Increased tower eigenfrequencies on floating foundations and their implications for large two- and three-bladed turbines

If the tower of a bottom-fixed turbine is put on a floating foundation, such as a spar, semi-submersible, or barge, its eigenfrequency increases. In the investigated case, the tower eigenfrequency rose to about twice its previous value. For bottom-fixed applications, the comparatively high blade-passing frequency of a three-blade turbine (3P) leaves a great bandwidth to design a light and soft tower with a low-enough eigenfrequency. On one of those floaters, however, the eigenfrequency of the same tower might neither be high nor low enough to avoid eigenfrequency excitation by 3P around rated speed. Due to the lower blade passing frequency of a two-bladed wind turbine (2P), the same tower eigenfrequency is high enough by default, enabling considerable material savings or at least eliminating the severe eigenfrequency excitation issue of its bottom-fixed version. Additionally, cost benefits in the whole life-cycle of two-bladed turbines remain. 20MW two- and three-bladed turbines were analyzed numerically on an upscaled version of the UMaine VolturnUS-S semi-submersible, confirming the reasoning.


Introduction
Large three-bladed floating offshore wind turbines (FOWTs) with tubular towers often suffer from a tower eigenfrequency that would naturally be very close to their blade-passing (3P) frequency above and just below rated conditions [1,2,3,4].From that starting point, neither a bottom-fixed typical soft-stiff nor a stiff-stiff tower configuration, with the tower eigenfrequency respectively beneath or above the 3P-frequency, is easy to achieve and requires massive additional amounts of steel either way.In combination with increased tower loads compared to bottom-fixed turbines, the thick-walled stiff-stiff configuration is currently a prominent but not necessarily a pleasant choice for FOWT designers.
In contrast, two-bladed turbines of a similar size offer a much lower starting point of the stiff-stiff configuration region because of their lower blade-passing (2P) frequency.This low 2P-frequency has been an enormous drawback for bottom-fixed installations since it opens an almost unfeasible small soft-stiff frequency gap between 1P and 2P (see Section 2 for details).As a result, the bottom-fixed tower loading has been comparatively high [5,6], but a stiffening of the tower above the 2P-frequency would have required uneconomical amounts of steel.For floating wind turbines, this interrelation might be a considerable advantage.If proven right, two-bladed turbines could then fully exploit their multiple benefits compared to classical threebladed machines, which are: • one blade less to produce, transport, erect, maintain, replace and decommission, along with one pitch bearing and actuator, • a beam-like rotor that allows for a safe and cheap pre-assembling of the rotor (and nacelle) in port, along with a higher vessel stacking and the usage of smaller weather windows, • structural advantages due to a wider blade chord and thickness, thus material and inertia savings in the rotor, and reduction of gravity and tip-to-tower clearance issues [7,8], • potentially reduced loads in hurricane conditions, • better accessibility by helicopter or drone.
The absent need for a stiffened, hence heavy tower, together with the reduced turbine head masses, has the potential to alleviate inertia forces from the floating platform's motion and simultaneously facilitates a lower center of gravity.
The paper at hand supports this hypothesis theoretically and numerically.The following section starts with a background on tower frequency design options.Then, the reasoning is showcased for an about twice as high tower eigenfrequency when comparing floating and bottomfixed turbines with the same tower.The third section shows the methodology of upscaling a 15 MW VolturnUS semi-submersible foundation [1] for the three-bladed INNWIND 20 MW reference turbine [9] and a most similar two-bladed 20 MW version [5,8,10,11,12,13].Finally, results for tower eigenfrequencies, their implications, and tower fatigue load examples are given.

