optimal thrust allocation for semisubmersible oil rig platforms using improved harmony search...

526 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 39, NO. 3, JULY 2014

Optimal Thrust Allocation for Semisubmersible OilRig Platforms Using Improved Harmony

Search AlgorithmParikshit Yadav, Student Member, IEEE, Rajesh Kumar, Senior Member, IEEE,

Sanjib Kumar Panda, Senior Member, IEEE, and C. S. Chang

Abstract—Deep-water offshore drilling vessels, such as asemisubmersible drilling rig, use the dynamic positioning (DP)system and the thruster-assisted position-mooring system formaintaining a stationary position. In the DP system, the thrustallocator is used to distribute the desired generalized forces com-puted by the motion controller among the thrusters. However, toensure safe operation of the vessel despite the thruster failure,the vessel is equipped with redundant thruster configuration and,therefore, is overactuated. For overactuated vessels, the choice ofa particular solution for thrust allocation is found using some kindof optimization process. In this paper, the thrust allocator triesto minimize the power consumption and takes forbidden/spoilzones into account. The formulated thrust allocation problembecomes nonconvex due to thrust direction constraints on azimuththrusters. The conventional methods get trapped in local minimaand fail to find the optimum solution for the formulated nonconvexthrust allocation problem. In this paper, an improved harmonysearch (IHS) algorithm for solving the nonconvex thrust allocationproblem is proposed. IHS is a variant of the harmony search (HS)algorithm. The HS algorithm is a music-based meta-heuristicoptimization method, which is analogous with the music improvi-sation process where a musician continues to polish the pitches toobtain better harmony. Obtained numerical results show that theIHS algorithm has better convergence property when compared tothe HS algorithm and the genetic algorithm (GA). Moreover, thepower consumption for thrust allocation using the IHS algorithmis lower when compared with HS, GA, and Mincon (sequentialquadratic programming) algorithms. The percentage savings intotal power consumption for thruster allocation as compared tothe Mincon algorithm for GA, HS, and IHS methods are 44.96%,48.39%, and 51.58%, respectively.

Index Terms—Dynamic positioning (DP), genetic algorithm(GA), harmony search (HS), optimization, thrust allocationproblem.

Manuscript received June 22, 2012; revised May 18, 2013; accepted June 14,2013. Date of publication January 17, 2014; date of current version July 10,2014.Associate Editor: H. Maeda.P. Yadav, S. K. Panda, and C. S. Chang are with the Department of Electrical

and Computer Engineering, National University of Singapore, Singapore117576, Singapore (e-mail: [email protected]; [email protected];[email protected]).R. Kumar is with the Department of Electrical and Computer Engineering,

National University of Singapore, Singapore 117576, Singapore and also withthe Department of Electrical Engineering, Malaviya National Institute of Tech-nology, Jaipur 302017, India (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JOE.2013.2270017

I. INTRODUCTION

I NCREASING industrial and social development on aglobal scale has led to unprecedented demand for en-

ergy, the vast majority of which continues to be met throughthe exploitation of the world’s finite reserves of fossil fuels.This motivates exploration and exploitation at continuouslyincreasing water depths. However, offshore drilling units fordeeper water depths, such as a semisubmersible drilling rig,face unique challenge. During drilling, when the drill pipecasing is connected to the oil well, if the rig is displaced dueto environmental disturbances, the drill case can collapse orfracture, resulting in huge financial losses. Since the depth ofthe seabed is more than 3000 m, conventional mooring systems,like a jack-up barge or an anchored rig structure, cannot beused. Rather, the dynamic positioning (DP) system and thethruster-assisted position mooring system are used to keepthe oil rig platform at the stationary position. The DP systemautomatically controls the position and heading of the oil rigsubjected to environmental and external forces, using largerating azimuth thrusters fitted at its pontoon level [1], [2].In this paper, a semisubmersible oil rig platform equipped

