. Wave loads on lattices of marine structures, such as wind parks with shared mooring systems, are considered. Modal excitation induced by second-order low-frequency wave loads is derived, leading to modal quadratic transfer functions. The consequence for the modal loads, of applying Newman’s approximation is investigated. 1. Introduction Forthcoming generations of offshore power grids might be largely floating and composed of hundreds of subsystems harvesting, converting and storing energy. As an example, floating wind parks, currently in the multi-MW power range, might eventually reach a multi-GW capacity, and be combined with e.g. floating photovoltaic cells, and hydrogen production units. In this context, offshore energy installations might change character, from single moored floating structures to large lattices of floating structures. Lattices are here defined as two-dimensional arrangements of identical structures, interconnected by soft connections such as mooring lines. As the size of the lattice increases, the number of eigenmodes and the range of associated eigenfrequencies will increase, with many possible resonances in the low-frequency (LF) range, i.e. below the wave-frequency range. When operating near these frequencies, it is critical to quantify the excitation forces and the amount of damping in the system to detect possible resonance, and accurately estimate responses of interest for the lattice. LF excitation loads are mainly due to wind and waves. The present paper focuses on the latter. LF wave loads result from nonlinear interactions between frequency components in the wave spectrum, and between the waves and a moving structure. Difference- frequency quadratic transfer functions (QTFs) have traditionally been used to describe these effects on individual floaters. Several authors have investigated the behaviour of lattices of floating wind turbines. Static analyses [1, 2, 3], eigenvalues analyses [4] and dynamic time-domain analyses [5, 6, 7, 8, 9] at various fidelity levels have been reported. However, no frequency-domain study targeting specifically nonlinear wave loads has been carried out yet. Frequency domain approaches, despite their limitations when considering nonlinear systems, often yield a good understanding of the physical phenomena at play. The main contribution of the paper is a derivation of the modal excitation from second-order (LF) wave loads, and of the lattice response under simplifying assumptions. The consequence of 0 0.2 0.4 0.6 0.8 1 1.2 1.4 [rad/s]0246810121416S [m2.s]Figure 1. Left: grid type of lattice that will be considered in the present study ( N=2). Anchored lines (represented in green) have a length and stiffness denoted laandka, respectively. Shared lines (represented in black) have a length and stiffness denoted lsandksrespectively. Right: wave spectrum used in the example: JONSWAP spectrum with Hs= 7m, Tp= 10s and γ=3.3 Table 1. Lattice parameters Notation and value N= 2 number of FWTs on one row of the lattice m= 14175 + 8300 t mass (mass + asymptotic LF added mass) of each FWT ks= 60 kN/m linear stiffness of the shared lines ka= 120 kN/m linear stiffness of the anchored lines ls= 1260 m projected length on waterplane of the shared lines la= 726 m projected length on waterplane of the anchored lines T0= 950 kN pre-tension of the lines γ1= 5×10−3s−1mass coefficient of the Rayleigh damping γ2= 3×10−1s stiffness coefficient of the Rayleigh damping using Newman’s approximation instead of a full QTF is also investigated. Theory is presented in Sections 3 to 7, illustrated by an example introduced in Section 2. 