Page 293 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 281

the network equivalent impedance. This is at least the case for well-designed TCSCs
operating in capacitive mode. Measurements conducted in both the Slatt and the
Kayenta TCSC systems support this observation. For instance, the Kayenta system
generates at its terminals, a maximum THD voltage of 1.5% when operated in
capacitive mode and firing at an angle of 147 (Christl et al., 1991). It should be

noted that there is little incentive for operating the TCSC in inductive mode since this
would increase the electrical length of the compensated transmission line, with
adverse consequences on stability margins and extra losses.
By recognizing that the thyristor pair in the TCSC module has two possible
operating states, namely off and on (Helbing and Karady, 1994) developed equations
for the TCSC voltage and current:
1. Thyristors off:

I M sin a I M cos a
V off (t, a) ˆ oC [1 sin (ot ‡ a)] oC cos (ot ‡ a) ‡ V st 0 (7:41)
C
where I M is the peak line current and V st is the voltage across the capacitor at
0
thyristor commutation time.
I C off (t, a) ˆ I M sin ot (7:42)
In this situation, the inductor and thyristor current are zero, and the capacitor
current equals the load current.

2. Thyristors on:
During conduction the inductor voltage equals the capacitor voltage.
2
o L cos a h p i
on 0 o 0
V (t, a) ˆI M o cos (ot a) o 0 sin o 0 t a
C 2
2
o o 0 o 2
2
o oL cos a h o 0 p i (7:43)
0
I M 2 sin (ot a) ‡ cos o 0 t a
2
o o 0 o 2
h p i
o 0
‡ V st cos o 0 t a
00
o 2
where V st is the capacitor voltage at the time of thyristor firing.
00

8 9
o p
2
oo sin a < sin o 0 t a 2 cos (ot a) =
on 0 o 0
I (t, a) ˆ I M 2 2 o
L
0
o o : o 0 ;
(7:44)
( )
o cos a o p
2
0
I M 2 cos o 0 t a ‡ sin (ot a)
2
o o 0 o 0 2
1
where o 0 ˆ .
p
LC
The capacitor current comprises the line current plus the inductor current
on
on
I (t, a) ˆ I (t, a) ‡ I M sin ot (7:45)
C L
These equations are useful for calculating on a cycle-by-cycle basis the currents and
voltages in the inductor, thyristor and capacitor. For instance (Helbing and Karady,