Page 105 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 93

A more familiar form of this equation is obtained for short lines, for which
p p p
sin y ' y ˆ ba ˆ oa (lc). Then Z 0 y ˆ oa (lc) (l/c) ˆ oal ˆ X L , the series
inductive reactance of the line. So
E s E r
P ˆ sin d (3:25)
X L
This equation is important because of its simplicity and wide-ranging validity. If E s and
E r are held constant (as is normally the case), the power transmission is a function of
only one variable, d. As noted earlier, there is a maximum transmissible power,

E s E r P 0
P max ˆ ˆ (3:26)
X L sin y
This is shown in Figure 3.10, which is usually plotted with d as the independent
variable; but in fact P is generally the independent variable and the power transmis-
sion has to be controlled to keep d within safe limits below P max . Typically d is kept
below 30 , giving a safety margin of 100% since sin 30 ˆ 0:5:

The reactive power required at the ends of the line can also be determined from
equation (3.23): thus
E r (E s cos d E r cos y)
Q r ˆ
Z 0 sin y (3:27)
E s (E r cos d E s cos y)
Q s ˆ
Z 0 sin y
If E s ˆ E r then
2
E (cos d cos y)
s
Q s ˆ ˆ Q r (3:28)
Z 0 sin y
If P < P 0 and E s ˆ 1:0p:u:, then d < y, cos d > cos y, and Q s < 0and Q r > 0. This
means that there is an excess of line charging current and reactive power is being
absorbed at both ends of the line. If P > P 0 , reactive power is generated at both ends.
If P ˆ 0, cos d ˆ 1 and equation (3.28) reduces to equation (3.19).

Fig. 3.10 Power vs. transmission angle.