A Mooring System Deployment Design Methodology for Vessels at Varying Water Depths

In this paper, a methodology for designing mooring system deployment for vessels at varying water depths is proposed. The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is combined with a self-dependently developed vessel-mooring coupled program to find the optimal mooring system deployment considering both station-keeping requirements and the safety of the mooring system. Two case studies are presented to demonstrate the methodology by designing the mooring system deployments for a very large floating structure (VLFS) module and a semi-submersible platform respectively at three different water depths. It can be concluded from the obtained results that the mooring system can achieve a better station-keeping ability with relatively shorter mooring line when deployed in the shallow water. The safety factor of mooring line is mainly dominated by the maximum instantaneous tension increment in the shallow water, while the pre-tension has a decisive influence on the safety factor of the mooring line in the deep water.


Introduction
With the rapid exploration and exploitation of ocean resources, the design and manufacture of floating structures have been greatly developed. Mooring system has become an essential choice for station-keeping of these floating structures (Chas and García, 2008) due to its simple structure, high safety and low energy consumption.
In recent years, much attention has been paid on the design and optimization of mooring systems. Huang et al. (2019) designed three types of mooring systems for a floating wave energy converter which is deployed at a water depth of 200−700 m with a sloping seabed. In their study, the performances of the three mooring systems, namely catenary mooring system, synthetic cable taut mooring system and suspended anchoring point mooring system, are investigated under a survival condition. The effects of features of the seabed condition, the mooring line material, size and layout, the installation and maintenance costs on three mooring systems are also specifically studied. Connolly and Hall (2019) presented a design algorithm for preliminary sizing of the shared mooring systems for pilot-scale floating offshore wind farms. They designed three alternative shared-mooring configurations for a four-turbine floating wind farm and compared their performance, safety and cost when deployed at four different water depths. Huang et al. (2018) designed a mooring system for a floating wave energy converter with the novel mixed superflex wire rope. They presented a strategy for the mooring system design and evaluation which involves the vessel motion response and the economy, safety and versatility of the mooring system. Campanile et al. (2018) designed and selected a mooring system for floating offshore wind turbines on intermediate and deep waters based on ultimate, accidental and fatigue limit states. In their study, 6-and 9-line mooring configurations had been selected as candidates to investigate the effects of the line number, platform admissible offset and line scope, and the mooring technology on relevant installation and maintenance costs. Benassai et al. (2014) dealt with the catenary mooring system design for tri-floater floating offshore wind turbine. They investigated the variation of line length and diameter of three candidate mooring systems at different water depths. The performance of the three mooring systems in ultimate limit states and accidental limit states were also examined in their study. The similar researches can also be found in Benassai et al. (2015), Wang et al. (2017Wang et al. ( , 2018Wang et al. ( , 2019 and Liang et al. (2019aLiang et al. ( , 2019b. According to these studies, the design of a mooring system deployment for the floating structure that has to be deployed at different water depths is complex, which usually involves many design variables and requirements. The mooring system deployment design is usually transformed into a multi-objective optimization problem, in which the mooring system parameters, such as line number and diameter, material and layout, are the design variables, and the requirements for each water depth, such as vessel motion response, mooring system safety and maintenance costs, are the objective functions. Several algorithms can be utilized to solve the multi-objective optimization problem for mooring system deployment design. Shafieefar and Rezvani (2007) described a mooring system optimization design method based on Genetic Algorithm (GA). They established a single-objective optimization problem for mooring system design by normalizing the objective functions based on the weighted sum method. Felix-Gonzalez and Mercier (2016) designed a statically equivalent mooring system for a target taut mooring system based on GA. In their study, three objective functions were normalized by utilizing several sets of weighting factors. It was found from their study that the selection of the weighting factor has a significant impact on the mooring system design results. Brommundt et al. (2012) described a new tool for the optimization design of catenary mooring system for a floating wind turbine. The optimization problem with four design variables and three constraints was established and solved by the Nelder-Mead simplex algorithm. Nevertheless, to minimize the mooring line length is the only design objective considered in their study. Monteiro et al. (2016) established a single-objective optimization problem for mooring system design by normalizing the objective functions with a kind of averaging method, and solved it with particle swarm optimization (PSO) and the differential evolution (DE) methods. However, the redesign of the mooring system deployment of the floating structures which have to serve at different water depths is a multi-objective problem and involves several design requirements such as mooring safety and floating structure motion response. The aforementioned studies and methods for single-objective mooring system design are not feasible for this task. Other than the above studies, there are several researches related to truncated mooring system design, such as Zhang et al. (2012Zhang et al. ( , 2014, Wei et al. (2017), among others. They established a multi-objective optimization problem of mooring system truncation and solved it by a multiobjective optimization algorithm. Different from the moor-ing system truncation, the mooring system properties such as maximum line length and line diameter are fixed in the optimal mooring system deployment at different water depths. The mooring system truncation method can only be used as a reference. With less design variables, it is more difficult for the mooring system deployment to be satisfied with the design requirements. In this context, this paper proposed a methodology for designing mooring system deployment for the vessels at varying water depths. The multi-objective optimization problem for mooring system deployment design is established and solved by the combination of NSGA-II and a self-dependently developed vessel-mooring coupled program, where the accurate dynamic characteristics of floating structure and mooring system can be selected as objective functions. Thus, the proposed methodology can design the mooring system deployment for floating structures at different water depths considering practical design requirements, such as the floating structure motion response, the safety factor of the mooring system, the length of mooring line lay down part and mooring system layout.
For the floating structures which have to serve at different water depths, the mooring system deployment should be redesigned to adapt to the corresponding water depth by considering mooring safety, vessel motion response and other practical requirements. In this context, this paper proposes a method for designing mooring system deployment for vessels at varying water depths. The multi-objective optimization problem for mooring system deployment design is established and solved by the combination of NSGA-II and a self-dependently developed vessel-mooring coupled program, where the accurate dynamic characteristics of floating structure and mooring system can be selected as objective functions. Thus, the proposed method can be used to design the mooring system deployment for floating structures at different water depths considering practical requirements, such as the vessel motion response, the safety factor of the mooring system, the length of mooring line lay down part and the mooring system layout.

