Proposals for a practical calibration method for mechanical torque measurement on the wind turbine drive train under test on a test bench

Correspondence Hongkun Zhang, Fraunhofer Institute for Wind Energy Systems IWES. Am Luneort 100, 27572 Bremerhaven, Germany. Email: hongkun.zhang@iwes.fraunhofer.de Abstract The mechanical torque input into the wind turbine drive train is a very useful measurement for tests performed on a test bench. To ensure the accuracy and the reliability, an accurate calibration of the torque measurement must be carried out and repeated within a certain period of time. However, owing to the high torque level and large structure size, such a calibration is both expensive and time consuming. To overcome this challenge, a new calibration method is proposed here. The method is based on the electrical power measurement, where a high level of accuracy is much easier to achieve. With the help of a special test process, a relationship between the torque-measuring signal and the electrical power can be established. The process comprises two tests with the drive train running in different operating modes. The calibration is possible by carrying out the same test process on several different torque levels. Detailed uncertainty analysis of the method is presented, whereby the uncertainty can be calculated by means of matrix operation and also numerically. As a demonstration, the implementation of the method on a test bench drive train that contains two 5-MW motors in tandem with the motors operating in a back-to-back configuration is also presented. Finally, some variations on the method and possible ways of achieving better accuracy are discussed.

(also known as "torque transfer standard"). 5,6 It is intended to be a movable reference torque measurement, which can be used to calibrate other torque-measuring devices, for example on a wind turbine test bench. The PTB 1.1-MW torque standard machine has a relative uncertainty of 0.1 %. 7 The partially calibrated and extrapolated 5-MN·m torque transfer standard (TTS) has been found to have a relative uncertainty of 0.15 %. 5 This level of torque accuracy would be more than sufficient for the efficiency determination of the drive train. However, in order to achieve this accuracy in the test, a state-of-the-art torque transducer similar to the PTB 5-MN·m TTS (or the TTS itself) has to be installed directly in front of the drive train. Since the design, manufacture, and calibration of such torque transducers require high levels of professionalism, and also because each transducer has to be individually customized for the specific application, the cost of such transducers can be a big challenge for industrial use.
As a fundamental variable, torque measurement is instrumented in almost all the nacelle and drive-train test benches for wind turbines.
However, there is no standard method for torque measurement. The solution for the measurement is mostly individually designed or specified for each of the test benches. Some discussions about the different methods used on the nacelle test benches are given in the work of Foyer et al. 6 For test benches that apply 6-DOF (degree of freedom) loads, the torque-measuring device is usually not located directly between the test bench and the turbine nacelle. Instead, in order to reduce the impact of the non-torque loads and protect the torque transducer against the shocks, the inherent torque-measuring devices of such test benches are often located behind the load application unit (LAU) that applies for example the bending moment. As a result, the friction torque of the LAU can introduce a major uncertainty to the input torque for the turbine, especially given that the friction torque is also highly temperature and load dependent. Research undertaken by Kock et al 8 has studied the friction behaviours of a specific LAU system on a 4-MW test bench, 9 aiming to quantify the friction torque and the uncertainty introduced by the LAU system. To measure the input torque to the turbine directly, the torque transducer needs to be very robust, in order to withstand the large non-torque loads. One possible way is to use the shaft adapter between the test bench and the nacelle as a torque-measuring device.
Certain strain gauge connections can be applied on the cylindrical adapter to measure the torque-introduced strain on the adapter. A calibration of enough accuracy is then necessary in order to convert the strain signal into torque.
For simpler test benches that apply only the torque, for example, the common end-of-line test benches of the turbine manufacturer, the accompanying non-torque loads are not as critical as above. In this case, torque measurement with professional torque transducers would be possible, but measuring with strain gauges on the adapter or connection shaft is still very attractive with lower cost of both funding and time.
On both types of test benches, accurate and regular calibrations are vital for the torque measurement where the accuracy is important, for example, to determine the drive-train efficiency. However, the high cost and great effort required for the calibration often make it impossible to achieve the desired high accuracy. Therefore, it makes good sense to seek a calibration process that is cost effective, practical to perform, and easy to repeat. Even for those applications where an accurate torque measurement is not necessary, for example when an uncertainty below 5 % is sufficient, such a calibration process is also an important means of validation. This paper proposes a new calibration method that establishes a relationship between the torque and the electrical power, so that the torque measurement is indirectly calibrated through the electrical measurement. The relationship is determined through a series of tests whereby the test bench and the turbine nacelle take turns to drive and generate power. Because no additional structural reconfigurations and only limited instrumentation are needed for carrying out the calibration, the method has a clear advantage in time and cost perspectives. Due to a parameter estimation adopted in the process, the method cannot be categorised as traceable calibration, which is strictly traced back to the national or international standards with known uncertainty. But for easier understanding, the term ''calibration'' is still used in this paper to describe the process of determining the linear relationship between the measured raw signal and the torque. As a countermeasure to the estimation, a detailed analysis of the uncertainty budget related with the method is presented in this paper.

