple prvleges the fttest ndvduals, who have a bgger chance of survvng and, consequently, a bgger chance of reproducng. Therefore, an ndvdual wth more descendents (offsprng) wll have more chances to perpetuate ts genetc codes to next generatons. Such codes are the denttes of every sngle ndvdual and they are represented n the chromosomes. Some mportant features of GAs are []: they work wth a codng of the desgn varables (genotypes) nstead of the varables themselves. They work wth multple concurrent search ponts, and not wth a sngle search pont. Moreover, they do not requre further nformaton as gradent of the objectve functon, hence the probablty of gettng stuck n a local mnmum s reduced and consequently the search wll have more chances of achevng the global mnmum. They also do not requre a wde knowledge of the specfed desgn problem, whch makes GAs capable of dealng wth dfferent types of problems. Fnally, they use probablstc nstead of determnstc transton rules. Accordng to Yeh [7], a GA s basc executon cycle can be descrbed by the followng steps: Step : Reproducton from an exstng populaton of ndvduals a new one s created, accordng to ther evaluated ftness functon. Step : Recombnaton to create a new populaton, crossover and mutaton operators are appled to ndvduals selected randomly from a populaton. Crossover s an operator responsble for the propagaton of the characterstcs of the fttest ndvduals by exchangng genetc materal [8]. Examples of ths operator are: one-pont crossover (X), two-pont crossover (X), three-pont crossover (3X) and unform crossover, amongst others. Mutaton s a reproducton operator that creates a new chromosome by modfyng the value of genes n a copy of a sngle parent s chromosome. Step 3: Replacement the exstng populaton s replaced wth the new one. The process of movng from the current populaton to the next one consttutes a generaton n the evoluton process. Fnally, f some convergence crteron s satsfed, stop; otherwse, go to Step. 7. Implementaton Detals In ths secton the man features of the proposed algorthm, together wth some mplementaton detals, are descrbed. Several algorthms found n the techncal lterature [, 8, 9] have been adapted and mplemented here n order to solve the moorng pattern optmzaton problem. Among them, the one that performed best was the verson of the steady-state GA, found n the work by Wu and Chow []. For ths reason, the algorthm mplemented n ths work s called SSGA (steady-state GA). 7.. Codng Desgn Varable There are many dfferent ways to represent the desgn varables, such as through bnary-dgt strngs, floatng-pont representatons, permutaton lsts, among others []. In ths paper the desgn varables were coded usng a fxed-length bnary-dgt strng representaton, constructed wth the bnary alphabet {0,} and concatenated head-to-tal to form one long strng. Ths concatenated structure represents the chromosome. Thus, every chromosome contans all desgn varables. The length L of the bnary strng s computed as [8]: 3

4 L = n λ (3) where n s the number of desgn varables and λ s the number of bts needed to represent the desgn varables n the search space. For example, consderng that a desgn varable x has the sde constrants 8 < x < 0, then the ampltude of the functon s doman s gven by: I = 0 8 =. Wth accuracy of two decmal places,.e. m =, the nterval I can be dvded nto equal sub-ntervals S n the followng way: m S = I 0 (4) In ths partcular case S = x 0 = 00 ponts; thus the bnary sequence must have at least 8 bts because 7 <00< 8. Consderng a problem wth only two desgn varables (n = ) and accordng to Eq. (3), the strng wll have a total length of L = x 8 = 6 bts. 7.. Decodng To obtan the real values of the desgn varables n the doman regon, each chromosome must be decoded [8]. Consder, for nstance, a chromosome C represented by the followng strng: C = [ b7b6... bbb 0a7a6... aa0 ]. Frst, the decmal value assocated wth the substrng composed by the b elements s obtaned as: m = decmal b = 0 x (5) Fnally, the real value of the desgn varable wthn ts doman s gven by: xmax xmn x = xmn + xdecmal λ (6) where x and x are the lower and upper bounds of the sde constrants, respectvely, and λ s the length of the bnary strng. mn max 7.3. Offset Computng A set of envronmental condtons relevant to the moorng problem conssts of waves, wnds and sea currents, and t may be transformed nto an equvalent external statc force (F ) actng on the floatng unt. When computng the unts dsplacements, moorng lnes are treated as non-lnear sprngs (see Fgure 4) that mpose restorng forces on the unts. Such forces, expressed as a functon of the horzontal dstance between the anchor and the connecton pont on the top of the lne, are obtaned through the restorng curves of each moorng lne, whch are generated usng the catenary equaton. Once the out-of-balance forces are obtaned, a new statc equlbrum poston of the floatng unt s computed by solvng the followng system of non-lnear equatons: K d = F R nt (7) where K s the global stffness matrx; d s the unknown dsplacement vector; F corresponds to the external forces due to each set of envronmental condtons; and R nt s the resultant of the nternal (restorng) forces consderng all moorng lnes. Tenson at the top x Horzontal dstance T T T d d d Restorng Curve Fgure 4. Representaton of moorng lnes by means of non-lnear sprngs. 4