2. Example case The example case is similar to the one studied by Connolly in [4]. A grid type of lattice is defined, as illustrated in Figure 1. The mooring lines are approximated by a linear stiffness with values taken from [4] and recapped in Table 2. The pre-tension in the mooring lines is 950 kN. The mass and LF asymptotic added mass properties for the floater correspond to the INO WINDMOOR 12MW design [10]. For this floater, a numerical QTF estimate is available, see Figure 5. The behaviour of the lattice will be investigated in a sea-state described by the JONSWAP spectrum S(ω), and characterized by a significant wave height Hs= 7 m and a peak period of Tp= 10 s, see Figure 1. The wave propagation direction is β= 30owith respect to the horizontal axis. Static wind loads are represented by a constant force of 1.5 MN, applied on all four floaters with an angle of 45owith respect to the horizontal axis. 3. Eigenmodes We consider the general case of a lattice with nddegrees of freedom. In the example case, only surge and sway motions are included for simplicity, so that nd= 2N2. Let us denote r∈Rnd the lattice response to an external nodal excitation F∈Rnd. Dynamic equilibrium yields M¨r+C˙r+Kr=F (1) where M,CandKarend×ndmass, damping and stiffness matrices. The response is defined relative to the equilibrium configuration. Note that system matrices for a statically loaded lattice can be very different from their counterpart obtained in an unloaded scenario due to large deformations under external static loading. In that case, the equilibrium configuration and the values of these matrices must be calculated with a nonlinear static analysis, which has been done here. Free vibrations of the lattice at a frequency ωare described by (−Mω2+K)r= 0 (2) Assume that Mis a real symmetric positive-definite matrix. Mcan be Cholesky-decomposed as a product M=LL∗where Lis a lower triangular matrix and∗denotes the Hermitian transpose. Defining q=L∗r, free vibration can then be re-written as ˜Kq=ω2q (3) where ˜K=L−1K(L∗)−1(4) is a symmetric matrix. Then, there exist a set of eigenvectors ( ϕi)i∈{1...nd}of˜Kthat form an orthonormal1basis of the Euclidean vector space Rnd. Each ϕiis associated to an eigenvalue Ω2 i. We define Φ = [ ϕ1, ..., ϕ nd] (5) and Λ =diag(Ω2 1, ...,Ω2 nd) (6) so that ˜KΦ = ΛΦ (7) Orthonormality yields Φ∗Φ =I. Note that all Ω iare not necessarily distinct. In our example, when no mean load is applied to the lattice, several modeshapes are associated with the same eigenfrequency ( degenerate eigenmodes), as shown in Figure 2. It is still possible in that case to define an orthonormal basis Φ using Gram-Schmidt orthogonalization. In the case of an arbitrarily loaded lattice, eigenmodes are in general non-degenerate, i.e. that all eigenfrequencies are distinct, as shown in Figure 3. 1In case of non-degenerate eigenmodes (all eigenvalues are distinct), orthogonality of eigenvectors is proved as follows. Let u∈ {1, ..., n}.ω2 uϕu=˜Kϕu,ω2 uϕ∗ u= (˜Kϕu)∗,˜Kbeing real symmetric we have ω2 uϕ∗ u=ϕ∗ u˜K. Letv∈ {1, ..., n}. Post multiplying by ϕv, we have ω2 uϕ∗ uϕv=ϕ∗ u˜Kϕv, i.e. ω2 uϕ∗ uϕv=ϕ∗ uω2 vϕv, yielding (ω2 u−ω2 v)ϕ∗ uϕv= 0. So if u̸=v, then ϕuandϕvare orthogonal. They can be normalized to yield an orthonormal basis. Figure 2. Modeshapes and associated eigenperiods for an unloaded 2x2 lattice. Red arrows indicate the nodal displacement for each mode. Segments of varying red color intensity represent the tension in each of the anchored and shared lines for the given mode (based on relative nodal displacements). Figure 3. Same representation of eigenmodes as in Figure 2, but resulting from a nonlinear static analysis, and where all nodes have been loaded with an 1.5MN force applied with a 45oangle with respect to the x-axis. Figure 4. Modal loads (top) and responses (bottom) in a bi-chromatic wave field with frequency ω1(x-axis) and ω2(y-axis). Each column correspond to the eigenmodes from Figure 3 above. The colorbar is similar to Figure 5: the blue color corresponds to relatively small levels, and yellow color to relatively large levels. 4. Modal load and response The modal load vector µ∈Rndand the modal response vector ξ∈Rndare defined by µ= Φ∗L−1F (8) ξ= Φ∗L∗r (9) The dynamic equilibrium (1) can be rewritten as M(L−1)∗Φ¨ξ+C(L−1)∗Φ˙ξ+K(L−1)∗Φξ=LΦµ. Premultiplying by L−1, leads to Φ ¨ξ+L−1C(L−1)∗Φ˙ξ+˜KΦξ= Φµ. Premultiplying by Φ∗, we get¨ξ+ Φ∗L−1C(L−1)∗Φ˙ξ+ Φ∗˜KΦξ=µ, and obtain the modal system of differential equations: ¨ξ+ Φ∗L−1C(L−1)∗Φ˙ξ+ Λξ=µ (10) where couplings between the modes exist due to the modal damping term. To proceed with the analysis, it is practical to assume Rayleigh damping, even if there is no indication that this assumption is fulfilled in reality. In that case, C=γ1M+γ2K (11) yielding L−1C(L−1)∗=γ1L−1M(L−1)∗+γ2L−1K(L−1)∗=γ1I+γ2˜K. Then ¨ξ+ Φ∗(γ1I+ γ2˜K)Φ˙ξ+ Λξ=µ, which leads to an uncoupled