Dynamic Response of Offshore Wind Turbine on 3×3 Barge Array Floating Platform under Extreme Sea Conditions

Offshore wind farm construction is nowadays state of the art in the wind power generation technology. However, deep water areas with huge amount of wind energy require innovative floating platforms to arrange and install wind turbines in order to harness wind energy and generate electricity. The conventional floating offshore wind turbine system is typically in the state of force imbalance due to the unique sway characteristics caused by the unfixed foundation and the high center of gravity of the platform. Therefore, a floating wind farm for 3×3 barge array platforms with shared mooring system is presented here to increase stability for floating platform. The NREL 5 MW wind turbine and ITI Energy barge reference model is taken as a basis for this work. Furthermore, the unsteady aerodynamic load solution model of the floating wind turbine is established considering the tip loss, hub loss and dynamic stall correction based on the blade element momentum (BEM) theory. The second development of AQWA is realized by FORTRAN programming language, and aerodynamic — hydrodynamic — Mooring coupled dynamics model is established to realize the algorithm solution of the model. Finally, the 6 degrees of freedom (DOF) dynamic response of single barge platform and barge array under extreme sea condition considering the coupling effect of wind and wave were observed and investigated in detail. The research results validate the feasibility of establishing barge array floating wind farm, and provide theoretical basis for further research on new floating wind farm.


