Time Domain Simulation of a One Line Failure for a DP-assisted Mooring System

Jianxun Zhu, Liping Sun, Shengnan Liu and Jichuang Kang

Deepwater Engineering Research Center, Harbin Engineering University, Harbin, 150001, China

1 Introduction1

The exploration and production in deeper water use may be made of semi-submersibles and ship-shaped floating structures. In the production phase the floaters are normally permanently moored by the mooring system to withstand the local extreme environment conditions.

Although a considerable amount of damping of the risers and mooring lines in deep water can be induced to counteract partly the low frequency motions, the mooring system needs to take the ultimate force to keep the vessel on station. In some water depths the length of the mooring lines may be considerable. The cost effect by reducing the mooring system in terms of less mooring lines or a lighter system by applying assisted DP is worth considering. And in deep and ultra-deep water, not only are the long mooring legs expensive, the installation costs considerably soar higher with the increasing of the water depth.

At the Marine Engineering Conference, the concept of the DP-assisted Mooring System, which indicated the reliability,security and economy of this system, was first proposed by Sargent >amp; Morgan (1974), Aalberset al. (1995, 1996),through the experiments in their marine laboratory,demonstrated that the system can reduce the mooring force and discussed the adaptability of different positioning styles to different water depth ranges. Wichers and Van Dijk (1996)illustrated the effectiveness and benefits of using DP to assist the mooring system in survival conditions. Wichers and Van Dijk (1999) studied the advantages of the DP-assisted Mooring System by using the programs DYNFLOAT and DPSIM. Fossen (1999) presented a model consisting of a rigid-body sub-model for the vessel, and a finite element sub-model for the mooring system. Fossen(2000) gave an overview of the methods for passive ship control and observer design. Jenman (2005) outlined the key elements of this new standard and then provided some feedback on an FMEA and full-scale trials of a thruster assisted system installed on a DP class 3 drilling semisubmersible. Stephens >amp; Meahan (2007) researched the motion performance of a new thruster assisted mooring system for global producer III. Nguyen >amp; Sorensen (2009)showed that the switching control could extend the weather window of the PM system. In this regard, Wang (2010)proved that the DP-assisted Mooring System can improve the position accuracy and safety margin of a platform. And Sun (2011) concluded that under a thruster failure mode, the DP-assisted Mooring System has better positioning accuracy and lower power consumption than the DPS. Yang >amp; Wang(2012) have assembled a comprehensive study of the DP-assisted mooring system. Fanget al.(2013) presented that the structural reliability criterion based algorithm ensures the safety of the mooring lines in a variety of external environmental conditions and also in situations where there is failure of a single line.

In this paper, for the purpose of studying the impact of a mooring line failure on the DP-assisted Mooring System, the mooring system and thruster’s arrangement were initially designed, followed by the positioning accuracy and power consumption of a semi-submersible platform under different mooring-line failure modes. These were further studied in a time-domain simulation program and some reasonable advice was put forward.

2 Horizontal motion equations for the platform

To establish a complete and reasonable platform motion mathematical model is the foundation of the response analysis. Under the complex environmental load and DP-assisted mooring system joint action, the platform is always in a circle of position deviation. The object we discuss in this paper is a semi-submersible platform with 6 azimuth thrusters and 12 mooring lines, and Figure 1 shows the underwater appearance of a platform and its arrangement of the DP-assisted mooring system. The platform is regarded as a certain quality and mass distribution rigid body, and the equation of the motion in the ship’s coordinate system is obtained by utilizing the dynamics theory of floating bodies.In order to simplify the equation of motion, which is applied to the research of the platform motion, the original point of the hull coordinate system is set at weight heart, then an equation of motion at a moment i can be inferred:

where m is the platform quality,Izis the rotary inertia about theOZaxis, the indexes likeH,E,T,Mrespectively stand for the hydrodynamic force, environmental load,propeller thrust and the horizontal mooring tension acting on the platform.

