Optimal strategies of deployment of far offshore co-located wind-wave energy farms

The most profitable offshore energy resources are usually found away from the coast. Nevertheless, the accessibility and grid integration in those areas are more complicated. To avoid this problematic, large scale hydrogen production is being promoted for far offshore applications. The main objective of this paper is to analyze the ability of wave energy converters to maximize hydrogen production in hybrid wind and wave far offshore farms. To that end, wind and wave resource data are obtained from ERA5 for different locations in the Atlantic ocean and a Maximum Covariance Analysis is proposed for the selection of the most representative locations. Furthermore, the suitability of different sized wave energy converters for auxiliary hydrogen production in the far offshore wind farms is also analysed. On that account, the hydrodynamic parameters of the oscillating bodies are obtained via simulations with a Boundary Element Method based code and their operation is modelled using the software tool Matlab. The combination of both methodologies enables to perform a realistic assessment of the contribution of the wave energy converters to the hydrogen generation of an hybrid energy farm, especially during those periods when the wind turbines would be stopped due to the variability of the wind. The obtained results show a considerable hydrogen generation capacity of the wave energy converters, up to 6.28% of the wind based generation, which could remarkably improve the efficiency of the far offshore farm and bring important economical profit. Wave energy converters are observed to be most profitable in those farms with low covariance between wind and waves, where the disconnection times of the wind turbines are prone to be more prolonged but the wave energy is still usable. In such cases, a maximum of 101.12 h of equivalent rated production of the wind turbine has been calculated to be recovered by the wave energy converters.