TEST PROCESS
In the previous study, a test process was proposed to determine the drive train efficiency with limited accuracy of the torque measurement. 1011 A further study has shown that the same test process can also be used for the calibration of the torque measurement. A relationship between the torque and the power measurement can be obtained through the test.
Two tests are necessary for the process, which are named test A and test B, respectively. The layout and the configuration of the tests are illustrated in Figure 1, where the power losses through the drive train are demonstrated using the level change of the power curves.
The factor k is introduced as the relationship betweenP loss.A andP loss.B , as expressed in Equation (2). The equation can be further transformed into the expression in Equation (3), with the relationship from Equation (1).
Other than drawing the relationship between the mechanical and electrical powers through the absolute drive train efficiency, Equation (3) forms the relationship based on the distribution of total power loss, which is depicted through a dash line in Figure 2. This is also the basis of the torque calibration method proposed in the present paper.

METHOD OF TORQUE CALIBRATION
The mean values of the mechanical powerP mech.A andP mech.B can be expressed as the accumulated energy divided by the corresponding duration, where the accumulated energy is the integrated torque in the angle domain. As a result, Equation (3) can be expressed further as Equation (4). In this equation, T denotes the torque, t A and t B are the durations of test A and test B, respectively, A and B are the corresponding accumulated angle displacements in test A and test B. Since the angular position is the raw signal from the encoder, it is more accurate to determine the power with the help of angle domain integration. Although encoders are often used on a wind turbine for measuring the rotational speed, the speed is in fact a derivative of the original position signal. For better accuracy, A and B can be chosen as integer multiples of 2 , so that the influence of the non-torque loads can be better compensated with signal from integer number of revolutions.
Replacing the theoretical true variables with the measured values, T and P elec can be expressed as in Equation (5), where • represents the measured raw signal from the the torque measurement channel. In many cases, the raw signal is the strain measured by the strain gauges. Depending on the measurement principle and the channel configuration, the raw signal can also be other variables.
• a and b are the sensitivity and offset parameters that define the linear relationship between the measured raw signal and the true torque T.
Due to the non-linear behaviour of the measurement and external influences, the real relationship can be more complicated than the linear one defined by a and b. The discrepancy from the linear behaviour is attributed to a time-varying quantity r(t), which also includes the repeatability error of the measurement, for example the error caused by temperature change and the noise from EMI (Electromagnetic Interference).
•P elec is the measured signal of the electrical power. c and d are the sensitivity and offset errors of the measurement with respect to the true electrical power P. Similar to the r(t) in the torque measurement, q(t) denotes here the non-linearity and the repeatability errors of the electrical power measurement.
Using the definitions in Equation (5), the mean torque values can be expressed as Equation (6), wherēA and̄B are the mean rotational speeds in test A and test B, respectively. For the mean rotational speed, no distinction is made here between the theoretical true value and the measured value. The high positional accuracy provided by the encoder and the relatively long test period make it possible that the error of the measured mean speed can be regarded as zero. The right-hand side of Equation (4) can be expressed as Equation (7), withP elec.A andP elec.B being the mean values of the measured electrical power in test A and test B. (4) can be transformed into Equation (8).
Denoting X and Y as expressed in Equations (9) and (10), the relationship defined in Equation (4) can be expressed in a concise form shown in Equation (11). A very clear linear relationship is seen in this equation, with a and b being the sensitivity and offset parameters. In fact, the aim of the calibration is exactly this-to determine the values of a and b.
It is obvious here that the parameters a and b can be determined with at least two sets of X and Y values at different torque levels. This means the test procedure described in Section 2 should be carried out on several different torque levels. If more than two sets of values are available, a and b can be determined through linear fit using the least squares method.
As can be seen in Equations (9) and (10), the X and Y values are expressions of the mechanical and electrical measurement as well as the factor k. Additionally, some error terms are also in the expression, including c, d, Q A , Q B , R A , and R B . The measured mechanical and electrical values are of course known for the calculation. The factor k must be determined through analysis to become a known value, denoted ask, with the uncertainty taken into account. Note that the k factor is not necessarily estimated as 1. In the calibration process, the error terms in the expression do not play any part, but they will be taken as uncertainty contributions in the following uncertainty analysis.
For the calibration, theX-Ỹ value sets can be used for determining the parameters a and b.X andỸ are defined in Equation (12). All the terms in the expressions are known values, and therefore, the calibration can easily be carried out. Suppose the test process in Section 2 is carried out on N different torque levels (N≥2), the estimated values of a and b, denoted as â andb, can be obtained through the linear fit, 12, p246 and the results can be expressed as Equations (13) and (14), where i is the index number of a certain torque level,X i andỸ i are the corresponding values obtained on the basis of that test. Note that the estimated valuek may be different for tests with different torque levels.
With the â andb known, and taking the definition in Equation (5) as the basis, the estimated torqueT can be calculated in Equation (15). The term r(t) contributes to the uncertainty of the torque calculation.T With r(t) being considered as a source of uncertainty, the expression can be simplified to Equation (16).

