Page 81 - Power Electronic Control in Electrical Systems

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70 Power systems engineering ± fundamental concepts

Now define the per-unit complex powerss 1 ˆ S 1 /S 1b and s 2 ˆ S 2 /S 2b where `b' means
15
the `base' MVA for each transformer. Also, the per-unit impedances are defined as
Z 1 S 1b Z 2 S 2b
z 1 ˆ ˆ Z 1 2 and z 2 ˆ ˆ Z 2 2 (2:53)
Z 1b V Z 2b V
1b 2b
so that
s 1 S 1 =S 1b Z 2 S 2b z 2 V 2 2b
ˆ ˆ ˆ 2 (2:54)
s 2 S 2 =S 2b Z S 1b z V
1 1 1b
If V 1b is chosen to be equal to V 2b , then
z
s 1 2
ˆ (2:55)
s 2 z
1
If the transformers are to be loaded in proportion to their ratings, then s 1 ˆ s 2 ,which
requires that z 1 ˆ z 2 . That is, the per-unit impedances of the transformers must be
equal, when evaluated on their own respective MVA bases and a common voltage base.
When three-phase transformers are connected in parallel, the requirement for
`correct polarity' is slightly more complicated. The phase shift between correspond-
ing primary and secondary voltages must be the same in both transformers. This
means that both transformers must belong to the same group. For example, a Yy0
transformer can be paralleled with a Dd0 transformer, because the phase shift is zero
through both of them. But a Yd1 cannot be paralleled with a Yy0, because the Yd1
has a phase shift of 30 .

2.11.4 Zero-sequence effects in three-phase transformers

In normal operation of a three-phase system, the voltages and currents are balanced
and

I a ‡ I b ‡ I c ˆ 0 (2:56)
This equation is satisfied not only by the line currents, but also by the line-neutral
voltages and the line±line voltages in balanced operation.
In a transformer core the voltages establish fluxes in the core. If each phase
winding has the same number of turns on each limb of the core, then the limb fluxes
will also be balanced: i.e.,
F a ‡ F b ‡ F c ˆ 0 (2:57)
In balanced operation, the flux through any limb at any instant is returning through
the other two limbs, so there is no tendency for flux to leakoutside the three limbs.
(Figure 2.44).
If the operation is unbalanced there may be a `residual' current, I 0 ˆ I a ‡ I b ‡ I c ,
and/or a residual voltage V 0 ˆ V a ‡ V b ‡ V c , and a residual flux F 0 ˆ F a ‡ F b ‡ F c .
These residual quantities are also called `zero-sequence' quantities. 16 Zero-sequence
components are all in phase with each other. Unlike positive/negative-sequence

15
See x2:13.