Page 157 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 145

supplied to the shunt converter, Re V vR I vR satisfies the active power demanded

by the series converter, Re V cR I m . The impedance of the series and shunt trans-
formers, Z cR and Z vR , are included explicitly in the model. The ideal voltage sources
and the constraint power equation given in equations (4.9)±(4.11) are used to derive
this UPFC model.
Based on the equivalent circuit shown in Figure 4.6(b), the following transfer
admittance equation can be written
2 3
V l

I l (Y cR ‡ Y vR ) Y cR Y cR Y vR 6 V m 7
ˆ 6 7 (4:86)
I m Y cR Y cR Y cR 0 4 V cR 5
V vR
The injected active and reactive powers at nodes l and m may be derived using the
complex power equation
" # " #" #
0 I
S l V l
l
ˆ

S m 0 V m I m
2 3
V l
" #" #6 7
V l 0 (Y cR ‡ Y ) Y cR Y cR Y 6 V 7
vR
ˆ vR 6 m 7 (4:87)
0 V m Y cR Y cR Y cR 0 6 V cR 5
7
6

7
4
V vR

V
2 3
l
" #" #6 7
V l 0 G ll jB ll G lm jB lm G lm jB lm G l0 jB l0 6 V 7
ˆ 6 m 7
6
7
0 V m G ml jB ml G mm jB mm G mm jB mm 0 6 V 7
4 cR 5
V vR

After some straightforward but arduous algebra, the following active and reactive
power equations are obtained
2
P l ˆ V l G ll ‡ V l V m jfG lm cos (y l y m ) ‡ B lm sin (y l y m )
jjj
jj
(4:88)
jjj
‡ V l V cR j G lm cos (y l y cR ) ‡ B lm sin (y l y cR )g
f
f
‡ V l V vR j G l0 cos (y l y vR ) ‡ B l0 sin (y l y vR )g
jjj
2
jjj
Q l ˆ jV l j B ll ‡ V l V m j G lm sin (y l y m ) B lm cos (y l y m )g
f
(4:89)
jjj
f
‡ V l V cR j G lm sin (y l y cR ) B lm cos (y l y cR )g
jjj
‡ V l V vR j G l0 sin (y l y vR ) B l0 cos (y l y vR )g
f
2
j
P m ˆ V m j G mm ‡ V m j V l G ml cos (y m y l ) ‡ B ml sin (y m y l )g
j
jjf
(4:90)
f
j
j
‡ V m j V cR j G mm cos (y m y cR ) ‡ B mm sin (y m y cR )g
2
j
f
Q m ˆ jV m j B mm ‡ V m kV l j G ml sin (y m y l ) B ml cos (y m y l )g
(4:91)
f
j
‡ V m kV cR j G mm sin (y m y cR ) B mm cos (y m y cR )g