optimal design of submarine pipelines by a genetic ...€¦ · mathematicalproblemsinengineering...

Research ArticleOptimal Design of Submarine Pipelines by a Genetic Algorithmwith Embedded On-Bottom Stability Criteria

Juliana Souza Baioco 12 Mauro Henrique Alves de Lima Jr1

Carl Horst Albrecht 1 Beatriz Souza Leite Pires de Lima 1

Breno Pinheiro Jacob 1 and Djalene Maria Rocha 3

1Laboratory of Computer Methods and Offshore Systems (LAMCSO) PECCOPPEUFRJ Civil Engineering DepartmentPost-Graduate Institute of the Federal University of Rio de Janeiro Avenida Pedro Calmon SN CidadeUniversitaria Ilha do Fundao21941-596 Rio de Janeiro RJ Brazil2Chemical and Petroleum Engineering Department (TEQ) UFF Rua Passo da Patria No 156 Sao Domingos24210-240 Niteroi RJ Brazil3Petroleo Brasileiro SA (Petrobras) ResearchampDevelopment Center (CENPES) AvenidaHoracioMacedo 950 Cidade UniversitariaIlha do Fundao 21941-915 Rio de Janeiro RJ Brazil

Correspondence should be addressed to Breno Pinheiro Jacob brenolamcsocoppeufrjbr

Received 27 July 2017 Accepted 22 January 2018 Published 11 March 2018

Academic Editor Maurizio Brocchini

Copyright copy 2018 Juliana Souza Baioco et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This work describes a computational tool based on an evolutionary algorithm for the synthesis and optimization of submarinepipeline routes considering the incorporation of on-bottom stability criteria (OBS) This comprises a breakthrough in thetraditional pipeline design methodology where the definition of a route and the stability calculations had been performedindependently firstly the route is defined according to geographical-topographical issues (including manualvisual inspectionof seabed bathymetry and obstacles) afterwards stability is verified and mitigating procedures (such as ballast weight) arespecified This might require several design spirals until a final configuration is reached or (most commonly) has led to excessivecosts for the mitigation of instability problems The optimization tool evaluates each candidate route by incorporating as softand hard constraints several criteria usually considered in the manual design (pipeline length bathymetry data obstacles)also with the incorporation of OBS criteria into the objective function stability becomes an integral part of the optimizationprocess simultaneously handling minimization of length and cost of mitigating procedures Case studies representative of actualapplications are presented The results show that OBS criteria significantly influences the best route indicating that the tool canreduce the design time of a pipeline and minimize installationoperational costs

1 Introduction

Submarine pipeline systems have been extensively used inthe oil and gas industry to transport the production betweenoffshore platforms andor to onshore processing facilitiesBeing one of the higher-cost items of the subsea layoutpipeline systems significantly affect the feasibility of anoffshore project thus demanding detailed studies to obtainefficient and low-cost designs comprising an iterative andvery complex process governed by several variables following

design recommendations addressed by codes such as DNV-OS-F101 [1]

In this context perhaps the most crucial step in thedesign of a submarine pipeline is the selection of its routeTraditionally this task has beenmanually performed by expe-rienced engineers by inspecting the seabed bathymetry andavailable information regarding obstacles (including subseaequipment flowlines and other preexistent pipelines) Manyenvironmental commercial regulatory or even geopoliticalissues may determine specific regions that should be avoided

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1781758 21 pageshttpsdoiorg10115520181781758

2 Mathematical Problems in Engineering

for instance corals geotechnical hazards or fields allottedto another oil company There are many other variables thatgovern the selection of a route thus the process has beentreated almost as an ldquoartrdquo being highly dependent on theexpertise of the engineer

Previous works have already recognized [6ndash8] that thetask of selecting a route with good performance and lowcost could be automated by devising its formal descriptionas a synthesis and optimization problem and building acomputational tool based on evolutionary algorithms (EAs)Such algorithms have been successfully used in many com-plex engineering problems and have been shown to beuseful for the optimization of offshore engineering problems[9ndash14]

In [15 16] we have presented results of preliminarystudies related to the development and implementation ofa computational tool for the synthesis and optimization ofsubmarine pipeline routes based on Genetic Algorithms(GAs) [17 18] The modeling of the optimization problem aspresented in [15 16] included only the basic geographical-topographical issues associated with the route geometry andwith the seabed bathymetry and obstacles Those issues wereconsidered for the representation of a candidate route in thecontext of the GA and for its evaluation in terms of criteriaincorporated into the objective function and constraintsThese criteria were defined considering only themain aspectsalready involved in the manual selection of a route includingbasically the total pipeline length interference with obstaclesminimum radius of curvature and declivity In [16] specialfocus was dedicated to the study and assessment of differentconstraint-handling techniques including the 120576-constrainedmethod [19 20] Therefore the developments presented in[15 16] could be seen as leading to a computational tool thatalthough innovative in itself merely automated the processof selecting an optimal route following the traditional designguidelines

Here the focus is on incorporating technicalengineeringcriteria into the route optimization tool related to the struc-tural behavior of the pipe under hydrostatic and environmen-tal loadings specifically the implementation of on-bottomstability (OBS) criteria The stability of a pipe is reflectedby its ability to remain within its outline in the installedposition (taking into account allowable tolerances) underenvironmental conditions of wave and current The DNV-RP-F109 code [21] presents criteria to check if a given pipe isstable or else to defineminimal values of submergedweight toreach stabilityThis idea of incorporatingOBS criteria into theroute optimization tool comprises a breakthrough in the tra-ditional pipeline designmethodology and has originally beenproposed in [22] where a preliminary implementation wassketched with the OBS criteria incorporated as constraintshandled by the classic static penalty technique This waytaking predefined values for the pipe submerged weight theldquooptimalrdquo route would seek areas and trajectories where thestability of the pipe is favored according to the predominantdirection of the environmental loadings

However that approach would possibly lead to longerroutes and require a ldquotrial-and-errorrdquo process where thedesigner should perform successive runs of the optimization

tool increasing the pipe weight in order to obtain smallerroutes Now this work describes an improved approach toincorporate theOBS criteria into the optimization tool wherean additional term is introduced into the objective functionrepresenting the ballast weight as an additional variable to beoptimizedThus the weight required for stability at each pipesegment is directly provided as a result of an optimizationrun

In the optimization procedure candidate routes arerepresented by specific geometric parameterization and areevaluated in terms of several criteria incorporated in theobjective function and in the constraint functions Thesecomprise the core of the route optimization model whichwill be described in Section 2 that focuses on the routeparameterization and encoding in the context of the GASection 3 presents the final form of the objective function anddescribes specifically the incorporation of the OBS criteriaSection 4 presents all remaining criteria that comprise theconstraint functions (including their distinction between softand hard constraints according to the consequences of theirviolation) Case studies are presented in Section 5 to illustratethe use of the optimization tool and assess the influence ofthe stability criteria on the definition of the optimal pipelineroute Lastly final remarks and conclusions are presented inSection 6

2 Modeling and Solving the RouteOptimization Problem

21 Objective Function A general constrained optimizationproblem is formally defined in an 119899-dimensional searchspace 119878 comprised by a vector of design variables x =(1199091 1199092 1199093 119909119899) The following expression mathematicallydefines the problem

minimize 119891 (x)subject to 119892119894 (x) le 0 119894 = 1 119898

ℎ119895 (x) = 0 119895 = 1 119901119897119896 le 119909119896 le 119906119896 119896 = 1 119902

(1)

The goal is tominimize an objective function119891(x) consid-ering inequality and equality constraints (resp 119892119895(x) le 0 andℎ119895(x) = 0) that define the feasible region The components119909119894 may have lower and upper bounds [119897119896 119906119896] In engineeringproblems the constraints are defined in terms of appropriatedesign criteria As will be seen later in the case of the pipelineroute optimization problem these criteria may be expressedas inequality constraints 119892119895(x) only so equality constraintsare not considered in this particular engineering applica-tion

Here it is assumed that the outer diameter and wallthickness of the pipe segments have already been selectedfrom previous design steps Usually the outer diameter isdictated by the amount of oil or gas to be transportedaccording to the yield of the well and the wall thicknessis dictated by strength constraints related to collapse underexternal hydrostatic pressure [23]Thus amongst the relevant

Mathematical Problems in Engineering 3

c

AC

AC

A B

x

y

C1

C2

R1

R2 R2

R1

PC1

PC2

０）1

０）2

PT1

PT2

T

T T

T

Figure 1 Planar representation of a route

A B

Obstacle

R1 1

1

０）1

p1

Figure 2 Primary parameters curvature radius (119877119894) coordinates of PI (120575119894 120572119894)

variables that should be considered in the design of apipeline route the first factor that comes to mind is the totallength which should be minimized to reduce material andinstallation costsTherefore as presented in [16] the objectivefunction 119891 could be defined simply as the ratio between thelengths of a given candidate route (119871119903119900119906119905119890) and of the straightline connecting endpoints 119860 and 119861 (119871119860119861)

119891 (x) = 119871119877119900119906119905119890119871119860119861 (2)

Considering the focus of the present paper which isto incorporate OBS criteria into the optimization proce-dure later in Section 3 we will introduce an additionalterm into this objective function to take into accountthe weight of ballast that would be required to assurethe stability of the pipeline Now before describing theother design criteria that comprise the constraints in theremainder of this section we will describe the geometricrepresentation of each candidate route in the context of theGA

