Modelling and Control of Thruster Assisted Position Mooring Systems for Ships

Copyright © IFAC Manoeuvring and Control of Marine Craft, Brijuni, Croatia, 1997

MODELLING AND CONTROL OF THRUSTER ASSISTED POSITION MOORING SYSTEMS FOR SHIPS

J ann-Peter Strand *, Asgeir J. S~rensen t and Thor I. Fossen *

* Norwegian University of Science and Technology Department of Engineering Cybernetics, N-7034 Trondheim, Norway. E-mail: [email protected]@itk.ntnu.no t ABB Industri AS, P. O. Box 6540 Rodel;1kka, 0501 Oslo, Norway, E-mail: [email protected]

Abstract: This paper aruesses the mathematical modelling and control design of an automatic thruster assisted position mooring (POSMOOR) system. Such control systems are applied to anchored floating production ofHoading and storage vessels (FPSO's) and semi-subs. The POSMOOR controller is designed using model based control with a LQG feedback controller in conjunction with a Kalman filter. The controller design is in addition to the environmental loads accounting for the mooring forces acting on the vessel. This is reflected in the model structure and in the inclusion of new functionality. Keywords: Ship control, moored ships, position mooring systems

In position mooring the thruster assistance will be complementary to the mooring system. Normally, most of the station keeping in surge and sway will be provided by the mooring system. This in contrast to dynamic positioning (DP) of nonanchored ships, where the station keeping is left entirely to the thruster system. Due to the physical properties of the mooring system, new control functionality and modes of operation are required. Slowly-varying excitation forces due to waves and wind may excite resonant oscillatory motions of the moored structure. Hence, an important feature of the POSMOOR system is to damp these motions by proper action of the thruster system.

Fig. 1. Moored vessel.

1. INTRODUCTION A position mooring (POSMOOR) system is a control system for automatic thruster assistance of moored structures, Fig. 1. Such systems are important for safe operation of floating production storage and off-loading vessels (FPSO's) and semi-submersible rigs. For the oil companies the FPSO concept has proved to be a cost efficient and flexible solution to extract oil from the smaller and marginal fields.

In rough weather situations a break in one anchor line may be critical for the overall safety. A mooring line break will result in higher load in the remaining lines. The subsequent large transient motion of the vessel increases the probability of yet another break, and even more important, possible break in the risers.

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where the right hand expressions represent generalized external forces 1 . 'T thr is a vector of generalized control forces provided by the thruster system. 'T wi is the mean wind load vector, 'T wa2 the second-order wave drift load vector and 'T mo is a vector of generalized mooring forces. Modelling of the mooring system is treated in the next section. M is the low-frequency mass matrix including added mass, CRB(V) and CA(v r ) are skewsymmetric Coriolis and centripetal matrix of the rigid-body and the added mass, respectively, DL is a strictly positive damping matrix, caused by linear wave drift damping and laminar skin friction and DNdvr,lr) is a nonlinear damping matrix. Vr is the relative velocity vector with respect to water current and Ir is the current direction relative to the vesseL For details regarding this model, the reader is referred to (S0rensen, et. al., 1996) .

Fig. 2. Reference frames. 2. MATHEMATICAL MODELLING

2.1 Kinematics Three different reference frames are used in position mooring, as illustrated in Fig. 2. The earthfixed reference frame XEYEZ E is located in the vicinity of the desired position of the vessel and defined as the field zero point (FZP) . A vesselparallel frame X v Yv Zv is fixed in the earth-fixed frame , rotated to the desired heading angle 1/J o and then translated to the desired Xo and Yo position coordinates. The body-fixed XY Z-frame is fixed to the vessel body with the origin located at the centre of turret (COT). It is assumed in the further modelling that the COT is located at the centre line of the vesseL

2.3 Mooring System

Generally, a spread mooring system consists of n pre-tensioned anchor lines connected to a turret of the ship, as illustrated in Fig. 1. The number of anchor lines may vary, typically from 6 to 12 lines. The location of the turret is decisive to the weather vaning abilities of the moored vessel (Aalbers, et. al., 1995) . When the turret is rotatable, the relative angle between the turret and the body-fixed frame is given by the turret angle, at. The length of the anchor lines are adjusted by winches and determines the pre-tension and thus the stiffness of the mooring system. In this paper it is assumed that the line length and the turret are manually controlled, and fixed around a working point.

The linear velocities of the ship in the body-fixed frame relative to the earth-fixed frame are related by the transformation:

r,

(1)

= J("7)v

where the earth-fixed vessel position and heading, and the body-fixed surge, sway and yaw velocities are defined by the vectors "7 =[x, y , 1/J]T and v =[u , v , r]T. The rotation matrix J("7) is:

J ("7) = J (1/J) =

COS [

1/J - sin 1/J 0] cos 1/J 0

sin 1/J

o

0

The anchor lines enters the turret through fairleads below the hull and the coordinates are defined as terminal points (TP). The anchor lines are composed of chain, wirelines or synthetic material, often partitioned into several segments of different types and properties.

