Dynamic Positioning Capability Analysis for Marine Vessels Based on A DPCap Polar Plot Program

Dynamic positioning capability (DPCap) analysis is essential in the selection of thrusters, in their configuration, and during preliminary investigation of the positioning ability of a newly designed vessel dynamic positioning system. DPCap analysis can help determine the maximum environmental forces, in which the DP system can counteract in given headings. The accuracy of the DPCap analysis is determined by the precise estimation of the environmental forces as well as the effectiveness of the thrust allocation logic. This paper is dedicated to developing an effective and efficient software program for the DPCap analysis for marine vessels. Estimation of the environmental forces can be obtained by model tests, hydrodynamic computation and empirical formulas. A quadratic programming method is adopted to allocate the total thrust on every thruster of the vessel. A detailed description of the thrust allocation logic of the software program is given. The effectiveness of the new program DPCap Polar Plot (DPCPP) was validated by a DPCap analysis for a supply vessel. The present study indicates that the developed program can be used in the DPCap analysis for marine vessels. Moreover, DPCap analysis considering the thruster failure mode might give guidance to the designers of vessels whose thrusters need to be safer.


Introduction
A dynamically positioned (DP) vessel is by the International Maritime Organization (IMO) and the certifying class societies (DNV, ABS, LR, etc.) defined as a vessel that automatically maintains its position and heading (fixed position or pre-determined track) exclusively by means of active thrusters (Sørensen, 2011). Dynamic positioning system (DPS) has been widely used in offshore engineering over the last five decades. Descriptions of DPSs including their early history can be found in Morgan (1978) and Faÿ (1990).
Operational safety is always the first consideration in the design and operation of a new DPS. To be able to plan a safe and efficient operation, it is important to know the window of the operation, and the maximum environmental conditions that the particular DP vessel can withstand. During critical operations such as drilling, oil production, and offloading, the requirements of the positioning precision are high regardless of the environmental conditions. It is thus important to know the positioning capability of the vessel in order to plan and execute operations in a safe manner (Pivano et al., 2012). A dynamic positioning capability (DP-Cap) analysis must be performed in designing a DP vessel.
When conducting a DPCap analysis, taking advantage of commercial softwares is often the first approach that comes to mind. DPCAP, developed by the Maritime Research Institute Netherlands, is a sophisticated software program which generates polar plots. However, only an old version of this software is available in the State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (SKLOE, SJTU). A minimum of 12 hours would be needed to conduct a complicated analysis with a suppressed iteration number. Developing one's own software program is an alternative approach when there is no efficient, accessible and commercial software available. Mahfouz and El-Tahan (2006) have also developed a software program as a marine tool for the selection of thrusters and their configuration, and for the use in preliminary investigations of the positioning ability of a newly designed vessel DPS. That is the motivation of this work.
The main objective of this paper is to present a newly developed software program capable of conducting the DP-Cap analysis for marine vessels. The accuracy of the DP-Cap analysis is determined by the precise estimate of the environmental forces as well as the effectiveness of the thrust allocation logic. The newly developed software program consists of two main parts: one is the estimation of the environmental forces; the other is the thrust allocation logic. Estimation of the environmental forces is based on the model tests, hydrodynamic computation, and empirical formulas (Sørensen and Ronass, 2001). The quadratic programming method is adopted to allocate the total thrust on every thruster of the vessel, due to the high efficiency and robustness of this method (De Wit, 2009). Yadav et al. (2014) has proposed an improved harmony search (IHS) algorithm for solving the nonconvex thrust allocation problem for a semisubmersible rig. In this paper, in order to maintain high efficiency of the program, a bisection search method is adopted to search the maximum environmental forces, as addressed by Xu et al. (2015a). The efficiency of a new software program DPCap Polar Plot (DPCPP) was validated by a DPCap analysis conducted on a supply vessel. Moreover, DPCap analysis which considered thruster failure mode was performed; the results may guide vessel designers who need increasing safety on their thrusters.