with eight azimuth thrusters is considered as a case study fora marine vessel. For vessels that are operating to DP class 2or 3 standards, the vessel should be left with sufficient powerand thrusters to maintain position after worst case failure.Therefore, the semisubmersible oil rig is equipped with re-dundant thruster configuration and is overactuated as per theguidelines of the International Maritime Organization (IMO)MSC Circ.645 and the International Marine Contractors As-sociation (IMCA) M 103 [3], [4]. Azimuth thrusters fitted atoil rig platform’s pontoon level can produce forces in differentdirections, leading to an overactuated control problem that canbe formulated as an optimization problem, to minimize thepower consumption, drag, tear/wear, and other costs related tothe use of control, subject to constraints such as actuator posi-tion limitations [5], [6]. In this paper, the thrust allocator triesto minimize the power consumption and takes forbidden/spoilzones into account. Forbidden/spoil zones are used to avoidthruster–thruster interactions, which reduce the efficiency ofthe thrusters [7]. The control allocation problem formulatedfor an oil rig equipped with azimuth thrusters is a nonconvexoptimization problem due to thrust direction constraints onazimuth thrusters [5]. In general, a nonconvex constrained opti-mization problem is hard to solve using state-of-the-art iterativenumerical optimization. Many methods, such as the linear or

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YADAV et al.: OPTIMAL THRUST ALLOCATION FOR SEMISUBMERSIBLE OIL RIG PLATFORMS USING IMPROVED HARMONY 527

quadratic programming [5], [8], [17], have been applied to solvethe thruster allocation problem. Classical optimization methodslike sequential quadratic programming (SQP) have proved tobe a feasible method to solve the nonconvex thruster allocationproblem, but these methods are highly sensitive to startingpoints and frequently converge to a local optimum solutionor diverge altogether. Therefore, conventional methods fail tofind the optimum solution for the thrust allocation problem andoften get trapped in local minima.Evolutionary algorithms eradicate some of the aforemen-

tioned difficulties and are quickly replacing the classicalmethods in solving practical problems. Evolutionary algo-rithms typically intend to find a good solution to an optimiza-tion problem by “trial-and-error” in a reasonable amount ofcomputing time. The most prominent evolutionary algorithmis the genetic algorithm (GA) that is based on natural genetics.John Holland developed genetic algorithms in the 1960s and1970s [18]. The essence of GAs involves the encoding ofan optimization function into arrays of binary or characterstrings to represent chromosomes. The population of GAsconsists of various sets of chromosomes. The initial populationthen evolves by generating new generation individuals viacrossover of two randomly selected parent chromosomes, andthe mutation of some random bits. Whether a new individualor offspring is selected is based on its fitness value, which isdetermined from the objective function [18]–[20].In this paper, the improved harmony search (IHS) algorithm

is used for solving the nonconvex thrust allocation problem.Harmony search (HS), developed by Geem et al. in 2001, isinspired by the music improvisation process [21]. The musiciansearches for a perfect state of harmony by adjusting the pitch.The effort to find the harmony in music is analogous to findingthe optimality in an optimization process. In other words, a mu-sician’s improvisation process can be compared to the searchprocess in optimization. The pitch of each musical instrumentdetermines the aesthetic quality, just as the objective functionvalue is determined by the set of values assigned to each deci-sion variable. The HS algorithm has been successfully appliedto many optimization problems in different areas [22]–[48]. TheHS algorithm is good at identifying the high-performance re-gions of the solution space in a reasonable time, but gets intotrouble in performing local search for numerical applications.To improve the fine-tuning characteristic of the HS algorithm,Mahdavi et al. [49] recommended parameters for pitch adjustingthat vary with the improvisation number. The IHS algorithm en-hances fine-tuning characteristic and convergence rate of HS.The IHS algorithm has the power of the HS algorithm with thefine-tuning feature of mathematical techniques and can performbetter than HS and other heuristic algorithms.The rest of the paper is arranged as follows. Section II

provides an overview of the three-degree-of-freedom (3-DOF)thrust allocation. The insight into these principles will helpthe reader in understanding the motivation for followingan engineering approach for optimal thrust allocation in anoveractuated marine vessel. The details and drawbacks of theLagrange multiplier method conventionally used for the thrustallocation in the case study of the marine vessel are discussedin Section III. The thrust allocator tries to minimize the powerconsumption and takes forbidden/spoil zones into account.

Fig. 1. Sign conventions used for thruster allocation.

The details of the formulated optimization problem are pre-sented in Section IV. The optimization problem formulated issubjected to both equality and inequality constraints. A novelconstraint handling method based on superiority of feasiblesolutions (SF) is presented in Section V. The brief overviewof the classical HS method and the IHS method is presented inSection VI. Section VII provides an overview of the practicalsemisubmersible oil rig platform. The results obtained fromthe case study of the marine vessel with detailed discussionsare presented in Section VIII. Finally, the research findings areconcluded in Section IX.

