Page 285 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 273

" p s 1 Z p s 2 #
1 Z ‡ 2 2 ‡ 2
2
b h ˆ sin (hot)dot ‡ sin (hot)dot
p p s 1 p s 2

2 2 2 2
(7:20)
2 hs 2 hs 1 hp
ˆ sin sin sin
hp 2 2 2
The harmonic switching vector can handle asymmetrically gated, reverse-parallel
thyristors since the conduction angles s 1 and s 2 can take independent values. Hence,
equation (7.15), incorporating the switching vector, as given in equation (7.16),
provides an effective means of calculating TCR harmonic currents for cases of both
equal and unequal conduction angles. This is a most desirable characteristic in any
TCR model, since TCRs are prone to exhibit imbalances due to manufacturing
tolerances in their parts.
7.4.2.2 Harmonic admittances
Similarly to the harmonic currents, the TCR harmonic admittances are also a
function of the conduction angles s 1 and s 2 . In cases when the TCR is fully
conducting, i.e. s 1 ˆ s 2 ˆ 180 , only the DC term exists. Smaller values of conduc-

tion angles lead to the appearance of harmonic admittance terms: the full harmonic
spectrum is present when s 1 6ˆ s 2 but only even harmonics and the DC term appear
in cases when s 1 ˆ s 2 .
The TCR harmonic admittances, when combined with the admittances of the
parallel capacitor and suitably inverted, provide a means for assessing SVC resonant
conditions as a function of firing angle operation. As an extension, the SVC harmonic
admittances can be combined with the admittances of the power system to assess the
impact of the external network on the SVC resonant characteristics.
The current derivative, with respect to the flux, is expressed as follows

1
0
f (Y) ˆ S R (7:21)
L R
In the time domain, the magnitude of the derivative is inversely proportional to the
reactor's inductance L R during the conduction period and it is zero when no con-
duction takes place. In the frequency domain, the harmonic admittances are inversely
proportional to the magnitude of the harmonic terms contained in the switching
vector.

7.4.2.3 Harmonic Norton and Thevenin equivalent circuits
Â
An incremental perturbation of equation (7.15) around a base operating point Y b ,I b
leads to the following linearized equation (Semlyen et al., 1988),

DI R ˆ f (Y) DY (7:22)
0
0
where f (Y) is a harmonic vector of first partial derivatives.
The evaluation of the rhs term in equation (7.22) may be carried out in terms
of conventional matrix operations, as opposed to convolutions operations, if the
0
harmonic vector f (Y) is expressed as a band-diagonal Toeplitz matrix, F R , i.e.