Mathematical modelling of the mooring line
The lumped mass method is generally used for modelling the mooring line (Low and Langley, 2006;Xiong et al., 2016). Based on the lump mass method, a vessel-mooring coupled program is developed by the authors to obtain accurate dynamic characteristics of the floating structure and the mooring system.

Lumped-mass model of the line
In the lumped-mass method, a mooring line is divided into several elements. Each line element is regarded as a massless spring while the mass and external force acting on it are lumped in its two end nodes. Fig. 1 illustrates the lumped mass model of three arbitrary nodes numbered from 1 to 3 which is connected to elements j and k. y 1 , y 2 , and y 3 are the coordinates of Nodes 1, 2, and 3 in the global coordinate system.
In the equilibrium position, the governing equation of each node can be expressed as: yẏ where m is the mass matrix of the nodes on the mooring line; , and y are the acceleration, velocity and position vector, respectively; F is the external force lumped in each node including tension, gravity, buoyancy, hydrodynamic force and seabed interaction force; the subscript N is the number of node on the mooring line.
2.2 Dynamic model of the mooring line Based on the lumped-mass model, the governing equation of the mooring line motion is written as follows: where m and a m are the mass and added mass matrix of the nodes on the mooring line; F T , F G , F sb and F DI are the tension vector, gravity and buoyancy vector, seabed interaction force vector and hydrodynamic force vector of nodes on a mooring line. β β The dynamic time integration scheme, the Newmarkmethod, is used to linearize the differential Eq. (2). The Newmark-method is expressed by: α β where and are used to adjust the accuracy and stability of the algorithm. Substituting Eqs. (3)-(5) into Eq. (2), yields: Δy is solved by Newton-Raphson iteration method from Eq. : ε where the subscript m is the number of iteration; is the correction to the iteration; J is the Jacobian matrix.

