1 Introduction

A major design objective for wind turbines (WTs) is low levelized cost of energy (LCOE). LCOE can be reduced by keeping capital and operational expenditures constant and increasing the annual energy production (AEP). Accordingly, to achieve high AEPs the WT control is optimized for maximum energy production in all wind conditions. At high wind speed, when the WT operates at nominal power, the power output is only limited by the maximum generator power and the efficiency of the rotor is actively reduced via blade pitching. In partial load operation, the power output is limited by the wind speed and an efficient energy conversion at the rotor is aimed [1]. The efficiency of the energy conversion from wind-speed v to output power P is described by the power coefficient, CP  and is calculated via Eq. 1 with the air density ρ and the area of the rotor A [2].

$$C_{P}=\frac{P}{\frac{1}{2}\rho Av^{3}}$$
(1)

The power coefficient is dependent on the aerodynamics at the rotor blades. Figure 1. shows a simplified representation of the inflow angle of a blade section.

Fig. 1
figure 1

Rotor aerodynamic according to [3]

Full size image

As can be seen, the inflow angle α, and the relative wind speed that is generating the aerodynamic force to turn the rotor is not only dependent on the wind speed v but also, the blade speed, that depends on rotor speed ω and rotor diameter R. Therefore, the tip-speed ratio λ is defined as:

$$\lambda =\frac{\omega \cdot R}{v}$$
(2)

Accordingly, the power coefficient CP  depends on λ. For each wind speed an optimal tip-speed-ratio λopt exists that leads to an optimal power coefficient CP,opt  . This is illustrated in Fig. 2.

Fig. 2
figure 2

Control curve for WT in the torque-speed map of the rotor, according to [4]

Full size image

Figure 2. shows the resulting CP, opt curve as function of rotor torque TR and λ for different v. To achieve CP,opt  in partial load range optimal torque control (OTC) is commonly used [5]. OTC represents a simple but effective method to run the WT close to the optimal operating point. The principle of OTC is shown schematically in Fig. 3.

Fig. 3
figure 3

Schematic OTC according to [6]

Full size image

With OTC only the shaft-speed ωR is measured at the rotor or is derived from the electric quantities measured in the converter. However, the converter varies the electrical generator currents to set the generator torque TG according to the shaft speed to reach optimal power coefficient Cp,opt.

In practical implementation the rotor torque is often approximated by merging Eq. 1 and 2 which leads to Eq. 3 and 4. Cp,opt at rated torque and optimal tip-speed ratio λopt depend on the specific aerodynamic behavior and are determined for each WT individually [6].

$$T_{G}=c\omega ^{3}$$
(3)

With

$$c=\frac{1}{2}\rho \pi R^{5}\frac{C_{p{,}\mathrm{opt}}}{{\lambda _{\mathrm{opt}}}^{3}}$$
(4)

As described above, TG is the control variable, but the actual control-target is the rotor torque TR, that can either be measured directly at the low-speed-shaft (LSS) or calculated from TG with the gear ratio and the drivetrain-efficiency. However, both approaches show significant errors.

On-site torque measurements can be performed with strain gauges at the LSS to measure TG. In torque measurement based on strain gauges, material parameters such as Young’s modulus and k‑factor, and geometry of the deformation body and the strain gauges themselves are usually insufficient. A correct torque measurement with a defined uncertainty interval can be guaranteed by calibration. Currently, however, calibration is only possible up to 1.1 MNm. For a non-calibrated torque measurement using strain gauges, an accuracy of only 5–7% can be achieved [7]. With the help of calibrated torque sensors, torque < 1.5 MNm on the LSS can be measured with an uncertainty < 0.5%.

