Nonlinear modelling of shared mooring concepts for floating offshore wind turbines

Shared mooring has been proposed as one of the innovative technologies to reduce mooring and anchoring expenses for floating offshore wind turbines (FOWTs). Despite cost saving benefits, shared moorings introduce more complexity for the connected FOWTs than the conventional moorings. In this paper, we propose a nonlinear modelling approach for mooring configurations in Simpack and investigate the impact of a shared line mooring configuration on floater motions and mooring tensions of two connected FOWTs. First, we validate the nonlinear mooring modelling method through a comparison between OpenFAST and Simpack simulations of a 15-MW semi-submersible FOWT. Second, two 15-MW FOWTs with a shared mooring line configuration are further analyzed in Simpack to investigate the shared mooring impact. Different performances of the wind-wave loaded and unloaded FOWTs in two cases reflects the importance of considering nonlinear mooring stiffness in the modelling approach. By comparing the simulation results of two FOWTs using shared moorings and the single FOWT using conventional moorings, we identify the challenges of applying a shared mooring configuration that the surge displacements of the connected FOWTs increase significantly.


Introduction
The high cost of floating offshore wind projects poses a significant challenge to the commercial deployment of floating wind farms in deep seas.Continuous technical innovations are proposed to achieve the mooring cost-reduction targets and to meet the requirements of rapid project scaling for floating wind projects [1].The study of shared mooring concept is expected to lower the mooring and anchoring expenditures for floating offshore wind turbines (FOWTs) in deep waters [2].And it is proved that the cost savings per single FOWT increase sharply with the deeper water for different mooring configurations [3].
While the shared mooring concept is promising to achieve the goal of cost-effective mooring design, it introduces more complexity to connected FOWTs than individual FOWTs with conventional mooring configurations.A static test of a dual-spar model with a shared mooring configuration shows that the mooring tension of the shared line is much larger than those of the conventional lines when the floater has sway offsets [4].In addition, the rapidly increasing sizes of wind turbines in floating projects, where wind plays a dominant role, bring challenges in maintaining the mooring structural integrity and request reliable and feasible mooring designs.Therefore, it is important to apply reasonable modelling methods for the mooring configurations to accurately predict mooring line tensions and floater motions under different environmental conditions.
Currently, the studies on shared mooring concept for FOWTs are still in early stages.It is expected that the implementation of shared mooring systems in wind farm array designs may begin around 2025 [2].Concerning the mooring modelling approaches, a linear design method is proposed to simplify the force-displacement relationship of mooring lines and treat them as the first-order linear springs [5].Only the horizontal stiffness is considered.This assumption is reasonable in the sense of motion amplitudes for the single FOWT cases of a 15-MW semisubmersible FOWT, when wind and wave have no directional misalignment, the surge motion has the largest amplitude and mainly contributes to the mooring tensions, compared to the other motions [6].
The linear method shows advantages of fast designs for shared mooring layouts [5], but it encounters with problems when the floater undergoes large motion fluctuations, requiring considerable effort to linearize the force-displacement relationship of mooring lines.For example, under normal operation scenarios of DLC 1.2, a 15-MW spar-type FOWT experiences significant surge variations, and the standard deviation of surge displacements reaches up to 2.3 m, whereas the ultimate surge offset is 10 m in DLC 6.1 [7].Also, linearizing the force-displacement of a mooring line can simplify the calculations.However, the ranges of floater motions under different environmental conditions are usually unknown in advance, which limits the effective application of the linear modeling approach.Therefore, we propose a nonlinear mooring modelling approach that is not constrained by the range of floater motions, to provide a comprehensive evaluation of the shared mooring concept.
The paper is structured as follows.Section 2 describes the 15-MW semi-submersible FOWT model, the mooring configuration and the environmental conditions.Section 3 introduces the theoretical background of nonlinear modelling method of mooring lines.The numerical results of fully coupled simulations of the single FOWT with a conventional mooring configuration and two FOWTs with a shared mooring configuration are presented in section 4 and section 5, respectively.Section 6 summarizes all results and outlines the future work.Based on the preliminary design model of a semi-submersible FOWT from the EU H2020 Project COREWIND, the tested FOWT consists of the IEA 15-MW wind turbine [8], the semisubmersible floater [9] and a new mooring configuration with practical mooring line properties [6], as shown in Table 1.The Reference Open Source Controller (ROSCO) developed by NREL [10] is used for turbine control in both OpenFAST and Simpack simulations.The current version of ROSCO is 2.6.0.