Vessel-mooring coupled program
A vessel-mooring coupled program based on the lumped-mass method is independently developed by the authors in Matlab environment. The static and dynamic responses of a mooring line are calculated by the program and compared with that obtained by the commercial software Orcaflex (Orcina, 2018). The right-handed coordinate system of the mooring line is shown in Fig. 2. The corresponding mooring line properties are listed in Table 1.
The comparison of mooring line static response is shown in Fig. 3. It can be seen that the self-developed program can simulate the static response of the mooring line accurately with a 10-m line element when applying an offset to the mooring line top end along X-axis. Further, the dynamic response of the mooring line is calculated by the program with 10-m line element and the comparison results are shown in Fig. 4. The forced oscillations are applied to the top end of the mooring line along X-axis. It can be obviously seen that the developed program with 10-m line element has excellent accuracy to predict the mooring line dynamic response as that of Orcaflex at both small and large water depths. Meanwhile, the motion response of the floating structure can be obtained in the time domain by coup-   XU Sheng-wen et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 185-197 187 ling the dynamic model of mooring line with the Cummins equation (Cummins, 1962) of the vessel: xẋ where M and A are the mass and the added mass matrix of the vessel, respectively; H is the retardation function matrix; K is the restoring force matrix; , , and x are the acceleration, velocity and position vectors of the vessel, respectively; F e is the external environmental force; F m is the mooring force which is obtained by Eq.
(2). The dynamic responses of a semi-submersible platform coupled with a 20-line mooring system at the water depths of 50 m and 1000 m are calculated by the Orcaflex and the self-developed program. The mooring line properties of the mooring system are the same with that listed in Table 1 except that the line lengths are 390 m and 1760 m for the water depths of 50 m and 1000 m, respectively. The dynamic response of the vessel-mooring coupled model obtained by Orcaflex and the self-developed program are compared in Figs. 5 and 6. It can be seen that the time series of the vessel six degrees-of-freedom motion responses and the top end tension of two representative mooring lines obtained by the two methods show a good coincidence, which indicates that the self-developed program can accurately predict the dynamic response of a vessel-mooring coupled model for designing of the mooring system deployment in both shallow and deep waters.

Mooring system deployment design methodology
For the floating structures which have to serve at different water depths, its mooring system deployment should be optimally designed to adapt to the corresponding water depth considering mooring system safety, vessel motion response and other practical requirements. In this scenario, a multi-objective optimization problem for mooring system deployment design is established, in which the self-developed program is adopted to calculate the dynamic characteristics of the vessel and the mooring system, and the NSGA-II (Deb et al., 2002) is utilized to find the optimal mooring system deployment.
For designing the deployment of a mooring system, its properties including the line number, maximum line length, line diameter, material grade and mooring line type are constant variables. Thus, the line length and mooring system layout for the corresponding water depth are the design variables. The line length is an independent design variable, while the mooring system layout depends on the mooring radius and azimuth angle of the mooring line. Hence, line  length L, mooring radius S and azimuth angle are three independent design variables.
The design of the mooring system deployment for vessels at varying water depths involves multiple requirements such as the station-keeping ability and the safety of the mooring system and the length of the mooring line lay down part. The self-developed vessel-mooring coupled program can effectively predict the mooring line dynamic response, such as the top end tension and the lay down part length through a time domain simulation. Hence, the design requirements can be expressed mathematically and the corresponding objective functions of the station-keeping ability and safety of mooring system deployment design are: where is the objective function related to vessel motion response; x is the time series of vessel motion response; is the safety factor which is the ratio of the mooring line maximum tension T max to the minimum breaking load T MBL (API, 2005); T pre and ΔT max are the pre-tension and the maximum instantaneous tension increment of the mooring line, respectively.
Meanwhile, it is also required to minimize the line length and the mooring radius when designing the optimal deployment of the mooring system at different water depths. The corresponding objective functions can be expressed as: where L and S represent the line length and the mooring radius, respectively. In addition to the objective functions listed above, other requirements, such as the fatigue damage, the lay down part length, installation and maintenance costs, are also feasible to be selected as objective functions. Therefore, the multiobjective optimization problem for mooring system deployment design can be established mathematically: where i indicates the i-th water depth; is the feasible range of design variables; represents the azimuth angle; can be the other objective functions such as the fatigue damage, the lay down part length, the installation and maintenance costs; is the corresponding constraints of the j-th design variables V j , including the mooring system safety factor, the vessel motion response limit, the mooring system layout limit, etc.
The multi-objective optimization problem (Problem 0) for mooring system deployment design is solved by the NSGA-II. A schematic diagram of the design methodology is illustrated in Fig. 7. The population (number of the optimal solution) and the generation (number of the total iteration steps) are set firstly. The rank and the crowding distance for each individual in the first generation are calculated by the non-dominated sort algorithm and the crowded comparison approach. The individuals with higher rank and larger crowding distance are selected for crossover and mutation to generate the new generation. Continue the process until the termination condition is met.
Since the NSGA-II algorithm is established based on Pareto optimality (Srinivas and Deb 1994), (Deb et al., 2002), the obtained optimal solutions have the same priority. More information is required for further selection of the optimal mooring system deployment. Eq. (15) combines the existing information (objective function used in the NSGA-II) with additional information (independent objective function), and is proposed as a criterion for the selection of the optimal mooring system deployment.
where Y k is the selection coefficient of the k-th optimal solution; represents the existing objective function used in the NSGA-II algorithm; is the independent objective function which is not used in the NSGA-II algorithm; i and j indicate the i-th and j-th objective functions used and not used η ε in the NSGA-II algorithm; and are the corresponding weighting factors.
The selection coefficient Y for each optimal solution can be obtained by appropriately setting weighting factors. Since the mooring system deployment design problem requires minimizing the objective functions, the mooring system deployment with the smallest selection coefficient Y is the desired mooring system deployment for each water depth.
4 Case study: mooring system deployment for a VLFS module The mooring system deployment design methodology is demonstrated in this section by designing mooring system deployments for a semi-submersible type very large floating structure (VLFS) module at the water depths of 50, 100 and 200 m, respectively.