Introduction
In today's world due to the rapid consumption of the traditional and nonrenewable energy, the demand for alternative energy has become one of the topics at the forefront of research. Seeking clean, green and renewable energy is an urgent task of top priority (Blanco et al., 2018). Wind energy is one of the most important renewable energy sources, which is growing rapidly all over the world. Compared with onshore wind energy, offshore wind energy resources are more abundant, and have the advantages of high wind speed, stable wind direction, high energy density, small wind shear and low turbulence intensity. In addition, with the continuous improvement of wind power generation technology, offshore wind energy has a very broad application prospect (Kim and Kim, 2018;Hildebrandt et al., 2019). According to the Global Wind Energy Report of 2019 (Lee and Zhao, 2020), the annual installed capacity of offshore wind energy would reach 15 GW by 2024, bringing its market share in global new installations to 20%. At present, offshore wind turbines are mainly concentrated in shallow water areas, but for deep water areas with more abundant wind resources and better wind conditions, floating platforms should be adopted for cost consideration (Hallowell et al., 2016). The offshore wind turbines are always subject to wave excitation in the service environment, which will not only reduce the power generation efficiency but also influence the life of unit. Obviously, the stability of floating platform is the basic guarantee for the safe operation of deepsea wind turbine.
Barge platform has the advantages of deployment, simple structure, and lower cost over others, but it is limited in terms of its high center of gravity and pitch stability (Butterfield et al., 2007). Furthermore, it is sensitive to wind and wave excitation, prone to overturning in complexity sea environment, which poses great challenges for practical engineering application. At present, there are few researches on barge platform, which is still in the primary stage of conceptual design and theoretical verification (Wayman, 2007). Several attempts have been made to improve the performance of barge platform for FOWT. Namik and Stol (2010) proposed an independent variable pitch control strategy applied to the barge-type floating wind turbine and investigated its control effect on platform motion. According to their conclusions, the roll and pitch fluctuation could be reduced by 39% and 43%, respectively. Shashikala et al. (1997) investigated the effect of flexibility of the mooring line, the point of mooring on the response of the barge platform. The results show that the surge and pitch responses are found to decrease as the attachment point is reduced below the mass centre level. Yang et al. (2019) substituted partial ballast of floating wind turbine on barge platform for tuned mass damper (TMD). The effect of TMD on vibration suppression and stabilization of wind turbine under five real ocean conditions were studied. The results show that TMD with optimal parameters can reduce the pitch standard deviation of floating platform to 47.95%. Hu et al. (2018) investigated load mitigation problem for a bargetype floating offshore wind turbine by using the inerterbased structural control system, and the results show that the overall performances can be improved by using inerters. Zuo et al. (2013) applied a memory compensated blade pitch control algorithm to the barge floating platform, which has the characteristics of short computation time and strong robustness. The simulation results show that the proposed method performs better in reducing power fluctuations, fatigue loads and platform vibration. Harriger (2011) investigated the motion of a 5 MW floating turbine subjected to ocean conditions. The results show that the barge platform is susceptible to greater surging and pitching motions than the spar platform when exposed to extreme wave conditions. Manmathakrishnan and Panneer Selvam (2018) investigated the pitch motion of barge floating platform by using gyrostabilizer. The numerical results imply that pitch motion is reduced considerably more than 90% under IEC 61400-3 Standard Design Load Case unidirectional random wave environment. Olondriz et al. (2018) adopted an advanced method based on an additional control loop to mitigate the platform-pitch motion on barge reference model. In the article, not only is the generator speed regulation improved, but also the performance deterioration caused by the hydrodynamic stiffness reducted.
According to the references mentioned above, it can be seen that the current researches on the dynamic response of the floating wind turbine on barge platform mainly focus on the active and passive structural control, blade pitch technology, improved design of the main body of the platform, new mooring system and so on. However, (1) the nacelle acceleration response is seldom involved. Modern wind tur-bines, as a typical structure with concentrated mass at the top, have obvious long and flexible characteristics. The nacelle acceleration indirectly reflects the force exerted on the tower. Therefore, the nacelle acceleration is an important parameter for judging the stability of the floating wind turbine; (2) unlike the traditional offshore platform, the rotation of the upper wind turbine rotor on the floating wind turbine platform will produce huge aerodynamic force. Therefore, the influence of wind load cannot be neglected, but many researchers have not considered wind load. In addition, although some researchers consider the effect of wind load, the wind load is only simplified to a steady axial thrust, or the thrust calculated by FAST is taken as an external excitation, ignoring the true motion form of the platform, so the research results are not rigorous; (3) their research objects are mainly concentrated on single floating platform, and there is no relevant literature on the dynamic response of multiple floating platforms with coupling effect between them.
This paper aims to present an innovative barge-type multi-platform with shared mooring system to improve the stability of the whole wind farm. As to the auther's knowledge, The design concept of multi-platform array floating wind farm was put forward for the first time in (Ding et al., 2018), several spar-type floating wind turbines are connected by mooring lines, and their dynamic responses in surge, heave and pitch are analyzed. The results show that the multi-platform with shared mooring system can mitigate the platform response to a certain extent.
However, when the platforms in wind farm move under the excitation of external wind and wave loads, the vertical force generated by mooring toward the sway direction on both sides of the platform cannot always guarantee the same magnitude. Therefore, the roll response of platforms is a characteristic parameter that needs special attention. In addition, the surge restoring force generated by mooring lines of adjacent platforms are changing dynamically, and the resulting torsional moment cannot be offset simultaneously. Thus, the yaw response of platforms is another characteristic parameter that needs to be analyzed. Futhermore, floating wind turbine platforms are always in dynamic motion, which will lead to the change of the relative inflow velocity, and eventually influence the aerodynamic load of the floating wind turbine. This means that it is not rigorous to simplify the wind load into a steady axial thrust and the induction effect of platform motion on the inflow speed can not be ignored, especially the platforms in wind farm with coupling effect between them.
The main purpose of this study is to investigate the feasibility of the proposed barge array and whether it could survive under extreme sea conditions. Therefore, a 3×3 array barge floating wind farm based on NREL's 5MW reference wind turbine model is established. In order to consider the coupling effect of wind and wave, the FORTRAN program-ming language is adopted to compile dynamic link library (DLL) file through AQWA reserved interface to solve the aerodynamic load. In addition, the extreme sea condition of the South China Sea (generally recognized, if the floating wind turbine can withstand the worst sea condition, it can withstand all the general sea conditions (Jonkman, 2007)) is adopted to conduct the 6-DOF platform dynamics of barge array floating wind farm. The rest of this paper is organized as follows: Section 2 provides the numerical method used here. Section 3 presents wind farm model proposed in this paper. Section 4 describes numerical models and proves its reliability. The time-frequency domain analysis for 6-DOF of platform in wind farm is performed in Section 5, where the motion performance and mooring dynamics analysis are presented. Finally, the conclusions of this work are given in Section 6. The study is expected to provide theoretical reference for the design, optimization and safe operation of floating wind farms.