(2)

1

2.2 Vessel Model The mathematical model describing the vessel dynamics is separated into a low-frequency (LF) model and a wave-frequency (WF) modeL The WF motions are assumed to be caused by firstorder wave loads. The WF vessel model is assumed to be decoupled from the mooring system and the reader is referred to (Faltinsen, 1990) and (S0rensen, et. al., 1996) for details.

Nonlinear Low-Frequency Model. The low-frequency model of the moored vessel in the horizontal plane may be written as:

My

+ CRB(V)V +

+DNL(Vr , lr)Vr

CA(vr)Vr

= 'Twi+'Tmo+'Twa2 +

1 Throughout this paper forces in surge and sway and moment in yaw will b e referred to as generalized forces or simply forces .

(3)

+ DLV

Single Line Modelling. In Fig. 3 a profile of an anchor line is sketched. The tangential force at the TP, often called the line tension, is denoted T, and its horizontal force component is denoted H. Ls is the suspended length, defined as the static cable length from the TP on the ship to the point where the line first touches the seabed. The latter point is called the touchdoum point. L tot is the total length of the cable, D is the water depth, Xh is the horizontal distance between the TP and the touchdown point, and X h is the horizontal distance between the TP and the anchor.

'Tthr

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T

x

D

I.

x.

"I Fig. 4. One anchor line in a mooring system.

Fig. 3. Line profile.

TPE

= x - cos(tj; + ac) yr PE = y - sin(tj; + ac)

Xi..

Mooring lines are subjected to three types of excitation (Triantafyllou, 1994): Large amplitude LF motions, medium amplitude WF motions and small amplitude, very high frequency vortex-induced vibrations. Since these effects occur at different frequencies, the excitations can be treated separately in order to make the modelling tractable. For the control system design the focus will be placed on the mooring lines' influence on the LF model (3).

. ( ) TPO lOtTPO + sm tj; + at Yir (6) cos(tj; + a,) Yi PO •

Xi PO -

The total restoring generalized forces represented in the earth-fixed frame can be written: k

E mo

~ . ,=1

= ~

[

C

Hi sm ':'S!3i ] Hi !3 i Hix sin!3 i - Hiy cos!3 i

(7)

where x = x - X;PE, y = y - y;PE and f3 i is an earth-fixed horizontal angle of line i given by:

Let T and H be modelled as a combination of the static_ terms T and fI and the dynamic terms andH:

t

f3 i

= atan2 (yf,

xn

xh, =x,A _ x ,TPE yf =yf

- y;PE

as illustrated in Fig. 4. The horizontal distance from the anchor to the TP of line i is:

When knowing the position of the line's upper end (TP) and the anchor position, it would be desirable to find the static tension and its horizontal component at the TP. This problem cannot be solved analytically. However, such distance/force relationships, often denoted as the line characteristics, can be found using specialized software (Marintek, 1995):

(8) Assume that the initial TP coordinates and the anchor coordinates are known. If a static line characteristics have been found for each line

(9) (5)

the static formulation of the restoring forces (7) becomes a function of vessel position, vessel heading and turret angle only:

The line characteristics are found numerically by solving the catenary equations, where the influence of hydrostatic effects and current are taken into account. For dynamic modelling of mooring lines and moored structures, the reader is referred to (Triantafyllou, 1994).

(10) The LF modelling of the mooring forces becomes quasi-static if disregarding the dynamic effects in the mooring lines and using (10) in the LF moored vessel model (3). According to (TriantafylIou, 1994) it can be assumed that the mooring system does not affect the 1st-order wave-induced motion of the vessel. On the contrary, however, superimposing the WF vessel motions with the LF motions yields a significant increase in the LF damping, due to the transverse response of the lines along the catenary profile. These slowlyvarying mooring damping forces varies in a nonlinear manner with the wave amplitude and the

Spread Mooring System. Let the horizontal force at the terminal point (TP) of each line be denoted Hi, i = 1 .. . n. Moreover, let (xt,y{,) denote the earth-fixed anchor position of line i and (x;PE, y;PE) the earth-fixed positions of the line's corresponding TP on the turret. The latter coordinates will depend on the vessel position, heading and turret angle. If (x;PO, y;PO) denotes the terminal point coordinates relative to the body fixed frame at initial turret angle, at = 0, one can write:

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frequency. For typical wave spectra the effective drag coefficients increases in a magnitude of 2 to 4, see (Huse and Matsumoto, 1989). Similar increased damping effects are also reported in (Triantafyllou and Yue, 1994). The following LF model of the mooring forces is proposed:

the generalized mooring forces (11) in a working point 1]0 = [xc, Ye, 1/JoJT can be approximated by: T mo (1],

v r ; ad ~

K M 1] + T~o (16)