Estimation of the environmental forces
The components of the environmental forces and moments on a vessel are generally wind, current, and waves. Each component produces six degree-of-freedom forces and moments, namely: surge force, sway force, heave force, roll moment, pitch moment, and yaw moment. However, only the three horizontal degree-of-freedom forces and moment (i.e. surge force, sway force, and yaw moment) are taken into consideration when conducting a DPCap analysis.
Wind, wave and current are assumed to be coincident in direction when conducting a DPCap analysis. The forces and moment due to each component are evaluated individually and summed to evaluate the total steady-state environmental forces and moment (API, 1987). A Computational Fluid Dynamics (CFD) method is implemented to evaluate wind loads (Gosman, 1999;Zhang et al., 2010). Second-order mean wave loads are obtained as quadratic transfer functions (QTF) (Newman, 1977;Faltinsen, 1990). Methods of estimating the current loads abound in the literature (Kim et al., 2009;Vaz et al., 2009;Leite et al., 1998). In the present study, the wind and current loads were estimated from the model tests. The vessel model was fixed in the wave basin connected by a six-degree-of-freedom (DOF) force gauge. Wind and current were generated separately with a constant speed. The wind and current force coefficient can be obtained by the measured wind and current force and velocity. The second-order mean wave loads were obtained by QTF methods. The required QTFs were obtained from software utilizing potential theory.
The coordinate system is fixed to the vessel body with the origin located at the mean oscillatory position in the average water plane with the x axis points towards bow, the y axis points towards port and z axis points towards upwards. The relative environment angle, α, is positive anti-clockwise starting from the stern. Moments are positive anticlockwise. The coordinate system is given in Fig. 1.
The precise estimation of the environmental forces and moment is essential to the accuracy of the DPCap analysis. Estimation of the environmental forces and moment is based on the model tests, hydrodynamic computation, and empirical formulas (Sørensen and Ronass, 2001). If there are no model test results available, forces should be calculated using accepted standards. The International Marine Contractors Association M140 (IMCA, 2000) details how to estimate the environmental forces.

Thrust allocation logic
The thrust allocation problem can be formulated as an optimization problem, of which the objective is typically to minimize the use of the total thrust (or power) subject to the thrusters' maximum thrust limit. A characteristic feature of thrust allocation problems is that there are more than enough thrusters to generate the commanded forces and moment, so the thrust system is over-actuated. Generally, it is sufficient to generate the thrust forces to counteract the environmental forces performing the DPCap analysis when a feasible solution is obtained. However, in order to yield the maximum environmental forces, an effective optimization allocation is of significant importance since it has to be ensured that the thrust system cannot counteract the environmental forces under the most optimal allocation.
Many optimization methods are presented in the literature, and the quadratic programming method has been demonstrated to be relatively effective and efficient. Therefore, the present study adopts the quadratic programming method to solve the thrust allocation problem.

3-DOF allocation
Referring to Johansen et al. (2004), if a marine vessel is τ ∈ R equipped with m thrusters, the generalized force vector produced jointly by the thrust system is where is the surge force, is the sway force and is the yaw moment. The vector contains the magnitude of the forces produced by each individual thruster in the bow and port direction. The columns of the matrix B is given by The location of the i-th thruster in the horizontal plane is in the adopted coordinate system.

Problem formulation
The thrust allocation problem can be simply formulated as: min Au ⩽ which can be solved via the QP method. u indicates the thrust force vector. A and b form the inequality constraints b, which contains the thrust region(s) for every type of thruster. The total power consumption is represented by the criterion, combining the power consumption coefficients W of the individual thrusters.

Thrust region constraints
Feasible zone approximations should be made to translate the nonlinear constraints to linear forms (De Wit, 2009). For each type of thruster, a set of inequalities is created to represent the thrust region. The thrust region will be formed by a finite intersection of hyper-planes, resulting in a convex polygon. A tunnel thruster has linear constraints since it can only generate force in one direction. However, for azimuthal thrusters and main thrusters with a rudder, constraints are absolutely nonlinear.

Tunnel thruster
The thrust region of a tunnel thruster can be represented by a line segment. Given its thrust limits T max and T min , the constraints (i.e. inequalities (4) and (5)) can be added to the QP problem. The thrust region of a tunnel thruster is illustrated in Fig. 2.

Azimuthal thruster
For an azimuthal thruster the inequality constraints are more complex. The thrust region for an azimuthal thruster without any forbidden zones has a circular shape with the radius T max . This can be written as: which requires linearization by approximating the circular region with a convex polygon. Taking forbidden zones into consideration, approximations of the thrust region of the azimuthal thruster can be divided into three categories, as illustrated in Fig. 3.
If the azimuthal thruster can be operated freely from 0 to 2π, the circular region with a radius can be approximated by an N-sided regular polygon ( ). The regular polygon divides the circular region into N sectors, each of which has a central angle . The approximated thrust region can be represented by the following inequality system.
for every If the azimuthal thruster has a feasible zone where , the thrust region is shaped like a Pacman. Because this shape is not convex in general, it must be split up into two disjunct convex thrust regions, as illustrated in Fig. 3. Approximation of the separated thrust region transfers to the next category.
If the azimuthal thruster has a feasible zone (i.e. ), the thrust region is a convex pie-like zone. The circular arc and the sector angle should be handled separately to approximate the pie-shaped region. The circular arc is represented by N hyperplanes, which integrate with the two sector angles to form a convex polygon, as illustrated in Fig. 3. The approximated thrust region can be represented by the following inequality system: For every with k = 0, 1, ..., N -1. Here and should be ordered counterclockwise. It should be emphasised that the premise is . Another point is that N should be large enough so the circular region or the pie-like region can be precisely approximated. An approximation error has been defined by De Wit (2009). With the approximation error smaller than 1%, the minimum interval of the arc required to achieve this accuracy is about 15°. Given the high computation efficiency of the QP server, a larger N is recommended.