Environmental load model
The environmental loads of floating wind turbines are very complex. Among many environmental factors, wind and wave are the most important ones in the analysis of floating bodies and also the most critically environmental conditions (Lin et al., 2018). In order to simplify the calculation without losing generality, this paper only considers the dynamic response of barge platform under the coupling of wind and wave.

Aerodynamic model
At present, there are three main methods to solve the aerodynamic loads of wind turbines: blade element momentum (BEM) theory (Hansen, 2007;Dehouck et al., 2017), vortex wake method (Gupta, 2006) and CFD numerical simulation method (Snel, 2003). Compared with the other two methods, the BEM theory model is simple, intuitive and efficient. Therefore, the BEM theory is used to calculate the aerodynamic load. Fig. 1 represents the airfoil cross-section to the rotor's plane of rotation.
In Fig. 1a, is the inflow wind speed, is the relative velocity of airflow, is the rotating angular velocity, is the inflow angle, is the pitch angle, is the angle of attack, is the axial induction factor, is the tangential induction factor, and are the lift and drag of blade element. (1) where and are the lift and drag coefficient respectively, is the chord length, and is the air density (assumed 1.29 kg/m 3 ). The aerodynamic force acting on the blade element can be identified by the following equation: (3) dF n dF t where is the axial aerodynamic force, which is corresponding to the force component being normal to the plane of rotation.
is the tangential aerodynamic force, corresponding to the one being tangential to the rotational direction. (5) C n C t where and are the axial and tangential force coefficients, respectively. dT dM dr r Therefore, the whole force and the torque acting on the blade element at the radius can be expressed as: where is the number of the blades. Based on the above BEM theory, the induction factors, aerodynamic parameters and the angle of attack of each blade element are obtained by the secondary development iteration algorithm based on AQWA, which is given in the following procedure: Step 1: Assume the initial values of axial induction factor and tangential induction factor.
Step 2: Calculate the wind inflow angle and the attack angle by Eqs. (9) and (10): Step 3: Calculate the blade-tip loss and hub loss according to Eqs. (11) and (12): ; (11) Step 4: Recalculate the axial induction factor and tangential induction factor by calling the lift force coefficient and drag force coefficient. ; ; σ where is the solidity of the blade element.
The iteration computation flow chart of induction factor is illustrated in Fig. 2. When the axial induction factor is larger than 0.4, the original BEM theory would be invalid, therefore, a correction theory to the axial induction factor developed by Buhl is applied in this paper (Buhl, 2005), which can be expressed as: In this paper, aerodynamic load includes two parts: the rotor thrust caused by the rotation of wind turbine rotor, and the wind resistance of tower and the structure above the waterplane of the platform relative to the wind direction.
Floating wind turbine platforms are always in dynamic motion, which will lead to the change of the relative inflow velocity, and eventually influence the aerodynamic load of the floating wind turbine. In order to accurately calculate the aerodynamic load, it is necessary to consider the induction effect of platform motion on the inflow speed in the following steps: Step 1: Extracting platform velocity at time t, and the velocity of the hub center at time t is obtained by the transformation matrix.

