Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 137

Since these components are constant, not time-dependent, they can be used in
Eqs. 8.60 and 8.61 only, i.e. in equations which apply orthogonal components at
the same instant (without delay). Using components ( 8.87a , b , c , d ) in these
equations one gets:

P 1 ¼ 0 : 5 ½ U 1C I 1C þ U 1S I 1S ;

ð 8 : 88 Þ

Q 1 ¼ 0 : 5 ½ U 1S I 1C U 1C I 1S ; ð 8 : 89 Þ

There are many methods allowing for magnitude and power measurements, as it
is seen from considerations above. Some very simple algorithms use either aver-
aging or Walsh orthogonal filters. When high immunity to noise is important then
FIR sine, cosine filters should be used. When fast measurement is required Kalman
filters, least square error or variable data window methods are recommended.
When not only fundamental but also components of other frequencies are required,
DFT is usually applied. In general one can always find a method adequate to the
requirements.

Example 8.9 Permissible measurement time equals 12 ms. Algorithms for power
measurement are to be proposed, with assumed sampling frequency 1000 Hz.

Solution For the assumed sampling rate the sampling period is equal to 1 ms, thus
maximum window length of applied filters amounts to 12 samples. Therefore it is
possible to propose the following algorithms fulfilling the requirements defined:

(a) application of FIR orthogonal filters with 12-sample window (e.g., sin, cos),

algorithms ( 8.82 ) and ( 8.84 ),

(b) application of algorithms ( 8.83 ) and ( 8.84 ) with half-cycle Walsh or sin/cos

FIR filters,

(c) usage of filters as in (b), application of algorithms ( 8.85 ) and ( 8.86 ), being

less susceptible to frequency changes (see Chap. 9 ),

(d) application of only two half-cycle filters, orthogonalization by single or

double delay.

The choice of one particular version depends on additional conditions that may
include possible spectrum of the input signals or algorithm computational burden.
The latter issue is illustrated in Fig. 8.6 where the schemes of obtaining orthogonal
components of current and voltage signals are depicted. It is seen that the cases (a-
c) are similar (signal delaying in principle does not introduce any computational
burden), while the case (d) is the simplest since the separated orthogonalization is
realized easier than any filter.
For assumed sampling frequency one gets: N 1 ¼ 20; N ¼ 12; X 1 ¼ 2p = N 1 ¼
0 : 1p :

(a) For the longest permissible window (N ¼ 12) the filter gains ( 4.34 ), ( 4.35 )

are equal: