DENG Ziwei, ZHANG Baocheng, *, MIAO Yu, ZHAO Bo, WANG Qiang,and ZHANG Kaisheng, *

Multi-Objective Optimal Design of the Wind-Wave Hybrid Platform with the Coupling Interaction

DENG Ziwei1), 2), ZHANG Baocheng1), 2), *, MIAO Yu1), 2), ZHAO Bo1), 2), WANG Qiang1), 2),and ZHANG Kaisheng1), 2), *

1) Department of Mechatronic Engineering, College of Engineering, Ocean University of China, Qingdao 266100, China 2) Key Laboratory of Ocean Engineering of Shandong Province, Ocean University of China, Qingdao 266100, China

An offshore wind-wave hybrid platform could consistently and cost-effectively supply renewable power. A multi-objective optimization process is proposed for a hybrid platform with hydrodynamic coupling interaction. The effects of various critical structural parameters, spacing values, and wave directions are studied for higher energy capture and offshore platform stability. Approximation models of various key parameters are established to optimize the hybrid system, with the objects of the power capture width ratio and the stability index of the platform. The optimization results are affected by the hydrodynamic coupling interaction, with a tendency to affect the higher frequency of hydrodynamic performance in the hybrid system. After the optimization, an appropriate spacing value effectively improves energy capture performance. The optimal array distance,, the optimal structural parametersR,r,d,r, andPTOare 11.57, 12.75, 5.1, 3.3, 1.5, 6.5 m, and 80436 Nm s?1, respectively. The peak value of the wave energy converter capture width ratio in the hybrid system increases by almost 50%, with a 54% decrease in the stability index.

wind and wave energy; hydrodynamic interaction; optimization

1 Introduction

The increase in the world population and limited fossil energy resources has resulted in a growing need for energy resources, especially sustainable renewable energy in the ocean. The percentage of wave energy available for con- tinued development exceeded 66% in the South China Sea (Wan, 2020). Zheng (2021) created a global oceanic wave energy resource dataset for the first time, compre- hensively including the temporal-spatial distribution, short- term forecast, climatic variation, long-term projection, clas- sification of wave energy, swell energy characteristics, wave climate, and wave energy of key nodes. This work made an excellent contribution to promoting the industri- alization of wave energy. It shows the wave energy re- source dataset has exhibited a significant increasing trend for the past four decades (Zheng, 2013). Although ocean energy with high energy density and predictability has been utilized rapidly in recent years, constructing a single kind of offshore wind turbine or wave energy con- verter (WEC) is costly for commercial implementation (Hu, 2020). Single power generation is always unstable. A promising way to improve the total energy harvesting per- formance is the combination of wind and wave energy pro-posed by Lakkoju (1996), which has been developed re- cently (Karimirad and Karimirad, 2014). To capture energy efficiently for the electricity needs of residents in off- shore cities and islands, research on optimizing the array of the hybrid system is necessary. Papers about optimiza- tion with the coupling effect are reviewed in three parts.

Most wave energy studies concentrate on the dynamic analysis and optimization of a WEC and its array. Numer- ous methods are available to enhance the power absorption of the wave converter. For example, increasing the draft and damping of converters (Chandrasekaran and Sricharan, 2020; Ruezga and Canedo, 2020), changing the number of floating energy output modules (Zhang, 2019), or in- creasing the number of WECs (Piscopo, 2016; Stans- by, 2017) may improve energy output performance. Controlling the electromagnetic force of the generator in a direct drive point absorber (Mendon?a and Martinez, 2016) and optimizing the hydrodynamic performances (McGuin- ness and Thomas, 2017) with maximizing yearly energy production (Piscopo, 2016) are effective for energy harvesting. The dynamics and power capture of the flap- type wave energy device are obtained using a frequency domain model, and it may be suitable for broadband wave conditions (Gunawardane, 2021). The array of WECs also affects energy capture efficiency (Mendon?a and Mar- tinez, 2016; Piscopo, 2016; Stansby, 2017). Sim- ulated floating-array-buoy WEC models are established to assess the efficiency performances of different buoy spac- ings, placement modes, and actuating arm lengths (Sun, 2021b). Appropriate values of damping of each WEC within the array are different from the optimal value of a single WEC (Bellew, 2009). In a time-domain model, several layouts of WECs are compared (De Andrés, 2014). The triangular array is optimal for multidirectional wave regimes (G?teman, 2020), and the array of point absorbers can be analyzed (G?teman, 2015) through an approximate model. In addition, a genetic al- gorithm is applied to obtain the optimal damping value (Sarkar, 2016). It is further developed to (Child and Venugopal, 2010) optimize the layout of the wave energy array to improve the interaction factors greater than one (Sharp and DuPont, 2018; Faraggiana, 2019) or min- imize the Levelized cost of energy.