-DMV -

where D M is the additional damping due to the mooring system. KM is found by linearization of (10), as demonstrated in the following. Let the horizontal forces at TP of each line i the working point be expressed as:

where the first term represents the additional mooring damping and the latter the static mooring stiffness. WF vessel motion in surge, sway and yaw is modelled by the 3-dimensional vector 1]w '

(17) Linearization of (10) yields:

Linear Mooring Model. As demonstrated in (Faltinsen, 1990) a mooring line will have a spring effect on a body attached at the line's upper end. Assume that the vessel moves in surge in a plane that coincides with the plane of the line profile. If the motion is not too large, one can write:

H = Ho + e.6.x

(18) where

(12)

where Ho is the average horizontal force in a working point Xo and

i=1

i=1

n

n

i=1

i=1

k23 i=1

e=

dfHi --(x dh i

= xo)

(13)

where the constant term about the linearization point in (16) is given by:

(14)

(20)

where d(u)u is a nonlinear damping term, m is the vessel mass, Xi< is added mass in surge and T x(t) a dynamic excitation force. From (14) the resonance period in surge becomes: _

.LO -

27r -_ 27r Wo

Jm e

Xi<

.

i=1

(19)

Let u = ~~ be the surge velocity of a ship in the direction of the line. Hence the surge equations of motion can be written:

'T'

= Lc,Nosin,BiO

and No iosin,Bio - yocos,BiO' The restoring forces can be transformed to body-fixed coordinates according to:

is the slope of the line characteristics (9).

(m - Xi<)u + d(u)u + ex = T x(t)

n

n

3. POSMOOR CONTROL SYSTEM In this chapter the POSMOOR functionality, modes of operation and control system design is addressed. The controller design is inspired by the conventional DP design, see (S!2Irensen, et. al., 1996) for details. The feedback controller is designed using LQG theory. As in the DP case, feedback is produced from low-frequency position and yaw angle estimates and corresponding estimated velocities. In the observer design, Kalman filter theory is utilized.

(15)

The resonance period in sway can be found analogously. Hence, when knowing the LF vessel mass matrix including added mass M and the static line characteristics (9), it is a priori possible to predict the resonance periods of the moored vessel. In a practical case resonance oscillations of a moored vessel will be excited by slowly-varying forces due to wind and waves. According to (Triantafyllou, 1994) a typical resonance period for moored vessels are 100 seconds or larger, depending on water depth and pretension. The motions may be in the magnitude of 5% of the water depth.

3.1 Control Modes of Operation

The three degrees of freedom (DOF) surge, sway and yaw can individually be controlled in the following modes of operation: • Manual Control. A joystick and a rotation knob generates force/moment setpoints.

Assuming fixed anchor line length, fixed turret angle and neglecting influence of the current field,

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• Damping. Feedback is taken from estimated vessel velocities. • Station Keeping. Feedback is produced from both estimated velocities and lowfrequency position/yaw angle estimates. • Marked Position/Heading. The vessel tracks a reference trajectory which is computed from the old to the new position and/or heading setpoint(s).

where Tthr is the estimated thruster force vector, based on RPM/ pitch and azimuth angle measurements, and 'e the total commanded thrust. In the state estimation, Kalman filter theory is utilized. Based on a linear vessel model, the linear quadratic controller (LQ) is derived. The LQ term of the commanded thrust is partitioned into: ILQ

The combination of damping in surge and sway and heading station keeping is often used, especially in bad weather. Effective damping will reduce possible large oscillatory motions. Moreover, the vessel will tend to an equilibrium position where the mean environmental forces are balanced by the mooring forces. Assuming a proper heading setpoint, this equilibrium position will be optimal with respect to the thrust usage and the fuel consumption.

=,y +''7 + 't

= -GLQe

= -Gye y - G'7e'7 - Gtet .

(23)

Integral action, l i , is added to the linear quadratic terms in order to avoid steady state offsets from desired position and heading.

3.5 Resulting POSMOOR Controller As discussed earlier, it is possible to change between different control modes of operation. The total commanded thrust in surge, sway and yaw is written:

3.2 Line Break Detection and Compensation Line break in one or several anchor lines are detected in order to have an immediate compensation by the thrusters. Irregular tension measurements triggers the line break detection algorithm. When a line break is detected, automatic feedforward thrust in surge, sway and yaw, 'lb , is computed. The purpose of this line break compensation term is to ease the load in the surrounding lines and thus avoiding yet another line break.

where the S matrices are control mode dependent diagonal matrices of zeros or ones along the diagonal. Sm is the logical inverse to the diagonal of Sm, such that no automatic control is possible when the system is operating in manual control. The problem of determining the thrust and direction of each of the thrust devices is handled by the thruster allocation algorithm, see (S0rdalen, 1996) for details.