Introduction
Offshore renewable energy is expected to play an important role in the transition towards strict sustainability of the electrical energy generation in the near future. Nowadays, wind is the most profitable and industrially mature offshore renewable energy source [1]. The offshore wind technology has undergone a remarkable development during the last years, with wind farms scaled up to 1000 MW [2]. On the one hand, in comparison to onshore technology, offshore wind turbines usually receive higher and less turbulent wind speeds and have no restrictions due to mechanical noise. Hence, they usually achieve a higher Annual Energy Production (AEP). On the other hand, the economical investment necessary for the installation of offshore wind turbines is higher, and difficulties related to their assembly and accessibility arise as well, due to their location in the sea.
In addition to offshore wind turbines, ocean energy (wave and tidal energy) is also presented in the literature as a growing and developing technology [3]. Under the assumption of an energy conversion efficiency of 40% and a wave power density above 30 kW/m in the 2% of the world's coasts, the power which can be achieved by ocean energy is estimated to be larger than 500 GW [4]. If the rest of the coastal line and the far offshore plants were additionally considered, the power production associated to ocean energy would significantly increase. In fact, the ocean energy sector plans to promote green energy generation projects of up to 100 GW for the year 2050 [1].
Regarding wave energy, Wave Energy Converters (WECs) are used to convert the kinetic or potential energy of the water into electrical energy. A detailed review of the existing WEC technologies is presented by Falcao [5]. These technologies can be divided in three main groups: Overtopping converters (the wave oscillations are used to store water in a reservoir above water level and then convert it into electrical energy), Oscillating Water Columns (OWCs) (wave oscillations produce an air flow in a chamber that, in turn, causes rotation of a turbine) and point absorbers (oscillating bodies which swing with the waves and transform their mechanical motion into electrical energy).
The design and performance of point absorbers with different geometries have been widely studied in the literature: One body linear point absorber [6], gyroscope-based inertial point absorbers [7], vertical axis pendulum based WEC [8] or a two-body linear point absorber [9] constitute nice examples of this kind of studies. The operation of point absorbers is regulated by means of a Power Take-Off (PTO) system, which controls the motion of the oscillating body and ensures the quality of the generated energy. Since the potential response of the PTO system is generally subject to constraints of different nature, it is important to consider the PTO when examining the operation and power generation of the WEC. Several PTO control strategies applied to WECs have been described in the literature, such as Model Predictive Control (MPC) [10], nonlinear control [11], robust control [12] or improved latching control [13] to mention a few.
The main drawbacks linked to offshore energy generation systems are the complicated accessibility, material transportation and the need of mooring/anchoring. Despite of them, the development and installation of multi-megawatt bottom-fixed [14] and floating offshore wind turbines [15] is already a reality. In case of installation of far offshore generators (>90 km from the coast), the main challenge is the energy storage, due to the difficulties associated to the grid integration. In this way, techno-economic feasibility studies that endorse the installation of such far offshore energy generation farms can be found in the literature [16]. The use of different energy storage methods is analysed in detail in the aforementioned study. The installation of far offshore wind turbines has also been addressed in wind energy related international industrial congresses [17]. To avoid the difficulties associated to the anchoring/ mooring in far offshore locations, mobile offshore energy harvesting solutions have been proposed as well. These mobile technologies can be divided in two main groups: Sailing wind turbines [16] (the power is produced by a wind turbine located in the ship) and energy ships [18] (the power is produced by a turbine located at the hull of a wind propelled ship).
The installation of integrated offshore wind turbine and WEC systems, considering their cross-couplings, has also been proposed in the literature. In fact, this topic is gaining much prominence among future developments of the offshore energy [19]. A detailed review of the economical feasibility of the wave energy is presented by Astariz et al. [20]. This review concludes that the installation of combined wind and wave energy systems is the best way to increase the efficiency of marine resource exploitation systems and increase their competitiveness. Ireland is already a well-known location due to its potential for colocated wind and wave farms. The low correlation between wind/ wave resources in Northwest and Southwest of this island can be successfully exploited, since co-located energy farms offer a more predictable, less variable renewable energy source [21]. This promising low variability of energy due to combination of wind and waves have been also detected in other island environments [22], continental nearshore areas in China or Denmark [23], and even in closed seas such as the Mediterranean or the Caspian sea [24]. In order to extend these studies, this paper takes these type of studies to the next level, analyzing the combination of wind and wave energies in far offshore locations in the Atlantic ocean.
The energy storage in far offshore locations, as shown by Babarit et al. [16], can be successfully and competitively accomplished using various energy vectors. Moreover, the use of energy vectors in standalone energy generation systems is known to help alleviate the intermittent nature of the offshore resource itself, which subsequently implies intermittent electricity generation patterns [25]. Many studies have highlighted the huge potential of hydrogen for the management and regulation of offshore [26] and far offshore [16] renewable energy generation systems. Due to its consideration as the fuel of the future, hydrogen [27] is being promoted by many governments and international companies. A recent study for France [28] suggests that a spatial planning at a regional level of different renewable sources might be needed to maximize the hydrogen production by 2035. The European Union published in 2020 a hydrogen promotion strategy by which the objective is to install in Europe 40 GW of renewable energy hydrogen electrolysers and an storage of up to 10 million tons of this fuel for years 2025-2030 [29]. In this context, many examples of hydrogen generation applied to offshore wind and wave energy farms have been found in the literature [30]. The main elements involved in the hydrogen generation process [30] are the power generator, sea water electrolyser, fuel cell and energy accumulator. In the case of far offshore applications, hydrogen production applied to wind turbines placed on sailing ships have also been analysed in the literature [31]. Drawbacks of hydrogen generation, and its subsequent use as a power source [32], lie especially in terms of storage and transportation safety. In response to that, recent developments also suggest that other energy vectors, such as methanol [33], could overcome some of these difficulties and improve the functionality of the hydrogen as a fuel [34].
The main objective of this paper is to make a quantitative assessment of the real performance and energy generation of different sized WECs (using a generic PTO) as auxiliary hydrogen producers in an hybrid wind-wave energy farm. The power produced by a WEC is usually significantly lower than the one produced by a wind turbine. Nevertheless, due to the variability and intermittent nature of the wind and the operational restrictions of the wind turbines (they are stopped if the wind speed is below the cut-in wind speed or above the cut-out wind speed value), WECs could play an important role as auxiliary power suppliers, thanks to their contribution during the disconnection times of the wind turbines. This way, the efficiency of the far offshore hybrid farm could be importantly improved with respect to an energy farm based solely on wind, thus reducing the Levelised Cost of Energy (LCOE).
The wind and wave energy resources in the Atlantic ocean have been analysed using ERA5 [35], which is a widely-used tool for the study of renewable energy resource data in locations worldwide [36]. In fact, many studies assessing the energy resource at different locations can be found in the literature: Meteorological measurements and wave resource data obtained with a buoy in the Icelandic Sea [37], a detailed analysis of the wave energy resource along the coast of Santa Catalina in Brazil [38], a study of the wave and tidal energy resource of the shelf seas in Uruguay [39] or the hybrid wind-wave energy resource in the northwest coast of the Iberian Peninsula [40] to name a few.
Previous evidence suggest that the large-scale teleconnection patterns affect the interannual variability of the wave height [41], the mean wave period and direction [42] and the available wave energy [43] at the regional level computed by means of regression analyses. Thus, the results from these studies will be compared with the ones obtained in this article using an alternative technique. The performance and subsequent hydrogen production of different sized WECs is calculated in the present analysis by considering realistic meteorological data and its covariance with the wave energy flux over the Atlantic. Hence, the contribution of the wave energy to improve the efficiency of a far offshore hybrid wind-wave energy farm can be assessed in a convenient way. As a result of the comparison of the performance of different WECs in this study, a more complex methodology for the selection of an optimal WEC in a defined location could arise, considering not only the generated auxiliary hydrogen, but also the size and the mass of the WEC and their affection in the integrity of the far offshore floating platform and its mechanical loads.
The modelling and simulation of the motion of the WEC, considering the displacement constraints and maximum force of the generic PTO system, has been accomplished by means of the open source Matlab toolbox WecOptTool [44], which allows to conduct simulations of a WEC at different sea states with high precision. Additionally, the hydrodynamic parameters of the designed oscillating bodies are obtained via simulations with the Boundary Element Method (BEM) code NEMOH [45], which is a consolidated and highly extended tool for the computation of hydrodynamic parameters of offshore structures. As the present analysis deals with an energy assessment, the selection of the mooring technology is considered to be out of the scope. Likewise, a generic PTO is considered, which could be implemented in the real device either as an hydraulic or as an electrical system. This paper is structured as follows: In Section 2 the wind and wave energy resource data obtained with ERA5, the analysis method based on the covariance and a detailed explanation of the design, modelling and simulation of the WEC are given. In Section 3 and Section 4, the results and the discussion are presented, respectively. Finally, the conclusions of the conducted analysis are explained in Section 5, including an initial assessment of the predictibility of the total wind plus wave energy production.

Methods
This section is organised as follows: In SubSection 2.1 the procedure to obtain the wind and wave resource data with ERA5 is presented. The theoretical basis of the Maximum Covariance Analysis (MCA) is presented in SubSection 2.2. The WECs considered for the calculations are presented in SubSection 2.4. The normalization process for the presentation of the obtained results is explained in SubSection 2.5.

Data
Hourly-averaged Wind Power Density (WPD) and magnitude of Wave Energy Flux (WEF) values have been derived from ERA5 Reanalysis [35]  Finally, the daily averaged fields have also been computed for the analysis of covariability between WPD and WEF.
Regarding ocean wave energy, the magnitude of WEF has been computed over the same region and period as WPD by using hourly data and considering the expression for deep water, see Eq. (2), where H s is the significant wave height of combined wind and swell and T is the mean wave period. The wave data present an horizontal resolution of 0.5 ∘ × 0.5 ∘ . Additionally, in order to correlate with the power matrix presented in Fig. 8, the peak period has also been downloaded from the ERA5 wave model. Finally, the hourly values of WEF, the significant wave height H s and the mean wave period T have also been seasonally averaged for diagnosis of the WEF field, and daily averages have been computed for the study of the coupled WPD and WEF variability.
The monthly values of the Artic Oscillation (AO) [46], the North Atlantic Oscillation (NAO) [47] and the East Atlantic pattern [48,49] covering the same period (2010-2019) have been downloaded from the Climate Prediction Center Web server.