Overview
The semi-submersible VLFS module is composed of ten cylinder columns, five pontoons and eight cylinder braces. The VLFS module is supposed to serve at the water depth of 400 m and its main particulars are presented in Table 2, where the BL and COG represent the baseline and the center of gravity of the module. As shown in Fig. 8, the longitudinal axis of the VLFS module is parallel to the X-axis of the global coordinate system.
The mooring system is symmetrical about the x and y axes in the body fixed coordinate system. The original lay- α β out when deployed at the water depth of 400 m is shown in Fig. 8, in which the dotted line is the angular bisector between two adjacent lines, and and are the azimuth angles of the angular bisector. The angle between the two adjacent lines linked in the same fairlead is 5°. It is stipulated that the azimuth angle of each mooring line is not changed at the different water depths.
The properties of the original mooring system at the water depth of 400 m are listed in Table 3.
The environmental condition is the same at the water depths of 50, 100 and 200 m, as listed in Table 4. The incident wave angle is 0° if the wave travels along the positive direction of the x-axis of the global coordinate system and increases in a counter clockwise direction.
4.2 Design of the mooring system deployment for the VLFS module at different water depths α β As the VLFS module is supposed to be operated in different sea areas at water depths of 50, 100 and 200 m, the deployment of its mooring system should be redesigned to adapt to the corresponding water depth. As described before, the azimuth angles of the mooring line in this case study, which are represented by and , are not changed at different water depths, thus the mooring line length L and mooring radius S are selected as design variables.
In this particular case, the station-keeping ability and the safety of the mooring system are not given when the VLFS module is deployed at water depths of 50, 100 and 200 m. Therefore, the station-keeping ability and safety of mooring system are selected as two constraints instead of design requirements. Meanwhile, the lower limit of the lay down part length is required to ensure the safety of the anchor. Thus, the specific constraints can be listed as follows: (1) The minimum lay down part length V min should be larger than 40 m. ϕ safe (2) The safety factor of mooring line should be smaller than 0.6. ϕ M (3) The VLFS module motion response (surge, sway and yaw) at the water depth of 50, 100 and 200 m can not be larger than that at the water depth of 400 m.
The solutions are unacceptable if they are not satisfied with the constraints. In this case, the unacceptable solutions are directly replaced by the acceptable solutions in each iteration to improve computational efficiency.
The multi-objective optimization problem is then established based on Problem 0: where is the feasible range; V min is the minimum lay down part length; superscript i can be 1, 2 and 3, which represent the water depth of 50, 100 and 200 m, respectively; represents the VLFS module motion response limit at the water depth of 400 m.