A small floating offshore wind-wave hybrid platform has a higher output per square meter, smooth power out- put, and lower construction costs. An offshore platform can provide shelter for operation and maintenance, and the hydrodynamic coupling effect of an additional wave energy device may increase the response amplitude (Ren, 2020). The WEC system attached to a floating plat- form may provide a bigger restoring moment and has an important impact on the control of platform pitch motions (Kamarlouei, 2022). A hybrid wind-WEC (Perez- Collazo, 2018) was introduced in 2014. Different types of hybrid platforms exist, such as floating tension- leg wind and wave combined energy systems (Cheng, 2019; Ren, 2020), semi-submersible platforms with cone-cylinder WECs (Sai, 2020), and floating sys- tems integrated with wind, wave, and tidal energy (Li, 2018). A platform encompasses three oscillating water column devices (Mazarakos, 2019). A three-pontoon semi-submersible platform is installed with a point absor- ber array. The model experiment of this system is con- ducted under regular and irregular waves (Sun, 2021a). A study observed a reduction in the motion am- plitudes of the hybrid floating system with the addition of WECs around the STLP floater (Rony and Karmakar, 2022). Floating vertical axis wind turbines have a better potential than horizontal axis wind turbines for reducing energy costs (Cheng, 2019). Yet, few publications have studied stability while studying the efficiency of en- ergy harvesting of the wind-wave platform. For instance, coupled aerodynamic-hydrodynamic analysis (Jiang, 2020) of two kinds of wind platforms showed that the heave plate positively affects stability (Yue, 2020). Similar to the offshore wind-wave hybrid platform, the hy- brid floating breakwater-WEC system is proposed, which generates electrical energy while providing coastal protec- tion (Martinelli, 2016; Favaretto, 2017; Zhao, 2019). A multi-floating-body coupled system is de- veloped to optimize the WEC array around the floating breakwater (Zhang, 2021). An oscillating water col- umn WEC bottom-mounted breakwater integrated system could improve adaptability in variable wave conditions (Guo, 2021).

The hydrodynamic performance of this kind of hybrid system could be calculated by using Star-CCM+ software (Zhang, 2020). Simulations under turbulent wind and irregular waves have been conducted (Li, 2018) to ensure survivability and evaluate power output perform- ance (Muliawan, 2013; Cheng, 2019). The power production is enhanced while the platform motion is reduced, and WECs are occasionally good for the total response of the platform (Jiang, 2020). However, a semi-submersible platform exhibits severe heave and roll response motion (Sai, 2020). Fully coupled simula- tions of the buoy and floating platform are performed un- der design load cases (Giassi, 2017; Youn, 2018). Only Hu (2020) further optimized a series of variables, in- cluding radius, draft, and layout of the cylindrical WECs of the hybrid system, with a numerical study in the initial design.

Most research focuses on optimizing the floating de- vices or the spacing of the WECs, respectively. Further studies are needed to focus on the capture efficiency of WECs, the stabilizing ability of the floating offshore plat- form, and the coupling effect between the platform and the WECs. In this work, the mathematical model and con- figuration of the hybrid system are shown in Section 2. The optimization process based on ISIGHT is proposed for the offshore wind-wave hybrid generation platform in Section 3. An automated analytical process is developed, which integrates structural parametric design, mass attrib- ute acquisition, and hydrodynamics response calculation. Sample points of the platform, WECs, and the spacing be- tween them are generated through an experimental design with the optimal initial values of the independent plat- form and the single WEC; the hydrodynamic response and the mean absorbed power of these points are calculated by using AQWA and MATLAB. An approximate model is constructed with the relationship between sample parame- ters and response values of the platform and the mean ab- sorbed power of WECs (Thiagarajan and Dagher, 2014; Asrasal, 2018; Giorgi, 2020). In the model, key parameters of the platform and WECs are optimized for better capture efficiency and platform stability by using the adaptive simulated annealing (ASA) algorithm, consider- ing the coupling effect between the offshore platform and WECs. The coupling effect on the initial optimization of the energy harvesting hybrid platform is analyzed by com- paring the optimization results in Section 4.

2 Methodology and Data

2.1 Model Description

The object of this study is a wind-wave hybrid power generation platform Fig.1a, which is a floating platform that has three vertical turbines (2 kW) and six point ab- sorber WECs. The mass of the small vertical axis fan is 917.30 kg, and the height and diameter of the tower stem are 3.5 and 0.2 m, respectively. It has three blades with a diameter of 2.6 m, a blade length of 2 m, and a rated power of 2 kW. Its center of gravity is located at 0.69 m of the tower stem. The momentum of inertiaI,I, andIis 1744.09, 1745.36, and 118.61 kg m2, respectively. The hy- drodynamic interaction of the hybrid system that consists of a floating platform and six WECs (Fig.1b) can be an- alyzed through AQWA. The wind-wave hybrid system can be seen as an axis system of structures of-th and-th in Fig.1c. The figure refers to the fixed axes (FRA)Owith the origin in the mean free surface. The local structure axes (LSA)and body-fixed axesGxyzare defined for the individual structure of-th and-th. The origin of the LSA is at the body’s center of gravity. Its axes through the center of gravity are initially parallel to the FRA. Po- tential flow theory is typically used for the initial design of floating offshore structures because it can estimate the performance of the hybrid system quickly and simply. The floating motion equation of the hybrid platform is estab- lished to derive the performance evaluation indexes of platform motion (Eqs. (4) and (5)) and energy capture per- formance (Eq. (9)) in the optimization.

