Verificación del diseño de elementos críticos

The total number of animals killed by wind turbines may be distributed among three categories: carcasses on the ground, G; carcasses that have been recovered by searchers, F; and carcasses that have been scavenged or are fully decayed, R. Animal deaths are assumed to occur at a steady rate of . Thus,

G(t) + F(t) + R(t) = t (A.1)

The rate animals that are killed by the wind turbines is found by taking the first derivative of Eq. (A.1) with respect to time. The rate that animals are removed either by scavenging or decaying is assumed to occur with a mean removal time of TR (i.e., ( ) ( ) ). In the absence of any searches, Eq. (A.1) may then be written as:

( )

( ) _.

(A.2)

The solution of this differential equation is:

( ) ( ) _{. (A.3)}

The integration constant is explicitly written as TRC to aid in the analysis of the models. Equation (A.3) predicts total number of animal carcasses on the ground will eventually reach a steady-state value, which is directly related to the mean removal time

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of the carcasses. The steady-state value may be determined by taking the limit of equation (A.3) as t → ∞:

( ) [( ) ] _{. (A.4)}

A.1.1 Simple-uniform model

This fatality model focuses on animals whose population density is temporally uniform in the vicinity of a wind farm. If steady-state conditions are established before searches begin (t < 0), then number of animals on the ground at the beginning of the survey is simply TR. Fatality searches occur at the beginning of each survey period, with the first search (i = 1) commencing at t = TS,1 = 0. The second search (i = 2) occurs at t = TS,2. Number of carcasses collected for a specific carcass class during the ith

search is denoted as Fi. Number of animals found depends on searcher efficiency for that survey, pi, and number of carcasses on the ground from the prior search period, Gi-1, that are available to be collected at the time of the ith search. Therefore, number of carcasses found for a given search event is . If total number of carcasses collected during fatality searches is relatively small, the act of searching will not significantly perturb number of carcasses on the ground from one search to another. In this case, G(t) can recover its steady-state value during the interval of time between each search.

It is possible that carcass removal rates may change during the course of the fatality survey (e.g., seasonal variations). Therefore, the steady-state value is modeled as TR,i. An individual fatality search is expected to yield:

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( ) ( ) (A.5)

The number of carcasses found depends on the search detection probability for a particular search and the steady-state number of carcasses on ground from the previous search interval. Total number of carcasses collected over the fatality survey is

∑ ( ) ∑ , (A.6)

where n is the number of fatality searches conducted during the survey. Total number of fatalities, NK, occurring over the same period of time (i.e., 0 ≤ t ≤ TS,n) is

∫

. (A.7)

Solving Eq. (A.7) for  and substituting the result into Eq. (A.6) gives an estimate for number of fatalities that occurred during the survey period:

∑ (A.8)

Equation (A.8) reduces to

( ) (A.9)

if searches occur in periodic intervals of equal duration (i.e., ( ) for the ith search) and if p and TR are constant over the course of the fatality survey.

The (n–1)/n factor in Eq. (A.9) appears because the number of search intervals used to evaluate the integral in Eq. (A.7) is one less than the number of searches being conducted. The simple-uniform model assumes there is one search occurring during

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each search period. Thus, duration of the entire survey is not merely the time elapsed between the first and last survey as this would require two searches to occur in one of the search periods. We defined the first search at t = 0, which is the beginning of the first search interval. This means the last search interval occurs immediately following the last search at t = TS,n. Duration of this search period is not clearly defined if the survey employed unequal search intervals. In the context of periodic searching, however, the amount of time that should be assigned following the nth search is simply

TS. Therefore, the value of TS,n in Eq. (A.8) is adjusted to (n – 1)TS + TS = nTS. This produces the “naïve” estimator that was frequently employed in many early bird and bat fatality surveys at wind power plants, with the result being

. (A.10 or Eq. (3))

The only difference between Eq. (A.9) and Eq. (A.10) is the amount of time counted in the total survey period; fatality rates calculated from these estimators will be identical.

A.1.2 Perturbed-uniform model

If searcher efficiency is relatively high or search intervals short, then G(t) may not have sufficient time to recover the steady-state value between searches. In these cases, the act of searching may exert significant perturbations on G(t) from one search to the next that may influence the total number of carcasses found. The overall strategy for developing a fatality estimator that accounts for these effects is to divide the search interval into discrete sub-intervals demarcated by the searching event. Number of carcasses found for each search is merely the number of animals present on the ground

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when searching is conducted multiplied by the searcher-detection probability, Fi(TS,i) =

piGi-1(TS,i). As before, searches occur at t = TS,i,, with the first search (i = 1) being at t = 0, and the period of time between each search does not have to be equal. The function

G(t) is discontinuous at t = TS,i. Equation (A.3) is valid for each of the Gi(t) segments:

( ) . (A.11)

The integration constants, Ci, may be determined by number of carcasses on the ground immediately following the previous search:

( ) ( ) ( ). (A.12)

Equation (A.12) allows a recursion relationship to be written for the integration constants, ( ) ( ) ( ) [( ) ] ( ). (A.13)

This enables each of the previously unknown integration constants to be determined from knowledge of initial conditions of the wind farm before searches began. Times before the first search (t < 0) have G0(t) = TR,0 since steady-state conditions are assumed to be in place before the fatality survey is initiated; C0 must be zero to be consistent with this requirement. Total number of animals collected for all searches conducted during the fatality survey is found by simply summing the number of carcasses found for all searches,

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∑ ( ). (A.14)

Solving Eq. (A.12) for Gi-1(TS,i) and substituting the result into Eq. (A.14) gives:

∑ { [ ( )]}

. (A.15)

Equation (A.15) may be combined with Eq. (A.7) to produce an estimator for total number of fatalities occurring during the survey period that explicitly takes into account the effect of searcher perturbations on G(t),

∑ { [ ( )]}

. (A.16)

For periodic search intervals that have constant p and TR, Eq. (A.16) simplifies to

(

( )∑ ( ( ) )

). (A.17) or Eq. (4)

The term in the parentheses modifies the simple-uniform model to account for searcher perturbation on G(t). The recursion relation in Eq. (A.13) is simplified under these conditions, ( ) (( ) ).

Equation (A.17) defines the survey period from the first search to the end of the last search period, including the time after the last search. A correction factor of (n‒1)/n may be applied if researchers wish to define the survey time period from the time of the first search to the time at which the last search occurs.

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