Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 142

X 1 ¼ u F1 ð n N 1 = 4 Þ i F1 ð n Þ u F1 ð n Þ i F1 ð n N 1 = 4 Þ
i F1 ð n Þþ i F1 ð n N 1 = 4 Þ

: ð 8 : 106 Þ

The advantage here are very simple algorithms and that one can use single
filters instead of pairs of orthogonal filters. However, disadvantage is longer
transient period and measurement time.
When standing phasor is used (DFT or correlation methods for instance) one
obtains:

Z 2 ¼ U 1C þ U 1S
I 1C þ I 1S

; ð 8 : 107 Þ

R ¼ U 1C I 1C þ U 1S I 1S
I 1C þ I 1S

; ð 8 : 108 Þ

X ¼ U 1S I 1C U 1C I 1S
I 1C þ I 1S

: ð 8 : 109 Þ

Example 8.10 Derive impedance measurement algorithms with permissible
measurement time equal 25, 20, 15 and 10 ms, respectively. Apply FIR sin, cos
filters or signal averaging, assuming the sampling rate 1000 Hz.

Solution Except for the case of shortest measurement time 10 ms, there exist a
possibility of using FIR filters with data window length corresponding to the per-
missible measurement time or slightly shorter, with the remaining time assigned to
signal delay—in order to obtain either very simple algorithm or the one with other
assumed/desirable features. This cannot apply for the case with 10 ms measurement
time, where the only reasonable solution is longest possible filtration (even with
10 ms filters the effects are not perfect) and usage of measurement Eqs. 8.107 - 8.109 .
Alternatively, one can propose application of half-cycle averaging (see below). The
following specific algorithms are feasible in the considered cases:
(a)

measurement time B10 ms:

(a1) half-cycle averaging algorithms ( 8.122 ), ( 8.123 ), m = 1; the reactance

value can be obtained from calculated resistance and impedance,

(a2) application of orthogonal components, algorithms ( 8.97 ), ( 8.98 ) and

( 8.99 ); the gains of half-cycle filters are identical and get cancelled in
final equations.

(b) measurement time B15 ms:

(b1) application of just one filter for current and voltage (10 ms, sine, cosine

or other window), and 5 ms delay for signal orthogonalization ( 8.104 -
8.106 ),

(b2) application of a pair of half-cycle orthogonal filters for current and

voltage and then the measurement algorithms with time delay ( 8.100 -
8.102 ), being less susceptible to frequency deviations.