II. 3-DOF THRUST ALLOCATION

The scope of the paper is to present a solution for a 3-DOFcontrol allocation (horizontal plane motion). However, the pre-sented approach can also be used for a 6-DOF control allocation.For horizontal plane motions (surge, sway, and yaw), the con-trol allocation problem is to select the control signals associatedwith the individual actuators to produce the commanded surgeforce , the sway force , and the yaw moment ,as shown in Fig. 1 [2]. In this paper, the origin is considered atthe midships position, at the centerline, and at the baseline. Pos-itive is toward the forward side of the oil rig, positive istoward the port side of the oil rig, and positive is upwards.The forces and moments demanded by the DP system are repre-sented by a force vector . The forcesand moments generated by the th actuator with state incylindrical coordinates are given by [2], [5]

(1)

(2)

(3)

where is the magnitude of thrust produced by the th actu-ator and is the azimuth angle of the th actuator. For each ofthe actuators, its position in the horizontal plane with respect tothe center of gravity of the ship (0, 0) is given by ,as shown in Fig. 2. Equations (1)–(3) can be combined andrewritten in a matrix notation, as given by [2], [6]

(4)


Fig. 2. Reference frame of the th actuator for the calculation of forces andmoments.

Assuming that the marine vessel is equipped with actuators,then the force vector demanded is jointly produced byactuators to keep the vessel at the stationary position, and it isgiven by

(5)

Here, is a vector of the magnitude of the forces pro-duced by each individual actuator. The thruster configurationmatrix is determined by the position of the indi-vidual thrusters. The th column in the configuration matrix cor-responds to the generalized force produced by the th thruster.However, each thruster has limited amount of thrust and lim-ited directions in which it generates thrust. Therefore, for eachthruster, the azimuth angle and the thrust force areconstrained to the sets given by

(6)

where and are the lower bounds on the azimuth angleand the thrust force for the th thruster, respectively, and, simi-larly, and are the upper bounds on the azimuth angle andthe thrust force for the th thruster, respectively. A set of allphysically realizable surge and sway forces for theth thruster is called the attainable thrust region (ATR) of theth thruster, and it is defined by [50]

(7)

An azimuth thruster is physically capable of generating thrust inany direction as it can rotate 360 . Typically, azimuth thrustersare not used to produce the thrust in a reverse mode, there-fore, the minimum thrust force is zero and the maximum thrustforce is limited to a rated thrust force. Therefore, ATR of theazimuth thruster is circular and, hence, convex. However, whentwo thrusters are positioned close enough, the thruster–thrusterinteractions can cause the thrust loss [7]. The thrust loss is very

Fig. 3. Shaded region represents the ATR of the th and th thruster.

significant in case of an oil rig because the two thrusters aremounted in pairs and are located on the same leg of the oil rig.The thrust generated by the thruster is given by [2]

(8)

where is the density of water, is the propeller diameter,and is the propeller speed. The nondimensional thrust co-efficient in general is affected by the change in the ad-vance speed and the cross-coupling drag caused by the otherthrusters [6]. To overcome the thruster–thruster interaction loss,the forbidden zones or spoil zones are defined and excludedfrom the ATR of the neighboring thrusters. The ATR for a set ofthrusters is given as a Minkowski set addition given by [50]

(9)

The Minkowski set addition for two sets andis defined by

(10)

Each thruster is assumed to have a forbidden angle of tothe closest thruster to prevent the thruster’s direct interaction, asshown in Fig. 3. With the port-forward pair, the center of angleof the outboard thruster is and for the inboard thruster

, it is . The positive angle is measured counterclock-wise while the negative angle is measured clockwise from thehorizontal axis. Fig. 3 shows the ATRs for the two port-forwardside thrusters and . Based on Fig. 3, the azimuth angle

and the thrust force are constrained to the sets givenby (11). The and for the th thruster are given by

(11)

(12)

360 (13)

Using (11)–(13), the ATR for the th thruster is given by

(14)

and the forbidden sector is .


