Operation in Point of Resonance

compared to results for a rigid setup. As the hydrodynamic damping is in the same order of magnitude for the chosen tip speed ratios and current speeds, the impact of the points of operation can be reduced to the excitation frequency. Nevertheless, the absolute values of the load amplitudes differ between the cases.

Fig. 4-25: Amplitude response ratio 𝐴/𝐴₀ of tower bottom bending moment 𝑀𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚

to excitation ratios 𝑓/𝑓_{𝑡𝑜𝑤𝑒𝑟}

The comparison of the results to the theoretical 1 DoF oscillator, cf. (4-2), gives a reasonable match. However, the agreement is not perfect, due to the higher harmonic oscillations. This behavior was also observed and analyzed for the case of 4𝑚/𝑠 current speed and tip speed ratio 2 ⋅ 𝜆_{𝑇𝑆𝑅 𝑜𝑝𝑡} in Section 4.3.5. As can be seen from Fig. 4-26, the impact of the higher order frequencies increases with increasing current velocity, and the time series of the load deviates from the sinusoidal shape at lower current speeds.

This analysis indicates a massive load increase in case of resonance; however, at the different locations in the turbine the load amplification differs. Fig. 4-27 shows the axial force for the resonance response at the tower top, 𝐹_{𝑥 𝑇𝑜𝑤𝑒𝑟𝑇𝑜𝑝}, and at the hub connection, 𝐹_{𝑥 𝐻𝑢𝑏}. As can be seen, moving from tower bottom to tower top, the resonance load factor of five is reduced to three. Moving on to the hub, there is no increase in the axial hub force loads, shown here, during resonance operation.

4.4. Operation in Point of Resonance 89

Fig. 4-26: Tower bottom bending moment 𝑀𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚 for 𝜆_𝑇𝑆𝑅 = 2 ⋅ 𝜆_{𝑇𝑆𝑅 𝑜𝑝𝑡} in fully flexible (dashed) and rigid (solid) setup normalized with the corresponding mean values 𝑀̅𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚|_{𝑟𝑖𝑔𝑖𝑑}(𝑣₁)

Fig. 4-27: Axial tower top load 𝐹_{𝑥 𝑇𝑜𝑤𝑒𝑟𝑇𝑜𝑝} (left) and axial hub force 𝐹_{𝑥 𝐻𝑢𝑏} (right) for 𝜆_𝑇𝑆𝑅 = 2 ⋅ 𝜆_{𝑇𝑆𝑅 𝑜𝑝𝑡} in fully flexible (dashed) and rigid (solid) setup normalized with the corresponding mean values

This location dependency of the resonance effect is mainly a result of the mass distribution.

Each mass of the system needs to be accelerated according to the system motion and causes a compensating load 𝐹 = 𝑚 ⋅ 𝑎. Furthermore, due to the motion of the rotor and considering the tower fore-aft mode, the rotor hydrodynamics result in lower thrust amplitudes compared to

the rigid case, decreasing the local loads further. In Fig. 4-27 also the load amplitude in the rigid case between the two locations seems to differ. However, this is only an artifact from the normalization with the respective mean value, which differs due to nacelle drag.

The steady operation at resonance causes a significant load increase. However, these are not necessarily spread evenly across the entire turbine system. A detailed analysis of the specific turbine system is required here. For the present case of the Voith HyTide^® turbine the tower resonance should be avoided to limit the fatigue and extreme loads on the tower, but is of minor importance for, e.g., the hub design.

4.4.2 Transient Ride-Through of Resonance

The issue of resonance is typically avoided on tidal current turbines by changing the rotational speed quickly in a short time to ride-through the resonance frequency. This approach is adopted from wind energy, [21]. To evaluate the loads during such a passage, the simulation setup is extended here to a variable speed case. For a current speed of 4𝑚/𝑠, the rotational speed of the turbine is decelerated from 2 ⋅ 𝜆_{𝑇𝑆𝑅 𝑜𝑝𝑡} to 𝜆_{𝑇𝑆𝑅 𝑜𝑝𝑡}. This change in rotational speed is described in the numerical setup as an additional rotational deformation of the rotor domain, which requires an adaption of the grid deformation algorithms, detailed in Appendix B, p. 145.

For the passage time, [32] calculated that a 1MW tidal turbine takes 1𝑠 to speed up from design condition to run-away condition upon a generator torque loss (𝑃_𝑒𝑙= 0𝑊). However, during normal operation this is unrealistically fast, as the allowed variation of electrical power output is limited. Hence, a passage time of 2𝑠 is chosen here with a linear change in rotational speed, Fig. 4-28. Resonance operation occurs at 𝑡 = 1𝑠, cf. Fig. 4-25.

The loads on the tower bottom and the tower top for this transient event are shown in Fig.

4-29. As can be seen, even with the quick ride-through procedure, the onset of resonance can not be fully prevented. However, the amplitude ratio for the tower bottom bending moment 𝑀𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚, which was about five for steady operation in resonance, cf. Fig. 4-25, is reduced here to approximately four. The same reduction in the load amplitude ratio applies also to the tower top load 𝐹_{𝑥 𝑇𝑜𝑤𝑒𝑟𝑇𝑜𝑝}.

Further changes in the loads occur due to the riding-through. Despite that the deceleration ends at 𝑡 = 2𝑠, in both, 𝑀𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚 and 𝐹_{𝑥 𝑇𝑜𝑤𝑒𝑟𝑇𝑜𝑝}, a variation of the loads can be still observed afterwards for the flexible and the rigid case. This is caused by the dynamic inflow.

4.4. Operation in Point of Resonance 91 With the change in rotational speed, the thrust coefficient 𝑐_𝑡ℎ, and thus the axial induction 𝑎_𝑎𝑥 change aswell. However, this change does not occur instantaneously due to the inertia of the fluid, resulting in a time delay of ca. 4𝑠 in the present case to reach steady flow conditions.

Taking also the flexibility and dynamic motions of the turbine structure into account, the flexible case requires several seconds more for the onset of the steady oscillation.

Fig. 4-28: Linear deceleration of rotational speed for passage of resonance

Fig. 4-29: Tower bottom bending moment 𝑀𝑦 𝑇𝑜𝑤𝑒𝑟𝐵𝑜𝑡𝑡𝑜𝑚 (left) and axial tower top load 𝐹_{𝑥 𝑇𝑜𝑤𝑒𝑟𝑇𝑜𝑝} (right) during passage of resonance normalized with the corresponding mean values at 𝑡 > 6𝑠

In the analysis of the flexibility shown in Section 4.3, the main shaft was found to have a negligible impact on the loads. In the here investigated transient load case however, a significant impact can be observed for the hub torque 𝑄_{𝑥 𝐻𝑢𝑏}, Fig. 4-30. In order to decelerate the turbine, the generator torque is increased to follow the prescribed rotational speed in the generator, Fig. 4-28. This causes a step response of the main shaft in its torsional mode. As this mode has a low hydrodynamic damping, the oscillation persists during the complete riding-through. However, this is only a local load and the main shaft’s torsional motion does not transfer the loads in a relevant extent to, e.g., the tower.

Fig. 4-30: Response of hub torque 𝑄_{𝑥 𝐻𝑢𝑏} to deceleration of generator rotational speed normalized with the mean value at 𝑡 > 6𝑠

A transient passage of resonance within a short period can therefore be used to limit the effect of the resonance. This affects mainly the fatigue load, as the number of high amplitude oscillations is reduced. The extreme loads are reduced with this maneuver, but the change in loads is small compared to a rigid structure.