Fig.1 Arrangement of the DP-assisted Mooring System

The magnitude of the vertical force (moment) is large and its cycle is short. To reduce the propeller machinery and fuel loss, the DP system often only makes an immediate response to the low frequency horizontal motion. Because the platform’s low frequency motion, to a great extent, is affected by the DP system, we could suppose that the floating body affected by the external effect has a small amplitude motion in the equilibrium position, and then the platform motion can be simplified:

Since the motion parameters of the increment are small,the parameters of the higher powersDu vDandDv rDare considered high order traces that can be ignored as well as the velocity and angular velocity of the cross coupling between items and the fluid inertia force (moment) caused by their quadratic terms likeand then the motion equation can be written as follows:

whereXu,Yv,Yr,NvandNrare the hydrodynamic derivatives (because the underwater part of the platform is symmetrical about the planeXOZ, but not completely symmetrical about the planeYOZ, and the surge motion is independent but has weak coupling between the sway and yaw);EandMrepresent the environmental load and the load of the mooring system.

3 Mathematical model for the whole motion system

3.1 Hydrodynamic model of the platform

The main components of the semi-submersible platform,and its principal dimensions are shown in Table 1~Table 4.

Table 1 Main dimension of the semi-submersible platform

Table 2 Parameters of pontoon m

Table 3 Parameters of column m

Table 4 Parameters of upper box deck m

The hydrodynamic model of the platform was built by using the Ansys as shown in Fig 2. Through the aqwa-line,we obtained the linear hydrodynamic coefficient and the added mass in thex,ydirections and the additional inertia moment around theOZaxis.

Fig 2 The hydrodynamic model of the semi-submersible platform

3.2 Mathematical model of the dynamic positioning system

The typical law for the positioning control system is the Proportional-Integral-Derivative (PID) control. It should be noted that the term ‘control forces’ refers to the control forces in the surge,sway and moment in the yaw. So the control instruction can be obtained:

whereεis the difference between the measured and target values;Kp,KIandKDrepresent the proportional gain coefficient, integral gain coefficient and differential gain coefficient, respectively;FW(αW,νW) is the wind feed-forward force;FMrepresents the force provided by the mooring system.

Another important part of this model is the distribution of the thrusting force, an optimization problem, which is multivariable and constrained. Based on the algorithm of the distribution optimization method, the control allocation problem can be transformed into including the objective function, the nonlinear constraint optimization mathematical model including equality constraints and inequality constraints, and then it can be solved by using the optimization algorithm. Since this platform uses azimuth thrusters, the thrust force is able to act on the platform in any directionαi(the angle is referred to as theXaxis). The thrust force can be divided into longitude force,tx,iin the surge and lateral force,ty,iin the sway. Obviously, there is a relationship amongαitx,iandty,i:

To simplify the problem, it can be proposed that the minimum total thrust is equivalent to the minimum total power consumption, and then the objective function can be written as follows:

whereNis the number of the effective propellers in the current. Equality constraints of this function include:

Specifically, the total thrust (moment) made up of the propeller component should be equal to the total thrust(moment) required by the controller. Inequality constraints include:

Namely, each propeller thrust should not exceed its maximum thrust. But the maximum thrustTmax,iusually accounts for 90% of the propeller’s maximum thrust due to the disturbance among the propellers and between the propellers and the hull.

The ban angle is realized by the following inequality constraints:

whereαl,iis the lower limit of the ban angle area whileαu,iis the upper limit.

In addition, according to the mechanical characteristics of the thrusters, the change rates of the thrust force and the azimuth thrusters’ rotation are limited in each time step.

According to these equality and inequality constraints, the thrust distribution problem can be successfully transformed into an optimization mathematical problem concerning the independent variable t and target function f.

The DPS is equipped with six azimuth thrusters with a total power of 3500 kW. The performance parameters of the thrusters are shown in Table 5.

Table 5 Main parameters of the thrustersModel Wartsila FS3500/NU

3.3 Mathematical model of the mooring system

The effect of the mooring system is similar to the control action from the positioning control system in terms of providing restoring, damping and mean forces. Among these,the dominant effect of the mooring system is to provide mean force compensating the mean drift loads arising from wind, waves and currents.

