If we examine the performance of the impedance like algorithms by taking the covariance of the estimate as a measure then Equation (5.22) implies a relationship between the length of the data window and the estimation error. For a measure-ment error covariance matrix which is a multiple of a unit matrix, W = σ2e I, the covariance of the estimation error is
E{( ˆY − Y)( ˆY − Y)T} = σε2
As in Figure 5.8, two quantities are being estimated, Yc and Ys. The diagonal entries in the inverse matrix in Equation (5.81) give the variances of each of the estimation errors while the off-diagonal entries give a cross covariance between the two. At multiples of a half-cycle the two estimation errors are independent.
The various sums are
For Kθ a multiple of π the estimation errors are independent and have equal vari-ances of 2σ2e/K. The determinant of the matrix in Equation (5.81) is complicated in
general but can be evaluated from the sums in Equation (5.82). If we add the two variances to obtain a variance for the phasor ˆYc+ j ˆYswe obtain
σ2Yˆ = 4σ2e
K (5.83)
Before we can reach any firm conclusion about the behavior of the estimation error we must examine the term σ2e. To clarify the notation let us consider that y represents samples of voltage. If, motivated by Section 5.1, we imagine that the non-fundamental frequency signals that are being sampled to produce the error ε in Equation (5.18) are a wide-sense stationary random process (Section 3.7).
Example 3.20, with ω1= 0 and ω2 very large) with a power density spectrum which is flat at a level Sv, i.e.
S(ω) = Sv for all ω
then we will have to filter the noise with the anti-aliasing filter before sampling. If we assume an ideal filter with a cut-off frequency of
ωc= πωo
8 (5.84)
one-half the sampling frequency where θ = ω0t then the noise before sampling has a spectrum
Ss(ω) = Sv; |ω| < πωo/θ
Ss(ω) = 0; |ω| > πωo/θ (5.85) where the sub s denotes sampling. From the definition of the correlation func-tion (Equafunc-tion (3.78) and the relafunc-tionship between the correlafunc-tion funcfunc-tion and the spectrum (Equation (3.81)) we obtain
σ2e = Rv(0) = 1 2π
πωo/θ
−πωo/θ
Svdω = Svωo
θ (5.86)
or
σV2ˆ = 4 Svωo
Kθ = 4 Sv
T (5.87)
where T is the length of the window in seconds. If the impedance to the fault is computed as a ratio of estimated voltage to estimate current and we assume that
DATA WINDOW
one cycle 1.0 1.0
per unit reach
σzper unit
reach
σz
Figure 5.25 The variance of the impedance estimate and the reach versus the data window
the errors are small then the variance of the impedance error is given by σ2Zˆ = σ2Vˆ + σ2ˆI = 4(Sv+ SI)
T (5.88)
Equation (5.88) is valid for T a multiple of a half-cycle of the fundamental fre-quency. The curve of the standard deviation (σz) of the impedance estimate obtained from Equation (5.81) is shown in Figure 5.25.
Several interesting observations can be made from Equation (5.88). The first concerns the sampling rate used in relaying algorithms. The sampling rate does not appear in Equation (5.88). The variance of the impedance estimate is independent of the sampling rate and is inverse to the length of the data window. The explanation lies with the anti-aliasing filter. At higher sampling frequencies the anti-aliasing filter has a larger bandwidth and each sample has a larger variance Equation (5.86).
On the other hand, there are more samples in the same time period T and K is correspondingly larger (see Equation (5.83)). This effect will hold until the sam-pling frequency is so large that the assumption that the noise before filtering is white (S(ω) = Sv for allω) is invalid. Since the noise is ultimately band-limited, for example, by the transducers, t here are sampling frequencies that are high enough to exceed the noise bandwidth. Laboratory and simulation results indicate that sampling frequencies in the kHz range would be required before this effect would be pronounced.5
There is an additional effect caused by the use of a non-ideal anti-aliasing filter.
The actual anti-aliasing filter (Chapter 1) has a phase shift which translates into a delay. This delay, for well designed filters, is roughly one sample time. Lower sampling frequencies mean longer delays introduced by the anti-aliasing filter.
This analysis has been for the Fourier algorithms but is appropriate for all of the algorithms described in this chapter. A longer window will produce better estimates for all of the algorithm types. The counter scheme used for the differential-Equation algorithms is an attempt to create a longer window. Higher sampling rates (with the
appropriate anti-aliasing filter) produce more noise per sample. In general, indepen-dent of sampling rate, the variance of the estimated fault location is inverse to the length of the data window.
The second curve in Figure 5.25 is also an important general result. If we accept that the estimated fault location is a random variable which has the correct mean but has a probability density with the variance given by Equation (5.88) then we must accept that there is a probability of false trip or failure to trip. The two situations are shown in Figure 5.26. In each case the density is drawn with its mean at the true fault location and the shaded area of the density corresponds to incorrect relay operation. If the density has tails, as shown in Figure 5.26, then there is no setting for the relay that can eliminate the failure to trip shown in Figure 5.26(a).
However, if we assume that almost all reasonable densities are concentrated within ±2.5σ then we can conclude that the maximum relay setting that can safely be used is 1 − 2.5σ. This would correspond to better than 99% confidence if the distribution were Gaussian and would guarantee no failures to trip if the density were triangular or uniform. The inverse time dependence of the variance of the estimates translates into the reach curve shown in Figure 5.25.
The implications of the speed-reach relationship are plain and are inherent in electromechanical relays. Close-in faults may be cleared quickly but faults near the zone-1 boundary require more processing. While the principle is physically appeal-ing and is recognized in relayappeal-ing practice,21 it has an impact on digital schemes which use a fixed data window for ease of computation. For example, a fixed one half-cycle algorithm cannot clear a close-in fault as quickly as might be justi-fied and cannot be set as near the zone-1 boundary as a longer window algorithm.
(a)
actual fault location
actual fault location
(b) l = 1
Figure 5.26 Distributions of estimated fault location. (a) Fault within zone-1. (b) Fault beyond zone-1
Evidently, what is needed is an adaptive scheme which adjusts the window length to the estimated fault location. An accumulator for this purpose has been suggested.22