III. LAGRANGE MULTIPLIER OPTIMIZATION METHOD

The Lagrange multiplier method is used for the thrust allo-cation in the case study of a marine vessel. The details of themethod are analyzed in this section. To solve the thruster allo-cation problem by the Lagrange multiplier method or quadraticprogramming (QP), the thrust vectors are converted to Cartesiancoordinates, also called the extended thrust formulation [51],to form a convex linearly constrained QP problem. In Carte-sian coordinates, forces and moments generated by the th actu-ator with state are given by reformulating (1)–(3).

and are the thruster force in the body- and -directions

(15)

(16)

(17)

Equations (15)–(17) can be combined and rewritten in a matrixnotation as given by

(18)

Then, the force vector demanded is jointly produced byactuators to keep the vessel in the stationary position, and it isgiven by

(19)where and are the vectors of ones and zeros, re-spectively, and and are the vectors of thethruster positions in the body - and -directions, respectively.The thruster configuration matrix is determinedby the position of the individual thrusters, and is thestate matrix of thrusters. For each thruster, the azimuth angle

and the thrust force can be obtained using

(20)

(21)

The thrust allocation problem is formulated as a quadraticequality constrained minimization problem and solved usingthe Lagrange multiplier method

(22)

subjected to

(23)

where

. . .

is the weight matrix to compute power from the thrust and. A subproblem is formulated by combining the fitness

function and the equality constraint using the Lagrangian andpenalty parameters as shown in

(24)

where are the Lagrangian multipliers. The solution ofthe subproblem can be found using the Karush–Kuhn–Tucker(KKT) condition. The KKT equations are necessary condi-tions for optimality of a constrained optimization problem[52]. If the problem is a so-called convex programmingproblem, then the KKT equations are both necessary andsufficient for a global solution point, and a solution is givenby . However, in the abovesolution, thrust limits of the thrusters are not taken into account.The solutions will contain thrust values that the thrusters arenot physically capable of producing. This problem can beovercome by adding saturation handling capability to the thrustallocator algorithm [53]. Apart from this, the formulation ofpower consumption of the thruster is not accurate. Traditionally,in the literature, using QP solvers, the power consumption ofa thruster is penalized by the term . However,since power is proportional to [5], [10], [15], [17], theformulated problem given by (22) can either overpredict or un-derpredict the power consumed by the thruster. The modelingof thruster power consumption will be discussed in more detailin Section VII.

IV. OPTIMIZATION PROBLEM FORMULATION

For vessels that are operating according to the DP class 2 or3 standards, the vessel should be left with sufficient power andthrusters to maintain its position after the worst case failure.Therefore, they are overactuated as per the guidelines of IMOMSC Circ.645 and IMCA M 103 [3], [4]. Azimuth thrusterscan produce forces in different directions leading to an overac-tuated control problem that can be formulated as an optimiza-tion problem. In this paper, the thrust allocator tries to minimizethe power consumption and takes forbidden/spoil zones into ac-count. Therefore, the optimization problem can be formulated asgiven by

minimize minimize (25)


subjected to

(26)

or (27)

where is the power consumed by the th thruster to pro-duce a thrust force . Since the marine vessel is assumed tobe equipped with thrusters, the total power is representedby . Equation (26) ensures that the error betweenthe demanded generalized force and the generalized forcejointly produced by actuators is close to zero to keep thevessel in the desired position. However, the thrust produced byeach thruster is restricted to its ATR; this is ensured by (27).

V. CONSTRAINT HANDLING USING SUPERIORITY OFFEASIBLE SOLUTIONS

In the formulated constraint optimization problem, the errorbetween the demanded generalized force and the general-ized force jointly produced by actuators must be close to zeroto keep the vessel in a desired position. A general constraint op-timization problem with parameters to be optimized is usu-ally written as a nonlinear programming problem as in [54]–[56]

minimize (28)

subjected to (29)

(30)

where is the objective function with as the solutionvector composed of decision variables and .The set is a search space, and it is defined as an -dimen-sional rectangle in with all possible ranges of values foreach decision variable , and it is given by

(31)

where and are the lower and upper bounds for the thdecision variable, respectively. is the feasible search spacesuch that and satisfies the additional constraints (29)and (30). The equality constraints can be transformed into aninequality form and can be combined with other inequalityconstraints

(32)

Most heuristic search methods use the penalty function ap-proach of handling constraints. The penalty function approachinvolves a number of penalty coefficients, which must be setright in any problem to obtain feasible solutions. However, theperformance of the search algorithm is greatly influenced by thepenalty coefficients [56]. Several sophisticated penalty func-tion approaches have been proposed, but all these approachesrequire extensive experimentation for setting up appropriateparameters needed to define the penalty function. In this paper,the SF method is used for constraint handling. The SF method

is based on the tournament selection operator, where twosolutions are compared at a time, and the following criteria arealways enforced [54]:1) any feasible solution is preferred to any infeasible solution;2) among two feasible solutions, the one having better objec-tive function value is preferred;

3) among two infeasible solutions, the one having smallerconstraint violation is preferred.