W ′
Step 2: Perform linear superposition of hub center velocity and inflow velocity at time t to obtain the actual inflow velocity .
Step 3: Solve aerodynamic force based on the BEM theory, and calculate the velocity of platform at the next time t+dt considering the effect of aerodynamic force.
Step 4: An interface in a dynamic link library (DLL) is created which communicates with and provides data translation functions from AQWA, so as to solve the aerodynamic load eventually.
When iteratively solving the axial and tangential induction factor, the lift-drag coefficients of airfoil of each section need to be called. The drag coefficient at the discontinuous angle of attack is obtained by linear interpolation.
Modern wind turbines are typically using variable speed pitch regulated as the wind speed exceeding the rated wind speed of designing to achieve the purpose of unloading. However, the purpose of this paper is to quickly solve the aerodynamic force at each time step. Therefore, the design pitch angle of NREL 5MW wind turbine blade is directly called and combined with linear interpolation to determine the pitch angle in this paper. In order to verify the reliability of aerodynamic load calculation, the calculated results of thrust at different wind speeds are compared with the calculated results of FAST, as shown in Fig. 3. It should be noted that the platform motion is not considered in the comparison work. It is observed that the calculation results were overestimated at a relatively high wind speed, this phenomenon was also verified in Tran and Kim (2016). Generally speaking, the calculated results are in good agreement with those of FAST code, which verifies the reliability of the aerodynamic load calculation in this paper. LIU Qing-song et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 186-200 189 2.2 Hydrodynamic model Morrison equation and diffraction/radiation method based on potential flow theory are usually used to solve the wave loads. Morrison equation is mainly used to solve the wave load of small-scale structure (Luo et al., 2015;Cai et al., 2000) (The ratio of incident wavelength to the characteristic length D of floating body ); the radiation/ diffraction theory is mainly used to solve the wave load of large-scale structure ( ). Since the main body of the floating wind turbine platform is very large, thus the radiation/diffraction theory is used to solve the wave load. y, z, t) According to the potential flow theory (Liu et al., 2016), velocity potential function can be regarded as linear superposition of incident potential , diffraction potential and radiation potential : Therefore, the hydrodynamic problem of the floating wind turbine platform under the action of linear micro-amplitude wave can be regarded as the linear superposition of the following two problems: (1) The force and moment are exerted on a fixed platform under regular incident waves. At this time, the wave load on the platform is the first-order wave excitation force, which is composed of Froude Kriloff (F−K) force and diffraction force. Both of them are obtained from the integration of the dynamic pressure field along the average wet surface of the platform. For small floating structure, the diffraction effect is not obvious, and only the F−K force can be considered. However, for large offshore platforms presented in this paper, the diffraction effect cannot be ignored, so the diffraction force should be calculated and corrected.
(2) The forces and moments on a platform are subjected to forced oscillation at the incident wave frequency, called radiation force. When the platform is in unsteady state, the inertial force acting on the platform is equivalent to the inertial force generated by the added mass attached to the platform, called the additional mass force, which is usually described by the added mass coefficient. According to Eq. (7), separate variables from velocity potential function can be obtained: where, i is the imaginary unit; is the radiation wave potential caused by the movement of the platform at the j-th degree of freedom, and is the movement displacement of the platform at the j-th degree of freedom under the action of the unit amplitude wave. Based on the linear micro-amplitude wave theory, the incident wave potential can be expressed as: According to the linearized Bernoulli equation, the firstorder linear pressure gradient can be obtained from the incident wave potential: After the water pressure distribution is obtained, the first-order wave excitation force acting on the platform can be obtained by integrating the pressure on the wet surface of the platform using Green's theory. It should be noted that the first-order wave excitation force of Green's theory depends on the discretization of the platform surface, it is necessary to divide the platform surface into panels.