In contrast to the rotor torque TR, calibration procedures are feasible and standardized in the measuring range of the generator torque TG. Calibration enables generator torque measurements with high accuracies of less than 0.1% [8]. To transfer the TG to TR, the drivetrain efficiency must be known. Existing approaches to calculate the efficiency of machine elements like gears and bearings determine the efficiency dependent on drivetrain loads as well as speed of rotation, and the viscosity of the lubricants. Nevertheless, efficiency calculations always show uncertainties too. Additionally, in practical use the efficiencies are often estimated as a constant value to calculate the rotor torque. Therefore, the scattering of the efficiency is neglected. Thus, it is to be expected that errors in rotor torque determination lead to control errors. These rotor torque control errors result in lower power coefficients and in lower AEP leading to higher LCOE for the WT. This paper quantifies the rotor torque control errors and their influence on the power coefficient in partial load operation.

2 Approach

To examine fluctuations in efficiency due to drivetrain speed, torque, temperature, and non-torque-loads (NTLs) an efficiency model for a WT drivetrain is build. This efficiency model of the WT drivetrain is then validated by calibrated measurements gathered on a system test bench. A special 5 MNm torque transducer from PTB with known uncertainty measures the rotor torque. With the calculated rotor torque errors, the impact on the power coefficient is determined.

2.1 Device under test

The study is conducted for the FVA (Forschungsvereinigung Antriebstechnik e. V.) research nacelle. The performance data of the device under test (DUT) are shown in Table 1.

Table 1 Performance data of DUT
Full size table

Among other components the drivetrain of the FVA research nacelle consists of a main bearing, a gearbox and a generator. The gearbox contains a planetary gear stage with three planets and two helical gear stages, resulting in the maximum speed at the HSS of up to 1100 rpm. Within the study the drivetrain efficiency is measured on the system test bench (see Fig. 4) of the CWD.

Fig. 4
figure 4

FVA research nacelle at the CWD test bench

Full size image

The test bench is equipped with a 4 MW PMSM permanent magnet synchronous machine (PMSM) prime mover to power the drivetrain. Additionally, the servo-hydraulic non-torque load (NTL) unit provides forces in three and bending moments in two load directions. Combined, the test bench can apply loads in six degrees of freedom. [9]. Overall, the test bench capacity exceeds the limits of the DUT allowing testing of the full operating range.

2.2 Efficiency model for WT drivetrain

The efficiency model consists of existing loss models for the drivetrain components. They enable loss calculations depending on the loads and temperature. The component loads are calculated with an existing validated multi-body-simulation (MBS) model of the nacelle on the testbench (see Fig. 5; [10,11,12]).

Fig. 5
figure 5

MBS model of the wind turbine drivetrain on the 4 MW test bench

Full size image

In the model, the losses due to tooth contacts and bearings are considered and implemented in the MBS model. Churning losses in the gearbox and sealing losses were neglected because of their minor influence [13]. The losses considered are displayed in Fig. 6.

Fig. 6
figure 6

Schematic illustration of the modelled drive train losses and the location

Full size image

The bearing losses are calculated acc. to Palmgren [14]. The total frictional torque Tfric. of a bearing is the sum of a speed-dependent friction torque T0 and a load-dependent friction torque T1.

$${T_{\mathrm{fric}.}}=T_{0}+T_{1}$$
(5)

This approach, in turn is implemented in the calculation model according to FAG [15]. The equations used are shown below. The parameters can be obtained from the bearing catalog.

$$T_{0}=f_{0}\cdot 10^{-7}\cdot \left(\nu \cdot n\right)^{\frac{2}{3}}\cdot {d}_{m}^{3}$$
(6)
$$T_{1}=f_{1}\cdot P_{1}\cdot d_{m}$$
(7)

The friction torque depends on the bearing specific parameters f0 and f1, the viscosity ν, the rotational speed n, the bearing diameter dm and the bearing load P1. In the MBS model, the rotational speed and load of each individual bearing in the MBS-model are measured at the bearing force element and the resulting friction torques are calculated accordingly. Hereby, losses on the LSS shaft, planetary stage, helical gear stages are considered.