Selection of the mooring system optimal deployment
The optimal solutions of the established multi-objective optimization problem are found by NSGA-II with the population and generation respectively set as 50 and 50. A time domain simulation is conducted based on the self-developed vessel-mooring coupled program with a total simulation time of 10800 s and a time step of 0.2 s.
The optimal solutions fulfilling the design requirements are found for different water depths. As shown in Fig. 9, each mark represents an optimal solution.
Since the VLFS module motion response and mooring system safety factor are not objective functions for the NSGA-II, they are selected as the additional information, or the independent objective functions, required for determining the desired mooring system deployment at different water depths. Meanwhile, the mooring line length L and mooring radius S are linearly dependent, only one of them needs to be selected as the existing objective function. Thus, the selection coefficient Y for this case is expressed as: In order to establish the connection between the selection coefficient Y and the mooring line length L, the maximum horizontal offset and safety factor are normalized and fitted as the function of L, which are represented by and . The selection coefficients Y for the optimal solutions with four sets of weighting factors are shown in Fig. 10.
From Fig. 10, it can be found that the setting of weighting factors has a significant impact on the selection coefficient Y. In this particular case, it is focused on the compromise between the VLFS module motion response, mooring system safety and mooring line length. Thus, the three weighting factors should be set to 1. Therefore, the desired mooring system deployment at each water depth can be selected, as listed in Table 5.

Discussion
The maximum horizontal offset of the VLFS module and the safety factor of the mooring system at three different water depths are compared in Fig. 11. As shown in Fig. 11a, the maximum horizontal displacement are smaller when the VLFS module deployed at the water depths of 50 and 100 m, which is 60% of the maximum horizontal displacement at the water depth of 200 m. It suggests that the mooring system can achieve a better station-keeping ability with short mooring line in the shallower water, though minimizing the VLFS module motion response is not a design objective. θ As shown in Fig. 2, the mooring line can achieve a large hang-off angle with the small mooring line length and mooring radius when deployed in the shallow water. With a large hang-off angle, the mooring line tension can be transformed into the horizontal restoring force more efficiently. Thus, compared with the other water depths, the mooring system deployed in the shallowest water (50 m in this case) is the most effective in restricting the motion response of VLFS module.
As shown in Fig. 11b, the safety factor shows significant variance when the mooring system deployed at the shallower water. In particular, the variation range of safety factor is close to 0.1 when the mooring system deployed at the water depths of 50 m and that is 0.04 and 0.02 when deployed at the water depths of 100 and 200 m. Further, the relation between pre-tension and safety factor of mooring system is shown in Fig. 12, in which the pre-tension is converted into the ratio of the minimum breaking load (MBL) of the mooring line. It can be found that the maximum tension is almost 6 times the pre-tension when the mooring system deployed at the water depth of 50 m. While the increment of the maximum tension on the basis of the pre-tension at the other two water depths is much smaller, which is only 2−3 times the pre-tension. This phenomenon indicates that the mooring line tension has significantly increased in the shallow water. To illustrate this phenomenon, the top end tension of a single mooring line with a 109-mm diameter is calculated at the three water depths with a series of static offsets applied to its fairlead and the results are compared in Fig. 13. As shown in this figure, at the water depth of 50 m, the mooring line has the lowest pre-tension and its tension rises exponentially with the increase of its fairlead offset. With the increasing water depth, the impact of the fairlead horizontal offset on the mooring line tension is significantly weakened. Thus, for the mooring system deployed at the water depth of 50 m, its line tension increases quickly and can reach 20% to 30% of the MBL though the line pre-tension is only 4% of the MBL and the maximum horizontal displacement is smaller than 8 m. Therefore, the safety factor of the mooring system deployed at the water depth of 50 m is mainly dominated by the maximum instantaneous tension increment ΔT max .
As shown in Fig. 13, the mooring line has a relatively large pre-tension and the line tension surface becomes flat  . 9. Line length and mooring radius of mooring system deployments at three different water depths. with the change of the horizontal displacement when the mooring line is deployed at the water depth of 200 m. Thus, for the mooring system deployed at the water depth of 200 m, the VLFS module horizontal offset does not significantly impact the mooring line tension. In the simulations, the line tension increases to 26% to 28% of the MBL from its pretension (16% to 18% of the MBL). The maximum instantaneous tension increment ΔT max is much smaller than that at the water depth of 50 meters, which is nearly 10% of the MBL, though the VLFS module has larger motion response. Therefore, the pre-tension T pre , instead of the maximum instantaneous tension increment ΔT max , has a decisive influence on the safety factor of the mooring system deployed in the deep water depth.
The mooring system deployed at the water depth of 100 m can be regarded as the transition state from 50 m to 200 m. The VLFS module horizontal offset has a significant effect on the mooring line tension increment. The pre-tension of the mooring line is relatively large. In particular, the maximum instantaneous tension increment ΔT max is nearly 25% of the MBL, though the maximum horizontal displacement is smaller than 8 m. Meanwhile, when deployed at the water depth of 100 m, the mooring system also has a larger pre-tension, which is 13% to 17% of the MBL. Therefore, the pre-tension T pre and the maximum instantaneous tension increment ΔT max jointly determine the safety factor of the mooring system deployed at the water depth of 100 m.