22 Geometric Parameterization of a Route Themodeling ofthe route optimization problem requires geometric parame-terization to allow the representation of each individual can-didate solutionThe full formulation for this parametrizationhas already been detailed in [16] here we will present only abrief overview Figure 1 illustrates the planar representationof a route between endpoints 119860 and 119861 as a horizontal linedefined in the 119909119910-plane and comprised by a sequence ofstraight lines and curves the curved segments are defined ascircular arcs Figure 2 illustrates the three primary parame-ters associated with each curve that as demonstrated in [16]completely describe this representation

(a) the curvature radius (119877119894) of each curve(b) the radial (120575) and angular (120572) polar coordinates of

Points of Intersection (PI119894)mdashat which the prolongation ofthe straight lines (before and after each curve) intersectsThese coordinates are relative to base points (119901119894) uniformlydistributed along the straight line AB as illustrated inFigure 2 showing the position of one PI119894 associated with itscorresponding base point 119901119894


Fitness evaluation

Genetic operators(crossover mutation)

Selection for reproduction(roulette wheel)

Initial population

Updated population

End of evolutionStop criterion

Objective function

Soft criteria cost functions

Hard criteria violation functions

c =

p

sumj=1

rjcj(x)

=m

sumj=1

pjj(x)2

f (x) = kL middotL

LAB

+ kWb middotWb

WbAB

Routeroute

Figure 3 Schematic view of the main algorithm

To obtain the complete three-dimensional representationof the route in terms of the (119909 119910 119911) coordinates of a seriesof nodal points along each candidate route the optimizationtool incorporates facilities to import bathymetric data for agiven subsea region Such data are generated by specializedoceanographic vessels equipped with side-scan sonars andare usually available from design databases maintained by oilcompanies They are provided as contour maps of isobathy-metric curves or isobaths Having imported the isobaths astandard gridding technique is then employed to generate awireframe of quadrangular elements Along the optimizationprocess the planar representation of each candidate routeis divided into 119873 equal-length segments connecting 119873 + 1nodal points The (119909 119910) coordinates are determined usingthe formulation mentioned above and described in detailin [16] From these (119909 119910) coordinates the correspondingvertical 119911-coordinate is then obtained by interpolating on thequadrangular wireframe mesh

23 Encoding the Routes into the GA Thepipeline route opti-mization tool described in this work employs canonical GAimplementation Each individual candidate route is encodedwith real value representation in a chromosome with 119899 sets ofgenes Each set is associated with a curve and comprises fourgenes The last three genes correspond to the primary routeparameters described above radial coordinate 120575119894 angularcoordinate 120572119894 radius 119877119894 the first gene is an ldquoactivation keyrdquo119860 119894 that will be described shortly The full codification of achromosome can then be written as

119909119903 = (11986011205751120572111987711198602120575212057221198772 sdot sdot sdot 119860119899120575119899120572119899119877119899) (3)where 1198601120575112057211198771 are the genes corresponding to the firstcurve and so on

A population is represented by a set of 119873 individualsin general an initial population 1198750 = 11990911 11990912 1199091119873 is

randomly created where 119909119903119894 is the 119894th individual in the119903th generation Following Darwinrsquos evolution theory thefittest individuals have higher probability of surviving andreproducing and their descendants keep the good geneticmaterial in the species thus GAs involve mechanisms ofnatural selection genetic recombination and mutation Theldquofitnessrdquo is provided by the evaluation of each individual viathe objective and constraint functions The individuals areselected for mating and reproduction by selection operatorsthat generally follow probabilistic rules here the fitness-proportional roulette wheelmethod is employed

Mating is performed with crossover combining genesfrom different parents to produce offspring and generate anew population Here we consider single-point crossoverwith the breakpoint on the parentsrsquo chromosomes randomlyset according to a probability value equal to 06 (followingusual guidelines) Thus the offspring inherit features fromeach of the parents and may be submitted to mutationwhich confers innovative characteristics to the individual andprovide a better exploration of the search space Mutationalters each bit randomly with a relatively small probabilityin the present implementation the mutation rate is set to02

The implementation of the GA follows a generationalapproach where the population is updated by replacing allparents by their offspring which are made to compete witheach other Also here we consider one-individual elitismthat is the fittest individual from the previous population isdirectly injected into the new population The process endswhen a predefined stopping criterion is reached and theindividual with the best fitness is then defined as the solutionof the optimization problem

Figure 3 presents a schematic view of the basic steps ofthe GA including themain expressions that will be employed


Figure 4 Procedures to stabilize a pipeline [2ndash4]

to evaluate each individual candidate route that is theobjective cost and violation functions Those expressionswill be defined later in Sections 31 and 42

24 Selective Activation of Curves The number of curvesneeded to adequately represent a route may vary dependingon its length and the complexity of the scenario (in termsof seabed bathymetry and obstacles) Therefore besides thethree parameters for each curve described above (radius 119877119894coordinates of PI 120575119894 120572119894) the encoding of the route into theGA incorporates a fourth parameter the ldquoactivation keyrdquo 119860 119894With this parameter the number of curves actually employedto define each candidate route varies along the evolutionprocessmdashfrom zero corresponding to the trivial straight line119860119861 up to a user-defined maximum value 119899max That is thenumber of curves and their associated PI is also a variable ofthe optimization process

During the generation of the first population (with indi-viduals randomly created) a real value for the activation key119860 119894 corresponding to each curve in a route is randomly gen-erated within the range [0 1] This value indicates whetherthe corresponding curve is active or not associated with agiven user-defined activation threshold 119860 119905 that defines theprobability of activation of the curve as (10 minus119860 119905) times 100Thenthese 119860 119894 values are compared with the activation threshold119860 119905 If 119860 119894 gt 119860 119905 the corresponding curve is generated by theother three parameters its radius and the polar coordinates ofthe PI relative to the closest base pointOtherwise the curve isinactive and the corresponding section of the route is straight

For instance if the user feels that a given scenario isrelatively simple the user may define a higher value for thethreshold parameter for instance 119860 119905 = 06 this meansthat the probability of a given curve to be generated is only40 and therefore the generated candidate routes will tendto have fewer curves On the other hand the user may forceall 119899max curves to be permanently active for all routes simplyby defining 119860 119905 = 00 As the evolution of the GA proceedsthe selectionreproduction operators of the GA propagatethe values of the activation keys 119860 119894 favoring routes witheither fewer or more curves depending on the complexity ofthe scenario also the mutation operator may introduce newrandom values for these genes

3 On-Bottom Stability Criteria

31 Objective Function considering OBS Now to incorporatethe OBS criteria into the optimization procedure an addi-tional term is introduced into the objective function of (2)The goal is to guide the optimization process towards an opti-mal solution that complies with the on-bottom stability withlower intervention costs This term is defined as the weightof ballast required to stabilize a candidate route 119882119887119877119900119906119905119890normalized by the ballast 119882119887119860119861 required to stabilize thestraight route 119871119860119861 With this additional term the objectivefunction now reads as follows

119891 (x) = 119896119871 sdot 119871119877119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

(4)

Usually those two objectives (minimizing length andballast weight)may be conflicting routes with shorter lengthsmay require more ballast weight and vice versa Thusweighting factors 119896119871 and 119896119882119887 are inserted so that the user canspecify a relative measure of the importance of minimizingrespectively length and ballast As will be seen later in theresults of the case studies this feature presents a crucial rolein the definition of an optimal routemdashrecalling that thereare different alternative procedures to provide ballast weight(including concrete mattress trenching burying or rockdumping as illustrated in Figure 4) in a preliminary designstage (which is the focus of this optimization tool) actual dataregarding the costs of those different procedures to provideballast may not be easy to obtain This way the weightingfactors allow the user to adjust the relative importance of eachterm of the objective function (length and ballast) favoringthe minimization of either the pipeline length or the ballastweight or even obtaining an intermediate solution Thisallows the decision-maker to obtain different optimal routesthat satisfy the conflicting objectives of the optimizationThisfeature will be illustrated later in the case studies

The remainder of this section will begin by brieflysummarizing in Section 32 the main concepts behind theOBS criteria as defined in [21] focusing on the two designmethods that will be incorporated into the optimizationtool It will be seen that such methods depend on severalparameters related to not only the pipe segments themselvesbut also the soil properties and environmental loads All


those parameters may vary along each candidate route andalong the spatial domain it occupies on the seabed thusto consider these variations each route is discretized into agiven number of nodes (119899Nodes) and segments (nSegm) InSection 33 the OBS expressions of [21] will be rearrangedto provide for each segment the ballast weight (per unitlength)119908119887119901119894119901119890 required to stabilize the pipe (either directly orusing iterative procedures to comply with the selected safetyfactors) The expressions will also be extended to considerslopes in irregular seabed Then the total ballast weight forthe entire route119882119887119877119900119906119905119890 (to be incorporated into (4)) may beobtained by a summation for all pipe segments

119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)

32 Approaches for the Analysis and Verification of OBSThe Recommended Practice DNV-RP-F109 [21] describesthree different methodologies for analysis and verificationof the lateral stability of a given pipeline configuration (1)ldquoabsolute lateral static stabilityrdquo with zero displacement thatis ensuring that the hydrodynamic loads acting on the pipeare less than the soil resistance and that the vertical lift loadis lower than the submerged weight (2) ldquogeneralized lateralstability methodrdquo ensuring ldquono break-outrdquo for a ldquovirtuallystablerdquo pipe allowing small displacements (less than aboutone-half diameter) taking advantage of the passive resistanceof the soil and ensuring that the pipe does not move outof its cavity with maximum displacements independent oftime and (3) ldquodynamic lateral stability analysisrdquo allowingldquoaccumulated displacementsrdquo with the pipe able to breakout of (and return to) its cavity and the soil resistance isdependent on the time-history of pipe displacements