Fig.1 Definition of axis systems. (a), wind-wave hybrid power generation platform; (b), hydrodynamic model of the hybrid platform; (c), definition of axis systems.

2.2 Methodology

An optimization design process was established through the multidisciplinary design optimization system ISIGHT (Fig.2a). It consists of three parts: the experimental de- sign, the building of an approximate model, and the opti- mization based on the approximate model. The experi- mental design includes generating sample points, paramet- ric model reconstruction of the platform and WECs, the acquisition of mass properties, and hydrodynamic analy- sis. The experiment was designed by using the optimal La- tin hypercube method in the optimization process. Then, the approximation model was applied to fit the relation- ship between the objective functions and the design vari-ables of the offshore structure according to the results of the sample points generated by using the design of experi- ments (DOE). In the approximation model, the ASA al- gorithm was applied to perform global optimization to obtain the key parameters of the platform and WECs. Finally, the optimization design process is used to ana- lyze the hydrodynamic coupling interaction (Fig.2(b)), the performance of the optimal platform and WECs in the hy- brid system was compared with the platform and the WEC individually pre-optimization and post-optimization.

To complete the hydrodynamic calculation, AQWA set- tings in Fig.2a includes three parts: wave parameters, float- ing platform and wave energy converter parameters, and point quality parameters of the wind turbine. The floating platform and WEC parameters are input according to Ta- ble 1. The mass point parameters of the wind turbine are input according to the wind parameters mentioned in the model description in Section 2.1. Spacing parameters be- tween the offshore platform and WECs are input based on the wind-wave hybrid platform data in Section 2.3. Wave condition parameters are set according to the wave condi- tion data in Section 2.4.

Fig.2 Optimal flowchart of the wind-wave hybrid power generation platform. (a), the optimal flow chart; (b), the study of the coupling effect of the hybrid platform.

Table 1 Optimal results without coupling interaction

The optimization process of the spacing of the WECs and the platform in the floating hybrid system was first taken as an example to illustrate the application method of the optimization flow in Fig.2b. It was optimized con- sidering the hydrodynamic coupling interaction between the WECs and the platform. The single platform and WECs of the wind-wave hybrid platform system were individu- ally optimized in a previous study, which was taken as the initial structure. However, the parameters of the hybrid system were influenced by the coupling effect of WECs and the platform in the hybrid system. In this paper, the effect ofspacing on energy harvesting and platform stabil- ity was studied first (Sections 3.3). Then, with the optimal spacing calculated from the first part, the platform and WECs were optimized again to study the coupling inter- action effect on the structure parameters (Section 3.4). Third, the PTO damping was optimized for better hybrid system performance (Section 4.2). Finally, the effects of wave direction on the energy capture and stability of the device were investigated according to the above optimi- zation results (Section 4.4).

2.3 Data of the Hybrid System

The variables of the hybrid system in the optimization process are shown in Fig.3, and the platform was con- structed by using five parameters in Fig.3a. The float of the platform’s top and bottom radii isand, respec- tively, and both initial values were 4 m. The distance be- tween the left side of the float is1, the right side is2, and both initial values were 12 m. A represents the angle between1and2, and its initial value was 60?. In the pre- liminary design stage, the effect of the equipment on the platform needed to be taken into account, and the mass T-point represents the mass point of the vertical turbines and other devices. The position, mass, and moment of in- ertia were set as a fixed value according to the properties of the devices.

Fig.3 Geometric configurations, the layout of WECs, and DOE of the spacing. (a), structural parameters of a point absorber and the floating platform; (b), structure of the wind-wave hybrid power generation platform and the DOE method; (c), optimization of spacing parameters in the hybrid system (top view).

Additionally, the initial WEC array and the platform are axisymmetric. Experiment DOE values of spacing are de- signed according to the optimal Latin method (Fig.3b). In Fig.3c,mainly represents the distance between two WECs on the same side of the platform, andrepre- sents the vertical distance from the WEC to the line that connects the centers of the two floats of the platform. The initial spacingandwas 12 m.

2.4 Wave Condition Data

Wave condition is critical for the design of marine struc- tures. The JONSWAP wave spectral model of the area around Zhaitang Island is set as the environmental bound- ary condition, and the water depth is 50 m. The spectral ordinate at a frequency is given by

where the peak value α and spectral peak elevation factorof the JONSWAP spectrum are measured by Yang(2017) as= 0.062,= 1.9, the peak valueωis around 2.14 rad s?1, and the significant wave height is 0.6 m.

2.5 Motion Equation

Three-dimensional potential flow theory is applied to establish the kinematic equations of a small floating plat- form, and the motion response of the platform is obtained. The indexes of the motion are the capture width ratio of the WECs and the significant values of the platform.

