Arboriculture & Urban Forestry 37(4): July 2011 the data (Brown 2009a). The Weibull distribution (Figure 3) represents probability, and the area under the curve equals 1. The Weibull distribution is a function of a scale parameter a, and a shape parameter b. It is able to assume various shapes to fit varying data sets (Brown 2009a). The a and b parameters can be established by trial and error, fitting the curve (Figure 3) to the data (Figure 2). Once the a and b values for the Weibull distribution were determined, the Weibull distribution was ad- justed first (Weibull 1, Figure 4) by minimizing the χ2 error, and secondly (Weibull 2, Figure 4) by application of a constant to reflect the upper limits of outage frequency. While it appears that the pattern of outage events (Figure 4) can be represented by a normal distribution, this was not statistically verified. The sample size was too small to meet the minimum bin frequency requirement for a Pearson’s χ2 149 for PSE could be said to cover wind speeds from 36 to 71 km/hr (x̃ ± s) (Figure 4). This may apply to other utilities also at winds goodness of fit test. Future work, with substantially larger outage incident data sets may serve to validate or reject this assertion. It will require inclusion of dis- tribution system data, which in PSE’s case would add more than 33,500 outage incidents to the 265 transmission outages. While modeling transmission tree-related outages via a Weibull distribution could not be statistically verified, the graph- ics produced in the process are revealing. Figure 4 shows that over the ten-year period most reported tree-related outages occur dur- ing winds ranging from 30 to 80 km/hr (estimated from Figure 4). Figures 2 through 4 present data based on a ten-year experi- ence. However, it is also important to know the distribution of outages in response to a single wind event of a particular inten- sity. To make this determination, an interruption frequency was calculated by dividing the number of reported tree-caused outage events that occurred at a wind speed over the ten-year period by the number of days that wind speed was experienced over the ten-year period. The interruption frequency presented in Figure 5 shows interruptions to be exponential to wind speed. The correla- tion between wind speed and interruption frequency was tested. This yielded a Pearson Product Moment correlation coefficient of 0.7257 with P(r = 0) of 0.0000. Regression, to explain outage fre- quency by wind speed, was performed. The exponential regression form provided an r2 the Hoerl’s regression form serves to minimize residuals and has a better fit (Figure 5). The Hoerl’s regression has an r2 [1] IF = 0.8716053773 * WS^-3.0114265384 * e^(0.2056078327 * WS) where IF is interruption frequency and WS is wind speed. The regression algorithm permits modeling of out- ages beyond the wind speeds for which data is avail- able. In Figure 6, interruptions for wind speeds up to 121 km/hr are modeled. At 121 km/hr interruptions are ap- proaching an asymptote. This projection fits with ob- servations made in the aftermath of Hurricane Hugo, where it was observed that tree damage increased rapidly with wind gust speeds in excess of 60 km/hr but did not worsen beyond 130 km/hr (Francis and Gillespie 1993). The ratio of expected interruptions was determined for 24 km/hr, 48 km/hr, 72 km/hr, 97 km/hr, 105 km/hr, and 121 km/hr (Figure 7). The expected outage frequency at 24 km/hr winds was used as the denominator to calculate the ratio of expected interrup- tions for the range of wind speeds. Normal operating conditions value of 0.8426 with P = 0.0000. However, of 0.8617 with P = 0.0000. The Hoerl’s regression produces the algorithm Figure 7. Tree failure ratio. over 71 km/hr, as some major storm damage has been experienced, which was the case with tropical storm Isabel (Michaels 2003). Total transmission system of danger tree exposure was found through trigonometry to be 4,276,395 trees. By divid- ing the upper limit Weibull modeled outage events (Weibull 2, Figure 4) by the system tree exposure, a ten-year failure rate for each wind speed was calculated. In a similar manner, divid- ing the Hoerl’s regression values by the total tree exposure pro- vides the means of calculating the expected tree-related outages for a particular wind event. In both cases, the failure rates are based on environmental conditions from 1998 through 2007. Assuming all else being equal, changes in system tree expo- sure will result in new outage expectations, which can be cal- culated by multiplying the tree exposure by the failure rate. A possible mitigation approach to PSE’s tree-related interrup- tion frequency that involved reducing the extent of tree exposure, ©2011 International Society of Arboriculture Figure 6. Transmission interruptions per wind incident. Figure 5. Interruption frequency.
July 2011
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