Main thruster with the rudder
The thrust region for a main thruster with the rudder operation in the forward mode is derived from the lift and drag curve (Molland and Turnock, 2011). Given the lift and drag curve and the bollard pull , the largest thrust and will be obtained for every rudder angle. The thrust region is represented by the myriad points for every continuous rudder angle. However, a finite of points with the origin of the coordinates define a polygon to approximate the thrust region, as presented in Fig. 4. Although the thrust region of main thruster with the rudder is continuous for every angle, it cannot be expressed by the formulas, which is different from the azimuthal thrust region.
The approximated thrust region can be represented by the following inequality system: for every couterclockwise succeeding pair of points and , where . is the start angle sector (couterclockwise) of the main thrust region, and is the end angle sector. Generally, and . Note that is the premise of Eq. (12), which is naturally satisfied since the operation angle of the rudder lies in a subinterval of in most circumstances.
The thrust region for a main thruster with the rudder operation in the reverse mode is obtained by changing the orientation of the tunnel thruster, as described in Fig. 4. Given the maximum thrust of the main thruster operation in the reverse mode, the thrust region for a thruster-rudder pair operation in the reverse mode is represented by the following inequalities: [

Handling non-convex thrust regions
The non-convex thrust region emerges when one handles a Pacman azimuthal thrust region and/or main thruster region which is generated in both forward and reverse modes. The thrust region for these types could be divided into separate convex thrust regions. A combination method is used to handle non-convex thrust regions when one conducts the QP search. If two thrusters which have two  separate convex thrust regions A/B and C/D, the QP method will search the following feasible regions, AC, BC, AD, and BD, respectively to yield the optimal solution. The number of the feasible thrust regions needed to be searched can be determined by where n i is the number of the separated convex thrust regions generated by the division of the i-th thruster's nonconvex thrust region. If the i-th thruster's thrust region is convex, n i =1. The optimal solution does not need to be a global optimal solution (i.e. the most optimal one in all feasible regions), whereas it can be an optimal solution in any feasible thrust region. When the QP method finds an optimal solution in a feasible thrust region, the QP search ends.

Comparison study
By using the newly developed software program DP-CPP, a DPCap analysis was conducted for a supply vessel. The obtained results were compared with those computed by a sophisticated commercial software program DPCAP. Parameters of the supply vessel are tabulated in Table 1. The thruster configuration is illustrated in Fig. 5. The maximum thrust of each thruster is tabulated in Table 2. Wind and current forces and moment were estimated by a model test conducted in SKLOE. Wave forces and moment were estimated via CFD method. The current velocity was 1 knot in this case, as recommended by IMCA M140. The relationships among the wind velocity, wave height, and period are tabulated in Table 3.
This study took into account the general thrust reduction due to the interactions among the azimuth and tunnel thrusters and the main propellers, and also the thruster hull interactions. The reduction ratio was assumed to be 0.9 for all thrusters. The main thruster with the rudder can be operated in both the forward and reverse modes. When the main thruster is operated in the reverse mode, 70% of the maximum thrust in the forward mode is regarded to be the maximum thrust limit. A combination method was used to handle the non-convex thrust region of the main thruster with the rudder.
The heading of the vessel was examined from 0° to 360°w ith an angle step of 1°. The increment of the wind velocity was 1 m/s to approach the maximum wind velocity. Fig. 6 describes the polar plot of the DPCap of the supply vessel obtained by the program DPCPP and DPCAP, respectively. It can be seen that the results of DPCPP coincide well with those of DPCAP. The thrust allocation logic used in this paper has better performance than the Lagrange method used in DPCAP. When the iteration number is suppressed to improve the efficiency, the accuracy somehow decreases. Therefore, the results of DPCPP are little larger than those of DPCAP. Table 4 gives the maximum wind velocities and the corresponding consumed thrusts for some specific headings by 8.17 Notes: 1 longitudinal center of gravity from centerline; 2 transverse center of gravity from centerline; 3 vertical center of gravity from baseline.    (2012). The DPCap analysis took less than half an hour on a 2 core 2.5GHz computer by use of DPCPP, while it took 12 hours to conduct the DPCap analysis by use of an old version of DPCAP. It can be concluded that the newly developed software program is able to conduct a DPCap analysis and the efficiency is relatively higher, compared with the old version of the commercial software. More cases considering thruster failure mode are investigated in Section 6.