Based on the above rules, the fitness function is evaluated using

if

otherwise.

(33)

where infeasible solutions are compared based on their con-straint violation only. The parameter is the objective func-tion value of the worst feasible solution in the population and

denotes the absolute value of the operand, if the operandis positive and returns a zero value, otherwise. Therefore, thefitness of an infeasible solution not only depends on the amountof constraint violation, but also on the population of solutions athand. However, the fitness of a feasible solution is always fixedand is equal to its objective function value. If no feasible so-lution exists in a population, is set to zero. However, themethod requires at least one feasible individual to be placed inthe initial population, which is similar to providing a startingfeasible point for the optimization process. If the initial popu-lation has no feasible solution , then there may beoccasions when . Therefore, thefirst rule of the tournament selection is violated. To overcomethis, we propose a modified SF method. The fitness evaluationonly is modified in the method, and it is given by

if

otherwise.

(34)

By doing this, the fitness function of the feasible solution is al-ways lower than the infeasible solution. The fitness functionvalue evaluated using (34) is assigned to in(37). The modified SF method ensures that the randomly gen-erated solution converges to the optimum solution of the entireoptimization problem.

VI. HARMONY SEARCH AND OTHER VARIANTS

In this section, an introduction to an engineering approachthat is used in the design and development of an algorithm in-spired by a musical process of searching for a perfect state ofharmony is presented. A brief overview of the classical HS andother variants is provided.


Fig. 4. Analogy between the music improvisation and the optimization [35].

A. Harmony Search Algorithm

A metaheuristic algorithm, mimicking the improvisationprocess of music players, has been recently developed andnamed the HS algorithm [21]. The HS algorithm has beenvery successful in a wide variety of optimization problems,presenting several advantages with respect to a traditionaloptimization technique. The HS algorithm imposes fewermathematical requirements and does not require initial valuesettings of the decision variables; however, few other param-eters need to be defined. As the HS algorithm uses stochasticrandom searches, derivative information is also unnecessary.In the HS algorithm, musical performances seek a perfect stateof harmony determined by aesthetic estimation, as the opti-mization algorithms seek the best state (i.e., global optimum)determined by the objective function value. The analogybetween the music improvisation and the engineering optimiza-tion is illustrated in Fig. 4.Each musician corresponds to each decision variable; a mu-

sical instrument’s pitch range corresponds to a decision vari-able’s value range; musical harmony at certain time correspondsto a solution vector at certain iteration; and audience’s aestheticscorresponds to an objective function. Just like musical harmonyis improved with time, the solution vector is improved withiteration.To understand the principle of the HS algorithm, let us first

idealize the improvisation process by a skilled musician. Whena musician improvises, he or she has three possible choices:1) playing any famous tune exactly from his or her memory;2) playing something similar to the aforementioned tune, thusadjusting the pitch slightly; or 3) composing new or randomnotes. In this section, various steps involved in the HS algo-rithm, describing how the HS algorithm is designed and applied,are presented.Step 1—Initialize the Optimization Problem and Algorithm

Parameters: To apply the HS algorithm, the problem shouldbe formulated in the optimization environment, having ob-jective function and constraints as in (28)–(30). In addition,the parameter values of the HS algorithm are specified in thisstep. These parameters are the harmony memory size (HMS),the harmony memory considering rate (HMCR), the pitchadjusting rate (PAR), and the number of improvisations oriterations (Maxiter).

Step 2—Initialize the Harmony Memory (HM): The initialHM consists of an HMS number of randomly generated solu-tions for the optimization problem under consideration. Eachcomponent of the solution vector in the HM is initialized witha uniformly distributed random number between the upper andlower bounds , where . The th component ofthe th solution vector is given by

(35)

where HMS, and is a uniformly dis-tributed random number between 0 and 1. Each row of the HMconsists of a random solution for the optimization problem, andthe objective function value for the th solution vector is de-

noted by and is computed using (33). The HM matrix

formed is governed by

HM (36)

HM (37)

and the HM with the size of HMS can be representedby

HM

......

......

...