Hydrodynamic coupling interaction concerns the influ- ence of the flow of one body field on another. The im- portance of the interaction depends on both the body sep- aration distances and the relative sizes of the bodies. The hydrodynamic interaction includes the radiation coupling and the shielding effects. In this hybrid system of the hy- drodynamic interaction case, the total degrees of rigid body motions are 6 ×.is the number of structures; the full unsteady potential is usually expressed as a super- position.

whereIrepresents the isolated incident wave,drepre- sents the diffraction wave potential, andxrepresents the-th degree of freedom motion amplitude of the-th structure.φrepresents the radiation potential due to the unit-th motion of the-th structure while other struc- tures remain stationary. It is defined by the boundary condition on the wetted hull surface mathematically.

To evaluate the performance of the WECs and the plat- form, response amplitude operators (RAOs) are chosen as an index, which is the transfer function between the wave amplitude spectrum and the motion spectrum. The set of linear motion equations of hydrodynamic interaction with frequency-dependent coefficients is obtained as

whereis a 6× 6structural mass matrix;M= [A,k] and= [B,k] represent the 6× 6hydrodynamic added mass and damping matrices, including the hydro- dynamic interaction coupling terms between different struc- tures; andhysrepresents the assembled hydrostatic stiff- ness matrix of each diagonal 6 × 6 hydrostatic stiffness submatrix corresponding to an individual structure.

2.6 Stability Index of the Platform

The motion response spectral analysis method is ap- plied generally to predict the response of the floating off- shore platform. The waves caused by wind approximately obey a normal distribution. On the basis of linear poten- tial flow theory, the instantaneous motion of the floating offshore platform satisfies a normal distribution with zero means, and the response amplitude satisfies the Rayleigh distribution.

where1/3and1/3are significant values of the pitch and roll motion of the floating offshore platform, respectively. A motion response spectrum corresponding to the wave spectrum exists. With all the motion amplitudes ranking from maximum to minimum, the significant value is the amplitude of the motion that takes a third of the total number. A small value indicates that the offshore platform is more stable and safer.

2.7 Energy Capture Equation of WEC

The objective function is established as an evaluation criterion to optimize the structure of the WECs. The power absorption spectrum is the dynamic response under the incident wave excitation per unit wavelength (() = 1). With a random wave, it can be expressed as Eq. (6)

With the above equation integrated into Eq. (6), the absorbed power of the device at-th frequency is

The objective function is the energy captured width ra- tioC, an essential criterion for evaluating energy har- vesting. It is defined as the ratio of the absorbed powerPto the incident wave energywaveover the width of the device

whererepresents the significant velocity.

The wave energy capture efficiency is a problem of matching the wave energy to the absorbed power spectrum. As a result of the coupling effect between the floating plat- form and the WEC,represents the ratio between Cof an individual WEC in the hybrid system and Cof the op- timum single WEC. The index(mean)is established to eval- uate the energy harvesting efficiency of the WEC’s array.

whereis the average captured width ratio of the array WECs, andrepresents the captured width ratio of the optimal single WEC. If(mean)> 1, then the hydrodynamic coupling interaction is positive for the power generation performance; if(mean)< 1, then the coupling interaction between the platform and WECs is negative.

2.8 Verification

The validation model is mainly intended to validate the calculation of the motion amplitude of the floating body with the coupling effect of multiple floating bodies, as mentioned in Section 2.1. The coupling model of five hem- ispherical floats was presented to calculate the energy capture width of the floats, which could verify the float motion calculation method mentioned in this paper. To validate the method of these functions, the hemispherical WEC array was simulated to validate the numerical model only in the heave motion. The radius and draft of the WEC wereand 0.15, respectively, and the mass was twice the displacement of the draft. The array spacing between WECs was 4, the sea depth was 7, and(mean)was cal- culated in the beam wave along the-axis. As shown in Fig.4a, the mesh number of each float was 155. Fig.4b shows that the simulated calculation results(mean)fit well with the calculation results in published papers (Bellew, 2011; Hu, 2020). With the coupling effect, the en- ergy capture width ratio calculated from the floating body motion amplitude agreed well with the calculation results in previous papers. Thus, the calculation method used in this study is reasonable.

3 Multi-Optimization of the Hybrid System

3.1 Design of Experiment for the Hybrid System

After the initial optimization of the WEC, the cylin- drical bottom was chosen as the shape of the WEC in the designed area. The optimal parameters of diameterd and draftrwere 7 and 1.8 m, respectively. The top and bot- tom radii of the float of the platform have significant impacts on its hydrodynamic performance in the optimi- zation of a single platform. To reduce the number of de- sign variables and improve the accuracy of the coupled optimization process.r,R,d, andrare chosen as the variables (Fig.3b), andwas set as a fixed value (4 m).

Fig.4 Comparison of numerical results. (a), mesh of five hemispherical WECs; (b), comparison of the mean inter- action factor q(mean) of this paper with results in published papers (Bellew, 2011; Hu et al., 2020).

Under the guidance of the flowchart in Fig.2, the kine- matic response significant values and their capture width ratios were calculated by MATLAB. An approximate mod- el was established with the root mean square valueSof the roll and pitch motion on the basis of the sample data. The mean capture width ratio(mean)of multiple WECs was set as the evaluation quantity. Dandrepresent the ratio ofandto the diameter of WEC, respec- tively. Error analysis was performed for the approximate model (Table 2).