FOWT model and environmental conditions
The conventional mooring configuration for the FOWT consists of three identical catenary lines and is spaced with 120 • angle between adjacent lines.Line 1 is parallel to the negative X-axis, line 2 and line 3 are symmetric about the positive X-axis.The total length is 750 m for each mooring line and the pretension force at each fairlead is given as 3 MN, around 15% of the chain MBL.The anchor radius is 745 m, from the anchor position to the floater center.
The reference site is Gran Canaria Island with a water depth of 200 m [11].The wind and wave point to the positive X-axis and no wind-wave directional misalignment is considered in this paper.The selection of the steady wind speed, the corresponding wave height and period is based on the design basis of the reference site [11].

Forces on the mooring line segment
For floating structures, mooring lines of steel chains provide restoring forces to the floater mainly by their weights, in order to restrain the floater displacements within allowable offsets.Classical catenary equations in Equation 1 [12] are commonly applied to calculate the forces on the mooring lines of steel chains.
where T denotes the mooring line tension force, ρgAz and ωs stand for buoyancy and mooring line weight in water, respectively.ds is the mooring line segment length, ϕ represents the spatial angle between the line segment and the horizontal plane.F and D are the hydrodynamic forces per segment length, and EA is the elastic stiffness of the mooring line.
Theoretically, the relationship of the floater displacement and the mooring line tension is a nonlinear function and hard to generate an explicit solution.In this paper, we neglect the hydrodynamic forces F and D in the mooring model, as we consider a static condition and a mild environmental condition with a steady wind and a regular wave, when the influence of hydrodynamic forces is limited on mooring tension forces.Also, as we use drag embedment anchors which are not designed for vertical lifting forces, a portion of the mooring line is laid down on seabed to avoid uplifting the anchors, the resulting contact forces and friction forces are neglected in the simulations.The nonlinear modelling method considers a quasi-static model of forces on the mooring line, including tension, chain weight, buoyancy and elasticity of mooring segments [12].Due to the high elastic stiffness of mooring chains used in this paper, we expect little impact of mooring elasticity on comparisons of the floater motions and mooring tensions.

Numerical simulations of one FOWT
In this section, the 15-MW semi-submersible FOWT model is simulated in Simpack applying a conventional mooring configuration.Simpack allows modelling the multi-body system and computes the aerodynamic and hydrodynamic forces through the internal force elements that interfaced to AeroDyn and HydroDyn, respectively.To calculate mooring tensions, we implement Simulink model and apply the MatSIM module in Simpack to achieve the signal processing for multi-domains in both Simulink and Simpack.In addition, we compare the results of one FOWT model with a conventional mooring configuration in OpenFAST and Simpack simulations to validate the nonlinear mooring modeling approach.

