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274 Harmonic studies of power compensating plant
. . . . . .
2 3
. . 6 . 7
2 3 . . .
6 . 7 6 . . j 0 j 1 j 2 7
6
7
j
6 7 6 . 7
6 2 7 6 . 7
j . j j j j
6 7 6 1 0 1 2 7
j j j j j j
6 1 7 6 7
6 7 ) 6 2 1 0 1 2 7 (7:23)
6 0 7 6 7
j
6 7 6 . 7
.
6 j j j j . 7
6 1 7 2 1 0 1
j
6 7 6 7
4 2 5 6 . 7
. 6 j j j . 7
.
. . 4 2 . . 1 . . . 0 . . . . 5
The alternative harmonic domain equation
DI R F R DY (7:24)
is well suited to carry out power systems harmonic studies. This expression may also
be written in terms of the excitation voltage as opposed to the excitation flux
DI R H R DV (7:25)
The following relationship exists between F R and H R
1
H R F R D (7:26)
joh
where D( ) is a diagonal matrix with entries 1/joh.
By incorporating the base operating point V b ,I b in equation (7.24), the resultant
equation may be interpreted as a harmonic Norton equivalent
(7:27)
I R H R V I N
where I N I b H R V b .
Alternatively, a representation in the form of a harmonic The  venin equivalent may
be realized from the Norton representation
V Z R I R V T (7:28)
where V T V b Z R I b and Z R 1/H R .
7.4.2.4 Constraint equations
In the presence of low to moderate levels of harmonic voltage distortion, the TCR is
a linear plant component, albeit a time-variant one, and the harmonic Norton
current source in equation (7.27) will have null entries, i.e. the TCR is represented
solely by a harmonic admittance matrix. Likewise, if the TCR is represented by a
The  venin equivalent, equation (7.28), the harmonic The  venin source does not exist.
The engineering assumption here is that the TCR comprises an air-core inductor and
a phase-locked oscillator control system to fire the thyristors. In such a situation, the
reactor will not saturate and the switching function will be constant.
For TCR operation under more pronounced levels of distortion, the switching
function can no longer be assumed to remain constant, it becomes voltage depen-
dent instead. This dependency is well represented by the periodic representation of
