Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 171

Fig. 9.1 Signal waveshape after measurement start (n = 0) and change of parameters

window and one reaches the actual steady state which will last until new distur-
bance appears. It means that filter transient lasts for the period equivalent to filter
window length. The transient period can last shorter when shorter filter window are
applied.
The situation is a little bit different when a filter starts its operation at the fault
instant. In such a case zero initial conditions may be assumed, i.e., signal samples
are equal to zero before the filter starts its operation (at the instant m = 0,
Fig. 9.1 ). The second sum in Eq. 9.2 is then equal to zero and output signal of the
filter is given by the equation:

y ð n 1 Þ¼ X

n 1

a ð k Þ x ð n 1 k Þ; ð 9 : 3 Þ

K ¼ 0

for 0 n 1 N 1 :
It is seen ( 9.3 ) that at the very beginning the first term in ( 9.2 ) has the initial sum
limit n 1 which is increasingly growing along the filtration process. The growth lasts
as long as n 1 reaches the value equal to filter window length (N) and then this value
becomes constant. At that time instant the new steady state of measurement begins.
Examinations and simulations concerning filters and measurement algorithm tran-
sients are easier to realize in the latter case since there is less parameters affecting
transients that should be considered. Though the measurement algorithms are usu-
ally nonlinear, it is possible to come to general conclusions important for some
arbitrary dynamical states. To reach this one may notice that each AC signal is
determined by three parameters: magnitude, frequency and phase shift. For constant
frequency of signals (as it is in power system) transients trajectory will depend
mainly on their phases since magnitude has influence on scaling factors only.