(38)Step 3—Improvise a New Harmony From the HM: After

defining the HM as shown in (38) for the optimization problem,the improvisation of the HM is carried out by generating a

new harmony vector . Eachcomponent of the new harmony vector is generated using

HM with probability HMCRwith probability HMCR

(39)

based upon the HMCR defined in Step 1, where HM is theth column of the HM. HMCR is defined as the probability ofselecting a component from the HM members, and HMCRis, therefore, the probability of generating it randomly from thepossible range of values. If is generated from the HM, thenit is further modified or mutated according to the value of theparameter PAR. The value of the parameter PAR determines theprobability of a candidate from the HM to be mutated and thevalue of PAR is the probability of doing nothing. The pitchadjustment for the selected is given by

with probability PARwith probability PAR

(40)


Fig. 5. Actuator layout of the Keppel’s B280 and corresponding dimensions with respect to the origin at main drill well.

where is the randomly generated number between 0and 1 and is the pitch bandwidth.Step 4—Update the HM: The newly generated harmony

vector is evaluated in terms of the objective function value.If the objective function value for the new harmony vector isbetter than the objective function value for the worst harmonyin the HM, then the new harmony is included in the HM andthe existing worst harmony is excluded from the HM.Step 5—Go to Step 3 Until the Termination Criterion Is

Reached: The current best solution is selected from the HMafter the termination criterion (Maxiter) is satisfied. This is thesolution for the formulated optimization problem.

B. Improved Harmony Search Algorithm

The HS algorithm is good at identifying the high-perfor-mance regions of the solution space at a reasonable time, butgets into trouble in performing local search for numericalapplications. The HMCR, PAR, and parameters introducedin Step 3 help the algorithm find globally and locally improvedsolutions, respectively [47], [57]. The performance of the HSalgorithm can be improved by fine-tuning the values of PARand . The choice of a small PAR value with a large valuecan deteriorate the performance of the algorithm and enhancethe computational time needed to find the optimum solution.Initially, the value of parameter must be large to increasethe diversification of the search. However, in final iterations,the algorithm must focus more on intensification, therefore, asmall value is preferred. To improve the performance ofthe HS algorithm and eliminate the drawbacks that lie with

Fig. 6. Percent error in power calculated using 0.176 T and0.011 T .

fixed values of PAR and , Mahdavi et al. [49] proposed animproved harmony search (IHS) algorithm that dynamicallyupdates PAR and with the number of iterations. IHS hasexactly the same steps as those in the classical HS with theexception of Step 3, where PAR is linearly increased andis exponentially decreased with the number of iterations. Themathematical expressions for PAR and are given by

PAR PAR PAR PAR (41)


TABLE ICOMPARISON OF POWER CONSUMPTION COMPUTED FROM 0.176 T AND 0.011 T WITH MEASURED POWER

Fig. 7. Shaded region represents the ATR for the corresponding thruster.

where PAR is the minimum pitch adjustment rate, PARis themaximum pitch adjustment rate, is iteration, andMax-iter is the maximum number of iterations

(42)

where is the minimum bandwidth and is the max-imum bandwidth.

VII. THE PRACTICAL OIL RIG

This section provides an overview of the semisubmersible oilrig considered as a case study of a marine vessel. The vessel isa four-column stabilized semisubmersible oil rig. Four rectan-gular-shaped stability columns and two pontoons provide the

buoyancy. The actuator layout of the Keppel’s B280 semisub-mersible oil rig is shown in Fig. 5 [58]. Eight azimuth thrusterslocated at the pontoon level of the oil rig are used by the DPsystem to control its position. The effective thrust (bollard pull)of each thruster is 780 kN at a motor power of 4000 kW at 600r/min. The thrust and the torque generated by a pro-peller are given by [2], [15]

(43)

(44)

The power consumption of a propeller is given by

(45)


Fig. 8. Commanded longitudinal resultant thrust , lateral resultant thrust , and moment at 50 time steps.

where is the density of water, is the propeller diameter,and is the propeller speed. and are the thrust andtorque coefficients, respectively. Using (43)–(45), the powerconsumption of a propeller can be rewritten as

(46)

Therefore, power consumed by the propeller can be computedusing T . However, conventionally, the relation be-tween the thrust (kN) and the power consumed (kW) is givenby T . This formulation is usually used to make thethrust allocation problem simpler. To validate both power for-mulations, the output torque (Nm) and the output power (kW)are measured at different load levels for the Keppel’s B280semisubmersible oil rig, as shown in Table I. The power cal-culated using both formulations is also shown in Table I.The error between the calculated power and the actual mea-

sured power for both formulations is shown in Fig. 6. It is ev-ident from Fig. 6 that the power calculated using Thas an error between 40% and 60%, whereas the power cal-culated using T has an error between 8% and 4%.Therefore, is a better formulation as compared to .Each thruster is assumed to have a forbidden angle of25 to the closest thruster with the port-forward thruster pair

center of angle of the outboard thruster as 63.44and for the inboard thruster as 116.56 . Fig. 7shows the ATRs for all the eight azimuth thrusters.Using (12)–(13), the corresponding and are computed

as shown in

38.44 141.56 91.56 88.44

38.44 141.56 91.56 88.44 (47)

