Research ArticleA Thrust Allocation Method for Efficient DynamicPositioning of a Semisubmersible Drilling Rig Based onthe Hybrid Optimization Algorithm

Luman Zhao1 and Myung-Il Roh2

1Department of Naval Architecture and Ocean Engineering Seoul National University Seoul 08826 Republic of Korea2Department of Naval Architecture and Ocean Engineering Research Institute of Marine System EngineeringSeoul National University Seoul 08826 Republic of Korea

Correspondence should be addressed to Myung-Il Roh mirohsnuackr

Received 7 April 2015 Revised 3 August 2015 Accepted 5 August 2015

Academic Editor Chunlin Chen

Copyright copy 2015 L Zhao and M-I Roh 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

A thrust allocation method was proposed based on a hybrid optimization algorithm to efficiently and dynamically position asemisubmersible drilling rig That is the thrust allocation was optimized to produce the generalized forces and moment requiredwhile at the same time minimizing the total power consumption under the premise that forbidden zones should be taken intoaccount An optimization problem was mathematically formulated to provide the optimal thrust allocation by introducing thecorresponding design variables objective function and constraints A hybrid optimization algorithm consisting of a geneticalgorithm and a sequential quadratic programming (SQP) algorithm was selected and used to solve this problem The proposedmethod was evaluated by applying it to a thrust allocation problem for a semisubmersible drilling rig The results indicate that theproposed method can be used as part of a cost-effective strategy for thrust allocation of the rig

1 Introduction

Research Background A dynamic positioning (DP) system isespecially useful to deep-sea working vessel such as drillingvessel and floating crane The DP system can maintain theposition (with a fixed location or on a predetermined track)and the heading of a working vessel exclusively throughthe use of active thrusters and propellers To keep theposition means to maintain the desired position within thenormal excursion from the desired position and headingin the horizontal plane For an offshore vessel which is asemisubmersible drilling rig illustrated in Figure 1 in thiscase the DP system generally consists of a position andheading reference system with sensor units a control systema power system and a thruster system [1] The positionand heading reference system along with the wind sensorsand gyrocompasses can provide the control system withinformation on the position of the vessel and the magni-tude and direction of the environmental forces that affect

the position All of the information obtained is then used toadjust the vessel itself

The DP system must be very responsive to changes in theconditions due to weather water depth and so on in orderto properly control the operating parameters of the vesselDespite the importance of the DP system to the vessel itsuse in a real working environment still facesmany challengessuch as a power failure and a thruster malfunction due toits complexity The fuel expenditure of the DP system isa considerable burden since it requires a large amount ofenergyThismeans that the DP system is overactuated whichrenders the thruster to have infinite allocation solutionsTherefore the thrust allocation can be formulated as anoptimization process under numerous constraints in order tominimize power consumption

Related Works In the literature many methods have beenproposed to achieve thrust allocation Johansen and Fossen[2] provided a recent review of thrust allocation methods

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 183705 12 pageshttpdxdoiorg1011552015183705

2 Mathematical Problems in Engineering

Azimuththruster

Optimizationthruster allocation

Terminal controlSystem of

dynamic system

Power system

Control systemxr yr Θr

120572i ni

Position andheading reference

system

Wind force

Wave force

Current force

Fx Fy Mz

Figure 1 Dynamic positioning system configured for 8 thrusters of a semisubmersible drilling rig

but most of these have focused on proposing optimizationalgorithms intended for thrust allocation for industries suchas automobile aerospace [3] and robotics and not for oceanengineering The relationship between the power consump-tion and the resulting thrust can usually be approximatedby using a quadratic function so some researchers haveformulated optimization problems for thrust allocation asquadratic programming problems and have solved these withsome variant of quadratic programming (QP) algorithms [4]

Some of the methods intended for ocean engineeringhave considered cases with ships or with offshore vesselsWit [5] proposed a method to achieve the optimal thrustallocation for a ship The thrust and azimuth directionof the thrusters were selected as design variables for theoptimization problem and the QP algorithm which is alocal optimization algorithm was used to solve the problemSimilarly Liang and Cheng [6] and Johansen et al [7] pro-posedmethods for the ship that include a similar formulationof an optimization problem and a solution through the useof a sequential quadratic programming (SQP) algorithmParikshit [8] also proposed a method to achieve the optimalthrust allocation for a drilling rig The thrust and azimuthdirection of thrusters were selected as design variables of anoptimization problem and various global optimization algo-rithms such as genetic algorithm (GA) and ITHS (IntelligentTuned Harmony Search) were used to solve this problemZhao et al [9] also proposed a method that includes a similarformulation of an optimization problem and solution for adrilling rig by using GA

In this manner most studies in the field of ocean engi-neering have determined the thrust and the azimuth direc-tion of the thrusters to be design variables for an optimizationproblem In fact the thrust can be expressed as a function ofthe speed of rotation the diameter and the thrust coefficientof a thruster In order to reflect the specifications of the giventhruster and to more easily control the thruster the speed ofrotation can be used as a design variable instead of the thrustMeanwhile a local optimization algorithm can be used as theoptimization algorithm such as the SQP algorithm to derivean accurate optimum However this method is very sensitiveto a starting point for optimization and sometimes finds alocal optimum On the other hand a global optimizationalgorithm such as the GA does not need a starting point forthe optimization but can only find a rough optimum [10]

This study proposes an optimal thrust allocation methodfor an offshore vessel with the aim of achieving the requiredgeneralized forces and moment for dynamic positioningwhile at the same time minimizing total power Findingsuch a method is a particularly important issue to operatean offshore vessel while using minimal energy The methodinvolves mathematically formulating an optimization prob-lem for thrust allocation and selecting and using a suitableoptimization algorithm to solve the problem not developinga new algorithm When the optimization problem is formu-lated the speed of the rotation and the azimuth direction ofthrusters are selected as design variables the total power isselected as an objective function and the required forces andmoment are selected as constraints A hybrid optimization

Mathematical Problems in Engineering 3

Table 1 Summary of related studies and comparison with this study

Studies Application Optimization algorithm Design variables forformulation

Consideration ofthruster interaction

Wit [5] Ship Local algorithm (QP) Thrust and azimuth direction ⃝

Liang and Cheng [6]Johansen et al [7] Ship Local algorithm (SQP) Thrust and azimuth direction times

Parikshit [8] Drilling rig Global algorithm (GA ITHS etc) Thrust and azimuth direction ⃝

Zhao et al [9] Drilling rig Global algorithm (GA) Thrust and azimuth direction ⃝

This study Drilling rig Global algorithm (hybridoptimization algorithm)

Rotation speed and azimuthdirection ⃝

T7

T8

T1

T2

T3

T4T6

T5

Port

Starboard

Afterward Forward

lx1

ly1

y

xsumMz

sumFy

sumFx

Figure 2 Schematic of thruster arrangement of a drilling rig with eight thrusters (Plan view)

algorithm that incorporates global and local optimizationalgorithms was used for the optimization

Table 1 shows a summary of related studies and a compar-ison of them with this study

2 Thrust Allocation for Dynamic Positioning

A thruster system is important for a drilling rig to maintainits position and heading since it can simultaneously providetransverse and longitudinal thrust An azimuth thrusterconsists of an electric podded drive that is fitted to thehull and can rotate 360 degrees along the horizontal angleto provide thrust in all directions The azimuth thrusterhas been selected in this study because it can providethrust in any direction in order to act as a propulsor fordynamic positioning and target optimization of a drillingrig However the thruster system is usually overactuated inpractice so a thrust allocation problem can be formulatedinto a constrained optimization problem for the system

Figure 2 depicts a schematic diagram for the thrusterarrangement of the drilling rig and the sign conventions forthe forces (119865

119909and 119865

119910) and moment (119872

119911) In this figure

sum119865119909 sum119865119910 and sum119872

119911represent the total forces required

in the longitudinal (119909) and transverse (119910) directions andthe total required moment about the vertical (119911) directionfrom the control system as shown in Figure 1 respectivelyNote that the positions of the thrusters are given in a 2D

coordinate system (only the horizontal forces and momentare considered) The coordinate system has its origin at thecenter of gravity (CoG) of the drilling rig

The environmental forces and the moment tend to movethe drilling rig away from its original position during oper-ation at sea In order to move the drilling rig back to itsoriginal or reference position the control system in the DPsystem of the drilling rig should first calculate the total forces(thrusts) andmoment required At this time thrust allocationshould be conducted to determine the thrust and azimuthdirection of each thruster so that the required forces (andmoment) are generated in each of the longitudinal (sum119865

119909)

and transverse (sum119865119910) and vertical (sum119872

119911) directions The

relationship between the thrust and the azimuth directionof each thruster and the total forces and moment that arerequired can be stated as in (1) to (3) These equations are thegoverning equations for the thrust allocation and can be usedas equality constraints in an optimization problem

sum119865119909= (1198651cos1205721+ 1198652cos1205722+ sdot sdot sdot + 119865

119899cos120572119899) (1)

sum119865119910= (1198651sin1205721+ 1198652sin1205722+ sdot sdot sdot + 119865

119899sin120572119899) (2)

sum119872119911= (1198651cos1205721sdot 1198971199101+ 1198652cos1205722sdot 1198971199102+ sdot sdot sdot