These approaches may be associated with analyticalexpressions precalibrated curves or dynamic FE analysesIn the case of the absolute lateral static stability methodstatic analytical expressions provide safety factors associatedwith the ratio between hydrodynamic loads and horizontalsoil resistance in the case of the generalized lateral sta-bility method precalibrated curves provide the minimumrequired weight for a given maximum allowable displace-ment

Finally the more complex dynamic lateral stability anal-ysis method is based on the generation of numerical modelsand the execution of dynamic analyses under environmentalloadings of current and irregular wave for each specificpipe configuration taking into account the appropriate soilresistance forces Differently from the previous methods itdoes not directly provide specific design values (in terms ofsafety factors or required weight) rather it provides resultsregarding the behavior of the pipeline in terms of motionsand stresses These results should then be compared with thelimit values of the respective design criteria in order to obtainthe corresponding safety factors

In this work we will study the implementation of the firsttwo methodologies (absolute static stability and generalizedstability) into the route optimization toolThe third approachis more adequate for more advanced stages of the design

of the pipeline its implementation would not be feasible inthe context of the optimization tool since it would requirethe dynamic analysis of each candidate route by a completemodel in a finite element simulation program demandingexcessively high CPU costs

321 Absolute Lateral Static Stability According to [21]ldquoabsolute lateral static stabilityrdquo is assured when both ofthe following conditions are met respectively for the lateraland vertical directions considering a given safety factor120574SC

120574SC sdot 119865lowast119884 + 120583 sdot 119865lowast119885120583 sdot 119908119904 + 119865119877 le 10

120574SC sdot 119865lowast119885

119908119904 le 10(6)

The first condition corresponds to the static lateral equi-librium where the ratio between the horizontal componentsof the hydrodynamic loads and the soil resistance is checkedagainst a safety factor 120574SC Hydrodynamic loads are incor-porated in the horizontal force 119865lowast119884 as well as the verticallift force 119865lowast119885 all calculated by the Morison formula takingthe current velocity at the level of the pipe and the watervelocity obtained by a linear wave theory considering a singleregular wave component In [21] specific considerations arepresented for the use of the Morison formula in terms ofcurrent velocity reduction factors to take into account the soilrugosity 1199110 and load reduction factors to take into account thepipe-soil interaction (in terms of the soil permeability pipepenetration and trenching) The soil resistance is calculatedin two parts one proportional to the normal force acting onthe soil being determined by a friction coefficient 120583 affectingboth the lift force 119865lowast119885 and the pipe submerged weight 119908119904 andanother that corresponds to the passive resistance 119865119877 due toan initial penetration of the pipe

The second condition which corresponds to the staticvertical equilibrium simply checks the ratio between the liftforce 119865lowast119885 and the pipe weight119908119904 against the same safety factor120574SC

322 Generalized Lateral Stability The generalized lateralstability method described in Section 35 of [21] is a relativelymore complex and less conservativemethodDifferently fromthe absolute lateral static stability method it now allows somepipe displacement under the action of a design spectrumof oscillatory wave-induced velocities 119880119904 at the pipelinelevel

This method depends on the level of allowable displace-ment specified as a value that does not result in excessivepipe deformations or stresses Two levels of allowable dis-placements are suggested (a) up to one-half pipe diameter(corresponding to the ldquovirtually stable piperdquo) and (b) up toten diameters According to the desired level the methodprovides values for the weight required for global stability ofthe pipeline in terms of a significant weight parameter L thatis 119871 stable and 11987110 respectively Intermediate displacementcriteria can be established by defining the required weight 119871119884


t

FlowastY

wsFR

V(z)

FlowastZ

(a) Up slope velocity

t

FlowastY

ws

FR

V(z)FlowastZ

(b) Down slope velocity

Figure 5 Hydrodynamic Forces considering slopes (adapted from [5])

for an allowable displacement 119884 according to the followingformula

log (119871119884) = log (119871 stable) + log (119871 stable11987110)log (05 (001 sdot 120591))

sdot log( 11988405) (7)

where 120591 is the number of oscillations in the design bottomvelocity = 119879119879119906 (with 119879 being the wave period and 119879119906 thespectrally derived mean zero up-crossing period)

Values for the parameters119871 stable11987110 are obtained throughempirical expressions and design curves calibrated from adatabase of results for a large number of dynamic analysesA detailed description can be found in [21] in summarythey depend on a set of dimensionless parameters includingparameters related to the type of soil (clay or sand) and tothe wave and current loadings (spectrally derived values ofthe oscillatory fluid flow due to waves and steady currentvelocity) To check if a pipe is stable the required value forthe significant weight parameter (119871 stable 11987110 or 119871119884) givenby the design curves of [21] may be compared with the 119871parameter calculated for the pipe by the following expression[21]

119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)

where 120588119908 is the mass density of water 119863 is the external pipediameter and 119880119904 is the spectrally derived oscillatory velocity(significant amplitude) for design spectrum perpendicular tothe pipeline If the value calculated by expression (8) is higherthan the required value (119871 stable 11987110 or 119871119884) then the pipe isstable

33 Calculating the Ballast Terms of the Objective FunctionThe expressions presented in Sections 321 and 322 allowthe verification of the OBS for a given pipeline configurationin the context of a conventional design procedure theengineer should then select a set of values for the pipe designparameters in order to obtain a minimum submerged weightthat complies with those expressions On the other hand inthe context of the optimization procedure these expressionsmay be rearranged to provide safety factor values for each

pipe segment into which a candidate pipeline route has beendiscretized in some cases they may provide directly theballast weight required to stabilize the pipe

The ldquoabsolute lateral static stabilityrdquo expressions (6) maybe rearranged to provide values for the lateral and verticalsafety factors 120574SC119910 and 120574SC119911 respectively as follows

120574SC119910 =120583 sdot 119908119904 + 1198651198771003816100381610038161003816119865lowast119884 + 120583 sdot 119865lowast1198851003816100381610038161003816

(9)

120574SC119911 =119908119904119865lowast119885 (10)

Similarly the ldquogeneralized lateral stabilityrdquo expressionsthat define the different ldquosignificant weight parametersrdquo119871 stable 11987110 or 119871119884 may be combined to provide a safety factor120574SC119910 as the ratio between the significant weight parameter 119871 ofthe pipeline and the significant weight parameter 119871 stable 11987110or 119871119884 required to obtain the stability in the case of 11987110 forinstance we have

120574SC119910 =11987111987110 (11)

The OBS expressions described above were originallyderived in DNV-RP-F109 [21] considering only horizontalseabed However since the optimization tool considers irreg-ular seabed defined by actual bathymetric data it is desirableto employ expressions that take into account the declivity ofthe bathymetric floor transversal to the pipe axis (Figure 5)This allows the optimization process to favor relatively leveledtrajectories (perpendicular to the pipe) to avoid the sliding ofthe pipe on regions with high transversal slopes

To reach this goal (9) that provides the lateral safetyfactor 120574SC119910 is altered to take into account the transversaldeclivity angle 120579119905 as follows

120574SC119910 =120583 sdot 119908119904 sdot cos (120579119905) + 1198651198771003816100381610038161003816119865lowast119884 + 119908119904 sdot sin (120579119905) + 120583 sdot 119865lowast1198851003816100381610038161003816

(12)

A similar expression for the vertical safety factor 120574SC119911 maybe derived from (10)

120574SC119911 =119908119904 sdot cos (120579119905)119865lowast119885

(13)


Regarding the generalized stability method the followingexpression can be defined to provide a safety factor consider-ing slopes

120574SC119910 =119871 sdot cos (120579119905)11987110 (14)

Taking these expressions for the safety factors a simpleiterative procedure has been devised to provide the ballastweight per unit length 119908119887119901119894119901119890 required to stabilize each pipesegment Basically this procedure gradually increases aninitial estimate for 119908119887119901119894119901119890 until it provides a safety factorthat meets a given minimum safety factor 120574SC such asthose listed in [21] Then the corresponding values for allnSegm pipe segments are summed to obtain the total ballastweight for the entire route 119882119887119877119900119906119905119890 as indicated in (5) Theballast119882119887119860119861 required to stabilize the straight route 119871119860119861 (thatnondimensionalizes the OBS term of (4)) is also calculatedfollowing a similar procedure

Considering particularly the ldquogeneralized lateral stabilityrdquomethod the expressions that define the different ldquosignificantweight parametersrdquo (119871 stable11987110 or119871119884)might provide directlythe submerged weight of the pipe (per unit length) requiredfor stabilitymdashsimply by equating (for instance) 11987110 = 119871 andthen rearranging (8) to obtain the total required submergedweight from which the desired ballast weight per unitlength may be obtained by subtracting the original pipeweight However strictly speaking this would apply onlyfor procedures that are not influenced by the calculation ofhydrodynamic loads which depend on the outer diameter119863of the pipe that is concrete mattress trenching burying orrock dumping as illustrated in Figure 4 For the most usualprocedures to apply ballast weight to a pipe which are basedon installing an additional layer of concrete around the pipeitself two terms of (8) would be simultaneously affected (119863and 119908119904) to solve this issue and provide the desired value for119908119887119901119894119901119890 an iterative procedure similar to that described aboveis also applied