3.2 Approximation Model and Optimization

An approximation model method is a mathematical ap- proach that approximates the relationship between a set of design points and its corresponding response values, as shown in Fig.5. The main types of approximation mod- els are the response surface model, the radial basis func- tion neural network model, and Kriging approximations.

The Kriging model was applied to fit the results of the sample points, with two spacing parameters as design vari- ables and the stability indexSand the capture perform- ance(mean)as objectives. It is an unbiased estimation with minimal estimated variance, and the correlation function is locally estimated in Eq. (11)

where() represents the global approximation;() re- presents the deterministic offset function of, which is typically expressed as a polynomial; and() is a random function with a mean value 0 and variance value2, pro- viding an approximation of the local simulated deviation.

The approximation error is evaluated by using the val- ues of the mean, the maximum, the root mean square, and the R-square. The R-square value measures the quality that fits the changes in the data and ranges from 0 to 1. The closer the value is to 1, the better the model fits the data. The approximation obtained by the Kriging method performed better than others for both the stability index of floating platformand the energy capture perform- ance(mean)in Fig.6.

Fig.5 Approximation model.

Fig.6 Approximation of C(mean) and Sm. (a), approximation of C(mean) versus Dfp and Dff ; (b), approximation of Sm versus Dfp and Dff .

Four indexes for these approximations are listed in Ta- ble 2. The error analysis of(mean)fitted well. In the Kri- ging approximation, the error probability of the(mean)predicted value was less than 10%, and that of thepredicted value was less than 30%. The model’s error is acceptable for the initial design of the hybrid system.

3.3 Optimization of the Array Spacing

Different arrays of spacing distance were analyzed in the approximation model, where the spacingandfitted the capture performance of WEC well. Therefore,and were still optimized in the approximation model with the mean value of the capture width ratios of multiple WECs and the platform stability index as the objective. Guided by the ASA algorithm and the optimi- zation Eq. (12), the optimal array parameters in the hy- brid system were calculated

where 0 ≤1/3,1/3≤ 15 represents the restriction of the platform motion, and the optimization was performed within the spacing range. As a result,andwere 11.57 and 12.75 m, respectively. The platform stability index and the average capture width ratio(mean)of WEC- 1 to WEC-6 represent the optimal object. They are used to visually analyze the hydrodynamic coupling effect on the structure of the platform and WECs.

Table 2 Error values for approximations of the spacing

3.4 Optimization of WECs and the Platform in the Hybrid System

An approximation model that considers the hydrody- namic coupling interaction was established, as shown in Table 1. The optimal spacingandwere set as fixed values; the radiusR,r, the radiusr, draft, and PTO dampingPTOwere set as variables; andof the plat- form andof the six WECs were set as the evaluation indexes. The approximation model was constructed based on the relationship between variables and the evaluation indexes. The hydrodynamic analyses of these samples were optimized under the guidance in Fig.2. The approxi- mation model obtained with Kriging fitted the experi- mental data better. The average, maximum, RMS error values, and the R-square of these approximations are listed in Table 3.

Table 3 Error values for approximations of the structure parameters

The R-squared value indicated that the predicted value of theand(mean)in the approximation model was very close to the actual value. The error probability values are less than 25% and 13%, which are acceptable. With the fixed values ofand, the approximation model fitted well withand(mean). The effects of different variables on the evaluation indexes are shown in Fig.7.Danddhad a more significant impact on energy capture efficiency(mean)thanRandr.PTOhad a weaker effect than that of other variables, as verified by Bellew(2009).Randrhad a greater effect onthan randd.PTOalso had a small effect on energy capture.

Furthermore, the platform and WECs were optimized in this approximate model with the hydrodynamic cou-pling effect taken into consideration under the ASA algo- rithm and the guidance of the optimization Eq. (13)

where the five variables areR,r,r,d, andPTO. Within the design range, the optimization is conducted in the limit of the platform motion1/3and1/3. Finally, the optimal structure of the platform and WECs changed in the hy- brid system due to the coupling interaction. R,,, and dwere 5.1, 3.3, 6.5, and 1.5 m, respectively.PTOhad less effect on(mean)compared with other variables in Table 3, which was optimized to 99999 Nm s?1. Thus, its effect may not be accurately reflected in multi-parameter optimiza- tion. Hence,PTOwas optimized further to ensure more accurate damping values of the hybrid system.

Fig.7 Global effects. (a), global effects on C(mean); (b), global effects on Sm.

4 Results and Discussion

4.1 Optimization Results of the Array Spacing

(mean)increased to the maximum value of most WECs asDwas 12.75, then decreased (seen in Fig.8). A large distance corresponded to a smaller capture width ratioC. The maximum values of WEC-1 and WEC-4, WEC-2 and WEC-5, and WEC-6 and WEC-3 were around 0.6, 0.3, and 0.1, respectively. The lower capture widths of WEC-3 and WEC-6 were caused by the sheltering of the platform. The variation of the coupling interaction factorof dif- ferentis shown in Fig.8b. The trend for each WEC is the same as the variation ofC. However, the platform has a negative impact, which leads to different levels of reduction in the energy capture efficiency of WECs.