+ 119865119899cos120572119899sdot 119897119910119899+ 1198651sin1205721sdot 1198971199091+ 1198652sin1205722sdot 1198971199092

+ sdot sdot sdot + 119865119899sin120572119899sdot 119897119909119899)

(3)

4 Mathematical Problems in Engineering

Supposing that the drilling rig is equipped with 119894 azimuththrusters 119865

119894is the thrust from 119894th thruster and 120572

119894is the

azimuth direction (angle) of 119894th thruster 119897119909119894and 119897119910119894are the

distances of 119894th thruster from the CoG of the drilling rigEquation (1) indicates that the longitudinal component ofthe force generated from the thrust and azimuth directionof each thruster should be equal to the total force requiredin the longitudinal direction Similarly (2) indicates thatthe transverse component of the force from the thrust andazimuth direction of each thruster should be equal to thetotal force required in the transverse direction Equation (3)means that the vertical component of the moment generatedfrom the thrust and azimuth direction of each thruster shouldbe equal to the total required moment about the verticaldirection

The goal of achieving optimal thrust allocation is toguarantee efficient power use when dynamically positioningthe drilling rig in the specific conditions of a given oceanenvironment Meanwhile the economic impact of minimiz-ing power consumption is also taken into account with theultimate aim of ensuring the stability and capability of thedrilling rig during operation

3 Thrust Allocation Method

31 Optimization Problem for Thrust Allocation This studymathematically formulated an optimization problem forthrust allocation and each component of the optimizationproblem is described below in detail

311 Design Variables Thrust allocation involves determin-ing the thrust (119865

119894) and azimuth direction (120572

119894) of each thruster

to generate the forces in longitudinal and transverse direc-tions and the moment about the vertical direction that arerequired for the DP system of the drilling rigThus the thrustand azimuth direction of each thruster can be used as designvariables in the thrust allocation optimization problem Theobjective function that is commonly used for this problem isto minimize the total power of the thrusters due to energysavings In order to use this power as the objective functionit should be mathematically formulated as a function of thedesign variables The total power will be proportional tothe sum of the thrusts for each thruster Some researchers[5ndash9] used the thrust and the azimuth direction of eachthruster as design variables and have formulated the objectivefunction (minimization of total power of thrusters) withthe corresponding design variables However the rotationalspeed (119899

119894) and the azimuth direction (120572

119894) of each thruster can

be used as design variables to reflect the characteristics of thegiven thruster tomore easily control the thruster and tomoredelicately represent the objective function Thus the designvariables for the thrust allocation optimization problem inthis study are as follows

119899119894 speed of rotation of each thruster

120572119894 azimuth direction of each thruster

312 Objective Function In this study the speed of therotation (119899

119894) and the azimuth direction (120572

119894) of each of the

thrusters were used as design variables and thus the objectivefunction (minimization of the total power for the thrusters)was mathematically formulated according to the followingprocedure where in fact the power consumed by the electricpropulsion system depends on the speed of rotation and theazimuth direction of the thrusters

The thrust 119879 and torque 119876 of a thruster can be expressedas functions of the speed of rotation (119899)

119879 = 1205881198634

1198701198791198992

(4)

119876 = 1205881198635

1198701198761198992

(5)

where 120588 is the density of the sea water 119863 is the propellerdiameter of the thruster 119870

119879and 119870

119876are the thrust and

torque coefficients respectively and 119899 is the speed of therotation of the thruster The expressions for 119870

119879and 119870

119876can

be obtained from the so-called openwater test of the thrusterThese nondimensional coefficients for thrust and torque werederived through a regression analysis that can be describedaccording to the following parameters [11]

119870119879= sum119904119905119906V

119862119879

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

119870119876= sum119904119905119906V

119862119876

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

(6)

119862119879119904119905119906V and 119862

119876

119904119905119906V coefficients and 119904 119905 119906 and V terms canbe found in a report of the Wageningen B-Series propellers[11] and were used in this study Here 119875119863 is the pitchdiameter ratio 119860

119864119860119874is the blade area ratio and 119885 is the

number of blades of the thruster The coefficient for theadvance 119869 is a nondimensional description of the thrusterperformance that can be expressed according to the speed ofrotation 119899 and the speed of the advance of the thruster definedas 119881119860 which can be represented by the speed near each

thruster (119881) and the azimuth direction of each thruster (120572)Since each of the thrusters has a different speed of rotation inpractice119870

119879and119870

119876of each thruster can be different

119869 =119881119860

119899119863=119881 cos120572119899119863

(7)

Now it is time to formulate the total power consumed bythe thrusters of the drilling rig The power consumption (119875)can be formulated as

119875 = 2120587119899119894119876119894 (8)

By combining (5) and (8) the power consumption of thethrusters can be rewritten as

119875 = 21205871205881198635

119873

sum119894=1

1198701198761198941198993

119894 (9)

where119873 is the number of thrusters

Mathematical Problems in Engineering 5

Outboard thruster

X

YPort

Starboard

Afterward Forward

Port-forward side thrusters

Y

X

120601i120572i

120572i + 1

120601i + 1 Inboard thruster((i + 1)th thruster)

120579f

120579f

(ith thruster)

Figure 3 Interaction of two closely spaced azimuth thrusters

Therefore the objective function can be written relatingthe speed of rotation and the azimuth direction of eachthruster In this study the total power (power consumption)of the thrusters isminimized by setting the objective functionfor the thrust allocation optimization problem as

Minimize 119875 = 21205871205881198635119873

sum119894=1

1198701198761198941198993

119894 (10)

313 Constraints To obtain a valid thrust allocation thegoverning equations shown in (11) to (13) should be satisfiedThe equations represent the thrust and azimuth direction ofeach thruster that generate the forces required in longitudinal(sum119865119909) and transverse (sum119865

119910) directions as well as themoment

required about vertical direction (sum119872119911) to dynamically

position the vessel If these equations cannot be satisfied thedrilling rig starts to drift Thus these equations are used asequality constraints for the optimization problem

The governing equation for the longitudinal force can bestated as

sum119865119909

= 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894)

(11)

Similarly the governing equation for the transverse forcecan be stated as

sum119865119910

= 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894)

(12)

And finally the governing equation for the yawing mo-ment can be stated as

sum119872119911= 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot

+ 1198992

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot

+ 1198992

119894cos120572119894119897119910119894)

(13)

where sum119865119909 sum119865119910 and sum119872

119911are the forces required in

the longitudinal and transverse directions and the momentrequired about the vertical direction respectively and are allgiven from the control system in the dynamic positioningsystem

It is essential to minimize the thruster interaction of asemisubmersible drilling rig to ensure effective DP operation[5 12] In order to find a solution to provide the mosteffective thrust for operation additional constraints shouldbe considered by taking into account the physical locationsof the thrusters Thrusters that are close together have aninfluence on each other and if one thruster is in the streamof another the efficiency would drop significantly To avoidhaving this happen certain angles must be prohibited forthe azimuth direction Each thruster is assumed to have aforbidden angle of 120579

119891for the closest thruster in order to

prevent the direct interaction of the thruster as shown inFigure 3 With the port-forward pair the center of the angleof the outboard thruster 119894 is 120601

119894and for the inboard thruster

(119894 + 1) it is 120601119894+1 The positive angle is measured clockwise

while the negative angle is measured anticlockwise from thehorizontal axis Figure 3 shows the ATRs (Attainable ThrustRegions) for the two port-forward side thrusters 119894 and (119894 + 1)

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Azimuththruster

Optimizationthruster allocation

Terminal controlSystem of

dynamic system

Power system

Control systemxr yr Θr

120572i ni

Position andheading reference

system

Wind force

Wave force

Current force

Fx Fy Mz

Figure 1 Dynamic positioning system configured for 8 thrusters of a semisubmersible drilling rig

but most of these have focused on proposing optimizationalgorithms intended for thrust allocation for industries suchas automobile aerospace [3] and robotics and not for oceanengineering The relationship between the power consump-tion and the resulting thrust can usually be approximatedby using a quadratic function so some researchers haveformulated optimization problems for thrust allocation asquadratic programming problems and have solved these withsome variant of quadratic programming (QP) algorithms [4]

Some of the methods intended for ocean engineeringhave considered cases with ships or with offshore vesselsWit [5] proposed a method to achieve the optimal thrustallocation for a ship The thrust and azimuth directionof the thrusters were selected as design variables for theoptimization problem and the QP algorithm which is alocal optimization algorithm was used to solve the problemSimilarly Liang and Cheng [6] and Johansen et al [7] pro-posedmethods for the ship that include a similar formulationof an optimization problem and a solution through the useof a sequential quadratic programming (SQP) algorithmParikshit [8] also proposed a method to achieve the optimalthrust allocation for a drilling rig The thrust and azimuthdirection of thrusters were selected as design variables of anoptimization problem and various global optimization algo-rithms such as genetic algorithm (GA) and ITHS (IntelligentTuned Harmony Search) were used to solve this problemZhao et al [9] also proposed a method that includes a similarformulation of an optimization problem and solution for adrilling rig by using GA