4 Other Design Criteria Constraints

41 Design Criteria Besides the total length and the OBScriteria that have been incorporated into the objectivefunction as described above the route optimization toolincorporates several other design criteria related to geo-graphicaltopographical aspects associated with the seabedbathymetry (such as slopes and interference with obstacles)These criteria and their respective violation functions hadalready been described in detail in [16] in the remain-der of this section we will present only a brief sum-mary

411 Interference with Obstacles Besides automaticallyimporting isobathymetric curves from design databases togenerate the three-dimensional representation of the route(as described in Section 22) the tool also incorporatesfacilities to selectively or fully import data describing thepositioning and specification of subsea obstacles that shouldbe avoided Such data is gathered by specialized vessels

equipped with a ROV (remotely operated vehicle) includingalso geophysical and geotechnical data obtained from thebathymetry and sonography with information about faciesand geotechnical hazards that identify critical areas such asgeological faults rocky outcrops natural reefs

Thus different types of obstacles may be consideredsubsea equipment wellheads flowlines other preexistentpipelines regions with corals or geohazards that shouldbe avoided To each one the user may associate a levelof severity according to the consequences of the possibleinterference Level 0 tolerated with low severity and easilymitigated Level 1 conditionally allowed with moderateseverity introducing relatively higher mitigation costs Level2 not allowed with severity being high Level 3 not allowedwith severity being critical During the optimization processthe number intersections for each severity level is countedand stored in variables nInterS and nInterH (according tothe classification of the interference as soft and hard aswill be seen later) interference is identified by verifying theintersections between the segments or volumes that definethe obstacles against nodal points and segments along eachcandidate route

412 Self-Crossing The geometric representation describedin Section 22 might eventually lead the optimization processto generate routes in which the pipeline passes over itselfTo identify such configurations with self-crossings or loopsthe optimization tool incorporates an algorithm that spansall nodes and segments of the route and counts the numberof self-crossings (nSelfCross)

413 Minimum Length of Straight Sections Pipeline launch-ing operations require straight sections between two consec-utive curves to allow proper space for the maneuvering ofthe launching ship Straight sections are already incorporatedin the geometric parameterization described before but if agiven minimum length 119871min is not respected the operationmay be infeasible To identify such situations the length 119871 119894 ofall straight sections (nStraight including the first and last atthe beginningend of the route) is checked against this limit119871min

414 Minimum Radius of Curvature During the pipelinelaunching operation the top connection device at the launch-ing ship applies a given tension to the pipe After thecompletion of the operation and the accommodation of thepipeline on the seabed the pipe segments still maintain acertain level of residual tension It is the balance between thisresidual tension and the lateral friction forces from the soilthat maintains the curved sections of the pipeline over thepredefined route On curves with smaller radius the frictionforces may not be sufficient and the pipe can slide sidewaysTherefore to assure the feasibility of a given route the curvedsections should present radius of curvature larger than agiven value 119877min function of the pipe-soil friction coefficient120583 pipe weight 119908119904 and residual pipe tension 119879residual at itsequilibrium configuration after installation


To identify such situations the optimization tool spanseach node of the route along the curved sections calculatesthe required limit 119877min and checks this limit against theradius of the curve 119877119894 Being a function of the soil and pipeparameters that are also variable 119877min may vary from curveto curve or even along each curve

415 Declivity The slope of the regions where the pipeline isset also comprises a design criterion for the definition of theroute Values for the longitudinal declivity 120579119894119871 along the nodesof a candidate route (nNodes) can be calculated taking the 119911-coordinate of two consecutive nodes which in turn are deter-mined from the grid of quadrangular elements interpolatedfrom the isobathymetric curves To implement this criterionthe optimization tool incorporates an algorithm that checkthose declivities 120579119894119871 against a user-defined limit 120579119871 119871119894119898

416 Attractors During the design of a pipeline route itmay be desirable to keep the route near some regions ofinterest referred here as ldquoattractorsrdquo for instance a manifoldconnecting other pipelines or flowlines or even productionplatforms allowing the pipeline to collect the production ofadjacent oil fields To identify such configurations the toolincorporates an algorithm that for a total number of nAttrattractors calculates the distance between the route segmentsclosest to a given attractor (119898119894119899119863119894119904119905119894) and checks if this valueis smaller than a user-defined limit expressed as the radius ofthe ldquoattractor circlerdquo 119877119860119905119905119903119894

42 Soft and Hard Criteria Handling of Constraints All theaforementioned design criteria should be incorporated asconstraints into the optimization process Since GAs andother evolutionary algorithms were originally designed todeal with unconstrained search spaces specific constraint-handling techniques are required to guide the evolutionarysearch process to feasible regions

Following the approach presented in [16] according tothe consequences of their violation the criteria describedabove are classified as soft and hard which will be handledseparately using different techniques The violation of the so-called soft criteria would not indicate that the correspondingsolution is infeasible since mitigation procedures could beadopted This is the case for instance of interference withLevel 0 and Level 1 obstacles such as preexistent pipelines inthis case there are specific installation procedures that allowthe new pipe to pass safely over the existing one Solutionsthat violate these soft criteria would still be feasible althoughrequiring additional costs therefore they do not formallycharacterize constraints in the search space

On the other hand violation of hard criteria cannot bemitigated and should be avoided Their violation will markthe solution as infeasible and therefore they should be for-mally considered as constraints to be treated by a particularconstraint-handling technique Amongst the criteria listed inSection 41 the ones that are classified as hard are interferencewith Level 2 and Level 3 obstacles (such as subsea equipmentor mooring lines) self-crossing minimum length of straightsections minimum radius of curvature

421 Soft Criteria Besides interference with Level 0 andLevel 1 obstacles the longitudinal declivity and attractorsmay be considered as soft criteria in the former casemitigation procedures can be adopted for route segmentswith declivity exceeding the specified limit Attractors canalso be considered a soft criterion if one assumes that routespassing far from such regions would not be infeasible butwould require higher costs

The soft criteria are handled by adding to the objectivefunction of (4) dimensionless cost functions 119888119895(x) associatedwith the violation of the 119895th soft criterion weighted bypositive constant values 119903119895 that provide a relative measureof the importance of the criterion This is equivalent to theclassic static penalty technique and leads to the followingexpanded expression for the objective function 119891

119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)

This objective function corresponds to a minimizationproblem where the goal is to reduce the value of 119871119903119900119906119905119890 theweight of ballast119882119887119877119900119906119905119890 and the cost terms related to the softcriteria For the ideal route one would trivially have 119871119903119900119906119905119890 =119871119860119861119882119887119877119900119906119905119890 = 0 and 119891 = 1 Solutions that violate one of thesoft criteria are not considered as infeasible adding the costterms only represents a handicap to favor the other possiblesolutions that do not violate them Mathematical expressionsfor the cost functions 119888119895(x) associated with those soft criteriawill be provided shortly in Section 423

422 Hard Criteria The 120576-Constrained Method The hardcriteria that formally characterize constraints are handledby the 120576-constrained method [19 20] In this technique theindividuals are ranked based on a lexicographic order ofsets of values (119891 120601) assigned to each individual where 120601is a function that indicates the level of constraint violationand V119895(x) are the violation functions associated with the119895th hard criterion weighted by positive constant values 119901119895that provide a relative measure of the importance of eachcriterion

120601 =119898

sum119895=1

119901119895V119895 (x)2

V119895 (x) gt 0 if constraint is violated V119895 (x) = 0 otherwise(16)

Considering two individuals 1 and 2 their ranking isperformed as follows

(1198911 1206011)lt120576 (1198912 1206012)

1198911 lt 1198912 if 1206011 1206012 le 1205761198911 lt 1198912 if 1206011 = 12060121206011 lt 1206012 otherwise

(17)


Table 1 Soft criteria and cost functions

Soft criteria Cost functionInterference with Level 0 and Level 1 obstaclesSeverity weight factor assigned by the specialist to each level 119888119900119887119904119905 =

119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)

Longitudinal declivityActivated for the nodes of the route with declivity 120579119894119871greater than the limit 120579119871 119871119894119898

119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)

AttractorEncourage routes near regions of interest activated whenever route does not passinto the ldquoattractor circlerdquo119908119894 weight factor indicating relative importance of each attractor

119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)

Table 2 Hard criteria and violation functions

Hard criteria Violation functionInterference with Level 2 and Level 3 obstaclesSeverity weight factor assigned to each level V119900119887119904119905 =

119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)

Self-Crossing V119878119890119897119891119862119903119900119904119904 = 119899119878119890119897119891119862119903119900119904119904 (23)Minimum length of straight sectionsActivated whenever one or more straight sections have length 119871 119894 lower than 119871min

V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)

Minimum radius of curvatureActivated whenever a curve has radius lower than 119877min

V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)

119877min = 119879residual120583 lowast 119908119904 (26)

In this expression 120576 is a specified value that representstolerance related to the sum of constraint violation Thefirst option means that if both solutions in the pairwisecomparison are feasible or slightly infeasible as determinedby the 120576 value they are compared using their objectivefunction values119891The second and third options refer to infea-sible solutions the second option means that two infeasiblesolutions presenting the same sum of constraint violation 120601will also be compared using their objective function valuesotherwise in the third option they will be compared basedon their sum of constraint violation

