A thrust allocation method was proposed based on a hybrid optimization algorithm to efficiently and dynamically position a semisubmersible drilling rig. That is, the thrust allocation was optimized to produce the generalized forces and moment required while at the same time minimizing the total power consumption under the premise that forbidden zones should be taken into account. An optimization problem was mathematically formulated to provide the optimal thrust allocation by introducing the corresponding design variables, objective function, and constraints. A hybrid optimization algorithm consisting of a genetic algorithm and a sequential quadratic programming (SQP) algorithm was selected and used to solve this problem. The proposed method was evaluated by applying it to a thrust allocation problem for a semisubmersible drilling rig. The results indicate that the proposed method can be used as part of a costeffective strategy for thrust allocation of the rig.
Dynamic positioning system configured for 8 thrusters of a semisubmersible drilling rig.
The DP system must be very responsive to changes in the conditions due to weather, water depth, and so on in order to properly control the operating parameters of the vessel. Despite the importance of the DP system to the vessel, its use in a real working environment still faces many challenges, such as a power failure and a thruster malfunction, due to its complexity. The fuel expenditure of the DP system is a considerable burden since it requires a large amount of energy. This means that the DP system is overactuated, which renders the thruster to have infinite allocation solutions. Therefore, the thrust allocation can be formulated as an optimization process under numerous constraints in order to minimize power consumption.
Some of the methods intended for ocean engineering have considered cases with ships or with offshore vessels. Wit [
In this manner, most studies in the field of ocean engineering have determined the thrust and the azimuth direction of the thrusters to be design variables for an optimization problem. In fact, the thrust can be expressed as a function of the speed of rotation, the diameter, and the thrust coefficient of a thruster. In order to reflect the specifications of the given thruster and to more easily control the thruster, the speed of rotation can be used as a design variable instead of the thrust. Meanwhile, a local optimization algorithm can be used as the optimization algorithm, such as the SQP algorithm, to derive an accurate optimum. However, this method is very sensitive to a starting point for optimization and sometimes finds a local optimum. On the other hand, a global optimization algorithm, such as the GA, does not need a starting point for the optimization but can only find a rough optimum [
This study proposes an optimal thrust allocation method for an offshore vessel with the aim of achieving the required generalized forces and moment for dynamic positioning while at the same time minimizing total power. Finding such a method is a particularly important issue to operate an offshore vessel while using minimal energy. The method involves mathematically formulating an optimization problem for thrust allocation and selecting and using a suitable optimization algorithm to solve the problem, not developing a new algorithm. When the optimization problem is formulated, the speed of the rotation and the azimuth direction of thrusters are selected as design variables, the total power is selected as an objective function, and the required forces and moment are selected as constraints. A hybrid optimization algorithm that incorporates global and local optimization algorithms was used for the optimization.
Table
Summary of related studies and comparison with this study.
Studies  Application  Optimization algorithm  Design variables for formulation  Consideration of thruster interaction 

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


Liang and Cheng [ 
Ship  Local algorithm (SQP)  Thrust and azimuth direction  × 


Parikshit [ 
Drilling rig  Global algorithm (GA, ITHS, etc.)  Thrust and azimuth direction  ◯ 


Zhao et al. [ 
Drilling rig  Global algorithm (GA)  Thrust and azimuth direction  ◯ 