In this manner most studies in the field of ocean engi-neering have determined the thrust and the azimuth direc-tion of the thrusters to be design variables for an optimizationproblem In fact the thrust can be expressed as a function ofthe speed of rotation the diameter and the thrust coefficientof a thruster In order to reflect the specifications of the giventhruster and to more easily control the thruster the speed ofrotation can be used as a design variable instead of the thrustMeanwhile a local optimization algorithm can be used as theoptimization algorithm such as the SQP algorithm to derivean accurate optimum However this method is very sensitiveto a starting point for optimization and sometimes finds alocal optimum On the other hand a global optimizationalgorithm such as the GA does not need a starting point forthe optimization but can only find a rough optimum [10]

This study proposes an optimal thrust allocation methodfor an offshore vessel with the aim of achieving the requiredgeneralized forces and moment for dynamic positioningwhile at the same time minimizing total power Findingsuch a method is a particularly important issue to operatean offshore vessel while using minimal energy The methodinvolves mathematically formulating an optimization prob-lem for thrust allocation and selecting and using a suitableoptimization algorithm to solve the problem not developinga new algorithm When the optimization problem is formu-lated the speed of the rotation and the azimuth direction ofthrusters are selected as design variables the total power isselected as an objective function and the required forces andmoment are selected as constraints A hybrid optimization

Mathematical Problems in Engineering 3

Table 1 Summary of related studies and comparison with this study

Studies Application Optimization algorithm Design variables forformulation

Consideration ofthruster interaction

Wit [5] Ship Local algorithm (QP) Thrust and azimuth direction ⃝

Liang and Cheng [6]Johansen et al [7] Ship Local algorithm (SQP) Thrust and azimuth direction times

Parikshit [8] Drilling rig Global algorithm (GA ITHS etc) Thrust and azimuth direction ⃝

Zhao et al [9] Drilling rig Global algorithm (GA) Thrust and azimuth direction ⃝

This study Drilling rig Global algorithm (hybridoptimization algorithm)

Rotation speed and azimuthdirection ⃝

T7

T8

T1

T2

T3

T4T6

T5

Port

Starboard

Afterward Forward

lx1

ly1

y

xsumMz

sumFy

sumFx

Figure 2 Schematic of thruster arrangement of a drilling rig with eight thrusters (Plan view)

algorithm that incorporates global and local optimizationalgorithms was used for the optimization

Table 1 shows a summary of related studies and a compar-ison of them with this study

2 Thrust Allocation for Dynamic Positioning

A thruster system is important for a drilling rig to maintainits position and heading since it can simultaneously providetransverse and longitudinal thrust An azimuth thrusterconsists of an electric podded drive that is fitted to thehull and can rotate 360 degrees along the horizontal angleto provide thrust in all directions The azimuth thrusterhas been selected in this study because it can providethrust in any direction in order to act as a propulsor fordynamic positioning and target optimization of a drillingrig However the thruster system is usually overactuated inpractice so a thrust allocation problem can be formulatedinto a constrained optimization problem for the system

Figure 2 depicts a schematic diagram for the thrusterarrangement of the drilling rig and the sign conventions forthe forces (119865

119909and 119865

119910) and moment (119872

119911) In this figure

sum119865119909 sum119865119910 and sum119872

119911represent the total forces required

in the longitudinal (119909) and transverse (119910) directions andthe total required moment about the vertical (119911) directionfrom the control system as shown in Figure 1 respectivelyNote that the positions of the thrusters are given in a 2D

coordinate system (only the horizontal forces and momentare considered) The coordinate system has its origin at thecenter of gravity (CoG) of the drilling rig

The environmental forces and the moment tend to movethe drilling rig away from its original position during oper-ation at sea In order to move the drilling rig back to itsoriginal or reference position the control system in the DPsystem of the drilling rig should first calculate the total forces(thrusts) andmoment required At this time thrust allocationshould be conducted to determine the thrust and azimuthdirection of each thruster so that the required forces (andmoment) are generated in each of the longitudinal (sum119865

119909)

and transverse (sum119865119910) and vertical (sum119872

119911) directions The

relationship between the thrust and the azimuth directionof each thruster and the total forces and moment that arerequired can be stated as in (1) to (3) These equations are thegoverning equations for the thrust allocation and can be usedas equality constraints in an optimization problem

sum119865119909= (1198651cos1205721+ 1198652cos1205722+ sdot sdot sdot + 119865

119899cos120572119899) (1)

sum119865119910= (1198651sin1205721+ 1198652sin1205722+ sdot sdot sdot + 119865

119899sin120572119899) (2)

sum119872119911= (1198651cos1205721sdot 1198971199101+ 1198652cos1205722sdot 1198971199102+ sdot sdot sdot

+ 119865119899cos120572119899sdot 119897119910119899+ 1198651sin1205721sdot 1198971199091+ 1198652sin1205722sdot 1198971199092

+ sdot sdot sdot + 119865119899sin120572119899sdot 119897119909119899)

(3)

4 Mathematical Problems in Engineering

Supposing that the drilling rig is equipped with 119894 azimuththrusters 119865

119894is the thrust from 119894th thruster and 120572

119894is the

azimuth direction (angle) of 119894th thruster 119897119909119894and 119897119910119894are the

distances of 119894th thruster from the CoG of the drilling rigEquation (1) indicates that the longitudinal component ofthe force generated from the thrust and azimuth directionof each thruster should be equal to the total force requiredin the longitudinal direction Similarly (2) indicates thatthe transverse component of the force from the thrust andazimuth direction of each thruster should be equal to thetotal force required in the transverse direction Equation (3)means that the vertical component of the moment generatedfrom the thrust and azimuth direction of each thruster shouldbe equal to the total required moment about the verticaldirection

The goal of achieving optimal thrust allocation is toguarantee efficient power use when dynamically positioningthe drilling rig in the specific conditions of a given oceanenvironment Meanwhile the economic impact of minimiz-ing power consumption is also taken into account with theultimate aim of ensuring the stability and capability of thedrilling rig during operation

3 Thrust Allocation Method

31 Optimization Problem for Thrust Allocation This studymathematically formulated an optimization problem forthrust allocation and each component of the optimizationproblem is described below in detail

311 Design Variables Thrust allocation involves determin-ing the thrust (119865

119894) and azimuth direction (120572

119894) of each thruster

to generate the forces in longitudinal and transverse direc-tions and the moment about the vertical direction that arerequired for the DP system of the drilling rigThus the thrustand azimuth direction of each thruster can be used as designvariables in the thrust allocation optimization problem Theobjective function that is commonly used for this problem isto minimize the total power of the thrusters due to energysavings In order to use this power as the objective functionit should be mathematically formulated as a function of thedesign variables The total power will be proportional tothe sum of the thrusts for each thruster Some researchers[5ndash9] used the thrust and the azimuth direction of eachthruster as design variables and have formulated the objectivefunction (minimization of total power of thrusters) withthe corresponding design variables However the rotationalspeed (119899

119894) and the azimuth direction (120572

119894) of each thruster can

be used as design variables to reflect the characteristics of thegiven thruster tomore easily control the thruster and tomoredelicately represent the objective function Thus the designvariables for the thrust allocation optimization problem inthis study are as follows

119899119894 speed of rotation of each thruster

120572119894 azimuth direction of each thruster

312 Objective Function In this study the speed of therotation (119899

119894) and the azimuth direction (120572

119894) of each of the

thrusters were used as design variables and thus the objectivefunction (minimization of the total power for the thrusters)was mathematically formulated according to the followingprocedure where in fact the power consumed by the electricpropulsion system depends on the speed of rotation and theazimuth direction of the thrusters

The thrust 119879 and torque 119876 of a thruster can be expressedas functions of the speed of rotation (119899)

119879 = 1205881198634

1198701198791198992

(4)

119876 = 1205881198635

1198701198761198992

(5)

where 120588 is the density of the sea water 119863 is the propellerdiameter of the thruster 119870

119879and 119870

119876are the thrust and

torque coefficients respectively and 119899 is the speed of therotation of the thruster The expressions for 119870

119879and 119870

119876can

be obtained from the so-called openwater test of the thrusterThese nondimensional coefficients for thrust and torque werederived through a regression analysis that can be describedaccording to the following parameters [11]

119870119879= sum119904119905119906V

119862119879

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

119870119876= sum119904119905119906V

119862119876

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

(6)

119862119879119904119905119906V and 119862

119876

119904119905119906V coefficients and 119904 119905 119906 and V terms canbe found in a report of the Wageningen B-Series propellers[11] and were used in this study Here 119875119863 is the pitchdiameter ratio 119860

119864119860119874is the blade area ratio and 119885 is the

number of blades of the thruster The coefficient for theadvance 119869 is a nondimensional description of the thrusterperformance that can be expressed according to the speed ofrotation 119899 and the speed of the advance of the thruster definedas 119881119860 which can be represented by the speed near each

thruster (119881) and the azimuth direction of each thruster (120572)Since each of the thrusters has a different speed of rotation inpractice119870

119879and119870

119876of each thruster can be different

119869 =119881119860

119899119863=119881 cos120572119899119863

(7)

Now it is time to formulate the total power consumed bythe thrusters of the drilling rig The power consumption (119875)can be formulated as

119875 = 2120587119899119894119876119894 (8)

By combining (5) and (8) the power consumption of thethrusters can be rewritten as