The objective function 119891 is given by 119891(x) of (15) thatincorporates the cost functions for the soft criteria Alsoassuming that the feasibility of a solution is more importantthan the minimization of its objective function we take120576 = 0 in this case the lexicographical ordering considersthat the minimization of the sum of constraint violation120601 precedes the minimization of the objective function 119891Feasible solutions are ranked based on their unconstrainedobjective function values119891 infeasible solutions are comparedbased on their sum of constraint violation 120601 If one of thesolutions is infeasible (120601 gt 0) and the other is feasible (120601 = 0)the latter will be selected

423 Summary of the Soft and Hard Criteria Tables 1 and 2summarize respectively the soft and hard criteria along withtheir associated cost and violation functions (119888119895(x) V119895(x))Although the expressions for these functions may seem sim-ple their evaluation is not trivial since it involves calculations

performed for possibly hundreds or thousands of pointsalong each candidate route using the three-dimensionalrepresentation that combines the planar formulation withthe vertical 119911-coordinates determined by interpolating theseafloor bathymetric data as described in Section 22 Laterin the presentation of the case studies the values assigned foreach parameter of these functions will be shown

5 Case Studies

51 Description of the Scenarios To assess the influenceof the stability criteria on the definition of the optimalpipeline route two different scenarios are considered Thefirst illustrated in Figure 6 corresponds to a relatively sim-pler subsea layout where a pipeline is designed to connecttwo neighboring platforms while the second (Figure 7) isassociated with a complex layout including more obstaclesand also attractorsThe straight line119860119861 connecting the pointsof the route to be optimized is depicted in pink Levels 0 1 2and 3 obstacles are indicated respectively in green yellowred and black

For scenario 1 (Figure 6) only Level 2 obstacles aredefined themooring lines of the floating platforms indicatedin red For scenario 2 (Figure 7) the mooring lines aretreated as Level 4 obstacles indicated in black Levels 0and 1 obstacles are also included (preexistent pipelines andflowlines) and another Level 4 obstacle is added the blackpolygon representing an area allotted to another oil company


S

EW

N

Platformmooring lines


0 2 4(km)

Figure 6 Scenario 1 simple layout

Field from another oil company

Flexible pipesMooring linesSteel pipesAttractors

0 7 14

S

EW

N

(km)

Figure 7 Scenario 2 complex layout with obstacles and attractors

The ldquoattractorrdquo regions close to other platforms are indicatedas purple circles

Table 3 presents the main geometric parameters definingthe endpoints of the routes for these scenarios while Table 4describes the properties of the pipeline for each scenarioTable 5 presents the values for the soil parameters requiredto evaluate the on-bottom stability criteria and also theminimum radius of curvature (see (26)) One can observethat scenario 2 comprises two different soil types defined

according to the water depth The pipeline properties alsovary with the water depth

52 Objective Function and Constraints Table 6 summarizesthemain parameters of the optimization algorithm employedfor each scenario described above those that define the searchspace of the problem (the maximum number of curves 119899maxand the threshold of the activating factor 119860 119905 as described inSection 24) along with the number of individuals in each


Table 3 Main geometric parameters

Scenario 119871119860119861(km)

Depth of endpoint 119861(m)


1 12359 14 5292 49393 96 1782

Table 4 Pipeline properties

Parameter Scenario 1 Scenario 2

Outside diameter (OD) 32385mm1234 in 457mm18 in

Wall thickness 119905119904 1905mm34 in

Depth (m) Thickness

0ndash300 175mm23 in300ndash800 191mm34 in800ndash1190 222mm78 in1190ndash1535 254mm1 in

1535ndash1800 286mm118 inSteel specific weight 120588119904 77000Nm3

Corrosion coating

Thickness Spec weight (Nm3)

Thickness (119905119901) 76mm3 in 015mm0006 in 141264020mm0008 in 8829

Spec weight (120588119901) 8826Nm3 365mm17 in 8829

48mm189 in 8829

Table 5 Soil properties

Parameter Scenario 1 Scenario 20ndash300m 300ndash1800m

Type Sand Sand ClayPipe-soil friction coefficient (120583) 07 06 02Dry unit soil weight (120574119904) - - 18000Nm3

Submerged unit soil weight (1205741015840119904 ) 13500Nm3 13500Nm3 -Undrained clay shear strength (119904119906) - - 10000Nm2

Table 6 Parameters of the optimization algorithm

119873119901119900119901 MaxGen 119899max 119860 119905

Scenario 1 60 200 4 015Scenario 2 100 300 6 015

population 119873119901119900119901 and the maximum number of generationsMaxGen comprising the termination criterion

To provide a relative measure of the importance ofminimizing respectively total length and ballast weightdifferent sets of optimization runs are performed varyingthe weighting factors 119896119871 and 119896119882119887 employed in the objectivefunction ((4) and (15)) Table 7 presents three sets of valuesthe first and third favors respectively the minimization ofthe pipeline length (possibly requiring more ballast weight)

and vice versa and the second corresponds to an intermediatevalue for the relative pipeline length weighting factor (119896119871)

The general form of the objective function is representedby (15) recalling that this expression incorporates the costfunctions 119888119895(x) for the soft criteria whose violation canbe mitigated The cost functions themselves were alreadypresented in Table 1 now Table 8 presents the cost factors119903119895 and the other user-assigned values for the parametersassociated with these soft criteria that will be employed in the


Table 7 Weighting factors associated with the objective function

Weighting factors Eq (15) Scenario1 2

Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001

Favoring ballast 119896119871 1 1119896119882119887 001 001

Table 8 Parameters of the cost functions associated with the soft criteria

Criterion Cost factors 119903119895 (15) Parameter ValuesScenario 1 Scenario 2 Scenario 1 Scenario 2

Interference with obstacles 1 1Severity values (19)

Level 0 (green) 02Level 1 (yellow) 05

Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘

Attractor -- 1 Radii of the three attractor groups (119877119860119905119905119903119894) (21) -- 2000m3000m4000m

Table 9 Parameters of the violation functions associated with hard criteriaconstraints

Criterion Violation factors 119901119895 (16) Parameter ValuesScenario 1 Scenario 2 Scenario 1 Scenario 2


Level 2 (red) 1Level 3 (black) 100

Self-crossing 1 10 -- -- --Min length of straight sections 1 1 119871min (24) 100m 500m

Min radius of curvature 1 1

119877min (26)Friction coefficient (120583) According to Table 5Residual tension

(119879residual) 500 kN

Pipe weight 119908119904 Calculated from pipeline properties (Table 4)

case studies Those values were tuned through preliminarystudies considering the characteristics of the respective sce-nario and the previous experience in the design of submarinepipeline routes

The objective function of (15) does not incorporate thehard criteria that formally characterize design constraints asshown in Section 422 these criteria are evaluated separatelyby the 120576-constrained method being incorporated into 120601(function of the sumof violations V119895(x) (16)) Table 9 presentsthe user-assigned values for the parameters associated withthese hard criteriadesign constraints also tuned throughpreliminary studies considering the previous experience ofthe designer

It is interesting to recall that the optimization tool mustperform checks for the whole series of nodes and segmentsinto which each candidate route is discretized not only theOBS criteria (as mentioned in Section 31) but also all othercriteria described in Section 4 whose parameters vary alongthe spatial domain depending on the water depth seabed

bathymetry and obstacles and the pipe and soil propertiesas seen in Tables 4 and 5

53 Additional Parameters for the OBS Criteria Besides thepipe and soil properties evaluation of the OBS criteria alsorequires the definition of a set of environmental loadingcases According to [21] two extreme loading combinationsof wave and current should be considered (1) 100-year returnwave combined with 10-year current and (2) 10-year wavewith 10-year current Wave and current are taken as alignedalong eight directions (N NE E SE S SW W and NW)Table 10 presents the JONSWAP spectral wave parameters(significant height 119867119904 peak period 119879119901) Table 11 presentsthe extreme near-bottom current velocities V119888 taken frommetocean data for the two scenarios Figure 8 presents polargraphs depicting the main current and wave parametersindicating that the predominant environmental loadings areacting along the NE-SW direction


Table 10 Wave parameters

Direction 10-year 100-year119867119904 (m) 119879119875 (s) 119867119904 (m) 119879119875 (s)

N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851

Table 11 Near bottom current velocities (ms)

DirectionAll scenarios Scenario 2Current 1 Current 2 Current 3 Current 4

10-yr 100-yr 10-yr 100-yr 10-yr 100-yr 10-yr 100-yrN 054 065 145 192 057 071 036 045NE 075 096 113 143 052 063 044 056E 062 074 151 206 052 065 039 05SE 065 077 16 203 05 064 029 04S 096 118 105 12 057 074 031 041SW 109 141 076 085 048 062 034 045W 063 075 092 108 038 047 056 077NW 055 061 111 137 041 05 042 054

0

05

15

N

NE

E

SE

S

SW

W

NW

Current velocity (ms) Hs (m)

0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years

Figure 8 Environmental loadings


Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)

Figure 9 Position of the current measurements

As seen in Table 11 for the more complex scenario 2 themetocean data indicate that the current values vary alongthe spatial domain according to four points of measurementdepicted in Figure 9 Thus the optimization tool must evalu-ate the current loading along each nodesegment of a routeby taking the values corresponding to themeasurement pointthat is nearest to the node

