#### B.1. Deriving Single Measurement for Inflow Velocity

The IEC 62600:200 technical specification sets out the approach for determining the power performance of a device using ADP inflow data. The ‘method of bins’ enables the velocities in the shear profile across the projected area of the rotor to be represented as an area weighted and power averaged single point measurement.

Figure A1 illustrates this method, the projected area of the turbine is slices into a series of sections representing the depth bins of the ADP. Meanwhile, the tubular sections represent each sample, or time. For the set-up in the TTT 3 project, each depth bin is 0.25 m deep and each time period is 1 s. Each section has an area

A_{K} and a velocity measurement

U_{i,j,k,n} (see nomenclature for definitions).

The IEC approach to obtaining a single value for the flow across the swept area of the turbine during a given test case follows this sequence of equations [

23]. Firstly, the power-weighted and area average for the projected area of the turbine for each period is calculated in Equation (A1), and illustrated by the thicker lines in

Figure A1.

The equation uses the instantaneous velocity measurement in each of profiler depth bins (${U}_{i,j,k,n}$), which was cubed and weighted by the area of the depth bin (A_{k}). The sum of these is then divided by the total swept area (A) and cube rooted. This provides the power-weighted current velocity (${\widehat{U}}_{i,j,n}$) for each period.

**Figure A1.**
Power-weighted current velocity calculation illustration.

**Figure A1.**
Power-weighted current velocity calculation illustration.

It is at this point that the adaptation of the IEC ‘method of bins’ is introduced. All ADP measurements are subject to uncertainty from Doppler noise. The IEC method requires each 1 s sample at each depth cell to be cubed, as shown in Equation (A1), thus cubing the measurement error from Doppler noise. To correct for this, the Doppler noise bias correction method has been developed. To demonstrate the analytical derivation for the Doppler noise bias correction; let us take m as the mean and σ as the standard deviation of a normal variate. The variable q is introduced, which is related to the variance of the distribution as shown in Equation (A2).

In the case of power measurements derived from ADP data, it is the distribution of the cube of samples, (

X_{k})

^{3}, which have been obtained from the normal distribution

N(

m,

σ) that is important. The effect of cubing a measurement with an included sampling error was first formally examined by Haldane in statistical biology [

31]. Haldane showed that the mean of this resulting distribution exceeds

m^{3} by the ratio

R, as follows in Equation (A3).

The values of q and m, as defined above, are readily obtainable from the ADP data. We have called the ratio R the ‘Doppler noise bias’, and R can be applied as a correction wherever the cube of a noisy velocity signal is sought and the ratio of m and σ is known. For the derivation of the power-weighted, area-averaged velocity, as previously described in Equation (A1), this can now be re-written to include the bias correction factor derived, as shown in Equation (A3). The outcome of this is an unbiased velocity measurement and reduced uncertainty in the propagation of the performance metrics.

The datasets were averaged over periods between 2 and 10 min for the Strangford Lough testing. For CNR-INSEAN, due to the limited length of the tanks, the maximum averaging period was approximately 90 s; however, given the controlled nature of the experiments in the laboratory this was not considered to be an issue. The mean velocity for the data set (

${\overline{U}}_{i,n}$) is calculated from the power weighted values

${\widehat{U}}_{i,j,n}$ over the time period from

j = 1 to

j = L, the length of the run is in seconds. Lastly, the average for all the velocities recorded in the given current velocity bin is calculated. The velocity bin increments were set to 0.10 ms

^{−1} and only flood phase of the tide is considered. These steps are described and equations defined in previous work [

13,

23].

The turbine’s instrumentation as described earlier includes a torque sensor and rotational encoder on the driveshaft behind the rotor. These outputs provide the mechanical power (P, W) of the rotor in advance of drivetrain losses. Power is calculated using Equation (A5).

The same velocity bin increments apply here also. When sampled over the same time period as the ADP data, the turbine data has more sample points (n), as it is sampled at 16 Hz as opposed to the 1 Hz sample frequency of the ADP. The mechanical power performance can then be calculated using the non-dimensional performance characteristic, C_{P} as previously described in Equations (1) and (2) inserting Equations (A4) and (A5). The water density was set to 1000 kg∙m^{−3} and 1025 kg∙m^{−3} for laboratory and field data respectively.

#### B.2. Propagation of Uncertainty

Understanding of the propagation of instrument uncertainties is crucial to determining the confidence intervals of derived performance characteristics. Similar studies into the propagation of uncertainty have been conducted in this area before [

32]. The previous work showed the significance of uncertainties in the torque, thrust and bending moments when propagated to derive the power performance coefficient. The propagation of inflow uncertainties was not specifically considered by Doman et al., due to the close control afforded by towing tank experiments. For experimental set-ups in real tidal flows, this exception can no longer be made. Therefore, the following section concentrates on the propagation of the velocity uncertainty from the ADP data, through the method of bins, used by the IEC technical specification (TS62600:200).

To ascertain the uncertainty of the derived performance indicators of a tidal turbine, the uncertainty of each variable in the derived performance indicator must be pooled. Equations (A6)–(A8) shows the propagation of uncertainty equation associated with each of the performance indicators.

As Equations (A7) and (A8) show, the propagated uncertainty is most sensitive to uncertainties in the inflow velocity. The uncertainty variables are derived from the Root Mean Squared (RMS) of bias and precision uncertainties, as shown in other work [

13,

32]. This is the case for all uncertainty parameters derived, with the exception of the inflow velocity uncertainty. The exception to the inflow velocity is due to the correction applied for Doppler noise bias in Equation (A6). This correction accounts for the bias uncertainty, leaving only the precision uncertainty, which is calculated in Equation (A9)