In this paper, the mooring system employed by the platform is called the catenary mooring system. It can be divided intoNsections to establish the equation of multi-component mooring lines and can be, according to the catenary equation carried out by Hu (2007), written as:

wherexiis the horizontal distance between the top point of section i and the anchor point of each line, whileziis the vertical distance;φ0irepresents the angle between the mooring-line and anchor;φirepresents the angle between the mooring-line and floater; Th represents the horizontal pretension;is the wet weight of section i.

The vertical tension near the floater can be accumulated as follows:

where t is the touchdown point;Rrepresents the vertical force on the mooring line provided by the anchor.

The elastic correction is needed for the elastic stretching of each section:

The cubic spline curve is referred to asxiandThcan be obtained through solving the equations (11)-(14). In the time domain simulation, we can obtain thexiin each time step.And then Th acting on the platform can be successfully obtained by the curve in each time step. According to the methods mentioned above, the mooring system can be successfully transformed into a mathematical problem.

The mooring system consists of 12 mooring lines and each line is divided into three segments with different materials. The mooring lines’ configuration is shown in Fig 2. The properties are listed in Table 6.

Table 6 Main properties of mooring lines

3.4 Mathematical model of the DP-assisted mooring system

The main objective of the DP-assisted Mooring System(PM) is to keep the vessel in a fixed position while the secondary objective is to keep the line tensions within an allowable range to prevent line breaks. Thus this system would be composed of a passive mooring system and a DP system.

The mathematical mode of the DP-assisted mooring system is a combination of the modes mentioned in 3.2 and 3.3. And then the whole motion system can be translated into a mathematical mode by solving the motion equation (3).

3.5 Calculation of the environmental load

The calculation of the wind load is based on the API recommended practice-the module method: the main structures above the platform waterline are dispersed, and the loads of each part are computed according to the wind area and shape. By the same token, the module method is also adopted during the flow load processing. In this paper,the buoy and post under platform waterline are dispersed,their flow load is calculated respectively and the connecting wing is processed as the Morison rod. It should be pointed out that because the module method simplifies the practical structure, the result is the approximation of the permitted precision scope.

The platform’s second order water drift force is obtained by the spectrum analysis method. Firstly the sea spectra is dispersed intoNequal parts, then the irregular wave is transformed into the form of a harmonic wave superposition,each wave band corresponds with wave frequencyωi, wave amplitudeAi.

The slow drift force is calculated approximately through the Newman’s simplified formula:

whereεiis the harmonic wave’s random phase angle. This method is not applicable for shallow waters, and the high frequency part produced from the calculation could be filtered through the filtering method.

In this paper, the 10-year wind domain is taken as the extreme sea conditions as listed in Table 7. Besides, it is assumed that the wind, wave-drift, and current act in the same direction as illustrated in Fig 2., so that the harshest environmental load would be applied to the semi-submersible platform in question.

Table 7 Environmental conditions of the platform

4 Time domain simulation of the one-line failure for a DP-assisted mooring system

The platform motions are numerically simulated in the time domain for different modes by using the Time Domain program. Whereas the time duration is taken at 2 000 s, and the statistical data is recorded at every 0.5 s interval which further describe the platform motion and power consumption etc. to the time history.

Considering the symmetry of the mathematical model and load direction, line numbers 2, 5, 8, and 11 are assumed failures separately to simplify the calculations. Thus the study is simply focused on the complete mode and Nos.2, 5,8 and 11 line failure modes, and then the difference comparisons among these modes will be made in respect to the position accuracy, power consumption and tension of the mooring lines.

The platform motion to the time history in each mode is shown in Fig.3. And the statistical data is presented in Tables 8, 9 and 10.

Fig 3 Offsets of the platform for different modes

Fig.3 and Table 8 reveal that the different failure modes bring different impacts on the positioning accuracy. For the No.2 failure mode, the surge decreases by 20%, while the sway increases by 8.4% and the yaw rises by 34.5% as compared to the values for the complete mode. Since the tension in line 2 is opposite to the environmental load in regards to direction, which means the No.2 failure is equivalent to the loss of ability against the environmental load, the thrusters have to increase their thrust to keep the position. Because the vertical component of the thrust is smaller than the reduction, the sway gets bigger. On the contrary, the horizontal component of the thrust is larger than the reduction, and the surge becomes smaller.