Dynamic mooring line system model
In traditional ocean engineering, it is generally considered that mooring lines cannot bear shear stress or bending moment. The axial stress along mooring lines can only be tension (Ji et al., 2011). The general form of the governing equation is as follows: where and represent the mass and additional mass per unit length of mooring line, and are the normal and tangential force of fluid acting on per unit length mooring line, is the mooring tension, is the net gravity per unit length of mooring line, and are the normal and tangential drag force coefficient of fluids, and are the diameter and length of mooring line, and are the normal velocity vectors of fluid and mooring line, respectively, and are the tangential velocity vectors of fluid and mooring line, respectively.
Eq. (23) is a complex time-varying dynamic equation. It has no analytical solution and must be solved by numerical method. At present, the commonly used numerical methods are qua-static model, dynamic catenary equation, lumped mass method and finite element method. The dynamic catenary equation can be used to model the catenary partially resting on the seabed and consider the geometric nonlinearity. By considering the weight and buoyancy, the non- linear relationship between the mooring restoring force and the fairlead locations of each mooring line is established, and the analytical solution of the mooring line shape and tension distribution is obtained (Jonkman, 2007). Because of its simplicity and high accuracy, this method has been applied to mooring system analysis by AQWA and WAMIT (Tran and Kim, 2015). When considering the hydrodynamic loads of waves and seabed friction effects on mooring lines, the catenary equation is discretized by dividing the catenary into a series of segments, and the numerical solution is performed by iterative method (Wang et al., 2010).
In this paper, the dynamic catenary model is adopted to perform the mooring dynamic analysis of floating wind turbine barge platform. The restoring force is applied on the fairlead point. The catenary model is shown in Fig. 4.
When a portion of the mooring line adjacent to the anchor rests on the seabed, the analytical equations are as follows: When no portion of the mooring line rests on the seabed, the analytical equation was given as follows: where ( , ) is the fairlead position relative to the anchor, and are the effective tension in the mooring line at the fairlead, both for horizontal and vertical component respectively, is the static friction coefficient between the mooring line and seabed. For each mooring line, is the extensional stiffness, is the mass of the line per unit length, is the total unstretched length, and , the unstretched portion resting on the seabed, . When the parameters of the mooring line and the fairlead position are known, the vertical and horizontal restoring forces of the mooring line at the fairlead can be obtained by the catenary formula, and the restoring moment of the mooring line can also be obtained. Jonkman has proven the accuracy of this modeling ).

Motion equation in frequency domain and time domain
Under the action of linear micro-amplitude wave, the floating wind turbine platform is in harmonic motion with the wave frequency. According to the potential flow theory and radiation/diffraction theory, it can be regarded as a simple harmonic vibration of a system with mass, stiffness and damping. Therefore, the frequency domain motion equation is established at the center of mass of the floating wind turbine platform as follows: where is the structure mass matrix, is the hydrodynamic added mass matrix, is the system linear damping matrix, is the total system stiffness matrix, is the external wave forces on the system (per unit wave amplitude), is response motions, and is the wave frequency.
The additional mass coefficient matrix and radiation damping coefficient matrix solved by frequency domain depend on the frequency of incident wave. Therefore, the impulse response function needs to be introduced in the form of convolution integral. By considering the wind, wave loads and mooring effects, the frequency domain equation Eq. (30) can be converted to its counterpart in time domain by transformation, which can be written as: where is the hydrodynamic added mass matrix at infinite wave frequency, is the velocity impulse function matrix, and , and are the external forces for wind load, wave load and mooring line force, respectively.

Barge-type floating wind farm model
The research object of this paper is a single barge platform and a 3×3 barge array floating platform for wind farm. The NREL 5 MW reference wind turbine model (Jonkman, 2009) was set above the tower. The main characteristic parameters are shown in Table 1. The design parameters of ITI Energy barge platform (Robertson and Jonkman, 2011) are summarized as shown in Table 2. Generally, in order to avoid the influence of upstream wind turbines wake on the aerodynamic performance of downstream wind turbine when building large wind farms, the spacing of horizontal axis wind turbines between the front and back is 7D−8D (Choi et al., 2013), and the spacing between the left and right is 3D−5D (D is the Rotor diameter of wind turbines, Stevens et al., 2014). However, due to the stable wind speed of offshore wind resources and the low turbulence, the influence of the wake on offshore wind turbines can be neglected. Furthermore, this paper does not involve the study of the aerodynamic performance and the wake of wind turbines. Therefore, the distance between wind turbines in floating wind farm is 500 m. The floating wind farm discussed in this paper consists of 9 floating wind turbines on barge platform (P1−P9). The calculation model of single barge platform and 3×3 barge array floating platform for wind farm are shown in Fig. 5.
The mooring system in wind farm consists of two types of lines, fixed catenary (connecting the platforms to the seabed, marked in red) and linked catenary (connecting platforms each other, marked in black) is illustrated in Fig. 6. The mooring line properties are summarized in Table 3.
In Fig. 6, it is noted that the incident direction of wind and wave toward the negative x-axis, and the mooring system is arranged symmetrically. Therefore, in single barge platform, only the mooring line No. 1 and No. 2 (both of them have the largest tension, shown in Fig. 6a) are considered. In barge array floating wind farm, only the mooring lines of lower side platforms and half of the mooring lines of middle side platforms (Line Nos. 1−9, marked with dashed lines) are selected to perform the mooring system's dynamics, as shown in Fig. 6b.

