Automated design of a ship mooring system The challenge: To investigate a mechanism to control and automate a mooring system between two ships at sea Maplesoft, a division of Waterloo Maple Inc., 29 Editor's Notes Introduction Problem Statement 1. Physical Model Representation 2. Modeling Wave Dynamics 3. Open-Loop Response 4. Closed-Loop Response Results ** This application was developed using Maple and MapleSim www.maplesoft.com/appbriefs 1 of 11
Editor's Notes The mooring of ships to harbors, terminals, and offshore structures is a common and essential procedure in most seafaring operations. Inadequate mooring can result in significant structural damage to the berthing vessel and moorings. To this day, most mooring operations are still performed in the same manner as they were decades ago; they are dependent on heuristics, or in other words, the captain or mooring master s experience. Unfortunately, the effects of global warming and climate change are altering the hydrodynamics of the sea, making manual mooring operations a very risky venture. The challenge: To design a mechanism to control and automate a mooring system between two ships at sea. MapleSim and Maple are used to: Create a physical model of the two ships and mooring cables Develop a realistic model of the external environmental forces and use the results to simulate conditions at sea Design and tune an appropriate controller to stabilize the system Developing a realistic model of the external forces was critical to the design of the controller. This allowed the engineer to tune the controller, eliminate vibrations, and stabilize the tension in the line. With the initial control system complete, further improvements can be made to the disturbance model by including irregular waves from multiple directions, wind gusts, wave drift forces, and the interaction effects from passing ships. Introduction The mooring of ships to harbors, marine terminals, floating terminals, and offshore structures is an essential part of most seafaring operations. Inadequate mooring can severely jeopardize the success of most ship-toship operations resulting in large operational downtimes. In addition, it can also result in significant structural damage to the berthing vessel and/or the moorings. Consequently, optimizing and stabilizing the mooring system is a high priority for most seafaring operations. Mooring operations today are still performed in the same manner as they were decades ago. That is, they depend on heuristics, that is, the captain or mooring master s experience. Unfortunately, the effects of global warming and climate change are altering the hydrodynamics of the sea, making un-mechanized mooring operations a very risky venture. This application investigates a preliminary design of a control system to automate the mooring process between two large vessels, such as an oil tanker, in the open sea. The hydrodynamic interactions of the waves and the presence of wind are modeled by a second-order linear partial differential wave equation. As expected, the open-loop response of the system results in severe vibrations on the tension in the mooring line. The incorporation of a PID controller to the open-loop system eliminates the vibrations on the line, and stabilizes the tension at 7 N/m. Problem Statement To investigate the hydrodynamic interaction of a ship mooring system and develop a controller to ensure that the mooring system can securely withstand external forces in the form of wind and waves. www.maplesoft.com/appbriefs 2 of 11
As can be seen in the diagram, the physical model is made up of a controller and plant model. The controller model adjusts the motor voltage driving the Mother Ship so that the tension in the mooring system remains at 72 N /m. The tension in the mooring line is fed back to the control system through a force sensor that is connected between the Mother Ship and the Mooring Cable. The Mother Ship submodel was modeled by a DC Motor and Gear submodel, where the DC Motor was modeled as an equivalent electric circuit and an ideal gear. The mass of the Mother Ship was assumed to be significantly larger than that of the Daughter Ship which is why it was modeled as a fixed, non-moving surface. The Mooring System which normally entails affixing the Daughter Ship to the Mother Ship with at least 6 cables, was modeled by a single stiff spring. The Daughter Ship submodel was modeled with a sliding mass component of 1 tons. For this model, it was assumed that the engine driving the Daughter Ship was turned off. The data used to model the Wave Dynamics submodel was derived in Maple. The details can be found in the next section. Each of these submodels, and the components that were used to model them can be seen in the.msim file. www.maplesoft.com/appbriefs 4 of 11