To model the influence of drivetrain temperature on the efficiency the viscosity of the gearbox oil was adapted in the component’s loss models. It is assumed that the temperature in the gearbox is homogenous and equal to the sump temperature. In practice the gearbox sump temperature is around 48 °C, on the other hand can reach up to 80 °C, before a shutdown is necessary. Bearings, on the other hand, e.g. at the HSS can reach higher temperatures of up to 90 °C. However, the effects of the local temperature variations mentioned are neglected in this study.

Tooth contact losses are caused by rolling friction and sliding friction on the tooth-flank. As rolling friction losses are comparably low, they are neglected as usual [16]. Sliding friction losses PVZP are calculated using Coulomb’s law, from the relative sliding speed νg(y), the friction coefficient μ(y) , and the normal force on the tooth flank FN(y) [17] (see Eq. 8)

$$P_{\mathrm{VZP}}\left(y\right)=\mu \left(y\right)\cdot F_{N}\left(y\right)\cdot \nu _{g}\left(y\right)$$
(8)

All parameters are dependent on the position of the tooth mesh. The relative sliding speed can be calculated using Linke [18] and Niemann and Winter [17] and results in a constant gear loss factor HV. The friction coefficient μ is simplified as a constant average μmz over the contact path [19]. For the normal load on the tooth flank, the simplified load distribution defined by Ohlendorf is used [20]. Thus, the tooth contact losses are implemented by Eq. 9 as a function dependent of the drivetrain power PA.

$$P_{\mathrm{VZP}}=\mu _{mz}\cdot P_{A}\cdot H_{V}$$
(9)

2.3 Experimental test campaign for model-validation

To validate the efficiency model, measurements were performed on the 4 MW system test bench at CWD to validate the efficiency model. Figure 7 shows the drivetrain and the test set up from above. The torque measurement points are located on the LSS (1) and the HSS (2).

Fig. 7
figure 7

Drivetrain of the DUT and location of measurement points

Full size image

The additionally installed 5 MNm torque transducer used in this work to measure generator torque on the LSS is a hollow-shaft transducer. It measures torque up to 5 MNm using strain gauges. Despite being only calibrated up to 1.1 MNm, it is used as a transfer standard to enable traceable torque measurements in nacelle system test benches. Above 1.1 MNm, it is metrologically characterized using an extrapolation method and assuming a linear behavior based on a linear calibration curve:

$$T_{\mathrm{TTS}}=3850\,\mathrm{kNm}\left(\mathrm{mV}/\mathrm{V}\right)^{-1}\cdot S_{\mathrm{TTS}}$$
(10)

STTS is the output signal in mV/V and TTTS is the corresponding calibrated torque output in kNm. The expanded relative uncertainty interval (k = 2) for the linear calibration curve up to 1.5 MNm is 0.290%. Figure 8 shows the torque transducer mounted at the DUT’s rotor hub flange.

Fig. 8
figure 8

5 MNm torque transfer standard mounted in the 4 MW test bench at CWD

Full size image

The torque signal is read out by a high-precision amplifier with a sampling frequency of \(f_{\text{sample}}=150\,\text{Hz}\) and a 50 Hz Bessel filter. To synchronize the autonomous data acquisition system DAQ of the 5 MNm torque transducer to the data acquisition system of the test bench, the test bench’s network time protocol (NTP) signal is used for the time stamps of the torque signal. The measured data is transmitted via WLAN using a rotating and a stationary access point. By completing the static torque calibration with additional influences under rotation, the overall relative expanded measurement uncertainty (k = 2) for rotor torque measurement on the LSS in the test bench was estimated for each load steps. It starts from 0.29% in higher torque range and goes up to 0.65% in lower torque range.

To measure generator torque on the HSS, which is between the gearbox and the generator, an 80 kNm hollow-shaft torque transducer (Fig. 9) is installed. The relation between the measurements output signal ST of the transducer in kHz and the applied torque TT in kNm, was defined with the help of a calibration and is described by the following equation:

$$T_{T}=\frac{2.666\,\mathrm{kNm}}{\mathrm{kHz}}\cdot S_{T}$$
(11)
Fig. 9
figure 9

80 kNm torque transducer on the HSS

Full size image

The relative expanded measurement uncertainty (k = 2) of the transducer amounts to < 0.1%. The data is transmitted via the transducer’s own telemetry system.