The near-bottom current velocities perpendicular to thepipeline are calculated for each node of each candidate routeby applying a factor to take into account the directionalityand also the effect of the bottom boundary layer For thispurpose the following expression [21] is considered wherethe directionality is taken into account by the angle betweencurrent direction and axis of the pipe segment 120579119888 and thebottom boundary layer is a function of the elevation abovesea bed 119911 bottom roughness parameter 1199110 (whose valuesare defined in [21]) and reference measurement height overseabed 119911119903

119881 (119911) = 119881 (119911119903) sdot ln (119911 + 1199110) minus ln (1199110)ln (119911119903 + 1199110) minus ln (1199110) sin (120579119888) (18)

This expression is employed for all directions of currentwith their respective values of velocity and relative angle 120579119888Taking all resulting perpendicular velocities the respectiveballast weights per unit length for the nodesegment arecalculated following the procedures described in Section 33that also takes into account the corresponding values oftransversal declivity the required ballast weight is then takenas the higher value amongst the obtained results (eventuallysmaller velocity values associated with higher declivitiesmight require more ballast)

The near-bottom wave-induced water velocities perpen-dicular to the pipeline must also be evaluated for each node

since they depend on the water depth Moreover recallingthat wave loadings are relevant only for pipelines in shallowerwaters the optimization tool automatically disregards waveeffects for pipe segments resting in water depths larger than auser-defined limit (for instance 100m) Thus as can be seenin Table 3 wave loadings are relevant only for the routes ofscenario 1 where a significant part of the pipeline rests onshallow waters

Moreover according to [21] the ldquogeneralized stabilityrdquocriterion is more adequate for shallower waters so a depthlimit may also be taken to allow the selection of this criterionThus while both the ldquoabsoluterdquo and the ldquogeneralizedrdquo criteriaare employed and compared for the shallower segments ofscenario 1 (the latter considering the allowable displacementas 10 times the pipe diameter) only the absolute stabilitycriterion is activated for the deeper segments of scenario 1 andfor the routes of scenario 2

54 Results The results for the case studies are comparedin terms of figures that show the geometry of the optimalroute indicating the ballast weight required for stability by acolor gradation scheme (where green yellow and red indicatesuccessively higher weight values) and in tables that presentthe corresponding values for the main parameters of theobjective function for example length 119871119877119900119906119905119890 total ballastweight 119882119887119877119900119906119905119890 and ballast weight per unit length 119908119887119901119894119901119890according to (4) and (5)

541 Scenario 1 For this simpler scenario two sets of opti-mization runs were performed employing respectively theldquofavoring lengthrdquo and ldquofavoring ballastrdquo weighting factors119896119871 and 119896119882119887 of Table 7 Each set consists of two runs the


Table 12 Scenario 1 parameters of the optimal routes

Absolute stability Absolute + generalizedFavoring length Favoring ballast Favoring length Favoring ballast

Route length (m) 12400 15060 12403 14792Total ballast weight (kN) 5208 3042 4812 2647Ballast weight per unit length (Nm)

Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW

Current velocity (ms)

Favoring ballast absolute

Favoring length

1 1

A

B

Favoring ballast Abs + generalized

0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4

500ndash4500 Nm100ndash500 Nm0ndash100 Nm

Hs (m)

(km)

Figure 10 Scenario 1 comparison of optimal routes

first using only the ldquoabsolute stabilityrdquo OBS criterion andthe second set considering both the absolute and generalizedcriteria automatically selected according to the water depth

Figure 10 shows the resulting optimal routes whileTable 12 compares the routes in terms of themain parametersof the objective function that is route length and ballastweight As expected the optimization runs with the ldquofavoringlengthrdquo weighting factors 119896119871 and 119896119882119887 of Table 7 generatedshorter optimal routes closer to the straight line betweenendpoints 119860119861 the OBS criteria did not affect their geome-tries so they are practically coincident and cannot be dis-tinguished in Figure 10 However they required considerablymore ballast weight mostly for the pipe segments closer toendpoint 119861 (located in shallower waters) as can be seen

from the lengthier segments depicted in red and yellow inFigure 10

On the other hand the routes generated with the ldquofavor-ing ballastrdquo weighting factors have lengths that are notdramatically greater (20 approximately) while requiringalmost half of the total ballast compared with the ldquofavoringlengthrdquo routes in this sense perhaps this choice of favoringballast minimization may lead to a more efficient designThis is mostly due to the straight section 1198611 located inshallower depths ranging from 20 to 150m where wave loadsare more significant in Figure 10 it can be seen that toreduce the resultant loads normal to the pipeline axis theoptimization algorithm tries to align the route with the mostsevere directions of environmental loadings (NE-SW and


B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast


Figure 11 Scenario 2 ldquofavoring lengthrdquo route


Favoring length Intermediate Favoring ballastRoute length (m) 53923 55966 56870Total ballast weight (kN) 14508 13564 13206Average ballast weight per unit length (Nm) 269 242 232

N-S directions) just enough to avoid interference with themooring line close to endpoint 119861

Differently from the ldquofavoring lengthrdquo routes the ldquofavor-ing ballastrdquo routes provided by the different stability criteriacan be clearly identified in Figure 10 (indicated as ldquofavoringballast absoluterdquo and ldquofavoring ballast Abs + generalizedrdquorespectively) Table 12 confirms that the ldquogeneralized stabil-ityrdquo criterion is indeed less conservative not only leading toshorter routes but also requiring less ballastThis latter aspectis also indicated by the shorter length of the segments thatrequire more ballast (depicted in red in Figure 10)

542 Scenario 2 The optimal routes obtained for scenario2 corresponding to the three different sets of values forthe weighting factors of Table 7 are shown next favoringlength (Figure 11) intermediate (Figure 12) and favoring

ballast (Figure 13) Figure 14 groups those three routes for abetter visual assessment

Table 13 presents the values for the main parameters ofthe objective function length and ballast weight Again asexpected increasing the weighting factor 119896119871 leads to thereduction of the route length and conversely to the increasein ballast weight Thus the ldquofavoring lengthrdquo route requiresmore ballast especially in the shallower region near endpoint119861 and also in the stretch depicted in red between points 3 and4 indicated in Figure 11 This stretch extends from the depthof 300m that sets the limit where the soil changes from sandto clay thus affecting the OBS calculations see Table 5 andFigure 15 where sand and clay regions are depicted in grayand green respectively

The other routes (ldquointermediaterdquo and ldquofavoring ballastrdquo)require slightly less ballast in this stretch Overall the


B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast


Figure 12 Scenario 2 ldquointermediaterdquo route

ldquofavoring ballastrdquo route is longer but requires less ballastwhile the ldquointermediaterdquo route is around 900m shorter butrequires slightly more ballast (4) Another advantage of theldquofavoring ballastrdquo route is that it deviates from the flowlines(depicted in light green close to point 3 in Figure 13)thus minimizing the requirements for intervention to avoidinterference with obstacles

In summary for this considerably complex scenario witha greater distance between the endpoints (approx 50 km)and irregular seabed bathymetry (including canyons) it canbe seen that the optimal routes successfully complied withthe main objectives and constraints of the optimizationprocedure Excessive declivities associated with the crossingof canyons have been minimized the obstacle indicatedby a black polygon (representing a hard constraint) hasbeen avoided and the proximity to the other platforms(corresponding to the ldquoattractorsrdquo depicted as purple circles)has been attained as close as possible

6 Final Remarks and Conclusions

It is well known that the costs of a given pipeline projectare dictated not only by the material of the pipe segmentsthemselves (that derives from the route length) but also

by installation and intervention procedures that mitigateproblems related to the structural behavior of the pipe underhydrostatic and environmental loadings Considering specif-ically on-bottom stability several methods to provide ballastand stabilize the pipe (including those described at the end ofSection 33) introduce substantial costs to the installation ofthe pipeline along the chosen route In the traditional pipelinedesignmethodology the definition of a route and the stabilitycalculations had been performed independently firstly theroute is defined according to geographical-topographicalissues including seabed bathymetry and obstacles thenstability is verified and the requiredmitigating procedures arespecified

Now with the incorporation of OBS criteria into theroute optimization tool following the approaches describedin this work stability becomes an integral part of the routeoptimization process This may be seen as a breakthrough inthe traditional design methodology allowing the designer tosimultaneously handle the minimization of route length andthe cost of the mitigating procedures

The optimization tool provides the decision-maker withprofitable information regarding the optimal routes consid-ering different aspects of the design and the costs associatedwith the different procedures that provide ballast to stabilize


B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast

300ndash500 Nm500ndash900 Nm

0ndash300 Nm

(km)

Figure 13 Scenario 2 ldquofavoring ballastrdquo route

a pipeline (mattress trenching burying concrete coatingor even employing a heavier pipe with additional steel wallthickness) For instance relatively high values for the weightof concrete might hinder the installation of the pipeline(due to limitations of the launching vessel andor excessivestresses on the pipe itself) In such cases the designermight then opt for a hybrid solution combining less concretecoating with a heavier pipe (with increased wall thickness)Taking the required ballast weights as indicated in Table 12or Table 13 different combinations of steel wall and concretecoating thickness may be provided by the iterative proceduredescribed at the end of Section 33