Case study: mooring system deployment for a semisubmersible platform
The mooring system deployment design for a semi-submersible platform is conducted to further demonstrate the proposed method. The semi-submersible platform is hypothetically deployed at water depths of 100, 200 and 300 m, respectively.

Overview
The semi-submersible platform is composed of four cylinder columns, two pontoons and two braces, and its main parameters are presented in Table 6. The semi-submersible platform is supposed to serve at the water depth of 500 m, and its longitudinal axis is parallel to the X-axis of global coordinate system as shown in Fig. 14.
The mooring system is symmetrical about the X and Y axes in the body fixed coordinate system. Fig. 14 shows the layout of the mooring system deployed at the water depth of 500 m. The angle between the two adjacent mooring lines linked in the same fairlead is 35°, the angle between the No.   1 mooring line and the X axes in the global coordinate system is 40°. Owing to structural limitations of the semi-submersible platform, the azimuth angle of each mooring line cannot be changed when the mooring system deployed at water depths of 100, 200 and 300 m.
The properties of the mooring system for the semi-submersible platform at the water depth of 500 m are listed in Table 7.
The environmental condition is the same when the semisubmersible platform is operating at the water depth of 100, 200 and 300 m, as listed in Table 8. The incident wave angle is 0° if the wave travels along the positive direction of the x-axis of global coordinate system and increases in a counter clockwise direction. The direction of the current is consistent with the wave. 5.2 Design of the mooring system deployment for the semisubmersible platform at different water depths As has been stipulated, the azimuth angles of the mooring line are not changed when the mooring system deployed at water depths of 100, 200 and 300 m. Thus, the mooring line length L and mooring radius S are the two design variables in this case.
The station-keeping ability and the safety of the mooring system are not given in this case when deployed at water depths of 100, 200 and 300 m. Thus, the station-keeping ability and the safety factor are selected as two constraints instead of objective functions. For designing the mooring system deployment for the semi-submersible platform, the established multi-objective optimization problem can be expressed as follows: where is the feasible range; V min is the minimum lay down part length; superscript j can be 1, 2 and 3, which represents water depths of 100, 200 and 300 m, respectively; represents the platform motion response at the water depth of 500 m. Similar to the first case, the unacceptable solutions are directly replaced by the acceptable solutions to improve computational efficiency.

Selection of the mooring system optimal deployment
The NSGA-II is utilized to solve Problem 2 with the generation and population both set as 50. The semi-submersible platform motion response and mooring system performance are obtained by the self-dependently developed vessel-mooring coupled program. The total simulation time is 10800 s with a time step of 0.2 s.
The optimal solutions fulfilling the requirements are found for each water depth, as shown in Fig. 15. Similar to the first case study, the semi-submersible platform motion response and mooring system safety factor are the two independent objective functions required for selecting the desired mooring system deployment at different water depths, while the mooring line length L is selected as the existing objective function. Thus, the selection coefficient Y can be written as: where the maximum horizontal offset and safety factor are normalized and fitted as the function of L, which is represented by and , to establish the connection between the selection coefficient Y and the mooring line length L. The selection coefficients Y for the optimal solutions with four sets of weighting factors are shown in Fig. 16.
The safety factor of the mooring system is very important in this case, since the mooring system contains only 4 mooring lines. The weighting factor of should be large enough to emphasize the importance of the mooring system safety. Therefore, the desired mooring system deployment for each water depth should be selected based on the selection coefficient Y, in which the weighting factor is set as 10 and the other two weighting factors are set to 1. The selected mooring system deployments for the semi-submersible platform are listed in Table 9.