system of differential equations: ∀i∈ {1, ..., n d},¨ξi+ (γ1+γ2Ω2 i)˙ξi+ Ω2 iξi=µi (12) In that case, the complex modal response ¯ξito a harmonic modal load ¯ µiat frequency Ω can immediately be found by ∀i∈ {1, ..., n d},¯ξi=¯µi Ω2 i−Ω2+iΩ(γ1+γ2Ω2 i)(13) Nodal responses and loads can be recovered from their modal counterparts by r= (L−1)∗Φξ (14) F=LΦµ (15) 5. Second-order modal wave loads and response Consider now a lattice in an incident multi-chromatic wave field ℜhP jζjeiωjt−kju·xi , where ζj∈Cdescribes the amplitude of the wave component and its phase at origin, kjthe associated wave number, u= (cos β,sinβ)⊤is he wave propagation direction, and xthe position vector. The second-order wave load at a frequency ∆ ω≥0 on a floater located at x, and on the degree of freedom d∈ {1,2,6}(for surge, sway and yaw) is given by fd(∆ω, x) =X |ωi−ωj|=∆ω(ζie−ikiu·x)∗(ζje−ikju·x)Q− d(ωi, ωj, β) (16) where Q− dis the difference-frequency quadratic transfer function (QTF) for the considered degree of freedom d. Figure 5 shows the QTF for the floater considered in our example. For simplicity, the floater is here assumed to be axisymmetric, so that the QTF represented in Figure 5 is valid for all β. Note that this is not true for the actual INO WINDMOOR 12MW floater. Figure 5. Amplitude (left) and phase (right) of the surge difference-frequency quadratic transfer function of the INO WINDMOOR 12MW floater, obtained from potential flow theory. The red box illustrates difference-frequencies corresponding to the range of eigenfrequencies of the lattice. The black box illustrates mean frequencies at which there is significant wave energy in the chosen sea-state (10 and 90th percentile of wave energy, see Figure 1). Assuming deep water, and hence a dispersion relationship ω2=gk, the load on the floater reads fd(∆ω, x) =X |ωi−ωj|=∆ωζ∗ iζjeiω2 i−ω2 j gu·xQ− d(ωi, ωj, β) (17) The nodal load2vector F(∆ω) for the lattice can then be built by gathering fd(∆ω, x) for each degree of freedom dand each position x. For example, for a lattice of three structures located at (x1, x2, x3) and with two degrees of freedom each, the nodal load would be obtained by F(∆ω) = [f1(∆ω, x 1), f2(∆ω, x 1), f1(∆ω, x 2), f2(∆ω, x 2), f1(∆ω, x 3), f2(∆ω, x 3)]⊤(18) Once nodal loads are obtained, the modal loads µican be calculated from (8), respectively. They are shown in Figure 4 (upper row) for our example case. Note that by setting ζi=ζj= 1, modal difference-frequencies QTFs are obtained for the lattice. From this derivation, and by considering Figure 4, it becomes clear that the amplitude of the modal QTF at a given bi-frequency ( ω1, ω2) is influenced by three parameters. First of all, (1) the quadratic transfer function at ( ω1, ω2) represented in Figure 5, appears as a background pattern, providing the strength and phasing of the nodal load on each of the (identical) floaters. On top of that background, (2) fringes are observed, that are variations of the modal load over shorter bi-frequency ranges. These fringes are induced by the ( ω2 1−ω2 2)u·xfactor, and thus depend on the wave propagation direction, and the lattice layout. Finally, (3) the projection F(∆ω)·ϕi, which involves the eigenvector/modeshape ϕiacts as a uniform multiplying factor over the bi-frequency plane. It is for example noticeable that modes 2 and 7 are clearly less excited than the other ones, due to the fact that the corresponding modeshapes are nearly orthogonal to the wave propagation direction. 2Note that the wave load is real due to the Hermitian symmetry of the QTF. Indeed consider a bi-chromatic wave, the sum above reduces to ζ∗ 1ζ2eiω2 1−ω2 2 gu·xQ−(ω1, ω2, β) +ζ∗ 2ζ1eiω2 2−ω2 1 gu·xQ−(ω2, ω1, β) which is a sum of complex conjugates, and is then equal to 2 ℜ[ζ∗ 1ζ2eiω2 1−ω2 2 gu·xQ−(ω1, ω2, β)]. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 [rad/s]100101102103104S() [N2.m-1.s]S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 [rad/s]10410510610710810910101011S() [m3.s]S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8Figure 6. Modal load (left) and response (right) spectra in Hs= 7m, Tp= 10s (see Fig. 1) From the modal load µ, modal responses ξare simply obtained from (13). They are shown in the lower row in Figure 4. It is seen that for the present choice of damping model and damping ratio, the modal response is strongly enhanced at difference frequencies ( ω1, ω2) matching the eigenfrequencies Ω iof the mode (resonance). Modal response is relatively smaller for large difference-frequencies in spite of a larger floater QTF, as the lattice does not respond significantly to such high frequencies. Note that the properties of the response ξare very much affected by the assumed Rayleigh damping model and ( γ1,γ2) coefficients. These assumptions simplify the analysis and enables understanding the mechanisms at play, but there is no indication they are valid. 6. Modal load and response spectra in a sea-state We now denote Ω, the frequency of the nonlinear wave loading, consistently with Section 3. For a given mode i∈ {1, ..., n d}and a given frequency Ω, the spectral density of the modal nonlinear wave load Sµs(Ω) is obtained by integration over all combinations of sea-state components ω andω+ Ω that will generate a load at a (difference-) frequency Ω [11, eq (7.99)]: Sµi(Ω) = 8Z∞ 0S(ω)S(ω+ Ω)|µi(ω, ω+ Ω, β)|2dω (19) Similarly the spectral density of the modal response for mode i, denoted Sξi(Ω) is obtained: Sξi(Ω) = 8Z∞ 0S(ω)S(ω+ Ω)|ξi(ω, ω+ Ω, β)|2dω (20) Spectra of both modal load and response are shown on the left- and right-hand side of Figure 6, respectively, for the sea-state whose spectrum used in the example, and plotted in Figure 1. Considering Sµ, and leaving aside modes 2 and 7, which are not significantly excited with the given wave propagation direction, it is clear that all other modes are loaded over the whole range of frequencies of relevance for the lattice. Furthermore, large variations of the loading are observed among the modes and across the frequency range (note the y-log-scale of the plots). Modal response is as expected largest near the eigenfrequencies (0.7 and 0.10 rad/s), and then quickly tends to zero. Note however that linear wave loads, not included in the present study, are likely to generate a significant modal load and response at frequencies close to 0.63 rad/s. 7. Consequence of applying Newman approximation In Section 5, we showed how the modal QTF was dependent on the individual floater QTF. Despite the fact that second-order potential flow codes are now widely available (such as WAMIT S and Hydrostar), it is still common to consider applying the so-called Newman’s approximation, instead of using full QTFs. Newman’s approximation consists in approximating the QTF at a bi-frequency ( ω1, ω2) by the arithmetic mean of its diagonal terms at ( ω1, ω1) and ( ω2, ω2): Q− d(ω1, ω2) =Q− d(ω1, ω1) +Q− d(ω2, ω2) 2(21) The historical reason for applying Newman’s approximation is that linear potential solvers, which are less computationally expensive than their second-order counterparts, give access to the diagonal terms Q− d(ω, ω) of the QTF, under the denomination wave drift coefficients. However, a QTF being Hermitian symmetric, its diagonal terms have zero phase . So an important bi-effect of Newman’s approximation is that the phase information in the QTF, shown on the right hand side of Figure 5, is lost. This has several consequences for the modal loads on lattices. (1) Phase information between degrees of freedom of a given floater is lost. If a mode involves e.g. sway and yaw of a given floater, the modal strength will be affected by an error in the relative phases in the nodal load. Furthermore, (2) phase information between modal loads will be lost, which can have consequences on quantities of interests that are extracted by combining modal responses. This can be for instance mooring line tensions, estimated from the motions of the floaters. Finally, by construction, (3) Newman’s approximation provides more accurate estimates of the module/real-part of the QTF near the main diagonal (small difference- frequencies), rather away from it (large difference-frequencies, often of interest for lattices). It implies that Newman’s approximation might be less conservative when a QTF exhibits maxima at large difference-frequencies, compared to when such peaks occur near the main diagonal. The error induced by Newman’s approximation3can be estimated quantitatively for the example at hand. Denote ˆSµ(Ω) the