Background on tower eigenfrequency design and the implications for two-and three-bladed floating turbines
The core message of the paper at hand is that two-and three-bladed turbines possess different excitation frequencies, which can be beneficial or problematic depending on the application.While the natural frequency of a bottom-fixed tower fits better with a large three-bladed turbine, the two-bladed counterpart potentially enables an easier tower design on spars, semisubmersibles, or barges due to an increase in tower eigenfrequency.
The generic Campbell diagram in Figure 1 shows the excitation frequencies (1P-6P) and corresponding tower design fields, customized for two-bladed turbines in green and three-bladed turbines in red.Excitation and eigenfrequencies should not come close to each other or at least not coincide longer during operation to avoid severe (fatigue) loading [5,14].The diagonal excitation lines start with the common once-per-revolution (1P) alias rotation speed excitation, which, like the others, increases proportionally to the rotation speed.1P-excitations are caused by weight imbalances in the rotor or blade pitch angle and shape deviations [14].The influence on the dynamics can be relatively small but are not to be neglected.The second and fourth diagonals are the blade-passing frequency of a two-bladed turbine (2P) and its higher harmonic (4P).Since a three-bladed turbine has no opposing blade, it does not excite with 2P or 4P.Its blade-passing frequency is the higher 3P excitation and its harmonic 6P.Analogously, 3P and 6P are non-relevant for a two-bladed turbine.The excitations of the respective blade-passing frequencies are caused by tower shadow and wind unbalances, with a larger effect for a twobladed rotor due to the rotor's symmetry.In General, the closer a crossing of blade-passing and tower eigenfrequency occurs to rated speed Ω rated , the more severe the effect on dynamic tower loads [5,6].A similar but less severe excitation [6] can be caused by the higher harmonics 4P and 6P, respectively, which is the reason for the gaps in the red and green dashed fields.Figure 1 highlights that from cut-in Ω cut-in to rated speed Ω rated , the gap is much larger between 1P and 3P than between 1P and 2P.It is this difference which enables cheaper towers with fewer dynamics for bottom-fixed turbines with three blades, but it can be an obstacle for floating three-bladed turbines.The reason is the following: The standard for bottom-fixed turbines is a soft-stiff tower configuration, meaning that the tower eigenfrequency should be placed between the rated 1P-frequency and the cut-in blade-passing-frequency, which is 2P cut-in for two-bladed and 3P cut-in for three-bladed turbines.This can already be challenging for a three-bladed (3B) turbine in the complete partial load region.It can lead to the choice of a slightly higher, aerodynamically non-optimal cut-in rotation speed Ω cut-in,higher [9] to avoid tower eigenfrequency excitation by 3P even at low wind speeds with little occurrence and lower power.For a two-bladed turbine, the bandwidth between 1P rated and 2P cut-in (higher) is too small to place a tower eigenfrequency sufficiently.Increasing the cut-in rotation speed analogously would cause a great loss in energy yield.A tower adaptation to increase its eigenfrequency above 2P and to get a stiff-stiff design is much easier for a two-bladed (2B) turbine (see lower dashed green vs. higher red region).However, a standard tubular tower still requires too much steel to shift the eigenfrequency from soft-stiff to 2B stiff-stiff to remain economically competitive.The best alternative is to avoid the excitation of the tower eigenfrequency by avoiding the matching 2P rotation speed with a rotation speed exclusion zone and withstand slightly higher tower loads compared to a similar three-bladed turbine [5].In contrast, the tower of a large floating wind turbine meets the requirement for the two-bladed turbine's stiff-stiff region by default.For a three-bladed floating turbine, this does not necessarily apply, which will be explained in the following and showcased in Section 4.
The increased tower eigenfrequency, when comparing floating with bottom-fixed turbines, is caused by the missing fixation on the ground, as depicted in Figures 2 and 3.The system of a tower on a floating turbine can rather be compared with a non-fixated free-free oscillating beam because the foundation (and the nacelle) can move in all six degrees of freedom, as sketched in Figure 3.The non-clamped foundation adds a new platform pitch motion eigenfrequency below the first fore-aft tower eigenfrequency.To understand the effect of the herewith altered first tower mode, one can analyze the eigenfrequency equation of unvarying homogeneous beams [15] where ω 0 is the eigenfrequency, EI is the bending stiffness, consisting of the Young's modulus E and the second moment of area I. ρ is the density, A is the area of the cross-section, l is the length of the cantilever, and λl is the eigenvalue of the eigenvalue equation.The latter has an eigenvalue of 1.875 for a one-sided clamped cantilever, as shown in Figure 2, and an eigenvalue of 4.73 for a free-free oscillating beam [15], which is visualized in Figure 3. Considering the ratio of these values, the first eigenfrequency of the same homogeneous tower would thus be increased by a factor of 2.52 if only the mounting were changed from clamped to free.However, there are multiple differences for wind turbines due to the inhomogeneous variations in the tower diameter and wall thickness and the vast mass-inertia of foundation and rotor-nacelle-assembly (RNA).
On the one hand, the length l of a bottom-fixed turbine increases due to the tower extension by a jacket or monopile, which would reduce the eigenfrequency.On the other hand, the floater and the mooring lines add inertia as well as water and line tensioning stiffness.This antagonizes the free mounting of the lower beam end and, in total, reduces the eigenfrequency as well.A precise estimation of the change in eigenfrequency is challenging to calculate analytically and is highly dependent on the foundation and floater design.Thus, an example with a numerical evaluation of 20 MW turbines will be given in Section 4.  Several examples can be found in the literature for the need of a costly increase of the tower eigenfrequency.In [1] the tower diameter and wall thickness of the IEA 15 MW on a VolturnUS floater have been increased, resulting in a +55 % heavier tower to meet the stiffstiff requirements.Note that according to Equation (1), a change in wall thickness would not result in a shift of eigenfrequency because stiffness and mass are increased equally, which is also termed an equal Rayleigh ratio [15].However, Equation (1) does not account for the RNA mass.With thicker walls, the ratio between RNA mass and tower mass reduces, which is similar to an unchanged tower with a lighter RNA.Still, increasing the tower diameter is a more mass-efficient measure to increase the tower eigenfrequency, but it could cause higher manufacturing costs.In [2] Électricité de France (EDF) further stiffened the already stiffened tower of the mentioned VolurnUS 15 MW with an additional 11 % of steel because the hydrodynamic added mass had not been taken into account in the earlier tower design, which affects the coupled tower-pitch dynamics and thus reduces the tower eigenfrequency.In [3] the towers of both a WindCrete style spar and an Activefloat style semi-submersible with an IEA 15 MW on top, have been designed stiff-stiff.This is even more remarkable since they reduced the tower steel's Young modulus to compensate for the actual platform flexibility of the rigid modeled floating foundations, which softens the tower.
The state of the industry is challenging to quantify, but there seems to be a trend toward stiff-stiff towers, especially for larger floating turbines.A larger FOWT moves naturally slower, while the wave periods do not scale up.That facilitates achieving the required platform motion natural periods and enables a reduction in floater size and cost, reduces the platform's added and material mass, and thus raises the tower eigenfrequency [4].The trend to stiff-stiff has partly been confirmed by several interviews with engineers of floater design companies.