Wind and Wave Maximum Covariance Analysis
The covariability of the WPD and the WEF has been studied by means of the MCA, following [50]. The MCA identifies the linear combinations of the original fields which lead to maximum covariance between temporal expansion coefficients computed from both fields' anomalies. In previous studies found in the literature, the MCA method has been used to determine the leading coupling modes between different variables: The significant wave height and the peak period and the significant wave height and the surface wind over the Southern Atlantic [51], the sea-surface temperature and the surface wind over the Tropical Pacific [52] and the sea-surface temperature and the wind stress over the Peruvian coasts [53].
In this study, the MCA has been applied to daily anomalies of the WPD (left field, following the description of the method in [50]) and the magnitude of WEF (right field). Since the analysis extends through a wide range of latitudes (from southern to northern areas of the domain), the values at every grid point have been scaled by multiplying them by the renormalizing factor ̅̅̅̅̅̅̅̅̅ ̅ cosφ √ , being φ the corresponding latitude, see [54]. This ensures that polar regions, which are characterized by a higher number of grid cells for the same area, do not get oversampled. Additionally, as the default value of the grid spacing recommended by the CDS for wind data (0.25 ∘ ) and for wave data (0.5 ∘ ) is not the same, the WPD fields have been reprojected onto the WEF grid by means of a nearest-neighbour algorithm.
Before the covariance matrices used in the MCA are calculated, the daily averages of the WPD and magnitude of WEF values are computed and standardized, since both variables are not expressed in the same units. This preprocessing is intended to avoid giving more weight to the field with the highest variability. Furthermore, the expected errors (sampling errors) in the singular values computed by means of the MCA analysis have been estimated by means of bootstrap [55]. To that end, random samples with 0.67 times the number of elements in the original sample have been extracted and, from these ones, an additional random resampling with repetition has been carried out to arrive to synthetic samples with the total length equal to the length of the whole sample. The process has been repeated 2000 times and a two-sided 95% confidence interval for the singular values has been calculated from the random samples.
Finally, an analysis of the relationship between the teleconnection indices and the temporal coefficients obtained from the MCA analysis has been carried out, by computing monthly averages of the temporal left and right MCA-based expansion coefficients. The seasonal cycle in these expansion coefficients has been removed and the correlation between the expansion coefficients' monthly anomalies and monthly indices (anomalies themselves) has been computed, thus calculating the fraction of variance (R 2 ) explained by every linear model used. Those relations, described in SubSection 3.1, present a confidence higher than 95% of not being random according to a significance test. As some of the expansion coefficients and the data show significant low-frequency variability (seasonal cycle), this significance test is run by generating 1000 autoregressive AR1 random sequences with the same autocorrelation and standard deviation of the error term as the ones in the expansion coefficients and every WPD and WEF grid point.

Existing wave-wind co-located far offshore systems
Although Babarit et al. [16] cite several possible technologies for the production of far offshore wind energy and although this paper is basically a resource assessment study, the technological combination of wind energy with wave energy should be briefly analyzed for far offshore applications.
First, it should be mentioned that the analysis presented in this paper is intended for co-located farms, because the wave weakening due to the presence of WECs is not considered for the corresponding stability of the floating wind turbines [56]. Considering this beneficial interaction of wave energy absorption for wind energy production, an hybrid system, such as Poseidon Floating Power by the Netherlands' Floating Power Plant AS [57] or W2 Power, proposed by Pelagic Power AS [58], could be paradigmatic approximations of hybrid systems to which the present analysis is intended, see Fig. 1 for the first case.
This kind of hybrid co-located systems should, however, be adapted for far offshore applications. Floating islands can be a solution. These structures are large floating multipurpose platforms, where a combined harnessing of marine resources can be carried out, not only wave-wind based energy production. The 50 MW platform project proposed by the UK's Energy Island Ltd. is an example [56], as well as the sailing wind  farm proposed by Tsujimoto et al. [59], in which an array of simple point absorbers can be implemented, see Fig. 2. This sailing wind farm is composed of a semi-submersible floating structure, eleven windmills, four sails and a storage for hydrogen, which perfectly fulfils the perspective of the present study. Furthermore, an effective meteorological model to search for optimum marine routes was developed, the path of which can reach far offshore locations around Japan, respecting the water limits established by Mariana Islands, Marshal Islands, Micronesia and Russian Federation within approximately 300 km [59]. The installation of wave energy arrays constituted of simple point absorbers in these kind of sailed semi-submergible systems with several wind turbines seems to be a suitable technological approach for the far offshore resource assessment study proposed in this paper.
Regarding the survival of these far offshore structures, the construction, operation and management of floating oil rigs constitutes a referential industrial experience. For instance, Perdido oil rig in the Gulf of Mexico is about 300 km away from the coast and is moored in approximately 2400 m of ocean water [60]. In any case, the proposal of a specific technology for a far offshore hybrid wind/wave energy colocated farm is out of the scope of this work, in which the main contribution is based on the pioneering application of MCA to identify the optimum combination of both marine renewable energies, together with the analysis of various WEC geometries for the optimal production of hydrogen.