UNCERTAINTY CONSIDERATIONS
As a result of the calibration process, the torque measurement can be expressed as a function of the estimated â,b and the measured raw signal . A time-varying variable r(t) takes account of the repeatability error of the measured signal as well as the non-linearity error of the Trelationship. This situation is presented in Equation (15). The uncertainty inT is therefore dependent on â,b, and r(t).
The uncertainty of the measured torque is ultimately composed of two parts. One part originates from the raw measurement itself, the other from the calibration. The uncertainty in the raw measurement represents the quality of the measured raw signal, which is affected by the signal noise, non-linear and hysteresis behaviour, signal drift due to temperature change and humidity, as well as other possible factors that make the signal less repeatable. The uncertainty from the calibration indicates how accurately the sensitivity and offset parameters are derived from the calibration. This paper will focus on the calibration uncertainty. Although the raw measurement uncertainty is not of great interest here, it does exert an influence on the calibration uncertainty, because the raw signals have to be used in the calibration process. A better calibration cannot improve the quality of the raw measurement, but better raw signals can indeed produce a more accurate calibration.

Uncertainty of the raw measurement
Since the variable is defined as the value of the raw measurement, the uncertainty of itself should be considered to be zero. As a result, the uncertainty of the torque measurement directly owing to the raw signal is attributed to r(t), which is introduced in Equation (5) to represent all the error contributions that cause the measured signal to deviate from the ideal linear response to the true torque. The error contributions can be grouped into two categories, namely, the non-linear behaviour and the repeatability errors. The total uncertainty contributed by r(t) is denoted here as u T.r . It characterises the quality of the measuring channel.
The non-linearity of a measurement is strongly dependent on the principle used and the measurement chain. In the case of a strain gauge-based measurement, it depends on the material and structural non-linear behaviour of the part to which the strain gauges are applied, as well as the non-linearity of the Wheatstone bridge circuit. 13 Further, non-linearity comes also from the data acquisition (DAQ) system of the measurement.
The repeatability error of the measurement comes from a number of sources that cause the measurement to be time-varying. A common source is the background noise in the measured signal due to EMI and the thermal noise 14 in the strain gauge resistors. Creep effect of the material and temperature change both cause a signal drift in the measurement and therefore also contribute to the repeatability error. Another contributor is the so-called "cross-talk" effect, which means that the load in other directions exert an influence on the torque measurement.
Despite the effort and common methods available for the compensation against cross-talk, 15 a satisfactory compensation is usually very difficult on a wind turbine. 16 Fortunately, in some cases, the transient value of the measurement is of less interest, for example, in a stable operation on the test bench. The torque can then be determined by the mean of a period of time, or more precisely the mean value of an integer number of revolutions. This will be the case for a torque calibration test process as mentioned in Sections 2 and 3.