Withof 1.83, the maximum value of WEC-1 and WEC-4 was close to 0.68 and 0.73 in Fig.9a, respectively. The maximum value of WEC-2 was 0.43 atof 1.89, slightly higher than the maximum value of WEC-5, which was 0.26 atof 1.81. The values of WEC-3 and WEC-6 follow a similar tendency, and they were close to 0.1 at of 1.6. The value of WEC-1 was slightly higher than that of WEC-4 except for its peak value zone, and the value of WEC-2 was slightly higher than that of WEC-5.of the left side was slightly higher than that of the right side of the platform, mainly because the left side was longer than the right. The variation of the coupling factorfor the different arrays is shown in Fig.9(b), appearing the same as the variation ofC. Although the coupling interaction negatively influences energy capture, appropriate distance could increase the performance.

4.2 Optimization Results of PTO Damping

The approximation of a single WEC is visualized in Fig.10 withCas the evaluation index of the optimiza- tion. WhenPTOincreased to 62167 Ns m?1,Cof WEC reached the peak value of 1.15.

Fig.8 Cw and q-factor of different WEC-to-platform distance ratio Dfp. (a), Cw versus Dfp; (b), q versus Dfp.

Fig.9 Cw and q-factor of different WEC-to-WEC distance ratios Dff. (a), Cw versus Dff; (b), q versus Dff.

Fig.10 Cw versus BPTO of a single WEC.

PTOwas affected by the coupling interaction. The draft and diameter of WEC were 1.5 and 6.5 m, respectively.PTOwas optimized to study the hydrodynamic coupling effect with an upper limit of 30E+4 Ns m?1and a lower limit of 0 Ns m?1. Following the process in Fig.2, the ap- proximation ofPTOis shown in Fig.11a. The optimal damping of WECs increased to 80427 Ns m?1, and(mean)reached the peak value of 0.3. In Fig.11b, the variation of the-factor had the same trend as the capture width ratio.(mean)was less than 0.2, thus indicating that the cou- pled platform has a negative effect on the energy capture performance of the WECs. The maximum capture width ratio can be achieved with a higher damping value of WECs relative to the single WEC.

4.3 Comparison of Hydrodynamic Performance

The RAOs of a single WEC, a single platform, and the hybrid system considering the hydrodynamic coupling effect were compared to evaluate the performance of the wind-wave complementary power platform before and after optimization.

In Fig.12, the variation of heave RAO (RAO-) of WECs was concentrated around 1.5 rad s?1, and the variation out- side the peak frequency range exhibited the same trend and was small. In Figs.12a and 12b, the peak value of RAO was larger for WEC-1 and WEC-4, followed by WEC-2 and WEC-5, and the smallest ones were WEC-6 and WEC-3. The peak frequency distribution of each WEC pre-optimization of the hybrid system was scattered. Af- ter optimization, the peak frequencies of WECs were gath- ered at 1.51 rad s?1; the values increased by almost 50%. In Figs.12c and 12d, the peak RAO value of optimal single WEC increased from 1.61 m m?1to 2.01 m m?1by nearly 34%. By comparing Figs.12a and 12c, we found that the peak RAO values of the WECs differed from the single one considering the coupling effect of the platform, and the peak values of the WEC-1 and WEC-4 increased rela- tive to the individual WEC. In Figs.12b and 12d, heave RAO values for all WECs, considering the hydrodynamic coupling effect, were more significant than those for the individual one post-optimization. The RAO-z peak values of WEC-1 and WEC-4 increased from 2 m m?1to 3.5 m m?1, or almost 1.75 times. The peak frequencies of all WECs in the hybrid system reduced closer to the peak frequency of the wave condition.

In Fig.13, the variation of the stability index was con- centrated around 1 rad s?1. Figs.13a and 13b show the roll and pitch peak values of the platform pre-optimization, and the stability indexes were 8.66, 23.53, and 17.72? m?1. After optimization, the pitch RAO decreased by 20.83? m?1or about 89%, and the roll RAO increased by 2.34? m?1or about 27%. However, the stability indexes de- creased from 17.7? m?1to 8? m?1or about 54%. A compa- rison between Figs.13c and 13d shows that the maximum peak values of roll and pitch RAO and stability index de- creased by 6.08 and 4.35? m?1, respectively, which are less than 10%. The first peak frequency did not exhibit any change. In Figs.13a and 13c, the RAO values of roll and pitch and the stability index of the platform in the hybrid system were less than those of the individual plat- form. In particular, the peak value of roll reduced from 113.72? m?1to 8.66? m?1or about 92.4%. Moreover, the optimal peak frequency was 0.91 rad s?1, which was far- ther from the peak frequency of the wave condition and lower than the first peak frequency of 0.97 rad s?1of the single platform. In Figs.13b and 13d, the hydrodynamic coupling effect impacted the platform stability, and the peak value of roll RAO was reduced by 96.64? m?1or about 90% of the peak value of the single platform.

Fig.12 Heave RAO of WECs. (a), RAO-z of WECs in the hybrid system pre-optimization; (b), RAO-z of WECs in the hybrid system post-optimization; (c), RAO-z of single WEC pre-optimization; (d), RAO-z of single WEC post-optimization.