Mesh generation and reliability verification
In order to verify the reliability of the hydrodynamic model of the barge floating wind farm proposed in this paper, the hydrodynamic model of the floating wind turbine on single barge platform is modeled and calculated. The diffraction unit under the waterplane is refined. The maximum unit size of the barge platform is 1.5 m, the total number of diffraction units is about 3000, and the non-diffracting units is about 5000. The result of the mesh generation is shown in Fig. 7.   In order to verify the mesh accuracy, two methods, nearfield method and far-field method, are used to solve the second-order mean drift force of surge. When the solution of the two methods with the same trend and the magnitude order is close, it can be considered that the mesh could satisfy the requirement of hydrodynamic calculation accuracy. The solution results are shown in Fig. 8.
As can be seen from Fig. 8, the results of near-field method and far-field method for solving the second-order mean drift force of a floating wind turbine on barge platform are almost identical. Therefore, mesh generation can satisfy the requirement of hydrodynamic calculation accuracy. The near-field method solves the second-order wave force by integrating the pressure on the wet surface, which relies on mesh generation, while the far-field method obtains the second-order average drift force by solving the mo-mentum equation. This method does not depend on the panels, and the far-field method has higher accuracy and can obtain accurate calculation results. Therefore, when the results of the two methods are close to each other, it can be considered that mesh generation can meet the need of hydrodynamic calculation accuracy.

Analysis result of motion performance
The 6-DOF of the floating wind turbine platform are rotating (roll, pitch and yaw) around x, y and z axes and translating (surge, sway and heave) along each axis. The 6-DOF motion coordinates of the ITI barge platform are shown in Fig. 9.   LIU Qing-song et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 186-200 In order to study the dynamic response of platforms in wind farm more comprehensively, we consider all the 6-DOF motion responses in this analysis. The extreme sea states at the South China Sea test site (Chen et al., 1998;Yue et al., 2020) (as presented in Table 4) are used in the calculation. The wind is generated by Turbsim which was developed by the National Renewable Energy Laboratory (NREL) to provide a numerical simulation of a full-field flow. The random wave is simulated by Pierson−Moskowitz (P−M) wave spectrum. In addition, in order to obtain more accurate time domain statistics, the analyses are conducted based on different wave seed numbers, including 1, 500, 1000, 2000, 5000 and 10000. The whole simulation time is 10000 s, and the time step is 0.1 s. The wave spectral density and time history curves of irregular wave are shown in Fig. 10 and Fig. 11, respectively.

