Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 117

x ð n Þ¼ X 1m cos ð nX 1 þ u Þ¼ X 1C cos ð nX 1 Þþ X 1S sin ð nX 1 Þþ e ð n Þ; ð 8 : 15 Þ

where X 1 is a relative fundamental angular frequency, X 1C ¼ X 1m cos ð u Þ; X 1S ¼
X 1m sin ð u Þ are unknown parameters, e(n) is an error signal representing all other
frequency components and noise.
The signal magnitude and phase can be determined provided the signal model
parameters are estimated with use of a proper algorithm. The parameters
X 1C ; and X 1S can be used for the purpose, being themselves close equivalents of
the signal orthogonal components obtainable from FIR filter outputs.
The signal model ( 8.15 ) can be extended by inclusion of one (or more) addi-
tional frequency components which, on the one hand, enables direct estimation of
parameters of that component and on the other hand—improvement of the fun-
dament frequency component measurement accuracy. In case when one harmonic
component is taken into account the signal model becomes:

x ð n Þ¼ X 1C cos ð nX 1 Þþ X 1S sin ð nX 1 Þþ X kC cos ð nkX 1 Þþ X kS sin ð nkX 1 Þþ e ð n Þ;
ð 8 : 16 Þ

with X kC and X kS representing the parameters of the kth harmonic.
In general, depending on the needs the signal model ( 8.15 ) can be extended
theoretically without limits, however, this entails higher computational burden
during practical implementation of the method for signal parameters estimation.
Equations 8.15 and 8.16 can be generalized to the form:

T þ e ð n Þ:
ð 8 : 17 Þ

x ð n Þ¼ cos ð nX 1 Þ sin ð nX 1 Þ cos ð nkX 1 Þ sin ð nkX 1 Þ ... ½ X 1C X 1S X kC X kS ... ½

For a vector containing K components there are 2K parameters to be deter-
mined, which is possible if at least N C 2K signal samples are available, which
leads to the following matrix equation:

x N ¼ h N X þ e N ; ð 8 : 18 Þ

T where x N ¼ x ð 0 Þ; x ð 1 Þ; ... ; x ð N 2 Þ; x ð N 1 Þ ½

is a vector of N signal samples,

T

e N ¼ e ð 0 Þ; e ð 1 Þ; ... ; e ð N 2 Þ; e ð N 1 Þ ½ is a vector of N error samples, X ¼

T ½ X 1C ; X 1S ; ... ; X KC ; X KC is a vector of unknown 2K signal parameters and

2
4

3
5

cos ð 0X 1 Þ sin ð 0X 1 Þ ... cos ð 0KX 1 Þ sin ð 0KX 1 Þ

cos ð 1X 1 Þ sin ð 1X 1 Þ ... cos ð 1KX 1 Þ sin ð 1KX 1 Þ

h N ¼

cos ðð N 1 Þ X 1 Þ sin ðð N 1 Þ X 1 Þ cos ðð N 1 Þ KX 1 Þ sin ðð N 1 Þ KX 1 Þ

is a matrix of known model elements for different sampling instants.
The sought parameters X can be estimated with high accuracy if only the mean
square error for all the N samples taken is minimized, i.e. if: