In this paper, the load effect of torque ripple reduction of a wind turbine generator is analyzed on the high-speed shaft gear stage and high-speed shaft bearings, which are the nearest components to the generator. Two generator designs with different torque ripples for the NREL 5‑MW reference wind turbine are considered. A decoupled analysis method is used, where global loads and torque ripple loads are used as input to a multibody model of the drivetrain in order to analyze the gear and bearing load response. Two different multibody models are considered in this work; one traditional decoupled model without generator inertia and one model with additional coupling and generator. For the model without generator inertia, statistical changes are observed for the load case with the largest torque ripple. This subsequently causes a limited increase in gear root bending damage and bearing fatigue damage. For the load case with the smallest torque ripple a limited statistical change and no damage increase is observed. For the model that includes the coupling and the generator, no statistical changes are observed between the simulation with the largest torque ripple and the smallest torque ripple. This is due to the torque ripple load that is expended to overcome the large inertia of the generator.
1 Introduction
Traditionally, torque ripple (TR) reduction in the design process of a wind turbine generator (WTG) is performed to reduce loads and avoid vibration problems on the connected mechanical system. However, TR minimization techniques can reduce the average torque of the generator and add to the production costs. Sopanen et al. [27] analyzed the effect of cogging torque and TR of a permanent-magnet synchronous generator (PMSG) on the dynamics of a direct-drive wind turbine (WT). Their main target was to evaluate the maximum of the cogging torque and TR that can be tolerated in direct-driven WTs. The developed model was a lumped parameter model with 3 torsional degrees of freedom (DOF) and from their model they concluded that the cogging torque should not exceed 1.5%–2% of the rated torque even when the resonance region is outside the operating speed range. The properties and load effects of torque ripple on a geared WT differ from a direct-drive WT. First of all, the size and the number of pole pairs differ significantly. Where a slow rotating system, like a direct-drive WT, requires a significant amount of pole pairs, a geared WT would require significantly less pole pairs. This, in combination with the magnitude of the torque, changes the load effect of torque ripple on the WT. Furthermore, the WT drivetrain consists of components that are able to dampen torsional vibrations, like the gear stages, and consists of more internal excitation sources, like gear stages and bearings. Thus, analyzing the effect of torque ripple on a geared WT requires a higher level of fidelity than for a direct drive WT. Guo et al. [8] recommended modeling approaches and requirements for DOFs, which are based on gearbox modeling practices, dynamometer tests and field tests. Furthermore, at the National Renewable Energy Laboratory (NREL) dynanometer research facility the effect of low-voltage ride-through (LVRT) on a 750-kilowatt (kW) drivetrain has been analyzed by Keller et al. [16]. Duda et al. [7] synthetically generated a power converter short circuit on a 4MW geared test rig and concluded that the gearbox acts like a low-pass filter, as the high frequency TR is filtered out, while the low frequency component is transmitted to the low-speed shaft (LSS). In Röder et al. [24] the same load propagation due to a power converter fault has been investigated on a validated multi body simulation (MBS) model of the test rig nacelle. From their work it is shown that the torque ripple, caused by the fault, is filtered out mainly in the spur gear stage from the HSS to the intermediate speed shaft (IMS), leading to the same conclusion that the gearbox acts like a low-pass filter due to its damping properties, as previously mentioned by Duda et al. [7]. Further more, Röder et al. [25] examined a grid fault and a converter fault on a doubly fed induction generator (DFIG) concept and showed, via simulation, that the risk of smearing for cylindrical roller bearings on the high-speed shaft (HSS) increases due to the electrical faults.
State-of-the-art literature where the effect of constant torque ripple is analyzed due to generator design is limited. The objective of this work is to analyze the effect of generator torque ripples on gear root bending fatigue damage and bearing fatigue damage. The scope of this work mainly focuses on the gear stage and bearings closest to the generator.
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The rest of the paper is organized as follows: In Sect. 2 the methodology of this work is presented. Then, in Sect. 3 the results are shown and discussed. In Sect. 4 concluding remarks are made and future work is elaborated on.