Response of surge and sway
The trajectory of platforms in wind farm and single barge platform in the xoy plane is shown in Fig. 12. In sway direction, the displacement of single barge platform is 0 m, which is mainly due to that the mooring lines on both sides of platform are symmetrically arranged with respect to the windward side, so the force in the y direction is always in equilibrium state. There is a certain degree of sway motion for platforms in the wind farm, and the sway displacement of the platforms (P1, P2 and P3) at the middle part of wind farm is approximately 0 m. This is also due to the symmet-   rical layout of the middle platforms. However, the platform (P4, P5, P6, P7, P8 and P9) on other two sides has a large sway motion, due to the movement of platforms under the wind, and wave load will lead to the movement of catenary connected with the platform, the restoring force of the fixed catenary on the upper and lower sides and the linked catenary on the middle side of platforms are always changing dynamically, leading to the unbalance of horizontal restoring force component.
In surge direction, we can observe that the surge displacement of P5 and P8 is larger than that of P4 and P7 in the wind farm, which is mainly due to the fact that P4 and P7 are close to the fixed catenary on the windward side, while P5 and P8 are all connected with the linked catenary in the surge direction. Under the extreme sea condition, the large drift motion of the link catenary leads to the increase of surge displacement of P5 and P8. In addition, the surge displacement of single barge platform fluctuates between −10 m and 40 m along the x axis, and the surge response of platforms in the wind farm is smaller than that of a single platform. It shows that the surge motion of platforms in the wind farm is restrained to a certain extent. This confirms the advantage of wind farm for reducing the surge responses of platforms, which is favorable for extreme designs.
In order to obtain the time history characteristics of surge response of platform, the time and frequency response analyses (only the platforms on the middle and lower side are considered) are shown in Figs. 13 and 14. As shown in Fig. 13, the maximum drift position and average balance position of P1, P2 and P3 in the middle part of wind farm are slightly larger than those of platforms P4, P5 and P6. It may be explained that the platforms at the upper and lower sides connect more fixed catenary, and its surge recovery force is larger. In addition to the surge motion of the platform in the middle, the surge recovery force of the platform on other two sides will also be transmitted to the platform in the middle, so the superposition produces a greater surge response. It can be inferred that if a 5×5 array platform floating wind farm is adopted, the surge response in the middle part of wind farm will be the largest. Therefore, additional fixed catenary could be added to increase the surge stability of all platforms in the whole wind farm.
It is observed in Fig. 14 that the wave frequency components are the most prominent parts in this numerical result, due to the fact that the frequency range of energy concentration is gradually close to the natural frequency of  LIU Qing-song et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 186-200 surge response. The low-frequency components of P1, P2 and P3 in the middle part of wind farm are slightly larger than those of P4, P5 and P6, and the wave frequency response of the same column platform is almost the same. We can conclude that the effect of mooring system on the surge response of the platforms in the wind farm is mainly at low frequency.

Response of heave, pitch and nacelle acceleration
In time domain analysis, the input environmental load and the response of platforms are random time histories. Therefore, in order to obtain a proper maximum value from a time domain analysis, this paper adopts the statistical approach, and the time domain statistics are based on the average value of different wave seed results.
The time domain statistical value (fluctuation amplitude and standard deviation) of heave, pitch and nacelle acceleration of single barge platform and platforms in wind farm under extreme sea conditions are shown in Figs. 15−17 and summarized in Table 5. It is demonstrated that the heave and pitch response of platforms in the wind farm are smaller than those of S−B platform, while the nacelle acceleration of P3, P6 and P9 tends to increase. By observing Fig. 15b, Fig. 16b, and Fig. 17b, it is found that the standard deviation of pitch and nacelle acceleration response of platforms in wind farm are smaller than those of single barge platform, and the standard deviation of heave response is comparable to that of the single barge platform. In general, although the nacelle acceleration of some platforms has large fluctuation range under extreme sea condition, the overall stability is better than that of a single platform.

Response of yaw and roll
Under the extreme sea conditions, the time histories of roll and yaw response of platforms in wind farm are shown in Fig. 18. It can be seen that the roll and yaw response of P1, P2 and P3 in the middle part of the wind farm are always kept at 0°, since the mooring of the above three platforms is symmetrical with respect to the windward side, and the recovery force provided is mutually offset. Meanwhile, the roll and yaw responses of platforms in the upper and lower part of wind farm show symmetrical distribution, and not more than 2°. As a consequence, the roll and yaw response of the platforms in wind farm proposed in this paper can be almost ignored and has excellent stability.