3 Results

First, the variation of the efficiency due to drivetrain torque and speed are examined. For this purpose, the measurements of the 4 MW test-bench are evaluated. The results are then compared to the simulated efficiency to validate the model. Afterwards, the influences of NTLs, and temperature on the drivetrain efficiency are simulative quantified using the efficiency model. Finally, the control error of the rotor torque and the resulting Cp deviation can be determined.

3.1 Influence of the drivetrain torque and speed on the efficiency

Depending on the operating point the WT drivetrain speed and the drivetrain torque varies. The results of the efficiency measurement campaign on the 4 MW test bench for different drivetrain speeds are shown in Fig. 10.

Fig. 10
figure 10

Measured efficiencies for different drivetrain speeds

Full size image

Figure 10 shows that drivetrain efficiency is influenced by the drivetrain speed. At 1150 kNm the deviation between 12.5 rpm and 17.5 rpm is 0.4% while at 150 kNm the difference is 1.6%. From this, it can be concluded that the deviation is small especially in the case of high torques and therefore high powers.

Figure 11 shows the calculated and measured drivetrain efficiency for different drivetrain torques. The chart bars represent the deviation between measurement and simulation.

Fig. 11
figure 11

Comparison of simulated and measured drivetrain efficiency (12.5 rpm, 48 °C)

Full size image

The modeled efficiency and the measured efficiency match with a deviation of less than 0.7%. The only exception occurs for the simulated value at 170 kNm which shows a deviation of 4.1%. One reason is that for small torques and thus low power, the absolute errors in the loss calculation or measurement have higher impacts. It can be concluded that the model is valid for operation points above a torque of 330 kNm.

Furthermore, the efficiency increases with increasing torque until reaching its maximum asymptotically. Overall, the drivetrain efficiency varies between 94.1% for 330 kNm and 97.1% for 1200 kNm, and therefore has a range of 3%.

3.2 Influence of non-torque loads on efficiency

The influence of NTLs on drivetrain losses is examined using the efficiency model. To do this, bending moments around Y‑axis (My) and Z‑axis (Mz), as well as forces in axial (Fx) and radial (Fr) direction were applied to the rotor hub. A realistic load range was determined with a rotor load calculation using blade element theory. The loads investigated are listed in the table below (Table 2.).

Table 2 Range of examined NTLs
Full size table

The NTLs were changed, while the WT was operated at rated torque and rated speed. (17.5 rpm, 1550 kNm). The results are concluded in Fig. 12.

Fig. 12
figure 12

Influence of NTLs on the drivetrain efficiency

Full size image

The efficiency deviations for different hub forces are lower than 0.01%. The bending moments, on the other hand, have a greater effect, but the deviations are still less than 0.1%. The main reason is that NTLs mainly affect the component loads at the main bearing, while the impact on the gearbox loads, where most losses are generated, is small. Therefore, the overall impact of NTLs on the efficiency can be neglected and does not need to be investigated further.

3.3 Influence of drivetrain temperatures on the efficiency

The influences of the temperature variation are simulated with the efficiency model. In Fig. 13 the results are displayed.

Fig. 13
figure 13

Drivetrain efficiency for different oil temperatures and torques at 17.5 rpm

Full size image

Figure 13 shows the efficiency of the drivetrain at rated speed (17.5 rpm) for a relevant torque range between 500 kNm and 1550 kNm, with temperature variation. The deviation in drivetrain efficiency is particularly high at low temperatures. The minimal efficiency is 88.0%, while the maximum efficiency is 97.7%. At rated torque reduction in temperature from 80 °C to 20 °C in the gearbox sump temperature leads to an efficiency reduction of up to 2.6%. At a lower torque of 500 kNm this difference is up to 8.7%. The main reason are churning losses that decrease with higher temperatures. Additionally, churning losses have greater impacts on efficiency at lower torque, because they are only dependent on speed that was constant for this analysis. It can be concluded that the influence of the temperature on drivetrain efficiency is significant and should be further examined.

