Add preliminary uncertainty analysis to test plan

[petebachant]

Also note that we will be averaging over an integer number of blade passages, and that final uncertainty values will have their own columns in the processed data CSVs.

Question: How to determine statistical uncertainty of the sample mean for an approximately sinusoidal quantity? The standard deviation will be very high, but not necessarily due to high random error.

[bgunawan]

Which quantity are you referring to?

My understanding is that we are only interested on measurements with a constant tow tank speed. In that case, the uncertainty analysis will be the same as the one with SAFL.

[petebachant]

Take power coefficient for example. We use the measurements to estimate the true mean value. Even during steady operation, C_P varies like a sine wave with frequency equal to the blade passage frequency. We will compute a sample mean from this time series, which will include 10--15 revolutions per tow at TSR = 2.0.

To compute the confidence interval on the sample mean, we use [ref]

Where N is the number of samples, t is the value from the Student t distribution for N-1 degrees of freedom, and s is the sample standard deviation.

So, the question is how do we compute N and sample standard deviation? If we use the total number of samples (at 2 kHz) as N, we will get a large sample standard deviation. If we reduce N by averaging over blade passages or revolutions, the sample standard deviation will decrease.

I'm thinking the right way to do this is to consider the mean over each revolution one "sample" for computing the statistical confidence interval. I will do the calculation with some old data to see how this works. Once we take a look at this it may give us a reason to perform more tows for a single datapoint to shrink the error bars.

Note I am not yet considering but haven't forgotten the error of each of the 2 kHz samples, or the systematic error.

I am also concerned that this formula in the document Vince sent does not have a 1/sqrt(N) somewhere in it:

[bgunawan]

Uncertainty is not exactly the same as confidence interval. We used the uncertainty equation from ISO, showed in Coleman and Steele book (see chapter 2 pages 55-end). I'll send that to you shortly. This was what we used for the SAFL experiment.

"Even during steady operation, C_P varies like a sine wave with frequency equal to the blade passage frequency. " Just looked at my old notes again. We need to calculate the uncertainty of the mean quantity, not the instantaneous. For the SAFL ADV data we parsed the samples into n number of timeseries, calculated the mean of each of the timeseries, and calculated the standard deviation of the means (see fig below for example). We tried different averaging windows and decided to select 180 s. I think I have a matlab code for this - will look.

The sinusoidal fluctuation is assumed to be part of the uncertainty. For now, my thought on this is to use the same approach for this testplan. Then we can think if we need to revise it for the final report between now and March. FYI, Cp uncertainty is propagated from uncertainties of a set of measured qunatity, e.g., torque, omega, velocity. Did you guys use torque and omega measurements to calculate power?

"So, the question is how do we compute N and sample standard deviation? If we use the total number of samples (at 2 kHz) as N, we will get a large sample standard deviation. If we reduce N by averaging over blade passages or revolutions, the sample standard deviation will decrease." Since we only have good data for 10-15 revs for each tow, I'd parse the the data into timeseries for each rev.

[petebachant]

I started some calculations using RVAT data, which can be seen here.

Using the "dumb" method, i.e., treating all samples as independent, the confidence interval is lower than I expected for a single run (since the std is high), but the histogram does not look Gaussian, so this method may be flawed.

Using 270 separate runs, with the same conditions, the confidence interval is smaller, and things look Gaussian.

Next step is to try treating each revolution as an independent sample as discussed above.

[petebachant]

Did a calculation using each revolution during a single run to compute a confidence interval on the mean power coefficient, and it looks pretty good: +/- 0.002. See the updated notebook.

[petebachant]

Did some sample work with wake velocity measurements: http://nbviewer.ipython.org/github/UNH-CORE/RVAT-Re-dep/blob/master/Documents/IPython%20notebooks/uncertainty.ipynb#Uncertainty-of-wake-velocity-measurements

[petebachant]

I'm satisfied with the latest methods (from Coleman and Steele, treating each revolution as a measurement) in the RVAT-Re-dep uncertainty notebook. I will integrate these calculations into the data processing code, and include some sample values in the test plan report.

[petebachant]

Added to test plan doc by https://github.com/UNH-CORE/RM2-tow-tank/commit/f6f679bceec8a3e50b7f09f4118854c7179245ea.