Analysis result of mooring system
Mooring system is one of the key technologies to ensure the safety, reliability of the floating wind turbine and may also influence the stability of tower significantly. It is particularly important to analyze and evaluate the performance of mooring system in wind farm proposed in this paper. The time histories of mooring tension are shown in Fig. 19. It should be noted that the SB-Nos. 1−2, correspond to the mooring line No. 1 and No. 2 in single barge platform; the WF-Nos. 1−9, correspond to the mooring line Nos. 1−9 in barge array floating wind farm.
As can be seen from Fig. 19, the mooring tensiones of line Nos. 1−9 in wind farm are fluctuating intensively with time, and several abrupt peaks can be observed. On the contrary, the mooring tensions of the line No. 1 and No. 2 of single barge platform are relatively stable. In addition, we can observe that the mooring tension on the windward side (No. 1,No. 2 and No. 3) is the largest, and the farther away from the windward side, the smaller the mooring tension; which is mainly due to the intensification of the wave frequency response of the platform under extreme sea condition, which increases the fluctuation amplitude of the surge response, so the maximum and average mooring tension are larger than those of a single barge platform.
In order to present the mooring tension clearly, the statistical results and its safety factor under extreme sea conditions are summarized in Table 6.
For barge platforms in wind farm, it is demonstrated that the mooring tension of the line No. 1 is larger than that of line No. 2 and No. 3, the mooring tension of the line No. 4 is larger than that of line No. 5 and No. 6, and the mooring tension of the line No. 7 is larger than that of line No. 8 and LIU Qing-song et al. China Ocean Eng., 2021, Vol. 35, No. 2, P. 186-200 No. 9. This is mainly due to the large surge displacement of the platforms (P1, P2, and P3) linked to the mooring line No. 1,No. 4 and No. 7, resulting in the reduction of the mooring length of the lying part. Besides, the maximum and mean tension of each mooring line in wind farm increases sharply compared with single barge platform. Therefore, the mooring system has stricter design requirements under extreme sea conditions, especially the fixed catenary at the middle part of wind farm. According to API-RP-2SK criteria (API, 2001), the criteria and equivalent factor of safety for different conditions are listed in Table 7.
As shown in Table 7, the dynamic analysis method adopted in this analysis shows that the mooring system is intact, and the tension limit of mooring lines is 60% of the breaking strength, corresponding to that the equivalent  factor of safety is 1.67. According to the values in Table 6, the minimum safety factor of mooring dynamics is 1.69, satisfying the operating environment of extreme sea condition.

Conclusions
In this paper, a 3×3 multi-platform array floating wind farm with shared mooring based on the barge platform is presented in this work. By considering the coupling effect of wind and wave, the dynamic response of the barge platform in floating wind farm under extreme sea conditions is studied by numerical simulation. The following conclusions have been drawn.
(1) Taking the incident direction of wind and wave as a reference, the trajectories of the platforms on the upper and lower sides in wind farm are symmetrical, and the maximum and average values of surge displacement of platforms in the middle part of wind farm are larger than those on other two sides. In addition, the wave frequency response of the same column platforms is almost the same from the frequency domain analysis, and the differences are mainly at low frequency, which may explain that the mooring system primarily affects the low frequency response of the platforms in the wind farm.
(2) The roll and yaw responses of platforms in the middle part of wind farm are always 0°. However, there is a certain roll and yaw response of platforms on the upper and lower sides in wind farm, and the distribution is symmetrical. Furthermore, the roll response is not more than 2°, and the yaw motion fluctuates around 0°, which can be concluded that the barge array in wind farm has better roll and yaw stability.
(3) The heave and pitch responses of each platform in the wind farm are smaller than those of a single barge platform. Although the nacelle accelerations of P3, P6 and P9 are larger than those of single barge platform at some time, the standard deviations of them are smaller than those of a single platform. Therefore, the barge array in floating wind farm has better heave, pitch and nacelle vibration stability compared with a single barge.
(4) The mooring tension and safety factor of each platform in the wind farm are analysed in this paper, the results show that the mooring tension in the incident side of wind and wave is larger than that of the other side. The mooring tension of each platform is larger than that of the single barge platform, but the mooring safety factor is larger than the minimum value of CCS specification.
(5) For practical engineering design, if a 5×5 array platform floating wind farm is adopted, the surge response in the middle part of wind farm will be the largest. Therefore, additional fixed catenary could be added to increase the surge stability of all platforms in the whole wind farm.
In conclusion, compared with a single barge platform, the barge array in floating wind farm presented in this pa-per has better stability and is feasible in engineering technology, which can provide certain theoretical reference for the design and optimization of floating wind farm in practical engineering application.