Fig.13 Stability index of the platform. (a), RAO and Sm of the platform in the hybrid system pre-optimization; (b), RAO and Sm of the platform in the hybrid system post-optimization; (c), RAO and Sm of the single platform pre-optimization; (d), RAO and Sm of the single platform post-optimization.

As a result of the platform and other WECs in a hybrid system, the hydrodynamic coefficients may change, in- cluding the added mass, radiation damping, and exciting force. The added mass ratioR, the radiation damping ratiosR, and the exciting force ratiosRare introduced in Figs.14, 15, and 16 to illustrate the effects of the plat- form and other WECs. This condition indicates the ratio of the additional mass, radiation damping, and excitation force between different parameter structures of the wind- wave hybrid platform system. The added mass and radia- tion damping represent impedance to the motion of the platform and WECs. The difference observed in the three figures is related to the positions of the WECs.

Fig.14 shows that the Rof the individual platform and WEC reduced after optimization. The largest ampli- fication ofRin Fig.14a was almost 0.76 at 2.53 rad s?1for the pitch and roll directions, and the reduction factor was close to 0.69 near the resonance frequency 3.55 rad s?1for the two directions. The largest amplification factor ofRwas almost 0.813 at 3.16 rad s?1, and the reduction factor was close to 0.795 near the peak frequency of 2.9 rad s?1(Fig.14b). With the hydrodynamic coupling effect, the additional mass of the platform increased considera- bly, while the additional mass of the WEC decreased around mainly its natural frequency. In Fig.14c, the largest am- plification factor ofRwas close to 1.03 at 3.4 rad s?1of pitch motion, and the reduction factor was close to 0.94 near the peak frequency of 2.5 rad s?1of pitch motion, vary- ing more at frequencies greater than 1 rad s?1. The values mostly fluctuated around 1. The coupling effect on the additional mass was limited. Fig.14d shows that almost allRof the 6 WECs were close to 1.0. The largest am- plification factor ofRwas close to 1.17 at 1.12 rad s?1for WEC-6, and the reduction factor was close to 0.9 near the peak frequency of 1.63 rad s?1for the WEC-3. Rela- tive to the performance of a single WEC,Rincreased in the range of less than 1.63 rad s?1and decreased in the range of more than 3 rad s?1with the hydrodynamic cou- pling effect.

In Fig.15a,Rof the individual platform increased in the frequency range of more than 3 rad s?1, while the ratio values of individual WEC exhibited a smaller decrease in the wide frequency range after optimization. The maxi- mum R of roll and pitch was almost 6.02 and 4.98 at 3.58 rad s?1, respectively, and the reduction factor was close to 0.2 at almost 3.6 rad s?1for the two motions. The largest amplification factor ofRwas close to 1.2 at 3 rad s?1, and the reduction factor was 0.19 at almost 3.92 rad s?1(Fig.15b). With the coupling effect, theRof the plat- form increased considerably, while the radiation damping ratio of the WECs decreased mainly at almost 3.92 rad s?1. In Fig.15c, the largest amplification factor of radiation damping ratio was almost 4.88 at 2.50 rad s?1of pitch motion, and the reduction factor was close to 0.67 at al- most 2.69 rad s?1of roll motion. The increased radiation potential of the platform in the hybrid system increases the radiation damping of the platform. The radiation damp- ing of the platform in the hybrid system increased by around five times, and almost all ratios of radiation damp- ing of the 6 WECs exhibited the same trend (Fig.15d).

Fig.15 Rrd of WEC and platform. (a), Rrd post-optimization to pre-optimization of the single platform; (b), Rrd post- optimization to pre-optimization of the single WEC; (c), Rrd of the hybrid system to the single platform post-optimization; (d), Rrd of WECs of the hybrid system to single WEC post-optimization.

The maximum factor ofRwas close to 145 at 3.92 rad s?1, and the reduction factors of 6 WECs were close to 0.56 at almost 0.61 rad s?1, which indicates a great effect on the high frequency of WECs. The size of the platform and WECs is relatively smaller than the wavelength in the low frequency region; thus, the impedance of the platform in- creases more than the variation of the impedance of the WECs only at high frequencies with the coupling effect. This condition is beneficial for the safety of the platform and to improve energy harvesting.

In Fig.16,R of the platform increased mainly in the high-frequency range, whileR of WECs changed mini- mally in the wide frequency range after optimization. The largest amplification factor of the exciting force ratio in Fig.16a was close to 17.17 at 3.55 rad s?1for pitch mo- tion, and the reduction factor was close to 0.19 at almost 3.75 rad s?1for roll motion. In Fig.16b, the maximum factor of exciting force was close to 0.86 at 0.86 rad s?1, and the reduction factor was close to 0.34 at almost 4.9 rad s?1. With the coupling effect,Rof the platform in- creased at a low frequency and fluctuated more at a fre- quency higher than 2 rad s?1. In Fig.16c, the maximum factor for roll motion was close to 3.26 at 2.13 rad s?1, and the reduction factor was close to 0.42 at almost 2.32 rad s?1for roll motion.Rof WECs increased at a frequency higher than 2 rad s?1, and the variations of WEC-1, WEC- 2, WEC-4, and WEC-5 changed dramatically. The exis- tence of WECs changes the exciting forces at a frequency higher than 1 rad s?1. The maximum factor of WEC- 1 was 14.81 at 3.92 rad s?1; the reduction factor of WEC-3 was close to 0.73 near 4.94 rad s?1(Fig.16d). A more hydro- dynamic coupling effect occurred on the high-frequency wave exciting force. The excitation force of both the plat- form and WECs increased with the hydrodynamic cou- pling effect. This situation is favorable for the energy cap- ture performance but not for the stability of the platform. Then, we study the effect of the variation of wave direction.