Discussion
The maximum horizontal offset of the platform and safety factor of the mooring system for corresponding water depths are shown in Fig. 17. Similar to that of the first case study, the mooring system has a better station-keeping ability when deployed in the shallow water. In particular, the maximum horizontal displacement of the platform is smaller than 5 m at the water depth of 100 m, which is nearly 40% of that at the water depth of 300 m. Further, the station-keeping ability of the mooring system tends to decrease with the increase of the water depth.
The relation between the pre-tension and the safety factor of this case is shown in Fig. 18, in which the pre-tension is converted into the ratio of MBL. It can be seen that the pre-tension of the mooring system increases from 7% to 9% of the MBL at the water depth of 100 m, and from 16% to 18% of the MBL at the water depth of 300 m. While the maximum instantaneous tension increment ΔT max decreases from 10% to 4% of the MBL when the mooring system changes its deployment position from the water depth of 100 to 300 m. Further, the relation between the fairlead horizontal offset and top end tension of a single mooring line with 150-mm diameter is presented in Fig. 19. As shown in the figure, the effect of the fairlead horizontal offset on the mooring line top end tension is relatively smaller compared with the first case study, though the mooring line top end tension rises up rapidly at the water depth of 100 m than that at the other two water depths. Thus, the maximum instantaneous tension increment ΔT max of the mooring line at the water depth of 300 m is only half of that at the water depth of 100 m, though the maximum horizontal displacement of the platform at the water depth of 300 m is almost 3 times that at the water depth of 100 m. It can be found from-Figs. 18 and 19 that the pre-tension T pre and the maximum instantaneous tension increment ΔT max dominate the safety factor of the mooring system when it is deployed at the water depth of 100 m, while the pre-tension T pre has a decisive influence on the safety factor of the mooring system at the water depths of 200 and 300 m.
By comparing the results obtained in this case study with that of the first case study, it can be obviously found that the mooring system can achieve a better station-keeping ability with a shorter mooring line in the shallow water although the vessel motion response is not selected as an objective function. The safety factor of the mooring system  deployed in the shallow water is dominated by the maximum instantaneous tension increment, while the pre-tension gradually replaces the maximum instantaneous tension increment to dominate the safety factor with the increase of the water depth. Therefore, the safety of the mooring system deserves more attention in designing the mooring system deployment in the shallow water, while the stationkeeping ability and the mooring line pre-tension are two im-portant factors which should be concerned when designing the mooring system deployment in the deep water. Therefore, it is feasible to introduce a constraint into the multi-objective optimization problem to limit the lower limit of the pre-tension so that the stiffness of the mooring system cannot be too small to cause excessive motion of floating structure and to cause a large maximum instantaneous tension increment in the shallow water. Meanwhile, it is also feasible to introduce a constraint to limit the upper limit of the motion response of the floating structure and the mooring line pre-tension to ensure the station-keeping ability and the safety of the mooring system deployed in the deep water.

Conclusions
In this paper, a methodology for designing mooring system deployment for vessels at varying water depths is proposed, in which the NSGA-II is combined with a self-dependently developed vessel-mooring coupled program. Two case studies are presented to demonstrate the proposed methodology, in which mooring system deployments are designed for a VLFS module and a semi-submersible platform respectively at three different water depths. Based on the obtained results, it can be concluded that the proposed methodology is effective in designing the mooring system deployment for vessels at varying water depths. Meanwhile, owing to the combination of the NSGA-II and the vesselmooring coupled program, the proposed methodology can attain an optimal mooring system deployment considering more practical design requirements, such as the vessel motion response, the safety factor of mooring system, the mooring line lay down part length and the mooring system layout. Furthermore, some conclusions can be drawn as follows.
(1) The minimum required mooring line length and mooring radius decrease with the decrease of water depth. The mooring system can achieve a better station-keeping ability with shorter mooring line in the shallow water, although minimizing the motion response of floating structure is not the objective function.   (2) The safety factor of the mooring system is dominated by the maximum instantaneous tension increment in the shallow water, while is gradually dominated by the pre-tension with the increasing water depth.
(3) Since the horizontal motion of the floating structure has a significant effect on the mooring line tension in shallow water, a constraint can be introduced into the multi-objective optimization problem to limit the lower limit of the pre-tension to ensure the stiffness of the mooring system, so that the motion response of the floating structure cannot be too large to cause a large maximum instantaneous tension increment in the shallow water. The constraints of the upper limit of floating structure motion response and mooring line pre-tension should also be introduced into the multi-objective optimization problem to ensure the station-keeping ability and the safety of the mooring system in the deep water.