2 Methodology
2.1 Decoupled approach
To calculate the drivetrain dynamic load responses a decoupled analysis method is used, which is considered to be standard procedure in the wind industry [3, 19, 21]. First the global loads are obtained through an aero-hydro-servo-elastic simulation. The global forces and moments on the hub center of the drivetrain, together with the drivetrain motions, are then used as input to a drivetrain model in SIMPACK [5] to calculate the gear and bearing loads. Additionally to the generator torque coming from the controller, a zero averaged torque ripple, which torque waveform corresponds to the mean torque of the controller, is applied on the generator shaft.
First, the considered load cases are mentioned in Sect. 2.2. Afterwards, the WT model is discussed in Sect. 2.3. Then, in Sect. 2.4 the considered generator designs are discussed. In Sect. 2.5 the drivetrain model is presented, in Sect. 2.6 the environmental conditions are mentioned and in Sect. 2.7 fatigue damage theory is mentioned.
2.2 Load cases
A total of 3 load cases (LCs) are considered in this work and can be found in Table 1. \(\text{LC}_{0}\) is the baseline load case which does not contain any additional torque ripple. \(\text{LC}_{1}\) is the torque ripple associated to the design with the largest harmonic load variations, whereas \(\text{LC}_{2}\) is the torque ripple associated to the design with the smallest harmonic load variations. \(\text{LC}_{0}\)-\(\text{LC}_{2}\) have been used as input to the baseline drivetrain model in SIMPACK [6] without coupling and generator shaft and a drivetrain model in SIMPACK [6] that includes a coupling and generator shaft. The design of the drivetrain model is further elaborated in Sect. 2.5.
Table 1
Harmonic load variation of the torque ripple waveform for one electrical period
Harmonic number
Load case
\(\text{LC}_{0}\)
\(\text{LC}_{1}\)
\(\text{LC}_{2}\)
6th harmonic [Nm]
0
803.5
115.9
12th harmonic [Nm]
0
685.9
28.4
18th harmonic [Nm]
0
1471.4
282.9
24th harmonic [Nm]
0
224.8
22.2
30th harmonic [Nm]
0
118.9
2.9
36th harmonic [Nm]
0
210
0.9
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2.3 Wind turbine model
The NREL 5MW WT [13] supported on a spar floating substructure is used as the reference floating WT (FWT) [12]. The natural periods corresponding to the FWT can be seen in Table 2. The spar floating substructure is modeled as rigid body and is given stability by heavy ballast deep in the hull. Global analysis is performed with first order and viscous hydrodynamic forces combined with mean wave drift forces. In order to estimate the difference-frequency wave excitation, a Newman’s approximation is applied [19]. Mooring stiffness is modeled using catenary chain mooring system with delta lines and clumped weights [12, 19]. Above rated constant torque is applied.
Table 2
FWT natural periods [s]
Surge
Sway
Heave
Roll
Pitch
Yaw
Tower
129.5
129.5
31.7
29.7
29.7
8.2
2.1
2.4 Generator model
The generator has been designed and optimized based on the information and requirements given in “Definition of a 5-MW Reference WT for Offshore System Development” [14] by NREL. Figs. 1 and 2 show the 2‑D cross section of the designed permanent magnet (PM) generator. ANSYS Motor-CAD [1], a dedicated electrical machine design tool, is used for analysis of the generator performance. The WT generator is a surface-mounted PM machine with 6 poles and 54 slots. The designed WT generator can produce 5.24 MW electric power at rated speed with an efficiency equal to 98.45%. The permanent magnet blocks are segmented both radially and axially to reduce the eddy current losses. This will improve the efficiency and reduce the demagnetization risk.
Fig. 1
Radial cross-section of the generator
Fig. 2
Magnetic field distribution of the generator
×
×
The torque ripple of the original design (i.e. \(\text{LC}_{1}\)) is 10.72% which is fairly high. This is defined as the ratio between peak-to-peak and mean value of the electromagnetic torque signal. A high value of torque ripple can potentially lead to vibration problems in the drivetrain, and therefore minimization of the torque ripple is one of the design targets for electrical machines. Three different techniques are used here to reduce the torque ripple in the original design, including skew, short-pitching, and magnet shape optimization. The new design (i.e. \(\text{LC}_{2}\)), as a result, has a torque ripple of 1.98%, which is substantially reduced compared to the original design. This however has been possible at the expense of a reduction in the average torque (5.19%) and an increased cost of production due to the applied techniques. This highlights the importance of the trade off that needs to be investigated when designing wind generators.