119875 = 21205871205881198635

119873

sum119894=1

1198701198761198941198993

119894 (9)

where119873 is the number of thrusters

Mathematical Problems in Engineering 5

Outboard thruster

X

YPort

Starboard

Afterward Forward

Port-forward side thrusters

Y

X

120601i120572i

120572i + 1

120601i + 1 Inboard thruster((i + 1)th thruster)

120579f

120579f

(ith thruster)

Figure 3 Interaction of two closely spaced azimuth thrusters

Therefore the objective function can be written relatingthe speed of rotation and the azimuth direction of eachthruster In this study the total power (power consumption)of the thrusters isminimized by setting the objective functionfor the thrust allocation optimization problem as

Minimize 119875 = 21205871205881198635119873

sum119894=1

1198701198761198941198993

119894 (10)

313 Constraints To obtain a valid thrust allocation thegoverning equations shown in (11) to (13) should be satisfiedThe equations represent the thrust and azimuth direction ofeach thruster that generate the forces required in longitudinal(sum119865119909) and transverse (sum119865

119910) directions as well as themoment

required about vertical direction (sum119872119911) to dynamically

position the vessel If these equations cannot be satisfied thedrilling rig starts to drift Thus these equations are used asequality constraints for the optimization problem

The governing equation for the longitudinal force can bestated as

sum119865119909

= 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894)

(11)

Similarly the governing equation for the transverse forcecan be stated as

sum119865119910

= 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894)

(12)

And finally the governing equation for the yawing mo-ment can be stated as

sum119872119911= 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot

+ 1198992

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot

+ 1198992

119894cos120572119894119897119910119894)

(13)

where sum119865119909 sum119865119910 and sum119872

119911are the forces required in

the longitudinal and transverse directions and the momentrequired about the vertical direction respectively and are allgiven from the control system in the dynamic positioningsystem

It is essential to minimize the thruster interaction of asemisubmersible drilling rig to ensure effective DP operation[5 12] In order to find a solution to provide the mosteffective thrust for operation additional constraints shouldbe considered by taking into account the physical locationsof the thrusters Thrusters that are close together have aninfluence on each other and if one thruster is in the streamof another the efficiency would drop significantly To avoidhaving this happen certain angles must be prohibited forthe azimuth direction Each thruster is assumed to have aforbidden angle of 120579

119891for the closest thruster in order to

prevent the direct interaction of the thruster as shown inFigure 3 With the port-forward pair the center of the angleof the outboard thruster 119894 is 120601

119894and for the inboard thruster

(119894 + 1) it is 120601119894+1 The positive angle is measured clockwise

while the negative angle is measured anticlockwise from thehorizontal axis Figure 3 shows the ATRs (Attainable ThrustRegions) for the two port-forward side thrusters 119894 and (119894 + 1)

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

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Mathematical Problems in Engineering 3

Table 1 Summary of related studies and comparison with this study

Studies Application Optimization algorithm Design variables forformulation

Consideration ofthruster interaction

Wit [5] Ship Local algorithm (QP) Thrust and azimuth direction ⃝

Liang and Cheng [6]Johansen et al [7] Ship Local algorithm (SQP) Thrust and azimuth direction times

Parikshit [8] Drilling rig Global algorithm (GA ITHS etc) Thrust and azimuth direction ⃝

Zhao et al [9] Drilling rig Global algorithm (GA) Thrust and azimuth direction ⃝

This study Drilling rig Global algorithm (hybridoptimization algorithm)

Rotation speed and azimuthdirection ⃝

T7

T8

T1

T2

T3

T4T6

T5

Port

Starboard

Afterward Forward

lx1

ly1

y

xsumMz

sumFy

sumFx

Figure 2 Schematic of thruster arrangement of a drilling rig with eight thrusters (Plan view)

algorithm that incorporates global and local optimizationalgorithms was used for the optimization

Table 1 shows a summary of related studies and a compar-ison of them with this study

2 Thrust Allocation for Dynamic Positioning

A thruster system is important for a drilling rig to maintainits position and heading since it can simultaneously providetransverse and longitudinal thrust An azimuth thrusterconsists of an electric podded drive that is fitted to thehull and can rotate 360 degrees along the horizontal angleto provide thrust in all directions The azimuth thrusterhas been selected in this study because it can providethrust in any direction in order to act as a propulsor fordynamic positioning and target optimization of a drillingrig However the thruster system is usually overactuated inpractice so a thrust allocation problem can be formulatedinto a constrained optimization problem for the system

Figure 2 depicts a schematic diagram for the thrusterarrangement of the drilling rig and the sign conventions forthe forces (119865

119909and 119865

119910) and moment (119872

119911) In this figure

sum119865119909 sum119865119910 and sum119872

119911represent the total forces required

in the longitudinal (119909) and transverse (119910) directions andthe total required moment about the vertical (119911) directionfrom the control system as shown in Figure 1 respectivelyNote that the positions of the thrusters are given in a 2D

coordinate system (only the horizontal forces and momentare considered) The coordinate system has its origin at thecenter of gravity (CoG) of the drilling rig

The environmental forces and the moment tend to movethe drilling rig away from its original position during oper-ation at sea In order to move the drilling rig back to itsoriginal or reference position the control system in the DPsystem of the drilling rig should first calculate the total forces(thrusts) andmoment required At this time thrust allocationshould be conducted to determine the thrust and azimuthdirection of each thruster so that the required forces (andmoment) are generated in each of the longitudinal (sum119865

119909)

and transverse (sum119865119910) and vertical (sum119872

119911) directions The

relationship between the thrust and the azimuth directionof each thruster and the total forces and moment that arerequired can be stated as in (1) to (3) These equations are thegoverning equations for the thrust allocation and can be usedas equality constraints in an optimization problem

sum119865119909= (1198651cos1205721+ 1198652cos1205722+ sdot sdot sdot + 119865

119899cos120572119899) (1)

sum119865119910= (1198651sin1205721+ 1198652sin1205722+ sdot sdot sdot + 119865

119899sin120572119899) (2)

sum119872119911= (1198651cos1205721sdot 1198971199101+ 1198652cos1205722sdot 1198971199102+ sdot sdot sdot

+ 119865119899cos120572119899sdot 119897119910119899+ 1198651sin1205721sdot 1198971199091+ 1198652sin1205722sdot 1198971199092

+ sdot sdot sdot + 119865119899sin120572119899sdot 119897119909119899)

(3)

4 Mathematical Problems in Engineering

Supposing that the drilling rig is equipped with 119894 azimuththrusters 119865

119894is the thrust from 119894th thruster and 120572

119894is the

azimuth direction (angle) of 119894th thruster 119897119909119894and 119897119910119894are the

distances of 119894th thruster from the CoG of the drilling rigEquation (1) indicates that the longitudinal component ofthe force generated from the thrust and azimuth directionof each thruster should be equal to the total force requiredin the longitudinal direction Similarly (2) indicates thatthe transverse component of the force from the thrust andazimuth direction of each thruster should be equal to thetotal force required in the transverse direction Equation (3)means that the vertical component of the moment generatedfrom the thrust and azimuth direction of each thruster shouldbe equal to the total required moment about the verticaldirection

The goal of achieving optimal thrust allocation is toguarantee efficient power use when dynamically positioningthe drilling rig in the specific conditions of a given oceanenvironment Meanwhile the economic impact of minimiz-ing power consumption is also taken into account with theultimate aim of ensuring the stability and capability of thedrilling rig during operation

3 Thrust Allocation Method

31 Optimization Problem for Thrust Allocation This studymathematically formulated an optimization problem forthrust allocation and each component of the optimizationproblem is described below in detail

311 Design Variables Thrust allocation involves determin-ing the thrust (119865

119894) and azimuth direction (120572

119894) of each thruster

to generate the forces in longitudinal and transverse direc-tions and the moment about the vertical direction that arerequired for the DP system of the drilling rigThus the thrustand azimuth direction of each thruster can be used as designvariables in the thrust allocation optimization problem Theobjective function that is commonly used for this problem isto minimize the total power of the thrusters due to energysavings In order to use this power as the objective functionit should be mathematically formulated as a function of thedesign variables The total power will be proportional tothe sum of the thrusts for each thruster Some researchers[5ndash9] used the thrust and the azimuth direction of eachthruster as design variables and have formulated the objectivefunction (minimization of total power of thrusters) withthe corresponding design variables However the rotationalspeed (119899

119894) and the azimuth direction (120572

119894) of each thruster can

be used as design variables to reflect the characteristics of thegiven thruster tomore easily control the thruster and tomoredelicately represent the objective function Thus the designvariables for the thrust allocation optimization problem inthis study are as follows

119899119894 speed of rotation of each thruster

120572119894 azimuth direction of each thruster

312 Objective Function In this study the speed of therotation (119899

119894) and the azimuth direction (120572

119894) of each of the

thrusters were used as design variables and thus the objectivefunction (minimization of the total power for the thrusters)was mathematically formulated according to the followingprocedure where in fact the power consumed by the electricpropulsion system depends on the speed of rotation and theazimuth direction of the thrusters

The thrust 119879 and torque 119876 of a thruster can be expressedas functions of the speed of rotation (119899)

119879 = 1205881198634

1198701198791198992

(4)

119876 = 1205881198635

1198701198761198992

(5)

where 120588 is the density of the sea water 119863 is the propellerdiameter of the thruster 119870