This study  Drilling rig  Global algorithm (hybrid optimization algorithm)  Rotation speed and azimuth direction  ◯ 
A thruster system is important for a drilling rig to maintain its position and heading since it can simultaneously provide transverse and longitudinal thrust. An azimuth thruster consists of an electric podded drive that is fitted to the hull and can rotate 360 degrees along the horizontal angle to provide thrust in all directions. The azimuth thruster has been selected in this study because it can provide thrust in any direction in order to act as a propulsor for dynamic positioning and target optimization of a drilling rig. However, the thruster system is usually overactuated in practice, so a thrust allocation problem can be formulated into a constrained optimization problem for the system.
Figure
Schematic of thruster arrangement of a drilling rig with eight thrusters (Plan view).
The environmental forces and the moment tend to move the drilling rig away from its original position during operation at sea. In order to move the drilling rig back to its original or reference position, the control system in the DP system of the drilling rig should first calculate the total forces (thrusts) and moment required. At this time, thrust allocation should be conducted to determine the thrust and azimuth direction of each thruster so that the required forces (and moment) are generated in each of the longitudinal (
Supposing that the drilling rig is equipped with
The goal of achieving optimal thrust allocation is to guarantee efficient power use when dynamically positioning the drilling rig in the specific conditions of a given ocean environment. Meanwhile, the economic impact of minimizing power consumption is also taken into account with the ultimate aim of ensuring the stability and capability of the drilling rig during operation.
This study mathematically formulated an optimization problem for thrust allocation, and each component of the optimization problem is described below in detail.
Thrust allocation involves determining the thrust (
In this study, the speed of the rotation (
The thrust
Now, it is time to formulate the total power consumed by the thrusters of the drilling rig. The power consumption (
By combining (
Therefore, the objective function can be written relating the speed of rotation and the azimuth direction of each thruster. In this study, the total power (power consumption) of the thrusters is minimized by setting the objective function for the thrust allocation optimization problem as
To obtain a valid thrust allocation, the governing equations shown in (
The governing equation for the longitudinal force can be stated as
Similarly, the governing equation for the transverse force can be stated as
And finally the governing equation for the yawing moment can be stated as
It is essential to minimize the thruster interaction of a semisubmersible drilling rig to ensure effective DP operation [
Interaction of two closely spaced azimuth thrusters.
Considering the ATR, the constraints for the azimuth direction of thrusters
The optimization problem for thrust allocation can be summarized as follows:
Thus, this problem has one objective function, three equality constraints, and
The optimization algorithms are generally divided into two categories: global and local. Several classes of global optimization algorithms are now available, including the genetic algorithm (GA) [
Various attempts have been made to combine global and local optimization algorithms in order to overcome the challenges of using them separately [
General procedure of the hybrid optimization algorithm used in this study.
An experiment on the mathematical optimization problem was performed in order to verify the efficiency, accuracy, and applicability of the hybrid optimization algorithm that was used to optimize the thrust allocation in this study. The selected problem, which is Rastrigin’s problem, one of the benchmark problems, is being widely used to check the efficiency of optimization algorithms [
The following optimization procedure was established to solve the optimization problem that was presented formulated above. First, the initial values were assumed for the design variables. At this time, the values can be randomly generated or can be manually set or extracted from an existing design. Now, these values are transferred to the hybrid optimization algorithm, and the values of an objective function and constraints are then calculated. We then check whether the current values of the design variables are at an optimum or not. If yes, the optimization process finishes and the result will be shown, and if not, the above steps will be repeated until the optimum is found. Figure
Optimization procedure for finding optimal thrust allocation.
The optimization target for this study is a semisubmersible drilling rig equipped with eight azimuth thrusters. Figure
The longitudinal and transverse positions of eight azimuth thrusters (Plan view).
This drilling rig has eight thrusters. Thus, the design variables in this problem are as follows:
Now, the objective function of this problem can be stated by using (
The constraints for the required forces and moment can be stated using (
Equations (
In order to overcome the energy loss due to the thrusterthruster interaction, the ATR should be considered, excluding forbidden zones, by using (
Schematic diagram of thruster interaction layout (Plan view).
In addition, there are some limitations on the maximum thrust and azimuth direction for each thruster. Thus, such limitations were also considered as additional constraints in this study. The additional constraints about the maximum thrust and the azimuth direction of each thruster can be stated in the following equations, respectively:
Thus, this problem has 3 equality constraints and 24 inequality constraints.
The problem in (
Optimization result for 14 time steps.
Time steps  Input from the control system  Thrusters  GA only [A] 
Hybrid [B] 
Ratio  

Thruster 1  Thruster 2  Thruster 3  Thruster 4  Thruster 5  Thruster 6  Thruster 7  Thruster 8  
Required force  Required moment  RPM  Angle  RPM  Angle  RPM  Angle  RPM  Angle  RPM  Angle  RPM  Angle  RPM  Angle  RPM  Angle  Total power  Total power  
(KN)  (KN⋅m)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (rpm)  (deg)  (kW)  (kW)  






