Wave Energy Converter
A two body point absorber oscillating in heave has been considered as the reference WEC in this analysis. The geometry of the designed oscillating body is based on the shape of the Reference Model 3 (RM3) point absorber, see Fig. 3a. The RM3 is a reference device created by Sandia National Laboratories and financed by the Department of Energy of the United States, with the objective of developing an open-source code and methodology aimed to the design and analysis of Marine Hy-droKinetic (MHK) technologies.
Analog to the RM3 device, the reference WEC considered in this analysis is formed by two bodies: a floating surface buoy that oscillates in heave as a result of the excitation force induced by the waves and a submerged reaction plate. Additionally, there is a vertical column that connects the surface buoy with the reaction plate. The main objective of the reaction plate is to help damp the oscillations of the floating body. Furthermore, it is also used for the mooring of the point absorber, in order to keep its horizontal position fixed and avoid big translational motions of the device. Many studies analyzing the different mooring possibilities applicable to an in heave oscillating point absorber [61] can be found in the literature. In the present paper, the design of the mooring system of the device is considered to be beyond of the scope of the conducted analysis.
The main physical characteristics and operational constraints of the reference point absorber considered in this paper, from now addressed as WEC 1, are listed in Table 1. The performance and power capture of a WEC are directly related to the hydrodynamic forces induced by the incoming waves in both the floating and submerged body of the point absorber. Having defined the geometry of the reference point absorber WEC 1, see Table 1, the open source code NEMOH has been used to calculate its hydrodynamic parameters. NEMOH [45] is an open source code developed by the Ecole Centrale de Nantes. It is intended to calculate the first order loads on offshore devices due to their interaction with waves. As a result, this code can be effectively used to calculate the hydrodynamic parameters of point absorbers, which will define their performance and, thus, their power absorption. In this case, the definition of the mesh for the calculations of the hydrodynamic parameters associated to the reference WEC 1 is shown in Fig. 3(b).
The WEC Design Optimization Matlab Toolbox (WecOptTool) [44] has been used in the present paper for the analysis of the performance of the two body point absorber. WecOptTool, devoloped by Sandia National Laboratories, is an open source code intended to carry out frequency-dependent performance analyses and optimization studies of WEC devices, including control strategies of different nature and considering the actuation constraints of the PTO system. A demonstration of the application of the WecOptTool to the frequency-dependent analysis of a WaveBot WEC is presented in the work by Coe et al. [62]. The consideration of the frequency-dependent dynamics of the  point absorber is essential to conduct a realistic analysis, since the excitation forces induced by the waves and the response of the device do not relate linearly for the whole frequency spectrum of the waves.
Regarding the control strategy of the WEC (aimed to regulate the performance of the point absorber), a PseudoSpectral (PS) control strategy has been selected in this paper, applied by a generic PTO system. The PS controller [62] is based on the resolution of an optimization problem, in which the dynamics of the point absorber model, the motion of the WEC and the constraints corresponding to the maximum in heave displacement z max and maximum PTO force f PTOmax are considered. The objective of the optimization problem is to maximize the power absorption of the WEC, while not exceeding the performance constraints.
Hence, in this analysis, WecOptTool is used to model the performance of the designed WECs in variable but realistic wave resource conditions. The final objective is to estimate the power generation of the WECs in each one of the wave resource conditions, defined by the significant height H s and the peak period T p .

Normalization of Wave Energy according to Wind Energy
The power production of a floating offshore wind turbine is, in most cases, considerably larger than the one produced by a WEC, especially in far offshore applications, where the wind turbines are not subject to noise, transport or visual limitations. However, wind turbines are also susceptible to have disconnection times during their productive lives, as they might suffer from discontinuities due to the variability of the wind or maintenance tasks due to malfunctions of the wind turbine. Consequently, in the analysis conducted in this paper, the hydrogen production of variable sized WECs during the stopping time of far offshore wind turbines is to be computed and analyzed. In such cases, this hydrogen production is considered as additional and auxiliary, since it will improve the overall efficiency of the far offshore co-located wind-wave energy farm and help cover the energetic needs of the farm while the main wind based energetic generators are stopped. In the present analysis, the stopping times of the wind turbines due to mechanical issues are not considered, on account of the difficulty in their prediction without real operational data and the variability between individual wind turbines of the same model.
The Hywind National Renewable Energies Laboratory (NREL) 5 MW floating Horizontal Axis Wind Turbine (HAWT) [63] has been considered as the reference wind turbine in the present study. Based on the power production of the NREL 5 MW wind turbine, different WEC configurations are considered and described according to the normalized ratio of their maximum power generation value with respect to the rated power of the NREL 5 MW wind turbine. For instance, a WEC with 50 kW maximum power generation is to be characterized as a 1% power ratio WEC. Following this criteria, three WEC models with different size and power ratio are defined and listed in Table 2.
The design and simulation of the three different WEC models has been accomplished following the methodology described in SubSection 2.4. The corresponding meshes for the resolution of the hydrodynamic calculations of the variable sized WECs are presented in Fig. 4a and Fig. 4b, respectively. In comparison to the reference WEC 1, see Fig. 3b, the dimensions of the WEC 2 have been increased, while the dimensions of the WEC 3 have been reduced. The increase in the dimensions of the WEC 2 will cause an increment of its mass, while the mass of the WEC 3 will be reduced. Moreover, if the dimensions and the mass of a WEC increase, the force necessary to regulate its oscillation will have to be larger as well. Thus, in order to maintain the performance standards of the PTO control system in each one of the designed WEC models, the maximum force f PTOmax must be adapted accordingly, as seen in Table 2. As a result, the correct performance of the PTO control system is ensured for all three models, and a realistic comparison is conducted. It must be noted that an increase of force of the PTO system will cause an increment of its cost and dimensions.

Results
This section is organised as follows: In SubSection 3.1 the results associated to the selection of the optimal locations after the MCA are presented. The power matrix and hydrogen generation of the reference WEC 1 are presented in SubSection 3.2. The combined and normalized wind and wave hydrogen generation results are presented in SubSection 3.3.