Uncertainty associated with the calibration
The parameters â andb are the results of the linear fit of a number of (X i -Ỹ i ) points in the X-Y plane. This is schematically illustrated in Figure 3, where four points are available. For each point, bothX i andỸ i have uncertainties. The uncertainties are presented in the form of error bars, with Furthermore, the uncertainty covariances of each two values also exert influence on the overall uncertainties of â,b. These include the uncertainty covariance of each two differentX values, of each two differentỸ values, and the uncertainty covariance of eachX andỸ combination.
In fact, because the same k factor is used in the expression of bothX andỸ, and also since the measurements on different torque levels are obtained from the same measurement chain, the covariances are of considerable significance. The standard uncertainty inT due to the linear fit parameters â andb (which are also the products of the calibration) is denoted in the following as u T.cal .
The uncertainty u T.cal represents the uncertainty of the (â +b) part of Equation (15), where â andb are functions of N sets ofX i -Ỹ i values.
Following the principle of uncertainty propagation, 17 u T.cal can be determined, 12, p253 as expressed in Equation (17).
where sX i and sỸ i are the random standard uncertainties in the variablesX i andỸ i , respectively. X i andỸ i similarly denote the systematic standard uncertainty ofX i andỸ i . Y iỸk ,X iXk andX iỸk are the covariance estimators for the correlated systematic errors of the corresponding variables in the subscript. Examining the equation, it is seen that most of the terms are very easy to obtain: • The partial derivatives of̂T X i and̂T Y i can be estimated numerically on the basis of the finite difference method. According to Equations (13), (14), and (16), the estimated torqueT can be explicitly expressed with the estimatedX,Ỹ values and the measured signal , all of which are known or already determined.
• The random uncertainties sX i and sỸ i refer to the uncertainty components which are independent and have no correlation with other components. Based on the definition of X and Y in Equations (9) and (10), the random uncertainty sX i can be considered as zero, because the accuracy of the time and angle position measurements (and hence the mean speed as well) are very high, and as a result, their uncertainties are negligible. The value of sỸ i is mostly dependent on the random uncertainty of the electrical power measurement. • The systematic uncertaintiesX i andỸ i are introduced by systematic reasons and can therefore be related with the uncertainties from other measurements. For example, the uncertainty in the measured electrical power on a certain power level can be related to the uncertainty on another power level or on the same level from another test. The values ofX i andỸ i are dependent on the uncertainties from the estimatedk factor and the electrical power measurement.
• The effect of the correlations between the systematic uncertainties is quantified in the calculation with covariancesỸ iỸk ,X iXk , andX iỸk . The covariance can be further expressed as a function of two systematic uncertainties and the corresponding correlation coefficient between them, 17 as expressed in Equation (18). Therefore, the task of determining the covariances is converted to the determination of correlation coefficients.
Although Equation (17) The Hadamard product of matrix (v T v), ( T ) and R, denoted as U sys in Equation (21), contains all the systematic uncertainty components in u 2 T.cal as expressed in Equation (17). A summation of all the entries of matrix U sys is easy to achieve in the calculation and can be expressed analytically as eU sys e T , where e is a row vector with all elements being 1. Therefore, with the help of matrix operations, the expression of u 2 T.cal in Equation (17) can be converted into a concise form shown in Equation (22).
In summary, the systematic uncertainty can be obtained using the two vectors v and , together with the correlation matrix R. The calculation process is suitable for matrix operations, which can be effectively carried out with computer programmes or commercial mathematical software.
While v and can be easily obtained through numerical calculation, the matrix R requires more analysis of the uncertainty source. Due to space limitation, the analysis is not discussed in detail in this paper. Nevertheless, the matrix R is not difficult to obtain as most of its entries are close or equal to zero or one.

EXPERIMENT WITH THE METHOD ON A 10-MW NACELLE TEST BENCH
During a re-commissioning of the motors on the nacelle test bench ''DyNaLab,'' 18 the mechanical torque on the structure between the two tandem motors is needed for the validation of the electrical model of the motors, which is a basis of the motor controller. In order to measure the torque between the motors, strain gauges are applied to the intermediate shaft between the motors, and thus the measured signal from the strain gauges needs to be calibrated. One possible calibration method is the FEM analysis of the intermediate shaft, where the relationship between structure strain and torque can be established. However, the accuracy is considered insufficient due to the accumulated uncertainty from the strain measurement and the FEM analysis. Another method is to apply torque on one side of the drive train with a lift crane over a lever of known length. The problem with this method is the logistic expense and the difficulty to lock the drive train on the other side.
After evaluation of possible ways of calibration, the method proposed in this paper is considered to be the most convenient and accurate way of calibration. Additionally, the efficiencies of the motors at different operation points can be also determined as by-products. The method and test process are therefore carried out during the re-commissioning.    Table 1. In the normal operation, the two motors work together to provide the driving torque of the test bench. During the re-commissioning, the motors run in back-to-back mode instead. In application of the new method, motor I and motor II correspond to the test bench and the turbine, respectively, as described in Sections 2 and 3.