modal load power spectra obtained as outlined in Section 6, but now applying Newman approximation (21) instead of the full QTF Q− d. We define the error ˜Sin the modal load as ˜S=ˆSµ(Ω)−Sµ(Ω) Sµ(Ω)(22) ˜Sis represented in Figure 7. It is seen to vary from mode to mode and across the frequency range, but with a clear trend being that the error increases with the loading frequency. For the range of frequencies relevant for the lattices considered here (say Ω <0.2 rad/s), ˜Scan reach -40%, meaning that Newman’s approximation can lead to modal loads underestimated by 20%. From linearity, the same error would be made on the response spectrum. Since numerical tools that are practical to use in an engineering context, give access to full QTFs, it is therefore recommended to use them when studying lattices. In many cases, it is recommended to use empirical corrections to the QTF, derived with a cross-bi-spectral analysis [12, Section 4] or least-square based techniques [13, 14]. This has been done for the INO Windmoor 12MW considered here [15], and it was found that for moderate to high sea-states, potential flow predictions generally underestimate the QTF. Such discrepancies are likely to be related to viscous loads, not modelled in potential theory, or to higher than second order potential flow effects. 3Since axisymmetry of the floater has been assumed in the present paper, and since quantities of interests due to modal responses have not been investigated, estimates the error induced by points (1) and (2) above are not accessible within the scope of this paper, and we are only considering the effect of point (3). 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 [rad/s]-100-80-60-40-20020Figure 7. Consequence of applying Newman’s approximation on modal load and response spectra. Thin curves refer to each individual mode, with the same color code as in Figure 6. The thick black curve is the average error accross all modes. 8. Conclusion The present paper introduced modal quadratic transfer functions (QTF) for lattices of marine structures, such as wind parks with shared moorings. The procedure for obtaining modal QTFs consists in performing a nonlinear static (to account for large deformations), a modal analysis of the lattice, followed by the procedure outlined in Section 5. As a results, modal QTFs presented in Figure 4 are obtained. Their features have been explained in details in Section 5. For the example used in the paper, that is a grid-type of lattice with four floaters, it has been shown that applying Newman’s approximation could lead to an significant underestimation the modal loads. The modal response of the lattice has also been derived, assuming a Rayleigh damping model which led to uncoupled modes, and given the choice of Rayleigh coefficients, to resonant modes. This enabled understanding the mechanisms at play, but there is no indication that this damping model is applicable for shared mooring systems. If shared lines induced a large amount of damping, resonance might be avoided, even under wave loading occurring at the eigenfrequencies of the lattice. Further work is ongoing to model and quantify damping. This will enable extending the frequency domain analysis to study lattice responses, and derived quantities of interest such as tensions in mooring lines, and absolute/relative excursion of floaters. Understanding the role of specific modes on these quantities of interest is of interest for design. Finally, it should be stressed that the present paper only focused on second-order wave loads, of interest because they occur at typical eigenfrequencies of lattices. Wind loads and linear wave loads, usually significantly larger in magnitude, should of course be included in the dynamic analysis of a lattice. 9. Acknowledgments This research has been funded by the Research Council of Norway through project 326654 CYBERLAB KPN, a collaboration between SINTEF Ocean, NTNU, University of Aarhus, Equinor, Aker Offshore Wind, APL Norway, Sevan SSP and Delmar Systems. The author wishes to acknowledge the contribution of students Claire Monin and Cl´ ement Morhain who have performed preliminary studies that led to the present work, as part of their stay at NTNU and SINTEF, respectively.