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Calculation and Re-calculation of 60-minute Sigma Theta and Stability

Calculation and Re-calculation of 60-minute Sigma Theta and Stability. Dale Paynter Operations Management Group 9 October 2003 NUMUG Conference. Calculating Standard Deviation. Definition. Sigma Theta wind direction is not continuous

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Calculation and Re-calculation of 60-minute Sigma Theta and Stability

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  1. Calculation and Re-calculation of 60-minute Sigma Theta and Stability Dale Paynter Operations Management Group 9 October 2003 NUMUG Conference

  2. Calculating Standard Deviation • Definition • Sigma Theta • wind direction is not continuous • one pass to compute mean, second pass to compute sigma theta • requires large memory in datalogger where X is a set of samples and  is the mean of the X values • Single Pass Computation accumulate the sum of X, the sum of the squares, and the number of samples OMG

  3. Historical Summary Several one pass estimators were developed and compared Yamartino • developed a method based on assumptions of isotropic, symmetric wind distributions • developed interpolation factor based on Monte-Carlo simulations with same assumption • showed better performance against Ackerman-Verrall and Williams methods Turner • Compared the 3 methods against limited real data and more simulated data - Yamartino preferred OMG

  4. Historical Summary II • Fisher • challenged isotropic or simple wind models • argued for more extensive computations • suggested algorithms were within capability of existing minicomputers, but not dataloggers OMG

  5. Yamartino Method Definition Compute the average sine of wind direction, the average cosine, and epsilon Compute sigma theta as the arcsine of epsilon, and apply a correction factor Restrictions ANSI/ANS-3.11-2000 contains the following advice on the Yamartino method: "Note that calculations of sigma theta values for a one-hour sampling period ... could be inflated by the contributions from long-period oscillations associated with light wind speed conditions (e.g., plume meander). To minimize the effects of plume meander … four 15-minute values should be combined to represent an hourly sampling period as follows: OMG

  6. OMG

  7. Pasquil Stability Class • Characterizes atmospheric stability • More sensitive to sigma theta changes at low sigma theta (most stable) • Frequency Distribution report categorizes wind speed and direction by stability class OMG

  8. Average 15-minute and 60-minute stability class values for one year 60-minute consistently indicates more unstable For 200’ sensor, indicates more unstable by one half stability class Comparison of Stability Class OMG

  9. Comparison of Sigma Theta • Average 15-minute and 60-minute sigma theta values for one year, along with average RSS computation • 60-minute consistently higher sigma theta OMG

  10. Stability Class Difference • For each hour, difference in stability class between 60-minute sigma theta and RSS of 15-minute sigma thetas • Difference is positive where 60-minute value indicates more unstable than RSS value OMG

  11. Stability Class Change by Wind Direction Change Wind Swing (Estimated) Stability Class Difference Sectors Count -2 -1 0 1 2 3 4 1 7256 0 4 5436 1756 60 0 0 2 955 0 0 167 577 203 8 0 3 254 1 1 45 43 105 58 1 4 103 0 0 25 31 31 16 0 5 69 0 0 26 21 18 4 0 6 44 0 0 28 8 8 0 0 7 26 0 0 23 3 0 0 0 8 23 0 0 21 1 1 0 0 9 23 0 0 21 2 0 0 0 10 6 0 0 6 0 0 0 0 11 6 0 0 6 0 0 0 0 12 & up 0 0 0 0 0 0 0 0 Totals 8765 1 5 5804 2442 426 86 1 0.01% 0.06% 66.22% 27.86% 4.86% 0.98% 0.01% OMG

  12. Stability Class Change by Wind Direction Change 70 % 60 % 50 % 40 % 30 % 20 % 10 % 0 % Estimated Number of Sectors (22.5o) Observed During the Hour OMG

  13. Stability Class Change by Wind Speed Wind Speed (MPH) Stability Class Difference Category Count -2 -1 0 1 2 3 4 < 1.0 22 0 0 19 2 1 0 0 1.0 - 2 131 0 0 103 23 4 1 0 2.1 - 4 366 0 1 163 124 56 22 0 4.1 - 6 603 0 0 265 254 73 11 0 6.1 - 8 786 0 0 415 284 75 12 0 8.1 -10 808 0 1 464 280 51 12 0 10.1-13 1352 1 0 856 424 61 9 1 13.1-18 2369 0 1 1748 542 65 13 0 18.1-22 1026 0 0 787 224 13 2 0 22.1-30 878 0 1 655 199 21 2 0 30.1-40 370 0 0 289 76 4 1 0 > 40 54 0 1 40 10 2 1 0 Totals 8765 1 5 5804 2442 426 86 1 0.01% 0.06% 66.22% 27.86% 4.86% 0.98% 0.01% OMG

  14. Stability Class Change by Wind Speed 45 % 40 % 35 % 30 % 25 % 20 % 15 % 10 % 5 % 0 % < 1.0 1.0 - 2 2.1 - 44.1 - 6 6.1 - 8 8.1 -10 10.1-13 13.1-1818.1-22 22.1-30 30.1-40 > 40 Wind Speed in Miles per Hour OMG

  15. Re-Averaging and Editing Sigma Theta • preserve 15-minute sigma thetas if no one minute data has been invalidated during the period • compute 60-minute sigma thetas as the RSS of the 15-minute values • investigate re-computing 15 and 60 minute sigma thetas from 1-minute wind direction values using the Yamartino method • sampling rate is too low for accurate results • may be applicable in some cases computations for the one hour example shown at the beginning of the paper OMG

  16. Original Calculation of Sigma Theta • Suggestion in ANSI/ANS-3.11-2000 is validated • Can be implemented most easily in the database or data management software • Can also be implemented in Dataloggers • modified to perform the RSS calculation • implement multi-pass algorithms • Implement some of the techniques considered by Fisher OMG

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