2.5 Drivetrain model
In this paper the 5MW reference drivetrain developed by Nejad et al. [20] was used as baseline model. The drivetrain adopts a four-point support configuration and consists of two planetary stages and one gear-pinion stage. The topology and the design parameters of the drivetrain can be found in Nejad et al. [20]. Based on the 5MW reference model, a redesign of the HSS has been carried out for:
Two different drivetrain models are considered in the Results section, referred to as \(\text{DT}_{1}\) and \(\text{DT}_{2}\). Both models consist of a partial redesigned HSS. The description of both models can be seen in Table 3.
Table 3
Updated drivetrain model content
Controller
HSS bearings
Coupling
Generator shaft
\(\text{DT}_{1}\)
x
x
\(\text{DT}_{2}\)
x
x
x
x
The initial PI-controller has been modified to a PID-controller with a low-pass filter to limit high-frequency gain. Further more, the Proportional and Integral values are modified to minimize the rippling effect caused by the error function, \(e\), where \(e=\omega-\omega_{\text{ref}}\). Here \(\omega\) is the rotational speed of the multibody model and \(\omega_{\text{ref}}\) is the reference rotational speed from the global simulation, both in rad/s. This causes the initial generator torque to not have any high frequency oscillations, while still maintaining its function to control the rotational speed of the system. Both models require different PID values, since both models differ in torsional dynamics, due to the addition of the generator shaft for \(\text{DT}_{2}\).
A shaft layout with one cylindrical roller bearing (CRB) at the rotor side and a tapered roller bearing (TRB) pair at the generator side is chosen. A diagonal linear bearing stiffness matrix is considered, since an above rated load case is chosen which will give an almost constant rotor torque. Initial bearing stiffness values are calculated using Bearinx online [26] and tuned to acquire a load sharing in radial direction comparable to field measurements performed by NREL on a 750 kW gearbox [15]; Approximately 60% of the radial load is taken by the CRB bearing, while the remaining 40% is taken by the TRB pair. The axial load is taken by the TRB pair, since CRBs do not provide axial stiffness.
A coupling, brake disc and generator shaft have been added to the model to take into account the additional inertia of the generator. The coupling parameters and generator inertia can be found in Table 4. The coupling is considered to be extremely stiff in torsional direction, so no significant changes in dynamics due to the coupling are expected. The coupling has been modeled as 3 torsional spring-dampers in series with bodies in between [2, 17, 29], where each body has one degree of freedom around the axis of rotation. The torsional stiffness for the disk pack, \(k_{\text{diskpack}}\), and the glass fiber reinforced polymer tube, \(k_{\text{tube}}\), are taken from [4] with the corresponding inertial properties for a nominal torque of 43 kiloNewton meter (kNm), while the inertial properties of the generator are taken from Jonkman et al. [12, 13]. The generator inertia, \(I_{\text{gen}}\), is given a degree of freedom around the axis of rotation, so misalignment in non-torque directions does not have to be accounted for.
Table 4
Coupling and generator properties for \(\text{DT}_{2}\)
The environmental conditions are based on \(\text{EC}_{4}\) from Nejad et al. [19], which measurements are taken from the Cabo Silleiro Buoy of the coast in Portugal [18]. The mean wind speed at hub height, \(u_{\text{hub}}\), turbulence intensity, TI, significant wave height, \(H_{s}\), peak period, \(T_{p}\) and power law exponent, \(\alpha\) are shown in Table 5.