Identification of locations. Maximum covariance analysis.
The selection of a sample of some of the most suitable locations in the Atlantic ocean for a far offshore co-located wind and wave energy farm has been performed according to the far offshore site evaluation criteria introduced in SubSection 2.2, in combination with the wind and wave resource data obtained through the meteorological analysis presented in SubSection 2.1. Following these evaluation criteria, four locations, marked with red circles in Fig. 5a, have been selected. The distance to the closest coast has also been shown in this Figure in order to focus on points very far from coast (P-00 and P-01 beyond 1000 km), a point not so far (P-02, close to 750 km from the nearest shore) and a closer one (P-03, around 500 km from the nearest coast). It should be emphasized that the Island of Azores constitutes a central geographical point that permits an almost total areal exploitation of far offshore marine renewable energy for the area of the North Atlantic.
The selection of the locations has been carried out on the basis of the results of the MCA analysis of the WPD and the WEF fields. The fraction of covariance between the WPD and the WEF fields over the Atlantic basin explained by each of the spatial modes identified in the MCA analysis is shown in Fig. 5b. The red or blue vertical lines represent the 95% confidence interval on the covariance fraction explained by each mode derived from the bootstrap analysis. The red colour indicates that the error bar does not overlap with the one corresponding to the next mode, while the blue colour indicates that overlapping (and therefore, sampling degeneracy) appears. Therefore, the leading mode represents a 79% of the common covariance, the second mode a 8% and the third one a 4%, which yields a 91% of total covariance explained by just three modes.
The spatial modes obtained from the MCA analysis are represented as homogeneous correlation maps [50] in Fig. 6. Points P-00 to P-03 have been selected under the criteria that they represent areas of high covariance. As shown in Fig. 6, P-00 and P-01 are located close to the maximum correlation areas in singular vector 1 both for left (WPD) and right (WEF) fields. However, the paper addresses the relevance of areas where the wave and wind energy are not that tightly coupled. To achieve this goal, P-02 and P-03 are selected at places with high absolute values of the correlation but negative signs, as shown by singular vector 2 (Fig. 6, middle row). Moreover, all of them are located relatively far from the continental coasts, so far that the installation of an energy farm would be considered as far offshore (>90 km).
Regarding the meaning of the main spatial modes identified in the MCA, the leading mode represents basically the seasonal cycle of storminess in the Atlantic, with the corresponding temporal expansion coefficients showing a highest amount of variability concentrated in periods longer than thirty days. The corresponding homogeneous correlation map for this leading mode (Fig. 6, top) shows that the seasonal Table 2 Designed WEC models with different size and normalized power ratio. cycle affects with the same polarity every point in the North Atlantic basin, but it is most intense in central Atlantic areas.
On the other side, the second mode (Fig. 6, middle panels) represents a meridionally oriented dipole structure with positive correlations over northern areas and negative correlations over southern areas. The regression analysis of the monthly anomalies calculated from the temporal expansion coefficients for this second mode shows that it is tightly tied (R 2 = 0.5 for the WPD and R 2 = 0.53 for the WEF) with the AO [46] and less strongly (R 2 = 0.36 for the WPD and R 2 = 0.33 for the WEF) with the NAO [47].
Finally, the third mode (Fig. 6, bottom) shows alternate patterns of negative and positive spatial correlations over the Atlantic, with P-02 located at the highest negative correlation.
The monthly anomalies represented by the monthly averages of the MCA-based temporal expansion coefficients are affected both by the AO (R 2 = 0.17 for the WPD and the WEF) and (R 2 = 0.21 for the WPD and R 2 = 0.18 the WEF) by the East Atlantic [48,49] pattern.

Power Matrix and Hydrogen Production of the reference WEC 1
The temporal performance of the reference WEC 1 point absorber with a wave elevation input defined by the significant height H s = 6.5 m and the peak period T p = 3.5 s during a time interval of 30 s is shown in Fig. 7. The upper plot represents the wave elevation η and the in heave motion of the point absorber z. The upper middle plot shows the motion velocity of the point absorber ż. The lower middle plot presents the instantaneous force exerted by the PTO system f PTO . And, finally, the lower plot represents the instantaneous power generated by the WEC P.
The most computationally efficient way to convert long periods of energy resource data into generated power is using the power curve or power matrix of the corresponding energy converter. In such a matrix, the characteristics of the energy resource and the power generated by the converter are directly related, and thus, the performance of the converter gets fully characterized and its energy production over long temporal periods can be analyzed, avoiding the execution of a huge number of simulations. In this case, the frequency-dependent WEC 1 model described in SubSection 2.4 has been simulated over its whole range of operating conditions, defined by the significant height H s and the peak period T p of the incoming waves, and the power matrix have been calculated. The power matrix of the reference WEC 1 is shown in Fig. 8. The cut-in and cut-off significant wave heights in order to start and pause the production of the WEC have been set to 0.5 m and 6.5 m, respectively. Regarding the peak period T p , simulations have been restricted to the interval between 0.5 s and 14 s, being the power absorption of the WEC for peak periods outside this range considered negligible. The obtained power matrix is observed to be in line with similar analyses found in the literature [64].
Once the performance of the point absorber has been characterized and the power matrix of the reference WEC 1 calculated, the hydrogen generation that would result from the installation of this device in a far offshore energy farm located at the Atlantic ocean points defined in SubSection 3.1 has been computed. The main characteristics of the wind and wave energy resources at each one of these four locations is summarized in Table 3.
The energy generated by the reference WEC 1 point absorber at each one of the locations has been computed by interpolation of the power matrix, see Fig. 8, with the wave resource data obtained in SubSection 3.1. For the hydrogen production, an energy consumption of 4.2 kWh