119879and 119870

119876are the thrust and

torque coefficients respectively and 119899 is the speed of therotation of the thruster The expressions for 119870

119879and 119870

119876can

be obtained from the so-called openwater test of the thrusterThese nondimensional coefficients for thrust and torque werederived through a regression analysis that can be describedaccording to the following parameters [11]

119870119879= sum119904119905119906V

119862119879

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

119870119876= sum119904119905119906V

119862119876

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

(6)

119862119879119904119905119906V and 119862

119876

119904119905119906V coefficients and 119904 119905 119906 and V terms canbe found in a report of the Wageningen B-Series propellers[11] and were used in this study Here 119875119863 is the pitchdiameter ratio 119860

119864119860119874is the blade area ratio and 119885 is the

number of blades of the thruster The coefficient for theadvance 119869 is a nondimensional description of the thrusterperformance that can be expressed according to the speed ofrotation 119899 and the speed of the advance of the thruster definedas 119881119860 which can be represented by the speed near each

thruster (119881) and the azimuth direction of each thruster (120572)Since each of the thrusters has a different speed of rotation inpractice119870

119879and119870

119876of each thruster can be different

119869 =119881119860

119899119863=119881 cos120572119899119863

(7)

Now it is time to formulate the total power consumed bythe thrusters of the drilling rig The power consumption (119875)can be formulated as

119875 = 2120587119899119894119876119894 (8)

By combining (5) and (8) the power consumption of thethrusters can be rewritten as

119875 = 21205871205881198635

119873

sum119894=1

1198701198761198941198993

119894 (9)

where119873 is the number of thrusters

Mathematical Problems in Engineering 5

Outboard thruster

X

YPort

Starboard

Afterward Forward

Port-forward side thrusters

Y

X

120601i120572i

120572i + 1

120601i + 1 Inboard thruster((i + 1)th thruster)

120579f

120579f

(ith thruster)

Figure 3 Interaction of two closely spaced azimuth thrusters

Therefore the objective function can be written relatingthe speed of rotation and the azimuth direction of eachthruster In this study the total power (power consumption)of the thrusters isminimized by setting the objective functionfor the thrust allocation optimization problem as

Minimize 119875 = 21205871205881198635119873

sum119894=1

1198701198761198941198993

119894 (10)

313 Constraints To obtain a valid thrust allocation thegoverning equations shown in (11) to (13) should be satisfiedThe equations represent the thrust and azimuth direction ofeach thruster that generate the forces required in longitudinal(sum119865119909) and transverse (sum119865

119910) directions as well as themoment

required about vertical direction (sum119872119911) to dynamically

position the vessel If these equations cannot be satisfied thedrilling rig starts to drift Thus these equations are used asequality constraints for the optimization problem

The governing equation for the longitudinal force can bestated as

sum119865119909

= 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894)

(11)

Similarly the governing equation for the transverse forcecan be stated as

sum119865119910

= 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894)

(12)

And finally the governing equation for the yawing mo-ment can be stated as

sum119872119911= 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot

+ 1198992

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot

+ 1198992

119894cos120572119894119897119910119894)

(13)

where sum119865119909 sum119865119910 and sum119872

119911are the forces required in

the longitudinal and transverse directions and the momentrequired about the vertical direction respectively and are allgiven from the control system in the dynamic positioningsystem

It is essential to minimize the thruster interaction of asemisubmersible drilling rig to ensure effective DP operation[5 12] In order to find a solution to provide the mosteffective thrust for operation additional constraints shouldbe considered by taking into account the physical locationsof the thrusters Thrusters that are close together have aninfluence on each other and if one thruster is in the streamof another the efficiency would drop significantly To avoidhaving this happen certain angles must be prohibited forthe azimuth direction Each thruster is assumed to have aforbidden angle of 120579

119891for the closest thruster in order to

prevent the direct interaction of the thruster as shown inFigure 3 With the port-forward pair the center of the angleof the outboard thruster 119894 is 120601

119894and for the inboard thruster

(119894 + 1) it is 120601119894+1 The positive angle is measured clockwise

while the negative angle is measured anticlockwise from thehorizontal axis Figure 3 shows the ATRs (Attainable ThrustRegions) for the two port-forward side thrusters 119894 and (119894 + 1)

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Supposing that the drilling rig is equipped with 119894 azimuththrusters 119865

119894is the thrust from 119894th thruster and 120572

119894is the

azimuth direction (angle) of 119894th thruster 119897119909119894and 119897119910119894are the

distances of 119894th thruster from the CoG of the drilling rigEquation (1) indicates that the longitudinal component ofthe force generated from the thrust and azimuth directionof each thruster should be equal to the total force requiredin the longitudinal direction Similarly (2) indicates thatthe transverse component of the force from the thrust andazimuth direction of each thruster should be equal to thetotal force required in the transverse direction Equation (3)means that the vertical component of the moment generatedfrom the thrust and azimuth direction of each thruster shouldbe equal to the total required moment about the verticaldirection

The goal of achieving optimal thrust allocation is toguarantee efficient power use when dynamically positioningthe drilling rig in the specific conditions of a given oceanenvironment Meanwhile the economic impact of minimiz-ing power consumption is also taken into account with theultimate aim of ensuring the stability and capability of thedrilling rig during operation

3 Thrust Allocation Method

31 Optimization Problem for Thrust Allocation This studymathematically formulated an optimization problem forthrust allocation and each component of the optimizationproblem is described below in detail

311 Design Variables Thrust allocation involves determin-ing the thrust (119865

119894) and azimuth direction (120572

119894) of each thruster

to generate the forces in longitudinal and transverse direc-tions and the moment about the vertical direction that arerequired for the DP system of the drilling rigThus the thrustand azimuth direction of each thruster can be used as designvariables in the thrust allocation optimization problem Theobjective function that is commonly used for this problem isto minimize the total power of the thrusters due to energysavings In order to use this power as the objective functionit should be mathematically formulated as a function of thedesign variables The total power will be proportional tothe sum of the thrusts for each thruster Some researchers[5ndash9] used the thrust and the azimuth direction of eachthruster as design variables and have formulated the objectivefunction (minimization of total power of thrusters) withthe corresponding design variables However the rotationalspeed (119899

119894) and the azimuth direction (120572

119894) of each thruster can

be used as design variables to reflect the characteristics of thegiven thruster tomore easily control the thruster and tomoredelicately represent the objective function Thus the designvariables for the thrust allocation optimization problem inthis study are as follows

119899119894 speed of rotation of each thruster

120572119894 azimuth direction of each thruster

312 Objective Function In this study the speed of therotation (119899

119894) and the azimuth direction (120572

119894) of each of the

thrusters were used as design variables and thus the objectivefunction (minimization of the total power for the thrusters)was mathematically formulated according to the followingprocedure where in fact the power consumed by the electricpropulsion system depends on the speed of rotation and theazimuth direction of the thrusters

The thrust 119879 and torque 119876 of a thruster can be expressedas functions of the speed of rotation (119899)

119879 = 1205881198634

1198701198791198992

(4)

119876 = 1205881198635

1198701198761198992

(5)

where 120588 is the density of the sea water 119863 is the propellerdiameter of the thruster 119870

119879and 119870

119876are the thrust and

torque coefficients respectively and 119899 is the speed of therotation of the thruster The expressions for 119870

119879and 119870

119876can

be obtained from the so-called openwater test of the thrusterThese nondimensional coefficients for thrust and torque werederived through a regression analysis that can be describedaccording to the following parameters [11]

119870119879= sum119904119905119906V

119862119879

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

119870119876= sum119904119905119906V

119862119876

119904119905119906V (119869)119904

(119875

119863)119905

(119860119864

119860119874

)

119906

(119885)V

(6)

119862119879119904119905119906V and 119862

119876

119904119905119906V coefficients and 119904 119905 119906 and V terms canbe found in a report of the Wageningen B-Series propellers[11] and were used in this study Here 119875119863 is the pitchdiameter ratio 119860

119864119860119874is the blade area ratio and 119885 is the

number of blades of the thruster The coefficient for theadvance 119869 is a nondimensional description of the thrusterperformance that can be expressed according to the speed ofrotation 119899 and the speed of the advance of the thruster definedas 119881119860 which can be represented by the speed near each

thruster (119881) and the azimuth direction of each thruster (120572)Since each of the thrusters has a different speed of rotation inpractice119870

119879and119870

119876of each thruster can be different

119869 =119881119860

119899119863=119881 cos120572119899119863

(7)

Now it is time to formulate the total power consumed bythe thrusters of the drilling rig The power consumption (119875)can be formulated as

119875 = 2120587119899119894119876119894 (8)

By combining (5) and (8) the power consumption of thethrusters can be rewritten as

119875 = 21205871205881198635

119873

sum119894=1

1198701198761198941198993

119894 (9)

where119873 is the number of thrusters

Mathematical Problems in Engineering 5

Outboard thruster

X

YPort

Starboard

Afterward Forward

Port-forward side thrusters

Y

X

120601i120572i

120572i + 1

120601i + 1 Inboard thruster((i + 1)th thruster)

120579f

120579f

(ith thruster)

