The post-fault current and voltage waveforms fail to be pure fundamental frequency sinusoids for a variety of reasons. The most predictable non-fundamental frequency term is the decaying exponential which can be present in the current waveform.
For the series R-L model of the line shown in Figure 5.2, assuming zero pre-fault current and a steady state fault current of the form I cos(ω0t −ϕ), the instantaneous current for a fault at time t0 is given by
i(t) = I cos (ω0t −ϕ) − [I cos (ω0t0− ϕ)]e−(t−t0)R/L (5.15) The second term in Equation (5.15) decays exponentially with the time constant of the line. This term is the main cause of transient over-reach in high speed relays, and must be eliminated if computer relaying at fractional cycle speed is to be achieved. For a typical EHV line the time constant is in the range of 30– 50 ms.
The initial amplitude of the exponential component can be as large as the peak of the fault current, as shown in Figure 5.3. The situation can be even more complicated near a large generator. The exponential term is not an error to algorithms based on a differential equation description of the line, since the exponential satisfies the differential equation. If the time constant of the line is known, then the decay can be removed with an external filter or even in software (for algorithms which see the exponential decay as an error). The dependence of the time constant on fault resistance makes the removal less effective for high resistance faults.
L
R e(t)
i(t)
Figure 5.2 Series R-L model of a transmission line
steady state fault current
t0 t
Figure 5.3 Fault current. The dashed curve is the steady state fault current
Other non-fundamental frequency terms are not so easy to remove because they are not so easy to predict. The current and voltage transducers contribute some of these signals. For example, as shown in Section 2.6, a capacitive coupled voltage transformer has a transient response to the abrupt change in voltage shown in Figure 5.1. High frequency signals associated with the reflection of waveforms between the bus and fault may be present, as will be developed in Chapter 8. The nonlinear behavior of the fault arc may produce harmonic signals. In addition, as shown in Section 1.5, the A/D converter contributes errors due the least significant bit in the conversion and due to timing errors, i.e. the samples are not exactly t sec appart. Most of these signals have considerable high frequency content and can be reduced through the use of an anti-aliasing filter.
Since the waveform is being sampled at a rate of fs= 1/t Hz the Nyquist sam-pling theorem from Section 3.5 implies that the signal should be filtered with a filter having a cut-off frequency of fs/2 to avoid aliasing. As mentioned in Chapter 1, such a filter will remove the high frequency error signals described above but will con-tribute a transient response of its own; also, drift over time of the component values in such filters (particularly active realizations of such filters) are sources of error.
Finally, the power system itself is a source of non-fundamental frequency sig-nals. Consider the single phase model of three lines and two generators shown in Figure 5.4. It is assumed that the lines are identical but one source is strong and one is weak. The lines are assumed to be 100 miles of typical 765 kV line.
If a fault is applied at 60% of the protected line, the voltage seen by the relay is shown in Figure 5.5. The smooth curve is the voltage that would result if the capacitors were removed from Figure 5.5. It can be seen that the inclusion of the capacitors has produced at least two non-fundamental frequency signals. These non-fundamental frequencies are natural frequencies of the system which are excited by the application of the fault. Since the network is fixed if the fault location is
protected zone
kL kr
L r
C C kC
Figure 5.4 Single-phase power system model
1.0
−1.0
Voltage in per unit
Time
Figure 5.5 Voltage waveforms for a fault at 60% of the line length
held constant, it follows that the natural frequencies are determined by the fault location.