one FOWT without mooring
Dynamic tests of the FOWT model without mooring lines are conducted in OpenFAST as an initial reference to validate the set-up of the model without moorings in Simpack.The simulations include one test of a steady wind speed of 16 m/s, and one test of a regular wave with the height of 2 m and the period of 7 s, respectively.These tests are designed to validate floater motions in six degrees of freedom without mooring lines.The comparisons of OpenFAST and Simpack simulation results are presented in Figure 2.For the 15-MW FOWT model without moorings, the results of OpenFAST and Simpack simulations match well, showing the motion deviations of less than 5% in surge, heave and pitch direction.Apparent deviations in sway, roll and yaw motions are observed between the OpenFAST and Simpack simulations in the steady wind test under floater free-drift conditions.
The motion difference in sway, roll and yaw directions are caused by a slightly different yaw stiffness that implemented in OpenFAST and Simpack simulations.
However, as we consider no misalignment of winds and waves, the surge, heave and pitch motions play a dominating role in mooring line tensions, and the displacements in sway, roll and yaw have much smaller amplitudes than those in surge and pitch directions.Thus, the contributions from sway, roll and yaw motions are limited on mooring line tensions, and the motion deviations in sway, roll and yaw between the OpenFAST and Simpack simulations make a slight difference on the mooring tension predictions.

one FOWT with mooring lines
The nonlinear modelling method utilize Simulink to solve the catenary equations in section 3, and the Simulink model is coupled with Simpack via the internal module of MatSIM to achieve signal processing in multi-domains.At a time step of 0.005 s, the Simulink model receives the fairlead positions in the global coordination frame that exported from Simpack and passes the mooring tensions that derived from the catenary equations to Simpack.
For the simulations of one FOWT with mooring lines, the static conditions without winds and waves, and a steady wind and a regular wave conditions are considered.The steady wind speed is 16 m/s, the regular wave height is 2 m and wave period is 7 s.The comparisons of OpenFAST and Simpack simulations are presented in Table 2 for the mean values of surge, heave and pitch motions, and mean mooring tensions for line 1 and line 2. For the 15-MW FOWT with three mooring lines, the results of OpenFAST and Simpack simulations match well, with the deviations of less than 5% in floater motions of surge, heave and pitch and in the mooring line tensions.This comparison proves the effectiveness of the nonlinear modelling method in prediction of mooring tension forces and floater motions with acceptable correctness.
As for the deviations between OpenFAST and Simpack simulations, different mooring modules are implemented in OpenFAST and Simpack.Moordyn is coupled in OpenFAST to compute the dynamic mooring tension forces, including the drag and inertia forces on the mooring lines.Also, the internal damping forces on the mooring lines are considered in Moordyn.While in Simpack, the nonlinear modelling method neglects these mentioned force components.

Numerical simulations of two FOWTs
In this section, the verified FOWT model in section 4 is further analyzed for simulations of two FOWTs with a shared mooring configuration in Simpack, as illustrated in Figure 3.The upstream FOWT is named "Front FOWT" and the downstream one is "Main FOWT".Main FOWT initially locates at the origin of the coordinates and the horizontal distance from Front FOWT is 1490 m, which is equivalent to twice the anchor radius in the single FOWT tests and larger than five times the rotor diameter.
The shared mooring configuration consists of five lines, with the shared line being twice as long as the remaining conventional lines.Line 2 to line 5 have a length of 750 m and are anchored at seabed, while the shared line is connected with two floaters.The initial lay-down length of the shared line is around 500 m.Line 2 and line 3, as well as line 4 and line 5 are symmetric about the X-axis.The angles of line 2 and line 4 with the X-axis are 60 • and 120 • clockwise, respectively.The mooring properties of the five mooring lines are the same as in Table 1, with a pretension of 3 MN applied to each mooring line.Figure 4 compares the force-displacement relationships for the shared mooring line and a conventional line (see Figure 1).For the shared mooring line, a positive surge displacement means a expanded horizontal distance between two FOWTs.It is clearly seen that the shared line experiences a slower growth in mooring tension compared to line 1, at the surge displacement larger than zero.The comparison of shared line and line 1 on the change of lay-down length shows the benefits of shared moorings, as the lay-down length of line 1 quickly drops to negative, while the shared line can maintain positive at the maximum stretched length of 30 meters.
For numerical simulations of a shared mooring configuration, we assume that two FOWTs have identical substructures, but with different fairlead positions, using the center of each floater as a reference.The fairleads on Front FOWT are not located on the columns, which may result in construction problems of hinged type fairleads for chain winches.However, a study of fairlead positions on mooring line connections is beyond the scope of this paper.
To investigate the impact of a shared mooring line configuration on global performances of the connected FOWTs, two tests of different wind and wave conditions are undertaken in Simpack, where one of the two FOWTs is exposed to a steady wind speed and a regular wave, and the other is under static conditions.The tested wind speed is 16 m/s and the wave has a height of 2 m and a period of 7 s, the same as in the single FOWT tests.Due to the absence of wind-wave misalignment and the symmetrical mooring layout, the floater experiences small motions in sway, roll and yaw directions, so the conventional lines that connect the floater are symmetrically loaded.Therefore, we only present the surge, heave and pitch motions, as well as fairlead tensions on line 2 of Main FOWT and on line 4 of Front FOWT.