Energy Conversion and Management 251 (2022) 114914
Nm − 3 has been considered. This global energy consumption is itemized in detail in the work of Saenz-Aguirre et al. [65], where the global efficiency and energy consumption of a wave energy based offshore hydrogen production system are calculated. The global efficiency includes the individual efficiencies of the desalination, compression, electrolysis and additional processes. Using this energy consumption  value, the hydrogen volume generated by the WEC 1 point absorber at each one of the four selected locations and for the last ten years is presented in Fig. 9.
The yearly hydrogen production of the reference WEC 1 shows a rather constant tendency and limited variability between productions values at one same location. A maximum production variability of 8.91% is found at location P-03 between years 2015 and 2017. For the rest of the locations, maximum variabilities of 7.36%, 6.83% and 7.29% have been found in P-00, P-01 and P-02, respectively. As a result, the uniformity of the wave resource trend emphasizes the important contribution a WEC could make as an energy/hydrogen supplier in a far offshore hybrid wind and wave energy farm.
Regarding mean hydrogen production of the WEC 1, the highest generation volume is found at location P-02. Correlating the statistical wave resource data in Table 3 and the power matrix in Fig. 8, it can be observed that although the mean significant height H s of the wave resource in the location P-02 presents the lowest value, the maximum hydrogen production is achieved. The reason for that is that the mean peak period T p of the wave resource in the location P-02 is the closest to the value of maximum power absorption of the WEC, as it can be observed in Fig. 8. In fact, it presents the lowest mean value and the lowest standard deviation from all studied locations. Therefore, the hydrogen production is maximal in P-02. The location with the minimum hydrogen production is P-00, where the mean significant height H s of the waves is high, but the mean peak period T p is the highest.

Combined and Normalized Wind and Wave Hydrogen Generation
In order to get an accurate idea of the efficiency of a WEC in comparison to a wind turbine located in the same far offshore energy farm, the wave energy based hydrogen production is normalized in this Subsection with respect to the wind energy based hydrogen production. Three different analysis scenarios have been considered, each one corresponding to a WEC model listed in Table 2 (WEC 1, WEC 2 and WEC 3). Following the methodology in SubSection 3.2, the power matrices corresponding to the devices WEC 2 and WEC 3 have been calculated and are presented in Fig. 10a and Fig. 10b, respectively.
By comparing the power matrices corresponding to WEC 2 and WEC 3, it is evident that, as expected, the power absorption/production of the WEC increases when incrementing its dimensions. As a result, the power matrix of the point absorber WEC 2 shows remarkably larger values than the one of the point absorber WEC 3. Additionally, it is observed that by incrementing the dimensions of the WEC, the peak period T p corresponding to the maximum power absorption of the WEC is shifted to higher values, which results in a higher power absorption of the WEC 2 at greater values of the peak period T p . Due to the characteristic peak period T p of the wave resource at each location, the dimension of the WEC will determine its power absorption, and hence suitability, for that specific location.
For the calculation of the normalized hydrogen production, a 5/1 relation has been considered in the number of wave converters with respect to the number of wind turbines in all scenarios, i.e, five WECs have been considered for each wind turbine in the farm. Following the methodology described in SubSection 3.2, the hydrogen volume production has been calculated for each one of the three scenarios and the four far offshore locations selected with the MCA analysis. In addition, the produced total hydrogen volume has been divided with respect to its origin: wind energy based (produced with the power absorbed by the NREL 5 MW wind turbine) or wave energy based (produced with the power absorbed by the corresponding WEC). Subsequently, the normalized mass (with respect to the reference WEC 1) and the normalized hydrogen production of each one of the WEC scenarios at the four far offshore locations have been calculated and are represented in Fig. 11.
A summary of the hydrogen produced by the NREL 5 MW wind turbine and the normalized WEC based hydrogen production is presented in Table 4.
The results presented in Table 4 show that important additional hydrogen volumes can be achieved in a far offshore wind farm by installation of WECs. As expected after the results in SubSection 3.2, the highest hydrogen production volumes have been found at location P-02, where five WECs of the type WEC 1 could supply a 4.03% of the hydrogen produced by the NREL 5 MW wind turbine during the last 10 years. The lowest additional hydrogen volume production would result at location P-00, where a 3.32% of the hydrogen produced by the HAWT would be supplied by five WECs of the type WEC 1.
In case of the WEC 2 point absorber, a maximum additional hydrogen production of 6.28% is achieved at location P-02. However, this is  accomplished by increasing the mass of the WEC in a 1.16 %, see Fig. 11, which will cause an increment of the cost of the WEC, the PTO system and the mooring to the offshore platform. Regarding WEC 3, a maximum additional hydrogen production of 1.36% is achieved at location P-02. In this case, the mass of the WEC is reduced in a 0.72%. It is to be observed that the maximum difference between the various WEC configurations can also be found at location P-02. The increase in the mass of the WEC 2 causes a shift of its period of maximum absorption towards higher values, i.e. closer to the mean wave peak period at location P-02. As a result, its power production is maximized with respect to the other WEC configurations. This fact clearly stresses the importance of a WEC design procedure focused on the meteorological characteristics at a specific location.
Finally, the capacity of the WECs to provide auxiliary energy/ hydrogen has also been analysed in this Subsection. To that end, the hydrogen production of the different WEC scenarios during the disconnection time of the NREL 5 MW wind turbine in the corresponding far offshore location is presented in Table 5. In order to compute the stopping time of the HAWT, the wind speed data obtained in SubSection 3.1 (and statistically summarized in Table 3) has been analyzed to find the wind speed values below the cut-in (3 m/s) or above the cut-off (25 m/s) wind speed values that define the operation range of this wind turbine. Due to the lack of operational data of a real offshore wind turbine, the disconnection times as a result of maintenance and mechanical issues are not taken into account. Regarding the maximum admissible wave elevation for the disconnection of the NREL 5 MW floating wind turbine, this value is bigger than the cut-off wave elevation for the proposed WECs. Therehore, no auxiliary hydrogen could be produced by the WECs if the wind turbine would be stopped due to too high wave elevation.
According to the results of the MCA presented in SubSection 3.1, the wind and wave energy are highly coupled in the locations P-00 and P-01, but this coupling is weaker at locations P-02 and P-03. As it can be observed in Table 4, the normalized mean hydrogen production of the WECs do not show a considerable dependency on the coupling between the wind and wave energy in the particular far offshore location, see very similar production values at locations P-00, P-01 and P-02. Instead, the peak period T p of the wave resource has been found to be the most influential parameter for a maximized hydrogen production.
Nevertheless, according to the results provided in Table 5, the coupling between the wind and wave energy is an important factor to be considered when analysing the auxiliary hydrogen production of a WEC. In those locations where the coupling between the WPD and the WEF is weaker, the number of hours in which the wind turbine is stopped, but the wave resource characteristics are adequate to produce, are greater. As a result, the auxiliary hydrogen production of the WECs become more important and the significance of their role maximized. As it can observed in Table 4, out of the total disconnection time of the HAWT, 101.12 h of equivalent operation of the HAWT (producing at 5 MW rated power) could be recovered with five WECs of the type WEC 2 at location P-02. The maximum equivalent HAWT operation recovery hours found at locations P-00, P-01 and P-03 are 68.99 h, 72.03 h and 78.2 h, respectively.