2. Modeling Wave Dynamics The wave equation that was used to derive a somewhat realistic model of the wave dynamics present in rough, open waters is shown in (1). WaveEquationdK v2 v2 u x, t C 2 vt vx 2 u x, t K v2 vt 2 u x, t = cos t $sin t C v2 u x, t = cos t sin t 2 vx (1) The initial conditions for the wave equation are defined as: InitialConditions d u x, = sin 5$ π$x, u, t = sin.5$ t C sin 2$t, u 1, t = sin.5$ t C sin 2$t, D 2 u x, = 5$π$cos π$x u, t = sin.5 t Csin 2 t, u 1, t = sin.5 t Csin 2 t, u x, = sin 5 π x, D 2 u x, = 5 π cos π x (2) The 3D plot shown below is obtained by solving the equation defined in (1) with the initial conditions in (2). WaveEquationSolnd pdsolve eval WaveEquation, InitialConditions, numeric, time = t, range =.. 1 : WaveEquationSoln:-plot3d t =.5.. 2, x =..1, axes = boxed, orientation = 36, 42, color = blue, light = 145, 45,,, 1 24 14 4 6 16. 26 5.5.25 t 1 15 2 1..75 x www.maplesoft.com/appbriefs 5 of 11
The wave dynamics felt by the Daughter Ship when it is.5 m away from the Mother Ship can be seen in the following plot. WaveEquationSoln:-plot x =.5, t =.5.. 2 3 2 1 K1 2 4 6 8 1 12 14 16 18 2 t K2 K3 K4 According to research, the effects of wind on the wave dynamics defined above can be modeled by Gaussian white noise. Statistics RandomVariable Normal 1,.15 WindRandomVariable d 2 WindFunction d t/ Statistics Sample WindRandomVariable, 1 1 : WindSignal d seq WindFunction t, t =.5..2,.5 : plot seq t, t =.5..2,.5, WindSignal, title = "Wind Signal" : www.maplesoft.com/appbriefs 6 of 11
Wind Signal.6.5.4.3 2 4 6 8 1 12 14 16 18 2 The total wave dynamics, which take into account the effects of wind, can be seen below. WaveDataProc d WaveEquationSoln:-value x =.5, t =.5..2 : seq rhs WaveDataProc i 3, i =.5..2,.5 WaveDatad 8 TimeData d seq i, i =.5..2,.5 : TotalWave d zip x, y /x$ y C.1, WaveData, WindSignal : : plot TimeData, TotalWave, title = "Wave Input (wave C wind)" www.maplesoft.com/appbriefs 7 of 11
Wave Input (wave + wind).2.1 2 4 6 8 1 12 14 16 18 2 K.1 K.2 WaveDynamics d convert TimeData, Vector column 4 x 2 Matrix Data Type: anything Storage: rectangular Order: Fortran_order convert TotalWave, Vector column (3) www.maplesoft.com/appbriefs 8 of 11
3. Open-Loop Response As expected, the open-loop response to the system resulted in severe vibrations far surpassing the desired line tension value. # Without Controller WaveWithoutController Amplitude.2.1 K.1 K.2 Wave Input 5 1 15 2 Time OuptutWithoutController Tension N/m Cable Tension Output 5 4 3 2 1 K1 K2 K3 K4 5 1 15 2 Time Note: These plots were obtained by simulating the model with the parameter set defined in "OpenLoop. params" file. www.maplesoft.com/appbriefs 9 of 11
4. Closed-Loop Response By adding a PID controller, the tension of the line was stabilized at around 7 N/m. The PID gain values were obtained by tuning the model with a step pulse of 1m. # With Controller WaveWithController Amplitude.2.1 K.1 K.2 Wave Input 5 1 15 2 Time OuptutWithController Tension N/m Cable Tension Output 7 6 5 4 3 2 1 K1 K2 5 1 15 2 Time Note: These plots were obtained by simulating the model with the parameter set defined in "ClosedLoop. params" file. www.maplesoft.com/appbriefs 1 of 11
Results This application illustrates the preliminary design of a controller to stabilize the mooring system of a ship on the open sea. As mentioned previously, inadequate mooring could severely hamper ship-to-ship transfers of cargo and other goods. Moreover, in the worst case, it could result in significant, and costly damage, to both the mother and daughter ships. Developing a realistic model of the external forces in the form of wind and waves is critical to the design of the controller. That said, the next step towards improving the control system design is to make improvements on the disturbance model by including irregular waves from multiple directions, wind gusts, wave drift forces, and the interaction effects from passing ships. In this application, the wave dynamics are only applied to the daughter vessel; however, in the future, the wave dynamics should also be applied to the mother vessel. Legal Notice: Maplesoft, a division of Waterloo Maple Inc. 29. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities. www.maplesoft.com/appbriefs 11 of 11