271.56 451.56 218.44 398.44

271.56 451.56 218.44 398.44 (48)

VIII. RESULTS AND DISCUSSION

To evaluate the effectiveness and efficiency of the IHS al-gorithm, we compare its performance with other conventionaland evolutionary algorithms. The algorithms for comparison arelisted as follows:

Fig. 9. Convergence of IHS, HS, and GA for demanded 49.959 kN,60.481 kN , and 63787 kN-m.

• Mincon method (sequential quadratic programming);• HS;• GA.

The parameters for the HS and IHS algorithms are the same asthose reported in [59]. For the HS algorithm, HMS , HMCR

, PAR , and . For the IHS algorithm,HMS , HMCR , PAR , PAR ,

, and . According to [1]and [37], the default values of GA parameters are: 20 individ-uals per generation, 10% of elite individuals, crossover prob-ability is 0.5, rank fitness scaling, roulette selection, and scat-tered crossover operators. The detailed description of these al-gorithms can be found in [59]. In the case of theMinconmethod,the initial point plays a very crucial role in finding the optima.If the initial point is specified as the lower bound , theMincon method is unable to find the solution for some of thedemanded force vectors. It is similar for the case when an ini-tial point is specified as an upper bound . The initial pointis chosen after the trial and error as . Therefore,the conventional optimization methods are sensitive to startingpoints. Evolutionary algorithms do not need any initial startingpoint to be specified.The Keppel’s B280 semisubmersible oil rig is used as the case

study vessel; the oil rig was subjected to a sea load profile. Thedemanded longitudinal resultant thrust , the lateral resul-tant thrust , and the moment at 50 time steps are


Fig. 10. (a) Delivered thrust forces and (b) corresponding azimuth angles for thruster 1.



shown in Fig. 8. For the first step time, the demanded longi-tudinal resultant thrust 49.959 kN, the lateral resultantthrust 60.481 kN, and the moment 63787kN-m. The performance of the algorithms is evaluated basedon the convergence speed and the final numerical result. Sinceevolutionary algorithms are heuristic in nature, we performed50 trials to obtain the best solution for the evolutionary algo-rithms. The convergence of the best fitness value for the objec-tive function for the demanded force vector using IHS, HS, and

GA is shown in Fig. 9, where it can be seen that the IHS algo-rithm converges faster than HS and GA.For the demanded force vectors given by Fig. 8, the thrust

allocation is performed using the IHS algorithm, and the corre-sponding thrust and azimuth angles for the entire eight thrustermotors are shown in Figs. 10–17. The total power consumptionof the eight thruster motors for 50 time steps for the IHS, HS,GA, and Mincon algorithms is shown in Fig. 18. It is evidentfrom Fig. 18 that the power consumption of the IHS algorithm





is lower than the power consumption of the HS, GA, andMincon methods. In the case of the Mincon method, the totalpower consumption is almost constant because it usuallysolves the formulated optimization problem by varying az-imuth angles only and it does not vary the thrust much. Theevolutionary algorithms, on the other hand, find the solutionfor the formulated optimization problem by varying both thrustand azimuth angles of the thrusters. The IHS algorithm, similarto the HS algorithm, generates a new vector, after considering

all of the existing vectors, whereas GA only considers the twoparent vectors. These features increase the flexibility of the IHSalgorithm and produce better solutions. The selected harmonyvectors are random in nature. The randomization diversifies thesearch space and prevents premature convergence. However,an additional subcomponent for HS diversification is the pitchadjustment operation performed with the probability of PAR,which is an important factor for the high efficiency of the IHSmethod as compared to HS and GA [1], [60].




Fig. 18. Total power consumption during thruster allocation for 50 time stepsfor IHS, HS, GA, and Mincon algorithms.