Failure mode of the DPCap analysis
DPCap analysis considering thruster failure mode is essential for ensuring safe vessel operation at sea. For some operations, IMCA guidelines require that after the worst case failure, the vessel shall maintain sufficient capabilities within safe limits (IMCA, 2000). On the other hand, thruster failure mode of the DPCap analysis can give designers instructions on the role each thruster acts against the environmental forces. If one thruster takes more effects, it should be paid more attention by raising the safety factors or considering the thruster redundancy design. The same supply vessel in Section 5 was adopted to conduct the thruster failure mode of the DPCap analysis. Investigated cases are tabulated in Table 5. Thruster No. 1 to No.7 represent the thrusters deployed on the vessel, as illustrated in Fig. 5 indicates that the thruster is working, while 0 indicates that the thruster is in the failure state. DPCPP was adopted to conduct the thruster failure mode of the DPCap analysis. The polar plots of the dynamic positioning capability are given in Figs. 7-9.
The wind velocity envelopes of Cases 1111100 and 1111110 are inside the wind velocity envelope of Case 1111111, as shown in Fig. 7. It is demonstrated that the main thruster with the rudder takes on an important role in resisting environmental forces and moments. The maximum wind velocities on 180° and 0° headings have 20% reduction when one main thruster with the rudder fails, and an approximate reduction of 55% when two main thrusters with the rudders fail. Although the positioning capabilities have an obvious reduction due to main thrusters and rudders failure, the positioning capability (42 m/s on 180°h eading and 25 m/s on 90° heading) is acceptable when the vessel is operated in moderate sea states. Several conclusions can be drawn: in moderate sea states, main thrusters with rudders can be shut down to reduce wear and tear; in less serve sea states, one main thruster with the rudder can be shut down; in serious sea states, the two main thrusters with the rudders should work together to resist the high environmental forces and moment. These conclusions may give guidance to the operator of the supply vessel. If the supply vessel is operated under serious environmental conditions for a long period, more safety concerns must be given to the main thrusters with the rudders since the thrust system cannot supply enough thrust forces when they fail.
The wind velocity envelopes of Cases 0011111, 1011111 and 0111111 are inside the wind velocity envelope of Case 1111111, as shown in Fig. 8. Moreover, the  wind velocity envelopes of Cases 1011111 and 0111111 are almost the same. If both thruster No. 1 and No. 2 fail, it causes a positioning capability reduction of approximately 25% from 90° to 150° heading by comparing the DPCap of Case 0011111 with Case 1111111. It is dangerous for the vessel to be operated in these headings. An interesting phenomenon of Case 0011111 is that there is a notch in the polar plot on the 180° heading. Without thruster No. 1 and No. 2, chances are that the thrust system cannot provide enough thrust to balance the yaw moment, even though the thrust remains available to resist the surge and sway forces. It can be concluded that thruster No. 1 and No. 2 have the same important roles in the whole thruster system (i.e. thruster No. 2 is a redundancy design of thruster No. 1).
Azimuthal thruster failure is regarded in Cases 1111011, 1110111 and 1110011, as presented in Fig. 9. The wind velocity envelopes of Cases 1111011, 1110111 and 1110011 are inside the wind velocity envelope of Case 1111111. overlap with each other. The positioning capability of Case 1111011 is larger, ranging from 180° to 240° heading, but smaller from 320° to 360° heading and from 0° to 40° heading than that of Case 1110111. Azimuthal thruster failure causes more remarkable positioning capability reduction than tunnel thruster failure, especially on the 90° heading. However, one azimuthal thruster failure is acceptable in moderate sea states, which may be of help guide the operator of the supply vessel.

Conclusion
In the present study, the authors developed a new program which can perform a DPCap analysis. The software program is composed of two units: estimation of the environmental forces and moment and the thrust allocation logic. Methods to estimate the environmental forces and moment were briefly introduced. A thrust allocation logic was presented in detail, from the formulation of the problem, approximation of the thrust region to the handling of non-convex thrust regions. The thrust allocation logic can be adopted in the DPCap analysis for vessels with the thruster configuration composed of the tunnel thruster, azimuthal thruster and main thruster with the rudder. A DPCap analysis for a supply vessel was conducted to validate the software program. The results indicate that the software program is able to conduct a DPCap analysis with high efficiency. Finally, a DPCap analysis considering thruster failure mode was conducted to investigate the positioning capability of the supply vessel comprehensively. Based on the thruster failure mode DPCap analysis, the guidance was given to the designer and operator of the vessel.