Table 5
Environmental Conditions
\(u_{\text{hub}}\) [m/s]
TI [–]
\(H_{s}\) [m]
\(T_{p}\) [s]
\(\alpha\) [–]
12
0.15
5
12
0.14
2.7 Remaining useful life theory
The gear root bending and bearing fatigue damage failure modes on HSS are considered in this work. The dynamic equivalent gear root bending and bearing load is calculated using ISO 6336-3 [11] and ISO281 [10] respectively. The number of stress cycles are calculated using the Load Duration Distribution (LDD) method [9, 22]. The accumulated damage on the gear and the bearings are calculated using Miner’s rule [23] and damage for \(\text{LC}_{1}\) and \(\text{LC}_{2}\) is compared to the baseline load case, \(\text{LC}_{0}\).
3 Results and Discussion
First, in Sect. 3.1 the original model is compared to the updated model by analyzing the Campbell diagram of the gear-pinion circumferential load and the vibration characteristics. Then, in the subsequent Subsections the shaft responses (3.2), gear circumferential loads (3.3) and HSS bearing loads (3.4) are analyzed and compared to \(\text{LC}_{0}\) for \(\text{DT}_{1}\) and \(\text{DT}_{2}\). Finally, in Sect. 3.5 the fatigue damage results are compared to the baseline simulation for \(\text{DT}_{1}\) and \(\text{DT}_{2}\).
3.1 Model comparison
\(\text{DT}_{1}\) is compared to \(\text{DT}_{2}\), which includes a coupling and a generator shaft. A run-up simulation is performed until rated and a Campbell diagram of the gear-pinion circumferential load is analyzed. As shown in Fig. 3, the main frequencies that are excited for both DT models are the mesh frequency, \(\omega_{\text{ref}}N_{\text{pinion}}\), and mainly its fourth harmonic. Here \(\omega_{\text{ref}}\) is the HSS rotational speed in \(Hz\) and \(N_{\text{pinion}}\) is the number of pinion teeth, which can be found in Nejad et al. [19].
Fig. 3
Campbell diagram of the HSS gear teeth circumferential load
×
For both models the gear harmonics and sidebands are excited, where the amplitude of the excited frequencies in the updated model is higher compared to the reference model. This can be explained by the added generator inertia in \(\text{DT}_{2}\), which changes the system dynamics and thus causes changes in the gear meshing. In Table 6 the gear-pinion load characteristics are given. Here it can be observed that only the standard deviation has changed due to the model updating.
Table 6
Gear-pinion load characteristics at rated
Mean [kN]
Standard deviation [kN]
\(\text{DT}_{1}\)
254
6.4
\(\text{DT}_{2}\)
254
15.1
3.2 Shaft responses
The power spectral density of the HSS rotational speed of \(\text{DT}_{1}\) and \(\text{DT}_{2}\) is shown in Fig. 4. In green the torque ripple harmonics are indicated. For \(\text{DT}_{1}\) a response can be seen in frequency domain, whereas for \(\text{DT}_{2}\) the torque ripple cannot be identified on the HSS rotational speed. This is because the torque ripple is almost entirely expended to overcome the large inertia of the generator and therefore does not have sufficient energy to further excite the system [28]. In combination with a HSS coupling that has little damping, vibrations are not completely able to travel from the gearbox to the generator and the other way around. Thus limited responses can be expected on the HSS. The peak around 500 Hz and between 1250 and 1500 Hz can be identified as the mesh frequency and its third harmonic, respectively.
Fig. 4
PSD of the shaft responses in rotational direction
×
3.3 Gear-stage response
In Fig. 5 the power spectral density of the gear-pinion circumferential load between the IMS and the HSS is shown. Similar to the HSS response, peaks can be identified at the torque ripple harmonics for \(\text{DT}_{1}\), while for \(\text{DT}_{2}\) no significant change in the frequency content of the gear-pinion circumferential load is identified, which has been explained in Sect. 3.2.
Fig. 5
PSD of the gear-pinion circumferential load
×
3.4 High-speed shaft bearing responses
The pinion loads are transmitted to the gearbox through the HSS bearings. Thus a similar frequency content compared to the gear-pinion circumferential load is expected. Fig. 6 shows the frequency content of the \(\text{HS}_{A}\), \(\text{HS}_{B}\) and \(\text{HS}_{C}\) bearing on the HSS shaft. Again, peaks are identified at the torque ripple harmonics for \(\text{DT}_{1}\). No significant change of the frequency content of the bearings is identified for \(\text{DT}_{2}\). Peaks around the gear meshing frequencies and its harmonics, as described in Sect. 3.2, can be found. A large difference between the first harmonic of the gear meshing between \(\text{DT}_{1}\) and \(\text{DT}_{2}\) can be observed.