Discussion
The fact that the expansion coefficients associated to the leading WPD/WEF coupled mode identified in the MCA analysis show a strong seasonal cycle, implies that their temporal behaviour allows a relatively easy predictability. Even for the second or third modes, see Fig. 12, in which the AutoCorrelation Function (ACF) decays much faster, there seems to exist some predictability based just on persistence up to 5 days (ACF>0.25) for the three modes which, as has already been indicated before, explain up to a 91% of the total covariance of daily WPD and WEF. The application of this result to the operational management of the kind of combined energy generation system analyzed in this paper is out of the scope of this paper, but future studies must build on the shape of these ACFs.
Some of the large-scale atmospheric teleconnection patterns, such as the NAO or the East Atlantic pattern, are well known for affecting atmospheric processes over the Northern Hemisphere and, in particular, its Atlantic basin [66]. Thus, it was expected from the beginning that their influence could be detected in the results presented in this contribution. However, it is found that they are affecting large fractions of the covariance between WPD and WEF over the whole basin.
The results in this paper, which link the spatial patterns of the MCA modes (homogeneous correlation maps shown in Fig. 6), through their corresponding expansion coefficients are in line with previous findings calculated by local regression of teleconnection indices onto wave energy flux at every grid point [42]. Some studies show that the available wave energy has already changed [67] over the Atlantic in the past and might also change in the future as a result of climate change, depending on the Representative Concentration Pathway followed by society [68]. Similarly, there are evidences of substantial variations in wind speed [69] and renewable energy produced in Europe [70]. Teleconnection patterns also affect temperature [71] which is a factor in the determination of WPD [72] and it is also changing due to global warming [73]. Thus, future studies should also be addressed to the coupling between WPD and WEF under future climate scenarios.
All in all, the coupling between the WPD and the WEF has been proven to be an important factor to be considered during the selection of the optimal location for far offshore hybrid wind and wave energy farms, where the variability of the wind speed, and the consequent disconnection times of the HAWTs, could lead to reduced energy/hydrogen generation for time intervals of uncertain duration, see the results in Table 5. During these time intervals, the auxiliary hydrogen produced by the WECs could help improve the efficiency of the farm and cover its energetic needs. Hence, the installation of WECs would result more Table 4 Hydrogen produced by the HAWT and normalized hydrogen production of the different WEC scenarios.  essential in far offshore wind farms with high wind variability and low covariance between wind and waves. The selection of the WEC model for a far offshore energy farm should be particularized with respect to the characteristics of the wave resource in the corresponding location. While WECs of larger dimensions have their maximum power absorption at higher peak period T p values, WECs of lower dimensions absorb more power from wave resources with lower peak periods T p values. As a result, the characteristics of the wave resource, and the influence of the large-scale teleconnection patters on them, should be taken into consideration in the desing process of the WEC to be installed in a far offshore energy farm. For instance, in a location with low significant height H s and high peak period T p values, a larger WEC should be installed. On the contrary, in a location with high significant height H s and low peak period T p values, a smaller WEC would be the best device to be installed.
In economical terms, the identification of the optimal locations for the installation of a hybrid farm enables to maximize the amount of generated energy/hydrogen. Thus, providing a qualitative economic estimation of the generation costs. Additionally, the dimensions of the WEC are an important factor to be taken into account as well, since they will directly affect the economical cost of the device and, indirectly, the design of the mooring system and the far offshore platform. Therefore, although in the present paper the design and dimensioning of the far offshore platform and the mooring system have been taken for granted, a more complex methodology for the selection of the optimal location of a far offshore farm could arise in future works. A methodology in which the characteristics of the wave resource in a given location and its influence on the economical cost of the platform and the mooring system are related. Thus, a comprehensive economic evaluation based on specific technological solutions could be carried out.
In this paper, hydrogen production has been considered as the procedure for energy storage in the far offshore hybrid farm. Nowadays, many governments and international companies are promoting the production of hydrogen with energy based on renewable energies and its use as a renewable fuel instead of classic fossil fuels. Among the technological developments that may contribute in the future to the development of far-offshore hydrogen production is the use of seawater for the electrolysis process instead of the current use of distilled water [74]. The European Union considers hydrogen a key element in the ongoing efforts towards a cleaner economy, among other things, because it is an area where Europe is still leading [75]. In recent studies, on account of the low volumetric energy density of the hydrogen (in standard conditions of temperature and pressure), the use of methanol is proposed for energy storage in energy ships [18]. According to these studies, many advantages are related to the use of methanol instead of hydrogen, such as, a good efficiency in the production, a high market value and its liquid state in standard temperature and pressure conditions. Consequently, the development of the methanol as the energy vector in far offshore applications must be in sight for future works related to this topic. The results of this paper are aligned with some of the challenges identified by all the Member States of the United Nations in a Resolution by the General Assembly of the United Nations adopted on September 2015, commonly called the 2030 Agenda for sustainable development [76]. In this resolution, there were 17 main goals which were developed up to 169 different targets which addressed the need to achieve a sustainable development whilst, at the same time, erradicating poverty. In particular, Sustainable Development Goal (SDG onwards) number seven (SDG7) is devoted to ensuring access to affordable, reliable, sustainable and modern energy for all. This goal covers targets such as the need to increase the share of renewable energy, facilitate the access to renewable energy to the population or upgrade technology so that renewable energy can also be used in land-locked developing countries. The findings presented here are expected to contribute in the future to these targets, because the area of ocean in which offshore wind energy can be produced is greatly expanded, so that geographical, spatial planning or other ocean uses are not so heavily affected by offshore wind farms [77]. Besides the use of renewable energy sources, the use of hydrogen as an energy vector is also considered by some authors as potentially leading to a better integration of renewable energies in the electricity sector [78], which agrees with the findings presented in this paper. However, as is also widely recognized in the literature, energy is a factor which affects different sustainability goals at the same time [79]. In the case of this paper, the increase in the efficiency of production of renewable energy affects the SDG13 (Take urgent action to combat climate change and its impacts) by improving the ability to reduce emissions from fossil fuels. The spatial planning implicit in the results presented allows a better planning in the use of renewable energy resources, which is the Target 13.2 in this SDG13. Finally, SDG14 (Conserve and sustainably use the oceans, seas and marine resources for sustainable development) includes some targets such as 14.7 (increase the economic benefits to small island developing states and least developed countries from the sustainable use of marine resources). An increased use of renewable energy sources has already been identified by previous studies [79] as a positive feedback in terms of SDG14 because the reduction of the carbon dioxide emissions will slow rates of ocean acidification. Beyond this, the authors foresee that small islands in the middle of the oceans can benefit from the ability to serve as points used for managing the kind of floating infrastructures presented in this paper. Previous studies also show that a sustainable use of ocean areas, the so called Blue Economy [80], improves indicators in SDG15 (sustainable use of terrestrial ecosystems), for instance, leading to a smaller footprint of terrestrial wind energy farms with less affections to mountain ecosystems or forests, by increasing the number of offshore installations. In any case, there might appear both synergies and incompatibilities when a full analysis of particular technologies are analyzed from the point of view of multiple Sustainable Development Goals. Thus, whilst the use of hydrogen as an energy vector will reduce emissions, it might lead to negative impacts depending on the technologies used in its production [78]. Systematic analyses of the interrelationships which appear between energy (SDG7) and alternative sustainable development goals show that, in general, positive interactions are more important than negative ones, even though these negative interactions exist [79]. To mention a few, an increased use of renewable energy sources imply also positive synergies with SDG9 (industry, innovation and infrastructure) by the financial and technical support given to the development of renewable energy sources and energy efficient infrastructure. On the other hand, in some cases, negative impacts have appeared in the past, such as the relationship between large-scale bioenergy production and agriculture or food distribution, SDG2 (Food security and agricultural productivity) as shown in the bibliography [79]. Thus, a detailed analysis of the different relations between a proposal such as the one presented in this paper and the whole set of sustainable development goals is challenging because of all the different dimensions which must be analyzed such as ocean use planning, operational risk analysis, economical analysis of the solution to name a few, including also the positive and negative impacts due to alternative energy sources [79]. This is outside the objectives of this paper and there are different frameworks which are currently developing this kind of studies [81] which could be performed in the future, when a particular installation is being designed.