case 1:
Front FOWT experiences a steady wind and a regular wave Table 3 lists the mean values of the floater surge, heave and pitch motions, as well as the mean mooring tensions, for case 1, where Front FOWT experiences a steady wind and a regular wave and Main FOWT is under the static conditions.The relative motions between two FOWTs are shown as the motion deviations.Driven by wind and wave forces, Front FOWT moves with a positive surge excursion, resulting in a higher mooring tensions on line 4 than the pretension force.The horizontal distance between two floaters decreases with the increasing surge offset of Front FOWT, causing a decreased tension on the shared line.Meanwhile, for Main FOWT, as the shared line tension becomes smaller than the pretension force on line 2, a positive surge occurs on Main FOWT and the mooring tension on line 2 decreases with this increasing surge of Main FOWT.
Since Front FOWT withstands the environmental forces and Main FOWT is under the static conditions, line 4 that connects with Front FOWT, becomes the most heavily loaded line, with 30% higher tension than line 2 that connects with Main FOWT.Also, pushed by the wind and wave forces, the mean surge amplitude of Front FOWT is 120% larger than that of Main FOWT.
We also observe different heave motions for two FOWTs in case 1, where Front FOWT shows a negative displacement and Main FOWT has a positive one.The wind-wave forces induces a large surge excursion of Front FOWT, causing the total vertical mooring restoring force to rise, thus dragging the floater deeper below the waterline.As for Main FOWT, mooring line tensions on the shared line and line 2 decrease with the increasing surge, causing a reduction on vertical mooring restoring forces and a slight increase in the floater heave displacement.
The pitch motions on two FOWTs with a shared mooring configuration are different as well, because the wind thrust forces on Front FOWT results in a large bending moment on the floater and leads to a positive pitch motion.While Main FOWT has a negative pitch motion due to the self-weight of the turbine and the pitch motion is slightly smaller than that in the single FOWT case, as the pitch restoring coefficient changes with the heave displacement.
In addition, we compare the results of Main FOWT using a shared mooring configuration and the single FOWT using a conventional mooring configuration under static conditions, as illustrated in Figure 5, to evaluate the effect of a shared mooring line configuration on the connected FOWT in terms of floater motions and line tensions.The shared mooring configuration brings a remarkable increase in the surge motion when Main FOWT is under static conditions.The single FOWT test shows a zero mean surge offset, while Main FOWT with a shared mooring configuration experiences a mean surge excursion of 5 m in case 1.Also, compared to the single FOWT results, Main FOWT sees a 9% rise in the mean heave displacement due to the mooring tension reductions.