Methodology: Floating foundation selection, simulation, and scaling to 20 MW
To select the most promising standard floating concept for the future market is not straightforward and has not been achieved yet, despite a throughout market research.The choice of a suitable floating foundation concept to compare the dynamics of 20 MW two-and three-bladed FOWTs for our project "X-Rotor -two-bladed floating turbines" was done with a relatively simple idea: The baseline floater should possess dynamics that are common for most floaters.This led to a semi-submersible that obtains its up-righting force, according to Eq. ( 2), by the synergy from a barge-like water plane stability M waterplane , a minor catenary mooring stiffness C Lines and a relatively low center of gravity (CoG), but with the sum of the CoGs z i , in contrast to a spar, still above and not below the center of buoyancy (CoB) z CoB .A semi-submersible is also the most popular floating concept and is not limited to specific sites due to its low draft.Finally, the semi-submersible should be capable of carrying a standard tubular tower with a standard upwind turbine.It should be popular in the research community to provide data and validate our numerical simulation setup.This led to the UMaine VolturnUS 15 MW in steel [1], which already has a similar size and needs only a small, thus less error-prone upscaling.The turbines compared on top are the three-bladed INNWIND 20 MW reference turbine [9] and its most similar two-bladed version with an equal absolute power curve, equal airfoils, relative chord layout, and angle of attacks [10], equal strength and stability limits of the blades [8,12], and an identical control architecture, tuned with a control cost criterion [11] for objective control parameter values.The models are available at [13].
The upscaling of the VolturnUS from 15 MW to 20 MW was mainly done by increasing the platform pitch stability C55 [16] until a platform pitch angle β desired of about 5 deg was achieved at rated thrust [16,14].The main driver for a semi-submersible like the VolturnUS is the water plane righting moment M waterplane , which is proportional to the product of the water plane area of the three outer columns and the square of the lever (the distance to the tower center).It is important to include the horizontal CoG, e.g., the overhang x CoG,RNA of the RNA mass m RNA in the equation of the target value C 55,desired , because it might add a beneficial negative contribution to the heeling angle [14].
Instead of performing a complete floater optimization [16], all foundation dimensions, except the freeboard and pontoon height, but including the steel wall thicknesses, are scaled by a single factor f scale to ensure a similar design as for 15 MW.The workflow is depicted in Figure 4.It starts with (a), an analytical calculation of f scale for an equilibrium of Equations ( 2) and ( 3  with ballast water, same hub height for two-and three-bladed turbine) and turbine parameters, such as static thrust in constant rated conditions F rated thrust , masses, and center of gravities.
With the resulting dimensions, FEM models were compiled in DNV's SESAM tool GeniE and transferred to (b), DNV's HydroD, to validate whether the analytical stability results of C55 had matched numerically.An important boundary condition was that the natural periods of the floater movements in all six DoF were kept above 20 s to ensure reduced natural period excitation by waves, which has been fulfilled by default.The periods proved to be large enough (heave=21 s, pitch and roll=33 s) even without iron ore concrete as additional ballast, which thus was omitted in contrast to the 15 MW version.Subsequently, the boundary element method (BEM) first-order hydrodynamic properties were calculated by DNV's solver WADAM and transferred to DNV's hydro-aero-servo-elastic tool Bladed, where Morison drag and dynamic 10-piece multibody mooring with the properties of [1] was added.In (c), Bladed, the heeling angle was checked for constant rated wind speed and calm sea state.It exposed that the added stiffness from the mooring lines C lines in its rated displaced position had been underestimated, causing a refinement from (a) to (c).Afterward, a decay test with a platform position offset in calm conditions was simulated to extract the fully coupled tower eigenfrequency from a power spectral density (PSD).In the case of the three-bladed turbine, a tower eigenfrequency redesign was vital, which resulted in a change of tower mass and CoG, and thus into a new design loop.
A precise quantification of fatigue and ultimate loads for structural tower and mooring design requires beforehand reasonable controller adaptations to dampen the FOWT movements [17], which is out of scope and will follow in future studies.