Figure 3 Interaction of two closely spaced azimuth thrusters

Therefore the objective function can be written relatingthe speed of rotation and the azimuth direction of eachthruster In this study the total power (power consumption)of the thrusters isminimized by setting the objective functionfor the thrust allocation optimization problem as

Minimize 119875 = 21205871205881198635119873

sum119894=1

1198701198761198941198993

119894 (10)

313 Constraints To obtain a valid thrust allocation thegoverning equations shown in (11) to (13) should be satisfiedThe equations represent the thrust and azimuth direction ofeach thruster that generate the forces required in longitudinal(sum119865119909) and transverse (sum119865

119910) directions as well as themoment

required about vertical direction (sum119872119911) to dynamically

position the vessel If these equations cannot be satisfied thedrilling rig starts to drift Thus these equations are used asequality constraints for the optimization problem

The governing equation for the longitudinal force can bestated as

sum119865119909

= 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894)

(11)

Similarly the governing equation for the transverse forcecan be stated as

sum119865119910

= 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894)

(12)

And finally the governing equation for the yawing mo-ment can be stated as

sum119872119911= 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot

+ 1198992

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot

+ 1198992

119894cos120572119894119897119910119894)

(13)

where sum119865119909 sum119865119910 and sum119872

119911are the forces required in

the longitudinal and transverse directions and the momentrequired about the vertical direction respectively and are allgiven from the control system in the dynamic positioningsystem

It is essential to minimize the thruster interaction of asemisubmersible drilling rig to ensure effective DP operation[5 12] In order to find a solution to provide the mosteffective thrust for operation additional constraints shouldbe considered by taking into account the physical locationsof the thrusters Thrusters that are close together have aninfluence on each other and if one thruster is in the streamof another the efficiency would drop significantly To avoidhaving this happen certain angles must be prohibited forthe azimuth direction Each thruster is assumed to have aforbidden angle of 120579

119891for the closest thruster in order to

prevent the direct interaction of the thruster as shown inFigure 3 With the port-forward pair the center of the angleof the outboard thruster 119894 is 120601

119894and for the inboard thruster

(119894 + 1) it is 120601119894+1 The positive angle is measured clockwise

while the negative angle is measured anticlockwise from thehorizontal axis Figure 3 shows the ATRs (Attainable ThrustRegions) for the two port-forward side thrusters 119894 and (119894 + 1)

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Outboard thruster

X

YPort

Starboard

Afterward Forward

Port-forward side thrusters

Y

X

120601i120572i

120572i + 1

120601i + 1 Inboard thruster((i + 1)th thruster)

120579f

120579f

(ith thruster)

Figure 3 Interaction of two closely spaced azimuth thrusters

Therefore the objective function can be written relatingthe speed of rotation and the azimuth direction of eachthruster In this study the total power (power consumption)of the thrusters isminimized by setting the objective functionfor the thrust allocation optimization problem as

Minimize 119875 = 21205871205881198635119873

sum119894=1

1198701198761198941198993

119894 (10)

313 Constraints To obtain a valid thrust allocation thegoverning equations shown in (11) to (13) should be satisfiedThe equations represent the thrust and azimuth direction ofeach thruster that generate the forces required in longitudinal(sum119865119909) and transverse (sum119865

119910) directions as well as themoment

required about vertical direction (sum119872119911) to dynamically

position the vessel If these equations cannot be satisfied thedrilling rig starts to drift Thus these equations are used asequality constraints for the optimization problem

The governing equation for the longitudinal force can bestated as

sum119865119909

= 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894)

(11)

Similarly the governing equation for the transverse forcecan be stated as

sum119865119910

= 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894)

(12)

And finally the governing equation for the yawing mo-ment can be stated as

sum119872119911= 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot

+ 1198992

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot

+ 1198992

119894cos120572119894119897119910119894)

(13)

where sum119865119909 sum119865119910 and sum119872

119911are the forces required in

the longitudinal and transverse directions and the momentrequired about the vertical direction respectively and are allgiven from the control system in the dynamic positioningsystem

It is essential to minimize the thruster interaction of asemisubmersible drilling rig to ensure effective DP operation[5 12] In order to find a solution to provide the mosteffective thrust for operation additional constraints shouldbe considered by taking into account the physical locationsof the thrusters Thrusters that are close together have aninfluence on each other and if one thruster is in the streamof another the efficiency would drop significantly To avoidhaving this happen certain angles must be prohibited forthe azimuth direction Each thruster is assumed to have aforbidden angle of 120579

119891for the closest thruster in order to

prevent the direct interaction of the thruster as shown inFigure 3 With the port-forward pair the center of the angleof the outboard thruster 119894 is 120601

119894and for the inboard thruster

(119894 + 1) it is 120601119894+1 The positive angle is measured clockwise

while the negative angle is measured anticlockwise from thehorizontal axis Figure 3 shows the ATRs (Attainable ThrustRegions) for the two port-forward side thrusters 119894 and (119894 + 1)

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The ATR represents a set of all physically possible surge andsway forces for 119894th thruster

Considering the ATR the constraints for the azimuthdirection of thrusters 119894 and (119894 + 1) can be formulated as

120579119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

120579119891minus1003816100381610038161003816120572119894+1 minus 120601119894+1

1003816100381610038161003816 le 0(14)

where 120579119891represents the forbidden angle of the thrusters 120572

119894

and 120572119894+1

are the azimuth directions of the thrusters 119894 and(119894 + 1) respectively 120601

119894and 120601

119894+1are the center of angle for

the outboard thruster 119894 and for the inboard thruster (119894 + 1)respectively

314 Summary The optimization problem for thrust alloca-tion can be summarized as follows

Minimize 119891 (x) = 119875 = 21205871205881198635119899

sum119894=1

1198701198761198941198993

119894

Subject to 1198921(x) = sum119865

119909minus 1205881198632

119870119879119894(1198992

1cos1205721+ 1198992

2cos1205722+ sdot sdot sdot + 119899

2

119894cos120572119894) = 0

1198922(x) = sum119865

119910minus 1205881198632

119870119879119894(1198992

1sin1205721+ 1198992

2sin1205722+ sdot sdot sdot + 119899

2

119894sin120572119894) = 0

1198923(x)

= sum119872119911minus 1205881198632

119870119879119894(1198992

1sin12057211198971199091+ 1198992

2sin12057221198971199092+ sdot sdot sdot + 119899

2

119894sin120572119894119897119909119894+ 1198992

1cos12057211198971199101+ 1198992

2cos12057221198971199102+ sdot sdot sdot + 119899

2

119894cos120572119894119897119910119894)

= 0

1198924+119894(x) = 120579

119891minus1003816100381610038161003816120572119894 minus 120601119894

1003816100381610038161003816 le 0

(15)

where x = 119899119894 120572119894 119894 = 1 119873 (119873 number of thrusters)

Thus this problem has one objective function threeequality constraints and119873 inequality constraints

32 Hybrid Optimization Algorithm for Thrust Allocation

321 Overview The optimization algorithms are generallydivided into two categories global and local Several classes ofglobal optimization algorithms are now available includingthe genetic algorithm (GA) [13 14] and the simulated anneal-ing method These algorithms are suitable for large-scaleproblems withmany local optima However these algorithmsrequire further iteration to obtain an acceptable optimumand not an accurate optimum in contrast to the localoptimization algorithms Several classes of the local opti-mization algorithms also exist including sequential linearprogramming (SLP) [15] and sequential quadratic program-ming (SQP) [15] and themethod of feasible directions (MFD)[16] Each of these algorithms can effectively determine anaccurate optimum However in some cases these algorithmsfind the local optimum that is closest to the given startingpoint

Various attempts have been made to combine globaland local optimization algorithms in order to overcomethe challenges of using them separately [10 17ndash21] Mostof these studies used a combination of a global (eg GA)and a local optimization algorithm which can be referredto as a hybrid optimization algorithm In this study thehybrid optimization algorithm was selected and used byincorporating the GA and the SQP algorithm to solve an

optimization problem for thrust allocation as described inSection 31 In this case the SQP algorithm was used toimprove the acceptable global optimum obtained from theGA That is at the end of the GA the SQP algorithmwas executed with a starting point that corresponds to theoptimum obtained from the GA Figure 4 shows the generalprocedure of the hybrid optimization algorithm used in thisstudy

An experiment on the mathematical optimization prob-lem was performed in order to verify the efficiency accuracyand applicability of the hybrid optimization algorithm thatwas used to optimize the thrust allocation in this studyThe selected problem which is Rastriginrsquos problem one ofthe benchmark problems is being widely used to check theefficiency of optimization algorithms [22]More details aboutcomparative test of the hybrid optimization algorithm can befound in [10]

322 Optimization Procedure The following optimizationprocedure was established to solve the optimization problemthat was presented formulated above First the initial valueswere assumed for the design variables At this time thevalues can be randomly generated or can be manually setor extracted from an existing design Now these values aretransferred to the hybrid optimization algorithm and thevalues of an objective function and constraints are thencalculated We then check whether the current values ofthe design variables are at an optimum or not If yes theoptimization process finishes and the result will be shownand if not the above steps will be repeated until the optimum

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Evaluate fitness

Reproduction

Replace population

Until terminationcriteria are met

EndPerform

refinement(local optimization)