Instrumentation of the measurement system
The drive train layout of the two motors is shown in Figure 5.    Figure 7.
During the test, the manholes have to be closed with the covers. Therefore, the cables of the full bridge channels are led out through a small hole (available on another cross-section) to the data acquisition (DAQ) system, shown in Figure 6. The DAQ system is fixed to one of the manhole covers, with the help of the bolts for the cover. Since the strain gauges together with the DAQ system are located on the rotating part, a WLAN (Wireless LAN) connection is utilised for the communication between the rotating system and the stationary parts, including the data and time synchronisation server, as well as the remote measurement controller. Time synchronisation on the rotating DAQ system is realised through the WLAN connection in the network time protocol (NTP). The router of the WLAN system is also fixed to the cover. To ensure the quality of the wireless connection, three antennas are placed at equal separations along the circumference of the intermediate shaft.
In addition to the torque measurement, the speed and angular position are measured by an absolute encoder at the end of the test bench drive train. The encoder measures the rotation of the rear shaft relative to the adjacent bearing housing. This encoder is part of the test bench and has a resolution of 16384 CPR (counts per revolution). The signal is connected to a stationary DAQ system, with the same time reference as the rotating DAQ system.
To measure the electrical power, voltage and current sensors are installed on the input power cables of the two motors. A high sampling frequency DAQ system is used to measure the alternating voltage and current signals. The electrical power can be processed on the basis of the voltage and current measurements. The same source of time reference is also used in this system for the signal time stamps.

Back-to-back test and torque calibration
Back-to-back tests are carried out following the method explained in Section 2. The electrical measurement of motor II is chosen as the reference during the test process. Since motor I is connected to a number of other components in the front, including the moment bearing of the load The torque calibration is then calibrated in the range of 20 % to 85 % of the rated torque.
The raw data of the measured electrical power, the strain, and the rotational speed are shown in Figure 8, where the real time series of the complete test sequence are displayed. The segments corresponding to the test in mode A and mode B are highlighted in yellow in the figure. The system needs to come to a halt between the two operating modes because the drive train needs to change the direction of rotation. At the same time, the motor controllers also need to be reconfigured for the reversed operating mode. Although blank segments are seen in the signals of the electrical power and rotational speed, the test and the measurements are indeed carried out sequentially and continuously. Here, the speed signal is shown to provide a better indication of the operating condition, although the angle position instead of rotational speed is used in the calibration.
In Figure 8, the steps of different power levels can be clearly seen. For each step, the operation is kept on a stable state for at least 180 seconds, in order to have a stable measurement of sufficient length for the subsequent signal averaging. The power levels are marked on the plot of the electrical power. A positive electrical power stands for active power output (generator mode), while the negative power values indicate power input and that the machine is operating in motor mode. The motor speed stays at a very stable amplitude during both test A and test B.
The rotating direction in test B can be seen to be opposite to the direction of test A. This ensures that the mechanical torque measured by the strain gauges is in the same direction for both test modes.
Before the torque calibration is possible, the k factor that describes the relationship between the power losses in the different operating modes as defined in Equation (2) must be estimated. In this test, thek factor is assumed to be 1.0 for all five test levels. Afterwards, the estimatedX and Y can be obtained on each test level using the measured data as per Equation (12   -Ỹ points and the linear fit with the least squares method same unit as the raw measurement , a strain signal. The widely used unit micro-strain is adopted here as the unit ofX and expressed as m∕m in accordance with the common practice in the industry. TheỸ is the result of power divided by rotational speed and therefore has the dimension of torque. The unit kN · m is adopted here as the unit ofỸ. The values are listed in Table 3. Each set ofX -Ỹ values in Table 3 can be plotted as one point in the X-Y plane. Five points corresponding to the five test levels can be drawn as shown in Figure 9. A linear fit of the five points is then possible. The slope and intercept of the line can be determined as â = 150.78 kN·m∕( m∕m) andb = −6876.9 kN · m, as per Equations (13) and (14). The calibrated torque measurement can be expressed as in Equation (24), with the unit of estimated torqueT being kN · m and the unit of raw signal being m∕m.
Unlike the conventional calibration methods that generate an accurate reference torque, the method described in this paper uses the electrical power as the reference and is based on the fact that the electrical power can be measured easily and with good accuracy. To present a better, visual explanation of the principle, the mean values of the mechanical, and electrical powers will be calculated and presented below. As the torque measurement is already calibrated and expressed in Equation (24), the average mechanical powersP mech.A andP mech.B on a certain test level can be expressed in Equation (25). The mean measured electrical powersP elec.A andP elec.B can easily be calculated in the normal way of averaging over the time domain.
The mean mechanical and electrical powersP mech.A ,P mech.B ,P elec.A , andP elec.B can be calculated for each of the five test levels listed in Table 3.
For a better understanding, the values are plotted in Figure 10 together with level lines, similar to the procedure adopted in