— 
1  50  −600  −64,000  3.10  −93  0.62  −91  0.09  85  0.55  −103  2.2  37  0.18  178  0.22  17.3  1.66  167  9,859  8,250  0.84 
2  −500  1,500  −25,000  2.51  115  1.02  171  1.29  152  0.79  93  0.03  −64  1.90  78  1.52  81  2.28  114  12,882  10,824  0.84 
3  100  −800  17,000  0.78  −35  1.57  −78  1.02  −2  0.54  −13  2.61  −95  0.26  −15  0.17  −92  0.45  20  7,186  6,580  0.92 
4  750  −550  63,000  2026  34  0.61  −177  0.76  170  0.07  32  0.62  −78  1.99  −78  0.03  40  2.64  −48  11,800  9,172  0.78 
5  175  560  −32,500  0.08  13  0.92  147  0.05  41  0.73  −128  2.74  74  1.11  −35  0.15  −41  0.29  −146  7,121  7,107  1.00 
6  −50  −1,080  3,000  2.17  −116  0.04  129  0.89  −65  0.32  −136  2.45  −85  1.42  −70  0.09  −141  0.04  −4  8,840  7,360  0.83 
7  −300  1,560  12,000  0.58  129  2.18  68  2.29  97  0.26  54  1.98  108  1.28  −180  0.07  −65  2.29  112  13,749  9,522  0.70 
8  1,050  250  −36,000  0.80  −84  0.52  144  0.64  29  2.40  9  0.1  165  2.78  11  1.32  51  0.13  −143  9,623  7,519  0.78 
9  −600  −850  −22,000  2.09  −120  0.38  91  0.34  173  2.42  −102  0.21  17  1.97  −180  0.20  −106  0.8  −91  9,687  7,380  0.76 
10  −1,200  250  −11,000  1.27  −165  1.31  169  2.28  −176  0.03  39  2.08  146  0.29  134  1.82  161  0.07  −37  9,206  8,140  0.88 
11  −500  1,200  −10,000  0.03  125  0.94  148  2.47  97  0.13  −150  1.89  86  1.10  153  0.20  −9  2.32  138  11,280  10,277  0.91 
12  600  550  −7,000  0.36  −10  1.93  26  0.04  −72  2.26  1  2  111  1.38  139  1.4  0  0.24  139  9,575  4,855  0.50 
13  50  −930  1,000  1.10  −120  1.18  −92  1.72  −78  0.84  −89  2.07  −107  0.96  −88  1.4  2  0.16  62  6,541  3,949  0.60 
14  −700  75  −26,000  1.23  −137  1.04  57  0.18  −12  0.18  −34  0.53  −8  0.06  9  2.27  168  1.86  180  6,243  4,802  0.77 
—  —  —  —  —  —  —  —  —  —  —  —  —  —  —  —  —  —  —  —  — 


Convergence history of the optimization by the hybrid optimization algorithm for first time step.
To evaluate the applicability of the proposed method, the optimization results of this study were compared against those obtained in Parikshit’s study [
Comparison of the optimization results between the existing study and this study.
Time steps  ITHS [A]  Hybrid [B] 
Ratio 

Total power (kW)  Total power (kW)  



— 
1  7,500  8,250  1.10 
2  10,850  10,824  0.99 
3  7,100  6,580  0.92 
4  7,800  9,172  1.17 
5  7,900  7,107  0.89 
6  5,800  7,360  1.26 
7  9,800  9,522  0.97 
8  7,000  7,519  1.07 
9  7,500  7,379  0.98 
10  9,800  9,206  0.83 
11  8,900  10,277  1.15 
12  6,200  4,855  0.78 
13  4,000  3,949  0.98 
14  7,600  4,802  0.63 
—  — 


In this study, a thrust allocation method was proposed for a semisubmersible drilling rig in order to produce the generalized forces and moment that are required to dynamically position the rig while at the same time minimizing the total power consumed. First, a thrust allocation optimization problem was mathematically formulated with the corresponding design variables, objective function, and constraints. In terms of the design variables, the speed of rotation and the azimuth direction of each thruster were selected. As compared with some studies about optimal thruster allocation, the selection of the design variables can not only represent the objection function more precisely but also control the thruster more easily. The objective function aimed to minimize the total power of the thrusters, and, in terms of the constraints, the governing equations for the thrust and azimuth direction of each thruster were used to generate the forces required in longitudinal and transverse directions as well as the moment about the vertical direction to dynamically position the vessel. Some limitations were also used for each thruster, and additional constraints were introduced by considering the energy loss due to the thrusterthruster interaction and the maximum thrust and azimuth direction of each thruster. The hybrid optimization algorithm was then used to solve the formulated problem. Finally, the proposed method was applied to an example to find the optimal thrust allocation for the semisubmersible drilling rig with 8 thrusters. A comparative test was also performed as part of the current study. In this comparative test, the proposed method was observed to produce slightly better results (about 2% in terms of total power) for thrust allocation than existing study, and thus the proposed method could be used to better determine a strategy to allocate the thruster of the drilling rig.
In the future, the total forces and moment required from the control system, which was the input of this study, will be estimated by considering the current position and the equations of motion of the drilling rig. That is, a more general method to dynamically position the rig will be further studied. In addition, we will improve the present hybrid optimization method and apply the improved method to the formulated problem in the future.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by (a) Brain Korea 21 Plus Program (Education and Research Center for Creative Offshore Plant Engineers of Seoul National University) funded by the Ministry of Education, Republic of Korea, (b) Engineering Research Institute of Seoul National University, Republic of Korea, and (c) Research Institute of Marine Systems Engineering of Seoul National University, Republic of Korea.