Fig. 6
PSD of the HSS bearing load
×
3.5 Fatigue damage results
The gear-pinion (GP) and bearing mean, standard deviation and damage equivalent load normalized to \(\text{LC}_{0}\) can be seen in Tables 7, 8 and 9, respectively. For \(\text{DT}_{1}\) no significant change of the mean gear-pinion circumferential load and bearing loads is observed, while for \(\text{DT}_{2}\) an increase of the mean for the bearings can be seen. On the contrary, an increase in standard deviation for \(\text{DT}_{1}\) can be found for all components, with the largest increase in standard deviation of 52% observed for the gear-pinion circumferential load. For \(\text{DT}_{2}\) no significant changes in standard deviation are observed. The increase in standard deviation for the gear-pinion circumferential load and bearing loads of the HSS agrees with the Results from Sects. 3.1–3.4 and is caused by the torque ripple, whereas the increase in mean value for \(\text{DT}_{2}\) could also be caused by a change in the static equilibrium of the shaft, rather than the torque ripple.
Table 7
Change in mean, \(\mu\), in % compared to the reference load case, \(\text{LC}_{0}\)
GP
\(\text{HS}_{A}\)
\(\text{HS}_{B}\)
\(\text{HS}_{C}\)
\(\text{DT}_{1}\)
\(\text{LC}_{1}\)
\(1.2\times 10^{-4}\)
\(1.2\times 10^{-4}\)
\(3.2\times 10^{-4}\)
\(2.2\times 10^{-4}\)
\(\text{LC}_{2}\)
\(3.8\times 10^{-5}\)
\(3.6\times 10^{-5}\)
\(3.4\times 10^{-5}\)
\(4.4\times 10^{-5}\)
\(\text{DT}_{2}\)
\(\text{LC}_{1}\)
\(2.0\times 10^{-4}\)
0.25
0.30
0.32
\(\text{LC}_{2}\)
\(-1.4\times 10^{-5}\)
0.25
0.30
0.32
Table 8
Change in standard deviation, \(\sigma\), in % compared to the reference load case, \(\text{LC}_{0}\)
GP
\(\text{HS}_{A}\)
\(\text{HS}_{B}\)
\(\text{HS}_{C}\)
\(\text{DT}_{1}\)
\(\text{LC}_{1}\)
52
4.5
5.3
3.2
\(\text{LC}_{2}\)
1.8
0.13
0.15
0.09
\(\text{DT}_{2}\)
\(\text{LC}_{1}\)
\(1.4\times 10^{-2}\)
\(-6.7\times 10^{-3}\)
\(2.9\times 10^{-2}\)
\(1.1\times 10^{-2}\)
\(\text{LC}_{2}\)
\(1.0\times 10^{-2}\)
\(-1.0\times 10^{-4}\)
\(2.6\times 10^{-2}\)
\(8.5\times 10^{-3}\)
For \(\text{LC}_{1}\) of \(\text{DT}_{1}\) the gear root bending damage equivalent load, DEL, increases with 1.2% which is caused by the large change in standard deviation. The marginal increase in fatigue damage is caused by the fatigue calculation method that is applied on gears and bearings, which is based on the absolute value of the load, rather than the load fluctuation, since gears and bearings endure one stress cycle each rotation. In theory only the amplitude component of the harmonic load will change the fatigue damage on gears and bearings if the LDD counting method is considered. Furthermore, the increase in damage due to a harmonic load is dependent on the bearing geometric constant, \(a\), which is based on geometrical properties and does not change over time.
The increase in fatigue damage for both load cases of \(\text{DT}_{2}\) is caused by the increase in mean loads on the HSS bearings. When no changes in standard deviation are considered, the damage should scale with the bearing geometric constant, which is \(\frac{10}{3}\) for the concerning bearings.