Uncertainty of the torque calibration
The uncertainty of the torque calibration comes mainly from the raw torque measurement, the electrical power, and the estimated k factor. Basic parameters used in the calculation are listed in Table 4. The notations of the error components are given in accordance with the definition in   The electrical power measurement result from the three-phase voltage and current measurement, where high accuracy industrial sensors are used. The whole measurement chain is considered for the uncertainties of the voltage and current measurement channels, whereby the uncertainty of the sensors turns out not to play a major role. One of the important reasons is that the power factor of the motor is close to 1, which makes the otherwise very important phase error of the sensors less relevant to the overall power uncertainty. The miscellaneous errors of the electrical and mechanical powers Q and R are integrated over a period of time or angular rotation, as defined in Section 3. As mentioned in Section 5.2, the estimated factork is adopted as 1 for all the five test levels. A uniform error of k is also assumed for all the levels. Of course, both the estimated factork and its error k can be considered individually for each test level. Here, a uniform assumption is adopted for the sake of simplicity. As a result, a larger error range k is also taken into consideration.
With the error parameters in Table 4 and according to the expressions forX andỸ in Equations (9) and (10), the uncertainties ofX andỸ can be calculated as shown Table 5. The subscript i in the table denotes different test levels. The five levels from 20 % up to 85 % presented in Table 3 correspond to i being 1 to 5, respectively. In accordance with the discussion in Section 4.2, the systematic and random uncertainties are listed separately. Three contributions of the systematic uncertainty ofỸ are also listed in the table. Note that the random uncertainties ofX are all zero, because the relevant uncertainties are attributed to the Y side, as shown in the definitions of the X and Y in Equations (9) and (10).
The correlation coefficient matrix R used in the calculation is presented in Table 6. The matrix R is defined in Section 4.2, and each of the entries represents the correlation coefficient between the corresponding standard systematic uncertainties. In Table 6, the corresponding standard systematic uncertainties are indexed in the first row and the first column. The values are taken with conservative considerations, and therefore, most of the entries are 1 or 0.
The estimated torqueT is a function of the measured signal , with the estimated â andb as the sensitivity and offset parameters, shown in Equation (15). Therefore, the uncertainty ofT comes from the uncertainties of â andb, which are in turn dependent on the uncertainties of theX andỸ values. Although by definition no uncertainty is attributed to the raw signal , it does exert an influence on the uncertainty ofT by affecting and scaling the contribution of â. This influence is reflected in the partial derivatives ofT with respect toX andỸ in Equation (17). The partial derivatives have different values on different torque levels. An example is given in Table 7, where the measured raw signal equals 65 m∕m, X 1X2X3X4X5Ỹ1Ỹ2Ỹ3Ỹ4Ỹ5   For the entire uncertainty ofT, the component r(t) in Equation (15) also plays a role. It can be considered to be the uncertainty related to the signal quality. Again, since this paper focuses on the calibration method, only the uncertainty due to the calibration process will be analysed here.
As defined in Section 4.2, the standard uncertainty of (15) caused by the calibration is denoted as u T.cal . The expanded uncertainty with the 95 % confidence level is twice the value of u T.cal , and is denoted as U 95 . The uncertainty results ofT due to the calibration on several torque levels are given in Table 8, where u T.cal , U 95 as well as the percentage of the U 95 relative to the nominal torque are also presented.
It is seen in Table 8 that the uncertainty stays at a relatively stable level as the torque level increases. This is partly due to the conservative assumption made in Table 6 for the correlation coefficients. As the uncertaintiesX i andX k are assumed to be well correlated, this also means theX-Ỹ points in Figure 9 deviate more collectively from the true positions. Other reasons are the constant components that are not dependent on the torque level, for example, the offset error of the electrical power. Considering the percentage of the uncertainty (U 95 ) in relation to the measured torque, 0.66 % is achieved near the calibration upper limit around 3600 kN · m. At lower torque levels, the percentage of uncertainty will be higher. However, for the majority of the measuring range, an uncertainty of less than 2 % of the nominal torque is achieved.