Results: 20 MW VolturnUS Campbell diagram and fatigue load spectrum
The method utilized to obtain a 20 MW VolturnUS semi-submersible (see Section 3) with the possible tower frequency design regions described in Section 2 yields some notable insights into the tower eigenfrequency issues of a large three-bladed floating turbine.Illustrated in Figure 5 is a Campbell diagram.Marked in solid black is the first tower eigenfrequency of the bottom-fixed INNWIND 20 MW reference turbine [9].The natural frequency of the soft-stiff designed tower is already a bit too high above the once-per-revolution (1P) frequency.The crossing of the threebladed turbine's blade passing (3P) frequency with the black line is at a low 3.5 rpm rotation speed, corresponding to low and less occurring wind speeds, with only a small impact on tower fatigue loads.For the two-bladed turbine, the 2P crossing appears much closer to the rated rotation speed, resulting in large tower fatigue loads, even if the corresponding rotation speed from the crossing of 2P and tower eigenfrequency is avoided [5].There is almost no load-efficient soft-stiff frequency gap for a two-bladed turbine.Aside from structural thoughts [12], the design idea of an increased two-bladed rotation speed Ω rated,2B has been to reduce loads by enlarging the gap between 2P at rated speed and this turbine's bottom-fixed tower eigenfrequency.A similar approach had been taken for the three-bladed INNWIND reference by increasing the rated tip speed to 90 m/s [9].In general, an equal rotation speed of two-and three-blade turbines would also have been possible, which might also result in further blade mass reductions [6,7,8,10,18].However, if the same baseline tower and the same RNA is placed on a floating 20 MW VolturnUS, the natural frequency almost doubles from 0.174 Hz to 0.36 Hz, marked in solid blue in Figure 5, which fits well with the description in Section 2. For the two-bladed turbine, this tower eigenfrequency directly provides a stiff-stiff design by default, avoiding the necessity of adapting the tower for a change in eigenfrequency.In contrast, this blue-marked floating tower eigenfrequency crosses the 3P line just above its worst intersection: near rated speed.Such a design would not allow a reasonable operation of the three-bladed turbine.Consequently, the three-bladed turbine's tower needs to be adapted.The concept for stiffening the tower to its minimum stiff-stiff eigenfrequency was inspired by NREL [1] and EDF [2].In the lower half, the tower diameter stays maximal with 11m and has thick walls.In the upper half, the tower diameters and walls transition with a maximum sectional angle of 3 deg [2] to smaller dimensions towards the top, as visualized in Figure 6.The result is an increase in tower mass of 38.82 % for a stiff-stiff tower configuration for the three-bladed turbine, which is below the relative mass increases of [1,2].The gap between 3P at rated speed and the tower eigenfrequency is 26.5 %.This gap accounts for 15 % rotor speed variations around rated, with an additional security gap of 10 %, as in [1].It is worth noting that the stiffened tower design, marked in red, possesses not only a higher natural frequency, but also a higher strength for tower base bending loads, which are most likely to increase for FOWTs [14,19].
The main parameters for the compared turbines and resulting floating foundations are listed in Table 1.It can be seen that the large increase in tower mass has only a slight impact on the floater masses and was, in comparison, almost fully outweighed by the slightly higher thrust of the two-bladed turbine with an equal power curve and a 2 % larger rotor.To visualize the respective effect on the tower base fatigue loads, the fatigue design load case DLC 1.2 of IEC-61400-3-2 was evaluated with wind of class IC and waves of [1] aligned with 0 deg yaw error.Shown in Figure 7 is an analysis of the tower base fore-aft damage equivalent loads (DELs) of DLC 1.2 at 11 m/s accumulated over the frequency in dashed and non-accumulated in solid lines.The plot highlights how the turbines experience a first congruent increase in DELs at the very low platform natural frequencies, followed by another rise caused by wave motions around the wave peak frequency 1/T P .The two-bladed turbine (green) experiences an abrupt increase of DELs around its blade-passing (2P) frequency.The three-bladed turbine DELs diverge around 3P, where the passing blades severely excite the eigenfrequency of the baseline tower (blue lines).The magnitude of these DELs can be reckoned by the area under the peak of the solid blue line around 3P.If 3P and tower eigenfrequency had coincided, even higher loads could be expected.Finally, the three-bladed turbine with the stiffened tower (red line) does not experience a significant increase in tower DELs around 3P, magnitude-wise comparable to the two-bladed DELs increase at 2P.Interestingly, the solid green line shows a slight rise in loads between the higher harmonic of the blade-passing excitation (4P) and the baseline tower eigenfrequency.That might indicate a severe rise in fatigue loads if the tower gets extremely stiff with an eigenfrequency near 4P at rated speed [6].A similar but lower effect can be observed in solid red between 6P and the stiffened tower's eigenfrequency.The influence of different tower eigenfrequencies on the tower loads of large two-and three-bladed floating turbines (similar to [6]) should be evaluated in future studies to clarify whether much higher tower frequencies might become problematic for either turbine concept.Accumulated for all wind speeds, the described DLC 1.2 shows a small difference of +7 % for the tower base DELs of the floating two-bladed 20 MW VolturnUS with the baseline tower in comparison to the floating three-bladed turbine with the stiffened tower.It has been ensured that the bandwidth of the blade pitch PI-controller is low enough to avoid unstable platform pitch excitation [20].Yet, the controllers should be improved to dampen platform movements [17] in future work, to alleviate the currently large increase of tower base DELs of +160 % when comparing the bottom-fixed and floating three-bladed turbines.Then, a larger amount of environmental conditions and load cases should be simulated, followed by a loaddriven adjustment of the towers for fatigue and of the currently possibly undersized mooring system of the smaller 15 MW FOWT [1].Thus, these fatigue results should be seen as a first attempt.Nonetheless, it is remarkable that the tower loads of two-and three-bladed floating turbines are almost equal.The high loads identify the need to increase the tower strength and thus result in a tower eigenfrequency that might be stiff-stiff by default for a two-or threebladed turbine, likewise.In contrast, the industry usually has much softer towers with smaller diameters.After adapting for the loads, they might then struggle with a natural tower frequency in the above-shown unfavorable region of 3P at rated, as presented in [4].