5 The soluton of Eq. (7) leads to a new poston of the floatng unt. Ths computaton s repeated untl the resultant of the obtaned dsplacements s smaller than a gven tolerance (convergence crtera),.e., untl the unt reaches ts fnal statc equlbrum poston Ftness Functon Ftness s a qualty value that s a measure of the reproducng effcency of ndvduals n a populaton accordng to the prncple of the survval of the fttest []. Ths means that a chromosome wth a hgher ftness value wll have greater probabltes of beng selected as a parent n the reproducton process. Therefore, the mnmzaton problem must be transformed nto a maxmzaton problem of a ftness functon, usng the followng expressons: and F φ = (8) φ max φ = (9) avg where s the objectve functon (see Eq.()); and n order to exclude negatve values. avg s the average objectve functon value. The ftness functon s normalzed 7.5. Selecton Chromosomes are selected as parents to produce chldren and ths selecton depends on ther ftness values. In ths paper the rankng selecton technque has been adopted n accordance wth the work by Wu and Chow [], whch outperforms other selecton technques such as the roulette-wheel selecton scheme. Accordng to the authors the advantages of ths selecton technque are that t can prevent the domnaton of extra-ordnary ndvduals and, therefore, prevent a premature convergence of the soluton; and that t can prevent wanderngs among near-equals or even prevent stagnaton n a populaton Crossover Operator The standard crossover operator, whch s also known as smple crossover or one-pont crossover (X) [], has been adopted heren. In ths type of crossover a crossng pont s randomly chosen along the strng, then the two chromosomes selected as parents exchange partal genetc materal above the crossng pont n order to produce new chldren. For example, consderng the strngs and and supposng that the randomly chosen crossng pont s 3, the new chldren wll be 000 and 000, respectvely. 7.7 Mutaton Operator For bnary-dgt strng representaton, ths operator changes the bt (allele) from to 0 or vce versa. The mutaton probablty (Pm) must be carefully prescrbed. Accordng to Wu and Chow [], f t s set wth a low value, the algorthms often get stuck at a local mnmum; but f t s hgh, then the algorthms wll degenerate nto a random search method. It s recommended to adopt a value for the mutaton probablty n the followng range: 0.% < Pm < % [8]. Mutaton can not only prevent premature convergence but also restart an evoluton process that has become stalled Generaton Gap The generaton gap (G) s a parameter that controls the percentage of the populaton that wll be replaced n each generaton [0]. In tradtonal GAs ths parameter s often set as.0, whch means that all ndvduals n the populaton are replaced. Ths strategy may lead to a computatonal problem, due to the large number of objectve-functon evaluatons. In addton, valuable genetc materal can be lost. To avod these problems, a small generaton gap s recommended, accordng to Wu and Chow []. In the present work, the followng value of G has been adopted: G = (0) n where n s the populaton s sze. Accordng to Eq. (0), only two ndvduals are selected to reproduce and the two new offsprng replace the two worst chromosomes of the current populaton. Consequently, the reducton n the number of functon evaluatons s gven by []: (n-)/n. 5

6 8. Computatonal Procedures The man computatonal steps of the proposed SSGA are summarzed n Fgure 5 [].. Start. Seedng 3. Reproducton 4. Updatng Intalze parameters: populaton sze, crossover and mutaton probabltes, among others. Intal populaton s generated randomly (bnary-strng dgt codfcaton) Intal populaton s decoded Eqs.(5) (6) Ftness values of each ndvdual s computed by applyng Eq.(8) Two chromosomes are selected as parents (rankng selecton technque) Applcaton of the crossover operator Applcaton of the mutaton operator The two new offsprng chromosomes substtute the two worst chromosomes of the current populaton 5. Evaluaton The two new chromosomes are decoded Ftness of the two new chromosomes s computed 6. Stoppng crteron satsfed If so, then go to step 7; else, go back to step 3 (Note: the stoppng crteron used n ths work s the maxmum number of teratons) 7. Report 8. End Fgure 5. Computatonal steps of the proposed algorthm. 9. Numercal Example The computatonal procedures, as shown n secton 8, have been mplemented here n a computer program (usng the C language) to solve a moorng pattern optmzaton problem. Crossover and mutaton probabltes were set as.0 and 0.0, respectvely. 9.. Example - Floatng Unt wth 8 moorng lnes A floatng unt anchored by 8 moorng lnes as llustrated n Fgure 6 was consdered; each lne s composed by three dfferent materals whose propertes are shown n Table. The 8 lnes were dvded nto 6 groups wth the same sde constrants (see Table ). Table. Materal propertes. Materal (from unt to anchor) Ф (mm) Length (m) EA(kN) Ww (kn/m) Breakng Load (kn) Chan Polyester Chan Table. Sde constrants of the lnes groups. Group Lnes Lower bounds (degrees) Upper bounds (degrees),, , , ,9, ,,3, ,6,7,