3.4 Summary of the resulting control torque errors

As shown in the previous section’s efficiency varies depending on drivetrain speed and torque, NTLs, and temperature. Since in WTs controls typically constant efficiencies at rated speed and rated torque are used to calculate the rotor torque, the variations of the efficiency represent the error in the rotor torque control. As a reference the direct measurement of rotor torque measurement uncertainty for in field measurements with uncalibrated strain gauges is used. Finally, the influence of the efficiency model uncertainty on the rotor torque control is evaluated. All influences on control torque errors are summarized in the following Table 3.

Table 3 Summary of rotor torque control errors
Full size table

For the experimentally investigated variables drivetrain torque and speed, the control torque errors were based on the reference torque measurement on the LSS, which’s measurement uncertainty is 0.29% in higher torque range and 0.65% in lower torque range. The uncertainty range is sufficient to guaranty a valid determination of the rotor torque control errors.

The measurement results show that the rotor torque control error can reach up to 10%. As a conclusion influence of rotor torque control errors on the power coefficient needs to be further examined.

3.5 Influence of rotor torque control errors on the power coefficient

Using an aerodynamic model of the rotor, the influences of rotor torque control deviations are examined. Rotor load simulation is done by means of MBS using blade element theory as it is implemented in the AeroDyn force element code [21]. The loads acting on the rotor blades are calculated within respect to local wind speed and wind direction given by turbulent wind fields, which were generated using TurbSim [22]. The Rotor model is generic but shows realistic behavior. To estimate the influence on the power coefficient, first simulations were performed with OTC and the maximum power coefficient Cp,max is calculated. Then the rotor torque is to simulate the torque control errors. The result is shown in Fig. 14.

Fig. 14
figure 14

Influence of rotor torque control errors on the power coefficient at different wind speeds

Full size image

The calculations were carried out at different wind-speeds between 83% and 100% rated wind speed. Those operating points represent the partial load range of the WT examined, in which the influence on the power output is most relevant. At higher torque deviations the power output is lower than expected. For 83% of rated wind speed and a rotor torque control error of 10% the power output is reduced by 1.2% at the respective operation point. For 100% rated wind speeds the deviation is 2.4%.

4 Conclusions

In this paper an efficiency model for WT drivetrains was developed, which is based on existing component efficiency models. This model was validated with measurements at different operation points on a system test bench. The validated model was used to quantify the influences of drivetrain torque and speed on the efficiency. Furthermore, influences of NTLs and temperature on the drivetrain efficiency were simulated. On the basis of the results the influence of the rotor torque control error on the power coefficient was simulated.

Key findings of the paper:

  • The drivetrain efficiency was simulated with an accuracy of up to 0.7%. This accuracy is sufficient for most applications and slightly above the measurement uncertainty range of 0.65%. Thus, the simulation is valid for use in the assessment of this paper.

  • The efficiency calculation model shows that NTLs have no significant influence on the drivetrain efficiency since most of the losses appear in the gearbox and NTLs are mainly carried by the main bearing. Therefore, NTLs can be neglected for efficiency models.

  • The efficiency model results show that the temperature has a major impact on the drivetrain efficiency.

  • The variations in drivetrain efficiency lead to errors in rotor torque control. The influence of the rotor torque control error on the power coefficient was determined using an aerodynamic rotor model. The power coefficient decreased by up to 2.4%.

In the future the efficiency model needs to be validated in terms of varying temperature. In addition, the occurrence of load cases and temperature in field operation must be considered. This allows us to assess the impact of the decrease of power coefficient on the total energy production of the WT.

Finally, it can be concluded that the significant errors in rotor torque control result in suboptimal turbine control. Reducing the WT’s power coefficient leads to lower annual energy production, and higher LCOE. An optimized prediction of the drivetrain efficiency, by modeling or test bench measurements, can reduce the rotor torque control errors and ultimately the LCOE.