3.3 Feedforward Control Law In addition to the line break compensation term, wind feedforward control can be applied. By measuring the wind velocity and direction estimates of the wind forces, T wind ) can be found. Such wind feedforward can be enabled individually in each DOF by the operator. The estimates are multiplied by a diagonal gain matrix, and the feedforward term is taken to be:

4. CASE STUDY The POSMOOR system described in this paper has been implemented as a part of the ABB Integrated Thrust Control System (ITCS), see Fig. 8. This system was installed on the Balder FPU, a FPSO contracted by Esso. The Balder FPU is equipped with 3 azimuth thrusters, one at the fore and two at the aft, each designed to develop a maximum thrust of approximately 540 kN. The ship length between perpendiculars is approximately 200 m and the dead weight is approximately 73,000 tons. The mooring system considered consists of 10 anchor lines and the water depth at the field is 125 m.

(21)

3.4 Feedback Control Law The control deviation vector is defined as: (22)

Computer simulations are performed with significant wave height 6.8 m, wave peak period 10.5 sec., current velocity 0.45 m/s and wind velocity 20 m/so The earth-fixed wind direction is 205 deg, the current direction 205 deg and the mean wave direction 175 deg. Initially the control system is operating with only heading control. After 500

where e y is a vector of LF velocity estimates, e'7 is a vector of LF position and yaw angle estimates relative to the vessel-parallel frame and et is a thrust deviation vector:

191

acrusl

ea~hxed

north-poSlDon [mJ

:~ .~~ 500

.10 0

1000

r ~-

1500

actual sarth-fixed east-positIon [m]

.30

500

1000

~

J~J) _.-, :1 -U .

-f"----,···-

1500

acrual and reference headIng {de9]

Fig. 8. The ABB ITCS configuration for Balder FPU. 500

00

1000

1500

ome (sec]

Fig. 5. Earth-fixed vessel position and heading relative to FZP.

strates the performance of the POSMOOR system. ACKNO~EDGEMENTS

This work is sponsored by The Research Council of Norway and ABB Industri AS and they are gratefully acknowledged.

02 015

REFERENCES _01 5

Aalbers, A. Ho , Janse, S. A. W., Boom, W . C. de (1995) . DP assisted and Passive Mooring for FPSO's. 27th Annual Offshore Technology Conference, ppr. 7722, pp. 281-288, Houston, Texas. Faltinsen, O. M. (1990) . Sea Loads on Ships and Offshore Structures. Cambridge University Press. Fossen, T. L (1994). Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd. Huse, E. and Matsumoto, K (1989) . Mooring Line Damping Due to First- and Second-Order Vessel Motion. 21th Annual Offshore Technology Conference, Procs. VoL 4, ppr. 6137, pp. 135-148, Houston, Texas. Marintek (1995) . MIMOSA User 's Documentation. Marintek report No. 519616.00.02, Trondheim, Norway. S0rdalen, O. J. (1996). Thruster Allocation: Singularities and Filtering. IFAC World Congress, VoL Q, pp. 369-374, San Fransisco, CA. S0rensen, A. J ., Sagatun, S. Land Fossen, T . L (1996). Design of a Dynamic Positioning System using Model-Based Control. Journal of Control Engineering Practice 4(3) . Triantafyllou, M. S. (1994) . Cable Mechanics for Moored Floating Systems. BOSS'94 pp.67-77, Boston, MA. Triantafyllou, M. S. and Yue, D. K P. (1994). Damping of Moored Floating Structures. 26th Annual Offshore Technology Conference, Procs. VoL 3, ppr. 7489, pp. 215-224, Houston, Texas.

-02

·0 2S!:-0---'--500=---~'OOO;;:----:-!'SOO I)me(sec]

Fig. 6. Surge velocity. ~ne leMIon. mosIlO8OeO.ne

1kr'-1

,SOO',---__- - -__- - - ,

'.00

900

Ilme(sec]

Fig. 7. Line tension in the most loaded line. seconds damping is selected in surge. Sway is manually controlled by the joystick. After 1000 seconds a marked heading operation is performed, with desired heading of 10 degrees. Earth-fixed vessel position and heading relative to FZP are presented in Fig. 5. The corresponding surge velocity is presented in Fig. 6 and in Fig. 7 the line tension in the most loaded line is presented. The effect of damping in surge is evident, since both the surge motion and the line tensions are reduced.

5. CONCLUSIONS An automatic thruster assisted position mooring (POSMOOR) system is proposed. A mathematical model of a moored ship has been derived. Moreover, a POSMOOR control system has been developed and different control modes of operation are described. Computer simulations demon-

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