Fig.16 Ref of WECs and platform. (a), Ref post-optimization to pre-optimization of the single platform; (b), Ref post- optimization to pre-optimization of the single WEC; (c), Ref of the hybrid system to the single platform of the platform post-optimization; (d), Ref of the WECs in the hybrid system to single WEC post-optimization.

4.4 Variation of Wave Direction

The above optimization was only for the head wave along the-axis inverse direction. When the wave direc- tion changes, the offshore platform, which works with a multi-point mooring, can hardly adjust its position. There- fore, the motion performance of the platform and WECs in the frequency domain needs to be studied. In this part, 30? wave spacing was chosen to study the effect of the wave direction of 180? to 180? onandC, and the other wave parameters were set in Section 2.4. The procedure (Fig.2) guided the experimental design and the hydrody- namic analysis of the samples. Swas large for beam wave directions, showing that the platform had poor stability.

In Fig.17a, the capture performance of each WEC reached the maximum value at its head wave, and the main direction of energy capture was concentrated at four an- gles for this trilateral platform. The coupling interaction factoron WECs is presented in Fig.17b. The trend was the same as the variation of capture width ratio; all values were less than 1, and the variation of wave angles had less effect on(mean). Above all, the energy capture ef- ficiency of this wind-wave hybrid power generation plat- form could improve by adjusting the floating platform for- ward to the head waves or the beam waves as much as possible, according to the traditional short-term forecast factors of wave power-wave direction (Zheng and Song, 2021).

Fig.17 Cw and Sm versus wave directions. (a), Cw of WECs and Sm of the platform versus directions Wd; (b), q-factor versus Wd.

5 Conclusions

This study proposes an overall optimization process for improving the energy capture performance of the offshore wind-wave hybrid generation platform. Several factors that could influence the energy capture performance of the de- vice were identified. The coupling interaction factor(mean)of the hybrid system was proposed to evaluate the hydro- dynamic coupling effect. Numerical calculation of the wind- wave hybrid power generation system was performed un- der the sea region around Zhaitang Island of China. The effects of WEC-to-WEC spacing, WEC-to-platform spac- ing, structures of platform and WEC, and wave direction on the energy capture performance and offshore platform stability with their hydrodynamic coupling effect were stud- ied. The capture performance increased after the optimi- zation, and the coupling effect made the offshore platform more stable (Section 4.3). The research methodology was applied to a case, and the following conclusions are drawn:

1) The average capture width ratio(mean)reached the maximum values when ratiosD and were close to 1.8. The optimal and were 11.57 and 12.75 m, respec- tively. Although the coupling effect of the hybrid system may be negative for the wave power capture of the WEC array (Hu, 2020), the appropriate distance could in- crease the energy capture performance (Section 3.4).

2) In the Kriging approximation model, the-squared value indicated that the predicted values of theSand(mean)were close to the actual value. The error probabil- ity was less than 25% and 13%, respectively (Section 3.2). The optimalR, r,d, r, andPTOwere 5.1 m, 3.3 m, 1.5 m, 6.5 m, and 80436 Nm s?1, respectively. The energy cap- ture performance(mean)of WECs increased by 50%, and the stability performance Sincreased by almost 54% after optimization.

3) As a result of the hydrodynamic coupling effect, the variation of excitation force and radiation damping led to the optimal damping of the WECs in the hybrid system being greater than that of the single WEC (Section 4.3).

4) The additional mass of the individual platform and WEC was reduced after the hybrid system was optimized. In the system, the radiation damping and the exciting force change are concentrated in the high-frequency zone caused by the hydrodynamic coupling effect (Hu, 2020). In this case, the performance of the platform and WEC is mainly affected by the varying excitation force (Section 4.3).

5) The energy capture efficiency and the stability of the hybrid system improved by adjusting the position of the floating platform to head waves and beam waves as much as possible (Section 4.4).

The hydrodynamic coupling effect between the plat- form and the WECs needs to be taken into account in the preliminary design of the wind-wave hybrid power gen- eration platform. The proposed method could obtain the optimal layout of WECs in real applications and provide design guidance.

Acknowledgements

This project was funded by the National Natural Sci- ence Foundation of China (No. U2006229). We thank the Research on the Qingdao Science and Technology Devel- opment Projects (No. 18-1-2-20-zhc). This work is also supported by the Innovation Program approved by the Ministry of Industry and Information Technology of PR China ([2016]24). We thank Dr. Wan Liu for writing as- sistance.

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