Until temporarypopulation is full

Initialize population

Perform crossoverPerform selection

Evaluate fitness

Global optimization bygenetic algorithm

Local optimization bySQP algorithm

probability

Perform mutation

probability with mutation

with crossoverSelect parent 1Select parent 2

Figure 4 General procedure of the hybrid optimization algorithmused in this study

is found Figure 5 shows this optimization procedure to findan optimal thrust allocation

4 Application of the ThrustAllocation Method

41 Optimization Target The optimization target for thisstudy is a semisubmersible drilling rig equipped with eightazimuth thrusters Figure 5 shows the schematics of thisdrilling rig with a thruster system and its position Weassumed that each thruster has 4 propeller blades (119885) apropeller diameter (119863) of 36m a pitch diameter ratio (119875119863)of 11 a blade area ratio (119860

119864119860119874) of 07 and a maximum

thrust of 600KN In addition we assumed that the speed(119881) near each thruster was of 1ms The longitudinal andtransverse positions of each thruster can be seen in Figure 6

42 Mathematical Formulation This drilling rig has eightthrusters Thus the design variables in this problem are asfollows

x = 1198991 1205721 1198992 1205722 1198993 1205723 1198994 1205724 1198995 1205725 1198996 1205726 1198997 1205727

1198998 1205728

(16)

where 119899119894and 120572

119894are the speed of rotation and the azimuth

direction of 119894th thruster respectively and thus this problemhas 16 design variables

force governing equation for transverse force

Minimize f(x) = Power consumption

governing equation for yawing moment constraints for maximum thrust and azimuth direction

Hybrid optimization algorithm

Optimization result

Yes

No

Subject to g(x) = Governing equation for longitudinal

Initial values x = ni 120572i

x is optimum

Figure 5 Optimization procedure for finding optimal thrust allo-cation

Now the objective function of this problem can be statedby using (10)

Minimize 119891 (x) = 119875 = 212058712058811986358

sum119894=1

1198701198761198941198993

119894

= 21205871205881198635

(11987011987611198993

1+ 11987011987621198993

2+ 11987011987631198993

3+ 11987011987641198993

4

+ 11987011987651198993

5+ 11987011987661198993

6+ 11987011987671198993

7+ 11987011987681198993

8)

(17)

where 120588 is the density of sea water119863 is the propeller diameterof the thruster and 119870

119879and 119870

119876are the thruster and torque

coefficients respectively 119870119879and 119870

119876were calculated from

(6) by referring to the report on the Wageningen B-Seriespropellers [11]

The constraints for the required forces and moment canbe stated using (11) to (13)

1198921(x) = sum119865

119909minus 1205881198632

(11987011987911198992

1cos1205721+ 11987011987921198992

2cos1205722

+ 11987011987931198992

3cos1205723+ 11987011987941198992

4cos1205724+ 11987011987951198992

5cos1205725

+ 11987011987961198992

6cos1205726+ 11987011987971198992

7cos1205727+ 11987011987981198992

8cos1205728)

= 0

(18)

1198922(x) = sum119865

119910minus 1205881198632

(11987011987911198992

1sin1205721+ 11987011987921198992

2sin1205722

+ 11987011987931198992

3sin1205723+ 11987011987941198992

4sin1205724+ 11987011987951198992

5sin1205725

+ 11987011987961198992

6sin1205726+ 11987011987971198992

7sin1205727+ 11987011987981198992

8sin1205728) = 0

(19)

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

44749m

40810m

33500m

33500m25499m

25499m

Port

Starboard

Afterward Forward

44741m

40735m

T7

T8

T1

T2

T3

T4T6

T5

Figure 6 The longitudinal and transverse positions of eight azimuth thrusters (Plan view)

1198923(x) = sum119872

119911minus 1205881198632

(11987011987911198992

1sin12057211198971199091

+ 11987011987921198992

2sin12057221198971199092+ 11987011987931198992

3sin12057231198971199093

+ 11987011987941198992

4sin12057241198971199094+ 11987011987951198992

5sin12057251198971199095

+ 11987011987961198992

6sin12057261198971199096+ 11987011987971198992

7sin12057271198971199097

+ 11987011987981198992

8sin12057281198971199098+ 11987011987911198992

1cos12057211198971199101

+ 11987011987921198992

2cos12057221198971199102+ 11987011987931198992

3cos12057231198971199103

+ 11987011987941198992

4cos12057241198971199104+ 11987011987951198992

5cos12057251198971199105

+ 11987011987961198992

6cos12057261198971199106+ 11987011987971198992

7cos12057271198971199107

+ 11987011987981198992

8cos12057281198971199108) = 0

(20)

Equations (18) to (20) ensure that the forces generalizedby the eight thrusters are equal to the forces demanded fromthe control system and themoment should also be satisfied inthe same way to keep the drilling rig in the desired position

In order to overcome the energy loss due to the thruster-thruster interaction the ATR should be considered exclud-ing forbidden zones by using (14) as shown in Figure 7For this each thruster should satisfy the following equationssimultaneously

1198924(x) = 25∘ minus 10038161003816100381610038161205721 minus 6344

∘1003816100381610038161003816 le 0

1198925(x) = 25∘ minus 10038161003816100381610038161205722 + 11656

∘1003816100381610038161003816 le 0

1198926(x) = 25∘ minus 10038161003816100381610038161205723 minus 11656

∘1003816100381610038161003816 le 0

1198927(x) = 25∘ minus 10038161003816100381610038161205724 + 6344

∘1003816100381610038161003816 le 0

1198928(x) = 25∘ minus 10038161003816100381610038161205725 minus 6344

∘1003816100381610038161003816 le 0

1198929(x) = 25∘ minus 10038161003816100381610038161205726 + 11656

∘1003816100381610038161003816 le 0

11989210(x) = 25∘ minus 10038161003816100381610038161205727 minus 11656

∘1003816100381610038161003816 le 0

11989211(x) = 25∘ minus 10038161003816100381610038161205728 + 6344

∘1003816100381610038161003816 le 0

(21)

In addition there are some limitations on the maximumthrust and azimuth direction for each thruster Thus suchlimitations were also considered as additional constraints inthis study The additional constraints about the maximumthrust and the azimuth direction of each thruster can bestated in the following equations respectively

11989211+119894

= 119879119894minus 600 (KN) le 0 (22)

11989219+119894

=10038161003816100381610038161205721198941003816100381610038161003816 minus 180

∘

le 0 (23)

where 119894 = 1 8Thus this problem has 3 equality constraints and 24

inequality constraints

43 Optimization Result The problem in (16) to (23) wassolved by using the hybrid optimization algorithm describedin Section 32 and then the problem was also solved byusing the GA in order to compare their relative efficienciesWe assumed that the total required forces (sum119865

119909 sum 119865119910) and

moment (sum119872119911) are given from the control system at 14 time

steps as stated in [8] and as shown in Table 2 For example

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Port

Starboard

Afterward Forward

1206011 = 6344∘

1206014 = minus6344∘

1206018 = minus6344∘

1206015 = 6344∘

T7

T8

T6

T5

T1

T2

T3

T4

1206016 = minus11656∘1206013 = 11656∘

1206012 = minus11656∘

1206017 = 11656∘

Figure 7 Schematic diagram of thruster interaction layout (Plan view)

20000

10000

0

10000

20000

30000

40000

0 500 1000 1500 2000 2500 3000 3500

Objective functionConstraints violation

Optimization by GAup to generation 3000

Optimization by SQPfrom generation 3001 sim 3015

All constraints were satisfied

Tota

l pow

er co

nsum

ptio

n (k

W)

Max

cons

trai

nts

viol

atio

n

(P = 9859kW)

(P = 8250kW)from generations

Figure 8 Convergence history of the optimization by the hybrid optimization algorithm for first time step

the values for the first time step are 50KN minus600KN andminus64000KNsdotm For a total of 14 inputs for the total requiredforces andmoment the optimizationwas performed by usingthe GA and the hybrid optimization algorithm and the opti-mization results are shown in Table 2 As mentioned earlierthe hybrid optimization algorithm is a combination of theGAand the SQP algorithmThat is at the end of the GA the SQPalgorithmwas executedwith a starting point that correspondsto the optimum obtained from the GA With respect to theparameters for the GA the number of individuals in thepopulation was 200 the maximum number of generationswas 3000 the crossover probability was 08 and themutationprobability was 001 [23] As shown in Table 2 the hybrid

optimization algorithmoutperforms theGAalthough theGAyielded similar results for certain time steps Figure 8 shows aconvergence history of the optimization results for the firsttime step This figure shows that the objective function isminimized and that the constraints are satisfied at the sametime

To evaluate the applicability of the proposed methodthe optimization results of this study were compared againstthose obtained in Parikshitrsquos study [8] The same inputs wereused for the total forces and moment required from thecontrol system for the comparative test Parikshit used anITHS (Intelligent TunedHarmony Search) algorithm to solvethis problemThe ITHS algorithmmaintains a proper balance

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table2Optim

izationresultfor14tim

esteps

Time

steps

Inpu

tfrom

thec

ontro

lsystem

Thruste

rsGAon

ly[A

]Hybrid

[B]

(GA+SQ

P)Ra

tio(BA

)