Table 9
Change in DEL in % compared to the reference load case, \(\text{LC}_{0}\)
GP
\(\text{HS}_{A}\)
\(\text{HS}_{B}\)
\(\text{HS}_{C}\)
\(\text{DT}_{1}\)
\(\text{LC}_{1}\)
1.2
0.31
0.28
0.28
\(\text{LC}_{2}\)
\(3.3\times 10^{-2}\)
\(8.8\times 10^{-3}\)
\(7.6\times 10^{-3}\)
\(7.7\times 10^{-3}\)
\(\text{DT}_{2}\)
\(\text{LC}_{1}\)
\(2.5\times 10^{-3}\)
1.1
0.65
0.54
\(\text{LC}_{2}\)
\(1.0\times 10^{-3}\)
1.1
0.65
0.54
Previous work with decoupled analysis did not always consider the inertia of the rotor blades and the generator, since the global loads were already determined based on a global model including these inertia’s. This has a limited effect when the global loads and motions from the aero-hydro-servo-elastic model do not contain a high frequency component and if the results are analyzed in steady state. However, when analyzing the load response of a high frequency excitation, this assumption could cause wrong conclusions to be drawn. Thus in order to capture the dynamic of the system under torque ripples the inertia of the whole drivetrain must be considered.
4 Conclusion and future work
This study deals with the effect of torque ripple harmonics on the gear root bending fatigue damage and bearing fatigue damage. The scope of this work is the load effect on the high-speed shaft in a decoupled approach at rated wind speed. Two electromagnetic finite-element generator designs with different torque waveforms are considered, \(\text{LC}_{1}\) and \(\text{LC}_{2}\). \(\text{LC}_{1}\) is the torque ripple associated to the design with the largest harmonic load variations, whereas \(\text{LC}_{2}\) is the torque ripple associated to the design with the smallest harmonic load variations. The global loads are obtained through an aero-hydro-servo-elastic simulation and are used as input to the NREL 5MW drivetrain model. Two different models are considered. Both models are based on the drivetrain model by Nejad et al. [20], where the first model is minimally adjusted and referred to as \(\text{DT}_{1}\), while for the second model an additional coupling with generator shaft is added and is referred to as \(\text{DT}_{2}\). The load path of the torque ripple is analyzed and the fatigue damage is compared to the baseline simulation.
For \(\text{DT}_{1}\), changes in the frequency content of the high-speed shaft rotational speed, gear-pinion circumferential load and bearing loads are observed. In contrast, the torque ripple does not affect the frequency content of \(\text{DT}_{2}\), since the torque ripple is almost entirely expended to overcome the large inertia of the generator.
For \(\text{DT}_{1}\) of \(\text{LC}_{1}\) an increase in gear root bending and bearing fatigue damage is observed, while for \(\text{LC}_{2}\) the damage is negligible. This change in the fatigue damage is mainly caused by the increased standard deviation caused by the torque ripple. For \(\text{DT}_{2}\) a slight increase in fatigue damage can be observed for all bearings on the high-speed shaft, which is caused by an increase in bearing loads for \(\text{LC}_{1}\) and \(\text{LC}_{2}\). From \(\text{DT}_{2}\), which represents the turbine dynamics closest to reality, it can be concluded that no significant changes in gear root bending fatigue damage and bearing fatigue damage on the high-speed shaft can be expected from different generator torque waveforms on a geared WT.
In this work, a decoupled approach is considered. For future work, a fully coupled electro mechanical model will be developed which can be used to promote additional insights into the structural responses and corresponding mode shapes of the WT drivetrain. Additionally, the effect of generator and rotor inertia will be analyzed on the gear loads responses. Further more, in this work an operating window at rated is considered. To analyze the effect of torque ripple below rated, different operating windows should be considered.
Acknowledgments
The first and fourth author acknowledge the financial support via the MaDurOS program from VLAIO (Flemish Agency for Innovation and Entrepreneurship) and SIM (Strategic Initiative Materials) through project SBO MaSiWEC (HBC.2017.0606). The first and fourth author would also like to acknowledge the support of De Blauwe Cluster through the project Supersized 4.0. The second author wish to acknowledge financial support from the Research Council of Norway through InteDiag-WTCP project (Project number 309205).
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