Overall the computational costs associated with thestability calculations by the different OBS methods (ldquoabso-lute stabilityrdquo and ldquogeneralized stabilityrdquo) are relatively lowcorroborating the feasibility of their incorporation into theoptimization tool The results of the first case study con-firmed that the latter is more adequate for shallower waterswith increased influence of the environmental loads on thepipeline it leads to smaller routes andor less ballast whencompared with those obtained by the absolute methodThusalso observing that the computational costs associated withthe ldquogeneralizedrdquo criterion are not significantly higher thanthose for the ldquoabsoluterdquo criterion in the optimization tool the

former may be set as the default procedure for pipe segmentsbelow a given user-specified water depth value

Several new developments related to the route opti-mization tool are currently underway including proceduresto incorporate other technicalengineering criteria into theobjective function (for instance VIV-induced fatigue on freespans and multiphase flow) Also here the multiobjectiveproblem of optimizing the route length and ballast weighthas been dealt with by combining the respective terms intoa single objective function As observed in the results ofthe case studies those may be conflicting objectives sinceusually the optimal routes with shorter lengths have moreballast weight and vice versa While the present approacheffectively provides different optimal routes by adjusting theweighting factors 119896119871 and 119896119882119887 along different optimizationruns another promising approach would be to formallymodel the optimization problem with a multiobjective pro-cedure using specialized algorithms based on the Pareto frontconcept thus obtaining a set of optimal routes from a singleoptimization run Preliminary studies related to those topicshave been briefly outlined in conference papers [24 25]more detailed and conclusive results will be presented insubsequent works


Favoring length

Favoring ballastIntermediate

N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)

Figure 15 Scenario 2 ldquofavoring lengthrdquo route detail

To summarize the main feature of the developmentspresented in this work consists of anticipating to the stageof route definition engineering checks and calculationstraditionally related to subsequent design stages Thus theoptimization tool may reduce the design time of a givenpipeline not only for the route definition but also forthe other stages involving the detailed verification of thoseengineering criteria more important the tool can minimizeinstallation and operational costs of the pipeline

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The first and second authors would like to acknowledgethe Brazilian research funding agencies CNPq (Conselho


Nacional de Desenvolvimento Cientıfico e Tecnologico)and FAPERJ (Fundacao Carlos Chagas Filho de Amparoa Pesquisa do Estado do Rio de Janeiro) for the financialsupport of their doctoral studies at PECCOPPEUFRJThe fourth and fifth authors acknowledge the support ofCNPq and FAPERJ corresponding to the research Grantsnos 3048502012-8 3061042013-0 and E-262011842014respectively Finally all authors acknowledge Petrobras SAfor supporting and allowing the publication of this work

References

[1] DetNorskeVeritasOffshore StandardDNV-OS-F101 SubmarinePipeline Systems Det Norske Veritas Oslo Norway 2012

[2] httpsstoprustcomproducts-and-servicesretromat[3] httpallseascomactivitiespipelines-and-subseapipeline-pro-

tection[4] httpoffshore-fleetcomdatarock-dumping-vesselhtm[5] W T Jones ldquoOn-BottomPipeline Stability in SteadyWater Cur-

rentsrdquo Journal of Petroleum Technology vol 30 no 3 pp 475ndash484 1978

[6] H Meisingset J Hove and G Olsen ldquoOlsen Optimization ofPipeline Routesrdquo in in the Proceedings of the Fourteenth Inter-national Offshore and Polar Engineering Conference ToulonFrance 2004

[7] D H Fernandes B P Jacob B S L P de Lima and C HAlbrecht ldquoOffshore Pipeline Route Assessment via Evolution-aryAlgorithmsrdquo in in the Proceedings of the XXIX Iberian-Latin-American Congress on Computational Methods in EngineeringCILAMCE Maceio AL Brazil 2008

[8] D H Fernandes B P Jacob B S L P de Lima A RMedeiros and C H Albrecht ldquoA Proposal of Multi-ObjectiveFunction for Submarine Rigid Pipelines RouteOptimization viaEvolutionary Algorithmsrdquo in in the processdings of Rio PipelineConference amp Exposition 2009 Annals Rio de Janeiro Brazil2009

[9] S Percival D Hendrix and F Noblesse ldquoHydrodynamic opti-mization of ship hull formsrdquo Applied Ocean Research vol 23no 6 pp 337ndash355 2001

[10] B D S L P de lima B P Jacob and N F F Ebecken ldquoA hybridfuzzygenetic algorithm for the design of offshore oil produc-tion risersrdquo International Journal for Numerical Methods inEngineering vol 64 no 11 pp 1459ndash1482 2005

[11] R L Tanaka and C D A Martins ldquoParallel dynamic optimiza-tion of steel risersrdquo Journal of Offshore Mechanics and ArcticEngineering vol 133 no 1 Article ID 011302 2010

[12] A A de Pina CH Albrecht B S de Lima and B P Jacob ldquoTai-loring the particle swarm optimization algorithm for the designof offshore oil production risersrdquoOptimization and Engineeringvol 12 no 1-2 pp 215ndash235 2011

[13] I N Vieira B S de Lima and B P Jacob ldquoBio-inspired algo-rithms for the optimization of offshore oil production systemsrdquoInternational Journal for Numerical Methods in Engineering vol91 no 10 pp 1023ndash1044 2012

[14] M A L Martins E N Lages and E S S Silveira ldquoCompliantvertical access riser assessment DOE analysis and dynamicresponse optimizationrdquoApplied Ocean Research vol 41 pp 28ndash40 2013

[15] I N Vieira C H Albrecht B S L P de Lima B P Jacob D MRocha and C O Cardoso ldquoTowards a Computational Tool for

the Synthesis and Optimization of Submarine Pipeline Routesrdquoin in the proceedings of the Twentieth International Offshore andPolar Engineering Conference - ISOPE Beijing China 2010

[16] R R De Lucena J S Baioco B S L P De Lima C HAlbrecht andB P Jacob ldquoOptimal design of submarine pipelineroutes by genetic algorithm with different constraint handlingtechniquesrdquo Advances in Engineering Software vol 76 pp 110ndash124 2014

[17] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Boston MassachusettsUSA 1989

[18] K F Man K S Tang and S Kwong ldquoGenetic AlgorithmsConcepts and Applicationsrdquo IEEE Transactions on IndustrialEletronics vol 43 no 5 pp 519ndash534 1996

[19] E Mezura-Montes and C A Coello Coello ldquoConstraint-han-dling in nature-inspired numerical optimization past presentand futurerdquo Swarm and Evolutionary Computation vol 1 no 4pp 173ndash194 2011

[20] T Takahama S Sakai andN Iwane ldquoConstrained optimizationby the epsilon constrained hybrid algorithm of particle swarmoptimization and genetic algorithmrdquo in AI 2005 Advances inArtificial Intelligence LectureNotes inArtificial Intelligence pp389ndash400 Springer-Verlag Berlin Germany 2005

[21] Det Norske Veritas On Bottom Stability Design of SubmarinePipelines Recommended Practice DNV-RP-F109 Det NorskeVeritas Oslo Norway 2008

[22] MHADe Lima Jr J S Baioco CHAlbrecht et al ldquoSynthesisand optimization of submarine pipeline routes considering On-Bottom Stability criteriardquo in Proceedings of the ASME 201130th International Conference on Ocean Offshore and ArcticEngineering OMAE2011 pp 307ndash318 Netherlands June 2011

[23] B Guo S Song J Chacko and A Ghalambor Offshore Pipe-lines Gulf Professional Publishing Elsevier Houston TexasUSA 2005

[24] J S Baioco P Stape M Granja C H Albrecht B S L P DeLima and B P Jacob ldquoIncorporating engineering criteria to thesynthesis and optimization of submarine pipeline routes On-bottom stability viv-induced fatigue and multiphase flowrdquo inProceedings of the ASME 2014 33rd International Conference onOcean Offshore and Arctic Engineering OMAE San FranciscoCalifornia USA June 2014

[25] J S Baioco CH Albrecht B P Jacob andDM Rocha ldquoMulti-Objective Optimization of Submarine Pipeline Routes Con-sidering On-Bottom Stability VIV-Induced Fatigue and Mul-tiphase Flowrdquo in Proceedings of the Twenty-Fifth InternationalOcean and Polar Engineering Conference - ISOPE Kona HawaiiUSA 2015

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Submit your manuscripts atwwwhindawicom


for instance corals geotechnical hazards or fields allottedto another oil company There are many other variables thatgovern the selection of a route thus the process has beentreated almost as an ldquoartrdquo being highly dependent on theexpertise of the engineer

Previous works have already recognized [6ndash8] that thetask of selecting a route with good performance and lowcost could be automated by devising its formal descriptionas a synthesis and optimization problem and building acomputational tool based on evolutionary algorithms (EAs)Such algorithms have been successfully used in many com-plex engineering problems and have been shown to beuseful for the optimization of offshore engineering problems[9ndash14]

In [15 16] we have presented results of preliminarystudies related to the development and implementation ofa computational tool for the synthesis and optimization ofsubmarine pipeline routes based on Genetic Algorithms(GAs) [17 18] The modeling of the optimization problem aspresented in [15 16] included only the basic geographical-topographical issues associated with the route geometry andwith the seabed bathymetry and obstacles Those issues wereconsidered for the representation of a candidate route in thecontext of the GA and for its evaluation in terms of criteriaincorporated into the objective function and constraintsThese criteria were defined considering only themain aspectsalready involved in the manual selection of a route includingbasically the total pipeline length interference with obstaclesminimum radius of curvature and declivity In [16] specialfocus was dedicated to the study and assessment of differentconstraint-handling techniques including the 120576-constrainedmethod [19 20] Therefore the developments presented in[15 16] could be seen as leading to a computational tool thatalthough innovative in itself merely automated the processof selecting an optimal route following the traditional designguidelines