Thruste

r1Th

ruste

r2Th

ruste

r3Th

ruste

r4Th

ruste

r5Th

ruste

r6Th

ruste

r7Th

ruste

r8Re

quire

dforce

Requ

iredmom

ent

RPM

Ang

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leRP

MAng

leTo

tal

power

Totalp

ower

(KN)

(KNsdotm)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(rpm

)(deg)

(kW)

(kW)

Δ119905(s)sum119865119909sum119865119910

sum119872119911

1198991120572111989921205722119899312057231198994120572411989951205725119899612057261198997120572711989981205728

119875119875

mdash1

50minus60

0minus64

000

310minus93

062minus91

009

85055minus103

22

37018

178

022

173

166

167

9859

8250

084

2minus500

1500

minus25000

251

11510

2171

129

152

079

93003minus64

190

7815

281

228

11412882

10824

084

3100minus800

17000

078minus35

157minus78

102minus2

054minus13

261minus95

026minus15

017minus92

045

207186

6580

092

4750minus550

63000

2026

34061minus177

076

170

007

32062minus78

199minus78

003

40264minus48

11800

9172

078

5175

560

minus32500

008

13092

147

005

41073minus128

274

74111minus35

015minus41

029minus146

7121

7107

100

6minus50minus10

80300

0217minus116

004

129

089minus65

032minus136

245minus85

142minus70

009minus141

004minus4

8840

7360

083

7minus300

1560

12000

058

129

218

68229

97026

5419

8108

128minus180

007minus65

229

112

1374

99522

070

810

50250

minus36000

080minus84

052

144

064

29240

901

165

278

1113

251

013minus143

9623

7519

078

9minus60

0minus850

minus22000

209minus120

038

91034

173

242minus102

021

1719

7minus180

020minus106

08minus91

9687

7380

076

10minus12

00250

minus11000

127minus165

131

169

228minus176

003

39208

146

029

134

182

161

007minus37

9206

8140

088

11minus500

1200

minus10000

003

125

094

148

247

97013minus150

189

86110

153

020minus9

232

138

11280

10277

091

1260

0550

minus70

00036minus10

193

26004minus72

226

12

111

138

139

140

024

139

9575

4855

050

1350minus930

1000

110minus120

118minus92

172minus78

084minus89

207minus107

096minus88

142

016

626541

3949

060

14minus700

75minus26000

123minus137

104

57018minus12

018minus34

053minus8

006

9227

168

186

180

6243

4802

077

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

mdashMean

079

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 3 Comparison of the optimization results between theexisting study and this study

Time steps ITHS [A] Hybrid [B] Ratio(BA)Total power (kW) Total power (kW)

Δ119905 (s) 119875 119875 mdash1 7500 8250 1102 10850 10824 0993 7100 6580 0924 7800 9172 1175 7900 7107 0896 5800 7360 1267 9800 9522 0978 7000 7519 1079 7500 7379 09810 9800 9206 08311 8900 10277 11512 6200 4855 07813 4000 3949 09814 7600 4802 063mdash mdash Mean 098

between the diversification and intensification throughoutthe search process by automatically selecting the proper pitchadjustment strategy based on its harmony memory Table 3shows a comparison of the optimization results betweenthe existing study and this study As shown in this tablethe results of this study were slightly better than those ofParikshitrsquos study in terms of a mean value of the ratio

5 Conclusions and Future Works

In this study a thrust allocation method was proposedfor a semisubmersible drilling rig in order to producethe generalized forces and moment that are required todynamically position the rig while at the same time mini-mizing the total power consumed First a thrust allocationoptimization problem was mathematically formulated withthe corresponding design variables objective function andconstraints In terms of the design variables the speed ofrotation and the azimuth direction of each thruster wereselected As compared with some studies about optimalthruster allocation the selection of the design variables cannot only represent the objection function more precisely butalso control the thruster more easily The objective functionaimed to minimize the total power of the thrusters andin terms of the constraints the governing equations for thethrust and azimuth direction of each thruster were used togenerate the forces required in longitudinal and transversedirections as well as the moment about the vertical directionto dynamically position the vessel Some limitations werealso used for each thruster and additional constraints wereintroduced by considering the energy loss due to the thruster-thruster interaction and the maximum thrust and azimuthdirection of each thrusterThe hybrid optimization algorithm

was then used to solve the formulated problem Finally theproposed method was applied to an example to find theoptimal thrust allocation for the semisubmersible drillingrig with 8 thrusters A comparative test was also performedas part of the current study In this comparative test theproposed method was observed to produce slightly betterresults (about 2 in terms of total power) for thrust allocationthan existing study and thus the proposed method could beused to better determine a strategy to allocate the thruster ofthe drilling rig

In the future the total forces and moment required fromthe control system which was the input of this study willbe estimated by considering the current position and theequations of motion of the drilling rig That is a moregeneralmethod to dynamically position the rigwill be furtherstudied In addition we will improve the present hybridoptimization method and apply the improved method to theformulated problem in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by (a) Brain Korea 21Plus Program (Education and Research Center for Cre-ative Offshore Plant Engineers of Seoul National University)funded by the Ministry of Education Republic of Korea (b)Engineering Research Institute of Seoul National UniversityRepublic of Korea and (c) Research Institute of MarineSystems Engineering of Seoul National University Republicof Korea

References

[1] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[2] T A Johansen and T I Fossen ldquoControl allocationmdasha surveyrdquoAutomatica vol 49 no 5 pp 1087ndash1103 2013

[3] C Y Duan S J Zhang Y F Zhao and X R Kong ldquoRobustcontrol allocation among overactuated spacecraft thrustersunder ellipsoidal uncertaintyrdquo Abstract and Applied Analysisvol 2014 Article ID 950127 12 pages 2014

[4] O Harkegard ldquoDynamic control allocation using constrainedquadratic programmingrdquo Journal of Guidance Control andDynamics vol 27 no 6 pp 1028ndash1034 2004

[5] C D Wit Optimal thrust allocation methods for dynamicpositioning of ships [MS thesis] Delft University of TechnologyDelft The Netherlands 2009

[6] C C Liang andWH Cheng ldquoThe optimum control of thrustersystem for dynamically positioned vesselsrdquo Ocean Engineeringvol 31 no 1 pp 97ndash110 2004

[7] T A Johansen T I Fossen and S P Berge ldquoConstrainednonlinear control allocation with singularity avoidance usingsequential quadratic programmingrdquo IEEE Transactions on Con-trol Systems Technology vol 12 no 1 pp 211ndash216 2004

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[8] Y Parikshit Analysis design and optimization of offshore powersystem network [PhD thesis] National University of SingaporeSingapore 2013

[9] D-W Zhao F-G Ding J-F Tan Y-Q Liu and X-Q BianldquoOptimal thrust allocation based GA for dynamic positioningshiprdquo in Proceedings of the IEEE International Conference onMechatronics and Automation (ICMA rsquo10) pp 1254ndash1258 XirsquoanChina August 2010

[10] K-Y Lee S Cho and M-I Roh ldquoAn efficient global-localhybrid optimization method using design sensitivity analysisrdquoInternational Journal of Vehicle Design vol 28 no 4 pp 300ndash317 2002

[11] M M Bernitsas D Ray and P Kinley ldquoKT KQ and efficiencycurves for the Wageningen B-series propellersrdquo Tech RepUniversity of Michigan Ann Arbor Mich USA 1981

[12] D T Brown and L Ekstrom ldquoVessel thruster-thruster inter-actions during azimuthing operationsrdquo in Proceedings of the24th International Conference on Offshore Mechanics and ArcticEngineering pp 991ndash996 Halkidiki Greece June 2005

[13] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[14] L Davis Handbook of Genetic Algorithms Van Nostrand-Reinhold 1991

[15] J S Arora Introduction to Optimum Design Elsevier 3rdedition 2012

[16] G Vanderplaats Numerical Optimization Techniques for Engi-neering Design McGraw-Hill 1984

[17] C Stork and T Kusuma ldquoHybrid genetic autostatics newapproach for large-amplitude statics with noisy datardquo in Pro-ceedings of the 62ndAnniversary InternationalMeeting pp 1127ndash1131The Society of ExplorationGeophysicists NewOrleans LaUSA 1992

[18] M J Porsani P L Stoffa M K Sen R K Chunduru and WT Wood ldquoA combined genetic and linear inversion algorithmfor waveform inversionrdquo in Proceedings of the 63rd AnniversaryInternational Meeting pp 692ndash695 The Society of ExplorationGeophysicists Washington DC USA 1993

[19] K Albarado R Hartfield W Hurston and R Jenkins ldquoSolidrocket motor design using hybrid optimizationrdquo InternationalJournal of Aerospace Engineering vol 2012 Article ID 987402 9pages 2012

[20] E Deniz Ulker A Haydar and K Dimililer ldquoApplication ofhybrid optimization algorithm in the synthesis of linear antennaarrayrdquoMathematical Problems in Engineering vol 2014 ArticleID 686730 7 pages 2014

[21] F-Z Oujebbour A Habbal R Ellaia and Z Zhao ldquoMulticri-teria shape design of a sheet contour in stampingrdquo Journal ofComputational Design and Engineering vol 1 no 3 pp 187ndash1932014

[22] HWilli and S Klaus Test Examples for Nonlinear ProgrammingCodes Springer Berlin Germany 1981

[23] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of