7 North Floatng Unt Y X Z X Fgure 6. Floatng unt wth 8 moorng lnes. The floatng unt was subjected to a set of envronmental condtons that have been combned accordng to a collnear approach,.e., wth currents, wnds and waves actng smultaneously n the same drecton. For ths example, eght combnatons have been consdered. For each combnaton, an external force of 5000 kn was appled to the floatng unt n a partcular drecton, as shown n Table 3. Table 3. External forces actng on the floatng unt. Angle w.r.t. X axs Drecton (degrees) 0 E 45 NE 90 N 35 NW 80 W 5 SW 70 S 35 SE The total number of teratons adopted here was 0000 and the mnmum value of the objectve functon (90 m ) was reached at teraton 39. Table 4 presents the fnal azmuths of each moorng lne, and the fnal moorng pattern s llustrated n Fgures 7 and 8. Table 4. Computed azmuths of the moorng lnes. Lne Azmuth - α (degree) Lne Azmuth - α (degree)

8 Fgure 7. General vew of the moorng system wth 8 lnes. Fgure 8. Fnal moorng pattern. 8

9 0. Conclusons A steady-state genetc algorthm (SSGA) to solve moorng pattern optmzaton problems was presented. The man feature of ths algorthm s the substtuton of only one or two ndvduals per generaton. The rankng selecton technque was adopted. The SSGA mplemented n ths work had very good performance n the soluton of problems wth contnuous varables. A representatve example was provded, whch llustrated the robustness and effectveness of the mplementaton. An extenson of ths work to consder the dynamc analyss of the moorng systems to compute the dsplacements of floatng unts s currently under nvestgaton by the authors.. Acknowledgements The frst author acknowledges the fnancal support provded by CAPES, a Brazlan agency for research and development. Most of the computaton n the present work have been performed n Tecgraf/PUC-Ro (Computer Graphcs Technology Group) usng the Prea3D program []. The authors are especally grateful to the Prea3D development group for provdng the resources necessary for ths research, and CENPES/Petrobras for the fnancal support for the development of Prea3D.. References. S.J. Wu. and P.T. Chow. Steady-State Genetc Algorthms for Dscrete Optmzaton of Trusses. Computers & Structures Vol. 56, No.6, 995, J. Andre, P. Sarry and T. Dognon. An Improvement of the Standard Genetc Algorthm Fghtng Premature Convergence n Contnuous optmzaton. Advanced n Engneerng Software 3, 00, J.H. Holland. Adaptaton n Natural and Artfcal Systems. The unversty of Mchgan Press, Ann Arbor, MI (975). 4. H.J.C. Barbosa. Algortmos Genétcos para Otmzação em Engenhara: Uma Introdução aos Algortmos Genétcos. ª Escola de verão em computação centífca, LNCC, Ro de Janero, RJ, M. Mtchell, L.D. Davs. Handbook of Genetc Algorthms, Artfcal Intellgence 00, 998, N.D. Lagaros, M. Papadrakaks and G. Kokossalaks. Structural Optmzaton usng Evolutonary Algorthms. Insttute of Structural Analyss & Ssmc Research, Natonal Techncal Unversty Athens. Computers and Structures 80, Athens-Greece, 00, I.C. Yeh. Hybrd Genetc Algorthms for Optmzaton of Truss Structures. Computer-Aded Cvl and Infrastructure Engneerng 4, 999, F.P. Samarango. Métodos de Otmzação Randômca: Algortmos Genétcos e Smulated Annealng,. Socedade Braslera de Matemátca Aplcada e Computaconal, Ro de Janero, Brazl. 9. S. Rajeev and C.S. Krshnamoorthy. Dscrete Optmzaton of Structures usng Genetc Algorthms. 0. J.J. Grefenstette. Optmzaton of Control Parameters for Genetc Algorthms. IEEE Trans. Systems, Man, and Cybernetcs SMA- 6 (), 986, -8.. Masett, I. Q.; Rolo, L. F.; Slvera, E. S. S.; Carvalho, M. T. M.; and Menezes, I. F. M., Sstemas Computaconas para análses Estátca e Dnâmca de Lnhas de Ancoragem, Proceedngs of XVIII CILAMCE, , October,