Here the focus is on incorporating technicalengineeringcriteria into the route optimization tool related to the struc-tural behavior of the pipe under hydrostatic and environmen-tal loadings specifically the implementation of on-bottomstability (OBS) criteria The stability of a pipe is reflectedby its ability to remain within its outline in the installedposition (taking into account allowable tolerances) underenvironmental conditions of wave and current The DNV-RP-F109 code [21] presents criteria to check if a given pipe isstable or else to defineminimal values of submergedweight toreach stabilityThis idea of incorporatingOBS criteria into theroute optimization tool comprises a breakthrough in the tra-ditional pipeline designmethodology and has originally beenproposed in [22] where a preliminary implementation wassketched with the OBS criteria incorporated as constraintshandled by the classic static penalty technique This waytaking predefined values for the pipe submerged weight theldquooptimalrdquo route would seek areas and trajectories where thestability of the pipe is favored according to the predominantdirection of the environmental loadings

However that approach would possibly lead to longerroutes and require a ldquotrial-and-errorrdquo process where thedesigner should perform successive runs of the optimization

tool increasing the pipe weight in order to obtain smallerroutes Now this work describes an improved approach toincorporate theOBS criteria into the optimization tool wherean additional term is introduced into the objective functionrepresenting the ballast weight as an additional variable to beoptimizedThus the weight required for stability at each pipesegment is directly provided as a result of an optimizationrun

In the optimization procedure candidate routes arerepresented by specific geometric parameterization and areevaluated in terms of several criteria incorporated in theobjective function and in the constraint functions Thesecomprise the core of the route optimization model whichwill be described in Section 2 that focuses on the routeparameterization and encoding in the context of the GASection 3 presents the final form of the objective function anddescribes specifically the incorporation of the OBS criteriaSection 4 presents all remaining criteria that comprise theconstraint functions (including their distinction between softand hard constraints according to the consequences of theirviolation) Case studies are presented in Section 5 to illustratethe use of the optimization tool and assess the influence ofthe stability criteria on the definition of the optimal pipelineroute Lastly final remarks and conclusions are presented inSection 6

2 Modeling and Solving the RouteOptimization Problem

21 Objective Function A general constrained optimizationproblem is formally defined in an 119899-dimensional searchspace 119878 comprised by a vector of design variables x =(1199091 1199092 1199093 119909119899) The following expression mathematicallydefines the problem

minimize 119891 (x)subject to 119892119894 (x) le 0 119894 = 1 119898

ℎ119895 (x) = 0 119895 = 1 119901119897119896 le 119909119896 le 119906119896 119896 = 1 119902

(1)

The goal is tominimize an objective function119891(x) consid-ering inequality and equality constraints (resp 119892119895(x) le 0 andℎ119895(x) = 0) that define the feasible region The components119909119894 may have lower and upper bounds [119897119896 119906119896] In engineeringproblems the constraints are defined in terms of appropriatedesign criteria As will be seen later in the case of the pipelineroute optimization problem these criteria may be expressedas inequality constraints 119892119895(x) only so equality constraintsare not considered in this particular engineering applica-tion

Here it is assumed that the outer diameter and wallthickness of the pipe segments have already been selectedfrom previous design steps Usually the outer diameter isdictated by the amount of oil or gas to be transportedaccording to the yield of the well and the wall thicknessis dictated by strength constraints related to collapse underexternal hydrostatic pressure [23]Thus amongst the relevant


c

AC

AC

A B

x

y

C1

C2

R1

R2 R2

R1

PC1

PC2

０）1

０）2

PT1

PT2

T

T T

T


A B

Obstacle

R1 1

1

０）1

p1



119891 (x) = 119871119877119900119906119905119890119871119860119861 (2)






Fitness evaluation



Initial population

Updated population


Objective function



c =

p

sumj=1

rjcj(x)

=m

sumj=1

pjj(x)2

f (x) = kL middotL

LAB

+ kWb middotWb

WbAB

Routeroute


















119891 (x) = 119896119871 sdot 119871119877119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

(4)





119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)








120574SC sdot 119865lowast119885

119908119904 le 10(6)






t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









c

AC

AC

A B

x

y

C1

C2

R1

R2 R2

R1

PC1

PC2

０）1

０）2

PT1

PT2

T

T T

T


A B

Obstacle

R1 1

1

０）1

p1



119891 (x) = 119871119877119900119906119905119890119871119860119861 (2)






Fitness evaluation



Initial population

Updated population


Objective function



c =

p

sumj=1

rjcj(x)

=m

sumj=1

pjj(x)2

f (x) = kL middotL

LAB

+ kWb middotWb

WbAB

Routeroute


















119891 (x) = 119896119871 sdot 119871119877119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

(4)





119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)








120574SC sdot 119865lowast119885

119908119904 le 10(6)






t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









Fitness evaluation



Initial population

Updated population


Objective function



c =

p

sumj=1

rjcj(x)

=m

sumj=1

pjj(x)2

f (x) = kL middotL

LAB

+ kWb middotWb

WbAB

Routeroute


















119891 (x) = 119896119871 sdot 119871119877119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

(4)





119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)








120574SC sdot 119865lowast119885

119908119904 le 10(6)






t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018
















119891 (x) = 119896119871 sdot 119871119877119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

(4)





119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)








120574SC sdot 119865lowast119885

119908119904 le 10(6)






t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










119882119887119877119900119906119905119890 =119899119878119890119892119898

sum119894=1

119908119887119901119894119901119890 (119894) sdot 119871 119904119890119892119898 (119894) (5)








120574SC sdot 119865lowast119885

119908119904 le 10(6)






t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









t

FlowastY

wsFR

V(z)

FlowastZ


t

FlowastY

ws

FR

V(z)FlowastZ





sdot log( 11988405) (7)



119871 = 11990811990405 sdot 120588119908 sdot 119863 sdot 1198801199042

(8)






(9)

120574SC119911 =119908119904119865lowast119885 (10)


120574SC119910 =11987111987110 (11)




(12)



(13)



120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










120574SC119910 =119871 sdot cos (120579119905)11987110 (14)




















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018

















119891 (x) = 119896119871 sdot 119871119903119900119906119905119890119871119860119861 + 119896119882119887 sdot119882119887119877119900119906119905119890

119882119887119860119861

+119901

sum119895=1

119903119895119888119895 (x)10038161003816100381610038161003816119878119900119891119905119862119903119894119905119890119903119894119886 (15)



120601 =119898

sum119895=1

119901119895V119895 (x)2



(1198911 1206011)lt120576 (1198912 1206012)


(17)




119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018











119899119868119899119905119890119903119878

sum119894=1

119878119890V119890119903119894119905119910119894 (19)


119888119889119890119888119897119894V = sum119899119873119900119889119890119904119894=1 120579119894119871 minus 120579119871 119871119894119898

119899119873119900119889119890119904 (20)


119888119886119905119905119903 =119899119860119905119905119903

sum119894=1

119908119894119898119894119899119863119894119904119905119894

119871119860119861 (21)



119899119868119899119905119890119903119867

sum119894=1

119878119890V119890119903119894119905119910119894 (22)


V119871min119878119905119903 =119899119878119905119903119886119894119892ℎ119905

sum119894=1

119871min minus 119871 119894119871min

(24)


V119877min =119899119873119900119889119890119904

sum119894=1

(119877min minus 119877119894)119877min

(25)






5 Case Studies




S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









S

EW

N



0 2 4(km)




0 7 14

S

EW

N

(km)








Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










Scenario 119871119860119861(km)



1 12359 14 5292 49393 96 1782





Depth (m) Thickness



Corrosion coating




48mm189 in 8829






119873119901119900119901 MaxGen 119899max 119860 119905









Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018











Favoring length 119896119871 50 10119896119882119887 001 001

Intermediate 119896119871 ---- 5119896119882119887 ---- 001






Declivity 5 1 120579119871 119871119894119898 (20) 5∘ 8∘



















N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018











N 474 92 501 956NE 488 947 517 97E 434 99 487 104SE 572 1028 653 1163S 619 1354 71 1435SW 716 1478 784 1555W 357 822 388 851NW 357 822 388 851




0

05

15

N

NE

E

SE

S

SW

W

NW


0

2

4

6

8N

NE

E

SE

S

SW

W

NW

1

2

10 years100 years

10 years100 years



Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









Current 1

Current 2

Current 3

Current 4

0 7 14

S

EW

N

(km)















Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018












Average 420 202 388 179Min 100 0 100 0Max 4500 2900 4500 2500

Ballast

0

1

2N

NE

E

SE

SSW

W

NW



Favoring length

1 1

A

B


0

5

10N

NE

E

SES

SW

W

NW

N

S

EW

0 2 4


Hs (m)

(km)







B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









B

A

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast












B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









B

A

6

5

4

3

2

1

N

S

EW

0 5 10(km)

Ballast











B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









B

A

6

5

4

3 2

1

N

S

EW

0 5 10

Ballast


0ndash300 Nm

(km)







Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018









Favoring length


N

S

EW

0 5 10

Ballast


(km)


B

5

4

3

N

S

EW

0 5 10(km)





Acknowledgments




References

































Journal of












Journal of







Volume 2018






Volume 2018










References

































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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