Table 8 Statistical offsets

For the No.8 failure mode, it is observed that the surge decreases by 72.2%, while the sway increases by 7.2% and the yaw rises by 96.7% as compared to the values for the complete mode. The tension in line 8 is in the same direction as the environmental load, which means the No.8 failure is equivalent to the deduction of the environmental load and thus the surge of the platform is dramatically reduced. The yaw however increases significantly due to the reason that the failure of line 8 leads to the reduction of clockwise torque.

Table 9 The statistical data of power consumption

Table 6 shows that the power consumption for each failure mode, except for No.8, dramatically increases compared with the complete mode. The reason is that the one line failure will partially lose the ability to withstand the environmental load, and thus the thrusters have to output more power to compensate for such a loss. The only exception is the No. 8 failure mode，which reduces the demand of the thrust and the power consumption and therefore decreases accordingly.

Table 10 The tension of the mooring lines kN

Table 10 indicates the minimal effect on the mooring system for each one line failure mode. The explanation for thisphenomena is that the drift of the platform is relatively small, so that the tension in the mooring lines nearly remains invariable. It concludes, from this point, that the existence of the DP system can help maintain the normal operations of the mooring system even in the case of one line failure and further prolong the life of the mooring line.

5 Conclusions

Through the time domain analysis for different one line failure modes, the following conclusions can be drawn:

(1) In general, the failure of the windward line will result in the reduction of the position accuracy and the increase of power consumption. And the failure of the leeward line leads to the reduction of power consumption and longitudinal drift. Aiming at the reliability, economy and feasibility of the control measures, the leeward line,therefore, can be slackened.

(2) In the case of the leeward line failure, close attention has to be paid to the variety of yaw forces provided by the leeward line. Some measures, e.g. relocation of the fairlead,can be taken to reduce the force to a minimum.

(3) For the least impact on the mooring system, the number of mooring lines may be reduced to cut the costs of the mooring system.

Aalbers AB, Janse SAW, de Boom WC (1995). DP assisted and Passive Mooring for FPSO’s.Offshore Technology Conference,Houston, 281-288.

Aalbers AB, Merchant AA (1996). The hydrodynamic model testing for closed loop DP assisted mooring.Offshore Technology Conference, Houston, 40-43.

Fang Shaoji, Leira BJ, Blanke M (2013). Position mooring control based on a structural reliability criterion.Structural Safety, 41,97-106.

Fossen TI, Aamo, OM (1999). Controlling line tension in thruster assisted mooring system.Proceedings of the 1999 IEEE International Conference on Control Applications, Kohala Coast, Hawaii, 1104-1109.

Fossen, TI (2000). Nonlinear passive control and observer design for ships.Modeling Identification and Control, 3(21), 129-184.

Hu Xuejun (2007).Dynamic analysis of mooring cable. M.S. thesis,Huazhong University of Science and Technology.

Jenman C (2005). Mixing dynamic positioning with mooring.Dynamic Positioning Conference, Houston, 32-35.

Nguyen DT, Sorensen AJ (2009a). Switching control for thruster-assisted position mooring.Control Engineering Practice, 17(9), 985-994.

Nguyen DT, Sorensen AJ (2009b). Setpoint chasing for thruster-assisted position mooring.Oceanic Engineering, 34(4),548-558.

Sargent JS, Morgan MJ (1974). Augmentation of a mooring system through dynamic positioning.Offshore Technology Conference,Houston, 48-51.

Stephens RI, Meahan A (2007). Design and commissining of a new thruster assisted mooring system (TAMS) for global producer III.Dynamic Positioning Conference, 9-10.

Sun Pan, Wang Lei, Wang Liang (2010). Research on power consumption of position mooring system for a deep sea semi-submersible platform.Oceanic Engineering,28(3),53-57.Wang S (2010). On the assessment of thruster assisted mooring.

Offshore Technology Conference, Houston, 254-260.

Wichers J, Van Dijk R (1996). Benefits of using assisted DP for deepwater mooring system.Offshore Technology Conference,Houston, OTC 10781.

Yang Huan, Wang Lei, Li Xin (2012). Review of the research on thruster assisted position mooring system.Research and Exploration in Laboratory, 26(4), 23-26.