The total power consumed by the eight thruster motors duringthe entire load cycle is calculated by finding the area underthe curve in Fig. 18. For a step size , the total power isgiven by , where corresponds tototal power consumed by the eight thruster motors for the thstep. Assuming that the step size is 1, then the total powerconsumed eight thruster motors during the entire load cycle, as

TABLE IITOTAL POWER CONSUMPTION OF THE OIL RIG PLATFORM

FOR THE ENTIRE LOAD CYCLE

shown in Table II, for Mincon, GA, HS, and IHS methods is1039.2, 572.01, 536.3, and 503.22MW, respectively. Therefore,the percentage savings in total power consumption for thrusterallocation as compared to the Mincon method for GA, HS, andIHS methods are 44.96%, 48.39%, and 51.58%, respectively.

IX. CONCLUSION

In this paper, the thruster allocation problem is formulated asa constrained optimization problem. The thrust allocator triesto minimize the power consumption by the thrusters and simul-taneously ensures that the vessel is at the desired position. Inaddition, it also ensures that the thruster–thruster interaction isminimal. In this paper, an IHS algorithm is used for solvingthe nonconvex thrust allocation problem. The ability of the IHSalgorithm has been demonstrated, and its performance is com-pared with other optimization techniques. The optimal thruster


allocation using IHS reduces the power consumption of the rigas compared to Mincon, GA, and HS. The percentage savingsin total power consumption by thrusters as compared to theMincon method for GA, HS, and IHS algorithms are 44.96%,48.39%, and 51.58%, respectively. It is clear from the resultsthat the power consumption is minimal and the percentage sav-ings in the total power consumption for thruster allocation arethe highest using the IHS algorithm. In addition, the conver-gence for the IHS algorithm is faster as compared to the GAand HS methods. The IHS algorithm can avoid the shortcom-ings of premature convergence of GA and HS algorithms. Vari-able PAR and improve the convergence and also numericalresults as compared to the classical HS algorithm. Generally, itcan be concluded that the IHS algorithm’s simplicity of imple-mentation, high-quality solution, along with the lower numberof setting parameters makes it an ideal method when dealingwith complex engineering optimization problems.

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Parikshit Yadav (S’10) received the B.Tech.(honors) degree in electrical engineering fromMalaviya National Institute of Technology, Jaipur,India, in 2007. He is currently working towardthe Ph.D. degree at the Department of Electricaland Computer Engineering, National University ofSingapore, Singapore.His research interests lie in the fields of marine

power system, optimization, and offshore windpower generation.

Rajesh Kumar (M’08–SM’10) received the B.Tech.degree in electrical engineering from the National In-stitute of Technology (NIT), Kurukshetra, India, in1994, the M.E. degree in power engineering from theMalaviya National Institute of Technology (MNIT),Jaipur, India, in 1997, and the Ph.D. degree from theUniversity of Rajasthan, Jaipur, India, in 2005.Since 1995, he has been a Faculty Member in the

Department of Electrical Engineering, MNIT, wherehe is serving as an Associate Professor. He was Post-doctoral Research Fellow in the Department of Elec-

trical and Computer Engineering, National University of Singapore (NUS), Sin-gapore, from 2009 to 2011. His field of interest includes theory and practiceof intelligent systems, computational intelligence and applications to powersystem, electrical machines, and drives.

Sanjib Kumar Panda (S’86–M’91–SM’01)received the B.Eng. degree from the RegionalEngineering College, Surat, India, in 1983, theM.Tech. degree from the Institute of Technology,Banaras Hindu University, Varanasi, India, in 1987,and the Ph.D. degree from the University of Cam-bridge, Cambridge, U.K., in 1991, all in electricalengineering.Since 1992, he has been a Faculty Member in the

Department of Electrical and Computer Engineering,National University of Singapore, Singapore, where

he is currently serving as an Associate Professor and Area Director of the Powerand Energy Research Group. His research interests are in control of electricdrives and power electronic converters, energy harvesting, renewable energy,assistive technology, and mechatronics.

C. S. Chang received the Higher Diploma in elec-trical engineering from the Hong Kong Polytechnic,Hong Kong, in 1970, and the M.Sc. and Ph.D. de-grees from the Institute of Science and Technology,University of Manchester, Manchester, U.K., in 1972and 1975, respectively.He is an Associate Professor in the Department

of Electrical and Computer Engineering, NationalUniversity of Singapore, Singapore. His researchincludes the analysis and artificial intelligence (AI)applications in power and traction systems.

Dr. Chang was the Technical Chair of the Year 2000 IEEE Winter PowerMeeting in Singapore. He was conferred the degree of D.Sc. by the Universityof Manchester Institute of Science and Technology in 2001.

optimal thrust allocation for semisubmersible oil rig platforms using improved harmony search...

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