Environmental Engineering Reference
In-Depth Information
Then
I
B
is expressed as a function of the line voltages using Kirchoff's voltage law,
VV
X
BA
I
=
(4.7)
B
S
Substituting Equation (4.7) into Equation (4.6) gives
−
j
δ
2
SVI V
VV
−
VVe
X
−
−
j
V
X
j
V
V
X
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
=
*
BA
BA
B
A
B
=
=
*
=
V
−
e
−
j
δ
=
B
B
B
B
B
X
j
s
s
s
s
VV
X
V
X
⎛
⎜
⎞
⎟
=+
(4.7a)
AB
B
S
=−
sin
δ
+
j
(
VV
−
cos
δ
)
P jQ
B
BA
B
B
s
s
Hence
VV
X
AB
P
B
=−
sin d
(4.8a)
s
V
X
B
(
)
Q
=
VV
−
cos
δ
(4.8b)
B
BA
If the above analysis were to be carried out for terminals A in Figure 4.7, the results would
s
be
PP
=−
(4.9a)
A
B
and
V
X
A
(
)
Q
=
VV
−
cos
δ
(4.9b)
A
AB
s
Equation (4.9a) confi rms the trivial fact that as the system is lossless, power
P
B
coming
out
of terminals B is equal to power
P
A
fed into
terminals A. The scalar equations (4.8) and
(4.9) are important in power systems technology as they describe the fl ow of active and reac-
tive power of grid-connected synchronous generators.
4.2.6 Three
-
phase Equations
The synchronous machine equations were derived without any reference to its three-phase
nature. Assuming that the voltages in Equations (4.8) and (4.9) are phase to neutral in volts,
the equations will give the single-phase active and reactive powers in watts and VAR respec-
tively. If the voltages are in kV then the active and reactive powers - both functions of voltage
squared - will be in MW and MVAR.
In a balanced three-phase system the three-phase
P
and
Q
will be three times the per-phase
P
and
Q
. Applying this to Equation (4.8),
3
VV
X
3
VV
X
3
VV
X
AB
A
B
Al
Bl
P
=
sin
δ
=
sin
δ
=
sin
δ
3
ϕ
s
s
s
where
V
Al
and
V
Bl
are line voltages.