Figure 5.6 shows a family of voltage waveforms produced by altering the fault incidence angle. It can be seen that the phase of the non-fundamental frequency com-ponents is a function of the fault incidence angle. As the fault location is changed as shown in Figure 5.7, the frequency of the non-fundamental frequency components
1.0
−1.0
Voltage in per unit
Time
Figure 5.6 Family of voltage waveforms for faults at 60% of the line length
1.0
−1.0
Voltage in per unit
Time
Figure 5.7 Family of voltage waveforms for varying fault locations
changes. A similar effect can be produced by altering the network structure behind the fault. Experiments on a model power system combining the effect of changing the structure of the network feeding the fault and the fault type and location have been reported.4 The conclusion is that an important part of the non-fundamental frequency signalε(t) in Equation (5.14), at least for high voltage lines, is due to the network itself. These signals depend on the fault location and on the nature of the system feeding the fault and, as such, are not predictable. Example 3.20 provides a model of such a process. If we model the fault incidence angle as the random phase and the fault location and network structure as the mechanism producing the random frequency then we can think of the power spectrum of Example 3.20 as the power spectrum of the signal inε(t) in Equation (5.14). It should be recognized that each realization of such a process is a rather deterministic looking signal, such as that shown in Figure 5.5. (This is counter intuitive if one expects a realization of a random process to look noisy.) The randomness is present because, considering the ensemble of times that the relay is expected to operate, the frequency and phase of the signal cannot be predicted.
Considering the ε(t) in Equation (5.14) as a random process, it is reasonable to consider the anti-aliasing filter and the algorithm taken together as filtering the random process, as in Section 3.8. The frequency response of the algorithm is then an important part of the filtering process. To obtain the frequency response of the algorithm we should compute the response of the algorithm (the phasor for Equations (5.6) and (5.7), for example) when the input signal is of the form ejωt.
Example 5.1 If y(t) = ejωt
y−1= e−jωt yo= 1 y1= ejωt
and Equations (5.4) and (5.5) yield
Yˆc= ejωtcosθ + 1 + e−jωtcosθ 1 + 2 cos2θ
Yˆc= 1 + 2 cosθ cos ωt 1 + 2 cos2θ and
Yˆs= ejωt− e−jωt 2 sinθ Yˆs= jsinωt
sinθ
It can be seen, for example, that, if y(t) = Re{ejωot} = cos ωot, ˆYc= 1, and ˆYs= 0.
In general, if y(t) = Re{ej(ωt+ϕ)} = cos(ωt + ϕ) then Yˆc = cos ϕ
1 + 2 cosθ cos ωt 1 + 2 cos2θ
Yˆs= sin ϕ
sinωt sinθ
The two bracketed terms are plotted in Figure 5.8 along with
( ˆY2c+ ˆY2s)/2 (the magnitude for ϕ = 45◦) for θ = 30◦ (12 samples per cycle). Since two quantities are being computed, there are two frequency responses. The choice of angleϕ that is used in presenting the magnitude is somewhat arbitrary. We will useϕ = 45◦ for consistency.
Different frequency responses are obtained for different values of c1 and c2 in Equations (5.6) and (5.7). The Prodar 70 algorithm was specifically designed to
2.0
Hz 1.0
60 240 360
0
cos coefficient
Mann-Morrison magnitude
sin coefficient magnitude
Figure 5.8 Frequency responses of Equations (5.4) and (5.5) along with the Mann-Morrison algorithm
10
2 6
0 0 120 240 360 Hz
Figure 5.9 Frequency response of the Prodar 70 algorithm
reduce low frequency response. The frequency responses of the Mann-Morrison and the Prodar 70 algorithms at the same sampling rate of 12 samples per cycle are shown in Figure 5.8 and Figure 5.9.
The algorithms compared in Figures 5.8 and 5.9 represent three samples at a rate of 12 samples per cycle, or a quarter of a cycle. As such, they do not reject non-fundamental frequency components (particularly the third harmonic) as well as might be necessary in many applications. These algorithms are fast in that the short window will be entirely in the post-fault region quickly. We will see later that longer window algorithms have a greater ability to reject non-fundamental frequencies at the expense of a longer decision time (the longer window takes longer to clear the instant of fault inception). This is a general result which we will develop throughout this chapter. That is, there is an inherent speed-reach limitation in line relaying caused by the presence of random signals in the measured voltages and currents.
The reach of the relay (the setting of the boundary of the zone of protection) is directly related to the accuracy of the estimates formed by the algorithm.