DISCUSSION
The calibration process described in this paper focuses on the measurement and behaviour on the turbine side. Both the electrical and mechanical powers input and output of the the turbine are measured. In fact, the method can also be realised on the measurements on the test bench side with the electrical power of the test bench being measured. The method actually requires only two electrical machines that can both run in motor and generator modes and also in different directions. In this sense, the calibration method can also be used in other industries and applications where a similar test layout can be realised. One advantage of focusing on the test bench is that the properties of this object are stable regardless if a different turbine is under test. The disadvantage is the potentially greater uncertainty due to the large friction torque on the test bench.
The method can be carried out on turbines with or without a gearbox. As the turbine changes its operating mode and rotational speed at the same time in the reversed test mode, the gear teeth inside the gearbox always mesh on the same side. This eliminates the concern that the gearbox should not run under load in the opposite direction because the teeth are manufactured differently on the two sides. Nevertheless, it is recommended to consult the gearbox supplier regarding potential problems and the efficiency difference in the two test modes.
To increase the calibration accuracy, efforts should be taken on the whole measurement chains of all electrical and mechanical raw measurements. For the electrical power, special attention should be paid to the time delay and hence the phase errors between the voltage and the current signals. For the torque measurement, the signal drift resulting from temperature change and creep has a direct impact on the calibration results. Furthermore, a proper time-synchronisation is imperative, in order to avoid time and phase shift between different signal channels.
For the test process, several efforts can be taken in the preparation and during the test. First, a preheating prior to the calibration test process is necessary in order to reach a relatively stable temperature and lubrication status for the whole drive train and the measurement system.
Second, the ambient temperature should be kept as stable as possible during the whole test process, so that temperature changes in all systems are kept under control. Third, the electrical power input and output amplitudes should be kept as close to each other as possible. In the case shown in Figure 10, further improvement could be achieved by better controlling the electrical powers to be at the same level. Another effort is to try to have the power factor of the electrical machine as close to 1 as possible, which helps reduce the uncertainty in the electrical power.
The biggest contribution to the uncertainty for the calibration is in many cases the uncertainty in the assumedk factor. As the k factor cannot realistically be determined by a test, it must be chosen either with the help of detailed analysis of the drive train and the electrical machine or with an approximate assumption which results in a large uncertainty. Therefore, for a smaller uncertainty of thek factor, a detailed analysis of the power loss and the efficiency of the drive train is necessary.

CONCLUSIONS
The proposed new calibration method takes advantage of the high accuracy of the electrical power measurement while at the same time it greatly reduces the uncertainty introduced by the power losses in the drive train. Based on the fact that the drive train tends to have a similar power loss in both motor and generator mode, the effect of the power loss can be largely compensated by carrying out the test in both the normal mode and in the reversed mode. With measured data from both test modes, an equilibrium can be drawn between the electrical and mechanical powers.
The calibration can be carried out with the help of several such equilibriums on different torque levels.
No special instrumentation is needed for the method. All the sensors and equipment used in the calibration are produced in series and commercially available. The necessary test process can be carried out with the same mechanical layout as the normal test, which means that no additional structural reconfigurations or logistical effort is necessary. As a result, the calibration is low cost and time efficient. It can also be easily repeated whenever necessary.
One prerequisite of the calibration method is the ability of the test bench and the DUT (device under test) to run in reversed mode and operate in both directions up to the rated power. The calibration method is demonstrated on a torque measurement between two 5-MW driving motors of a nacelle test bench for the wind turbine, where both the calibration parameters and the accompanying uncertainties are determined.
With conservative considerations, the uncertainty due to calibration is, in this case, within 0.7 % at upper limit of the calibration range. Better calibration accuracy can be achieved by reducing the electrical power uncertainty, improving the quality of the raw torque-measuring signal, and most importantly, by a better analysis of the drive train and therefore a smaller uncertainty in the k factor estimation.