Conclusions
A methodology for the selection of the optimal location of a far offshore hybrid wind and wave energy farms is proposed in this paper, focused on maximizing the auxiliary power supply role of the WECs. Furthermore, an assessment of the hydrogen production of different sized WECs in those selected locations is presented. As a result of the conducted analysis, the importance of a joint analysis of both, wind and waves regimes throughout a huge area like the Atlantic for the selection of the best location is stressed. Likewise, the necessity of adapting the design process of the WECs to the characteristics of the wave resource data at the location selected is demonstrated.
The MCA is shown to provide an adequate initial criteria for the selection of the location of far offshore hybrid energy farms. The first singular vector presents areas characterized by positive covariances over the whole Atlantic basin. This explains a 79% of the total covariance, so that it is to be expected that the production of hydrogen is relatively constant for the different points, as shown in Fig. 9. However, the corresponding daily temporal expansion coefficient presents a very strong seasonal behaviour (Fig. 12), which means that this is strongly predictable over the full Atlantic basin. The second mode (8% of basin-wide total covariance of daily WPD and WEF) shows a meridional dipolar structure which is related to the AO or NAO oscillations. It indicates that during positive WPD and WEF anomalies over the northern Atlantic, negative anomalies are to be expected over the southern basin. The obtained results show that locations with low covariance between WPD and WEF, particularly P-02, are the most adequate for the installation of a wind and wave hybrid energy farm. As a result of the higher variability of the wind speed in those locations, the disconnection times of the HAWTs are more prolonged and the auxiliary hydrogen generation coming from the WECs is greater and more essential. In addition, in all selected and analysed far offshore locations, the wave energy has been found to have a rather constant trend and a limited variability during the last 10 years, which proves its utility to be used as a reliable energy resource.
Finally, the characteristics of the wave resource in a given location has been found to be an important factor to be considered in the design and the selection of the geometry of the WEC to be installed. The obtained results show a direct relation between the size of the WEC and the peak periods T p associated to the maximum power absorption of the device. Consequently, the design of the WEC should be directly related to the wave resource in that location. The installation of WECs with reduced dimensions could not only result in a lower economical cost of the device, but would also simplify its mooring and the design of the far offshore floating platform, the restraints of which are expected to be added to a more complex methodology to be developed in future works. Although the focus of this study has been the Atlantic, the MCA methodology could be applied to any other oceanic location of the world to identify an optimal location, while the conclusions regarding the appropriate dimensioning of the WEC would also remain valid for other basins with similar wave regimes.
An initial assessment of the predictability of the coupled wind and wave energy resource has been presented based on the cross-correlation functions of the expansion MCA coefficients. This opens a path for future studies in the field.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. MCIN/AEI/ 10.13039/501100011033. Authors also acknowledge financial support by the University of the Basque Country under the contract (UPV/EHU, GIU20/008).