Conclusion
It has been shown theoretically and numerically that the tower eigenfrequency increases significantly if the tower of a bottom-fixed turbine is placed on a floating foundation -in the investigated case, it doubled.The exact increase of eigenfrequency is, however, very dependent on the respective floater and tower design.If it is comparable to the situation described for the floating three-bladed 20 MW VolturnUS-S, it leads to an unfavorable match of the tower eigenfrequency and the blade passing (3P) frequency at rated and thus to huge tower loads if no design changes are done to the tower.In contrast, the same tower eigenfrequency is very suitable and fit-for-purpose for a (floating) two-bladed turbine.For the three-bladed turbine, the tower steel masses had to be increased by 39 % to avoid eigenfrequency excitation by 3P at and above rated conditions.It is not unlikely that design adaptations for the increased tower loads of a floating system will result in a stiff-stiff configuration for both the two-and the three-bladed turbine, especially if the tower is guyed.
However, even if the tower on a floater gets stiff enough for a three-bladed turbine without adaptations, the most important note remains: The largest drawback of a two-bladed turbine in the previous studies had always been the increase in tower loads due to the tiny soft-stiff tower design region [5,6,18].For floating foundations like spars, barges or semi-submersibles, or self-aligning concepts with guyed towers, the tower configuration is most likely stiff-stiff by default.A vital enhancement of the controller to reduce two-and three-bladed floating turbine's tower loads, along with a structural fatigue design of the tower and the mooring system, was out of scope for this work.Yet, if the relative load comparison between the two-and three-bladed VolturnUS turbines stays within the small 7 % deviation, as shown above, a two-bladed turbine would stand out with a vast variety of cost benefits in its whole life cycle.All models can be found on a public repository [13].