case 2: Main FOWT experiences a steady wind and a regular wave
In case 2, Front FOWT is under static conditions, while Main FOWT is exposed to a steady wind and a regular wave.Table 4 presents the mean displacements of surge, heave and pitch motions and mean mooring tensions for the dynamic tests of two FOWTs in case 2. And the relative motions between two FOWTs are shown as the motion deviations.
The shared line that connects Main FOWT is most heavily loaded to balance the wind and wave forces in case 2. The horizontal distance between two floaters is expanded with a higher floater surge excursion of Main FOWT.And the increased tension on the shared line drives Front FOWT by a positive surge offset.As a result, Main FOWT and Front FOWT undergo the mean surge offset of 12.5 m and 5.3 m, respectively.The negative heave motion of Main FOWT comes from the increased mooring vertical tensions that pulls the floater below the waterline.The positive pitch angle of Main FOWT is due to the wind thrust forces that cause a positive bending moment on the floater.Figure 6 demonstrates the comparisons of Main FOWT with shared moorings in case 2 and the single FOWT test with a conventional mooring configuration under similar steady wind and regular wave conditions.Compared to the single FOWT test, a significant increase of 102% in the mean surge displacement can be observed for the FOWT with shared moorings.Also, for the FOWT applying shared moorings, the mean heave displacement is uplifted by 0.27 m and mooring line tension is reduced by 9% on line 2, compared to the conventional mooring test.The motions of the wind-wave loaded and unloaded floaters in case 2 are different from those in case 1 where Main FOWT is under static conditions.For instance, the mean surge offset of the loaded floater changes from 11.0 m in case 1 to 12.5 m in case 2. The different performances of the wind-wave loaded and unloaded floaters in two cases are caused by the different restoring stiffness of the nonlinear shared mooring system.
Considering two FOWTs with a shared mooring line as a whole, mooring tensions on conventional lines (line 2 to line 5) withstand the wind and wave forces, as the mooring tension on the shared line acts as an internal force.In two cases with identical wind and wave conditions, the mooring restoring forces from conventional lines equally balance the environmental forces.Due to the symmetrical mooring layout, the total horizontal restoring force from the shared mooring configuration is determined by the tension deviation between line 4 and line 1, which depends on the relative floater motions.In case 1 and case 2, the relative surge displacement between two floaters is positive and negative, respectively, which results in a smaller mooring restoring stiffness in case 2 than in case 1, because the mooring line stiffness increases nonlinearly with the line elongation (see Figure 4).Therefore, the mean surge displacements of two floaters are higher in case 2 than in case 1.

Summary and future work
This paper outlines the theoretical basis of nonlinear mooring modeling approach, and validates the proposed method in Simpack through a well-matched comparison with OpenFAST simulations of a 15-MW semi-submersible FOWT.Two FOWTs with a shared mooring line are further analyzed in Simpack, where one FOWT is exposed to a steady wind and a regular wave, and the other FOWT is under static conditions.Different performances of the wind-wave loaded and unloaded floaters in two tested cases reflects the importance of considering the nonlinear mooring stiffness in the modelling approach.In addition, we evaluate the impact of applying a shared mooring line configuration on FOWTs, and observe a significant increase in the surge excursion of the connected floater, compared to the single FOWT using a conventional mooring configuration.
Future studies are recommended to consider the hydrodynamic forces on mooring lines, and the seabed contact and friction forces when evaluating shared mooring configurations.In addition, this paper is limited to steady wind and regular wave conditions, and future studies considering turbulent wind and irregular wave conditions are recommended to investigate the mooring structural integrity of shared moorings.

Figure 3 :
Figure 3: Two FOWTs with a shared mooring configuration.

Figure 4 :
Figure 4: The force-displacement relationships for shared line and line 1.

Figure 5 :
Figure 5: Comparison of Main FOWT in case 1 with a single FOWT test.

Figure 6 :
Figure 6: Comparisons of Main FOWT in case 2 with the single FOWT test.

Table 2 :
The comparisons of the OpenFAST and Simpack simulations for one FOWT tests.

Table 3 :
The floater motions and mooring tensions of two FOWTs in case 1.

Table 4 :
The floater motions and mooring tensions of two FOWTs in case 2.