Figure 1 .
Figure1.Generic Campbell diagram for two-bladed (green) and three-bladed (red) turbines, including a potential soft-stiff and respective stiff-stiff tower eigenfrequency bandwidths.The vertical black lines represent cut-in and rated rotation speeds for both turbines.The gaps in the stiff-stiff bandwidths indicate safety regions to avoid resonance at the first higher harmonic of 2P and 3P at rated conditions, respectively.

Figure 2 .
Figure2.The first mode shape of a clamped cantilever is similar to the first mode shape of a bottom-fixed turbine.

Figure 3 .
Figure3.The first mode shape of a free-free oscillating beam is similar to that of a floating turbine.
), taking into account the boundary conditions (e.g., same freeboard, pontoons completely filled floater criteria (C 55 ) boundary conditions rated thrust RNA + tower masses (d) update tower structure to match stiff-stiff configuration if needed (a) analytical scaling of floater dimensions by f scale Floater FEM in GeniE (b) validation of C55 and natural periods in HydroD (c) validation of tower eigenfrequencies, heeling angle and mooring strength in Bladed Results of Section 4

Figure 4 .
Figure 4. Scaling process for the UMaine VolturnUS-S from 15 MW to 20 MW.

Figure 5 .
Figure 5. Campbell diagram of the three-bladed 20 MW INNWIND reference turbine and a similar twobladed version, bottom-fixed on a jacket and floating on an upscaled VolturnUS semi-submersible, with the original tower as baseline and minimal stiff-stiff tower designs, respectively.See Section 2 for a generic Campbell diagram description.

Figure 6 .
Figure 6.Diameters and wall thicknesses of the INNWIND baseline tower[9], which is already stiff-stiff for the floating 20 MW two-bladed turbine and a stiffer tower version for the floating three-bladed 20 MW turbine to avoid eigenfrequency excitation at and above rated conditions.

Figure 7 .
Figure 7. Tower base fore-aft damage equivalent loads (1 Hz DELs, m=4) binned per frequency (solid) and accumulated over the frequency (dashed) for two-and three-bladed 20 MW turbines on upscaled VolturnUS-S foundations for DLC1.2 at 11 m/s with the 20 MW INNWIND reference turbine's tower as baseline (dashed-dotted blue) and a stiffened tower (dash-dotted red).

Table 1 .
Main parameters for the 20 MW two-and three-bladed turbines on an upscaled VolturnUS-S together with the bottom-fixed INNWIND reference turbine.