2.4
Formation of ȲBUS matrix in the presence of mutually
coupled elements
Let us consider Fig. 2.9. In this figure, the impedance Z̄c connected between nodes ‘u’ and ‘v’ is
mutually coupled with the impedance Z̄d connected between nodes ‘x’ and ‘y’ through a mutual
impedance Z̄m . The currents through the impedances, the voltages across the impedances and the
injected currents at all the four nodes are also shown in Fig. 2.9.
Figure 2.9: Two mutually coupled impedances
From Fig. 2.9, the relationship between the voltages and currents associated with the two
impedances can be written as,
Z̄c Z̄m I¯c
V̄c
][ ]
[ ]=[
Z̄m Z̄d I¯d
V̄d
Or,
−1
V̄c
Z̄c Z̄m
I¯c
V̄c
Z̄d −Z̄m
1
] [ ]=
[¯ ] = [
][ ]
[
2
V̄d
Z̄m Z̄d
Id
Z̄c V̄d
Z̄c Z̄d − Z̄m −Z̄m
Or,
I¯c
Ȳc Ȳm V̄c
][ ]
[¯ ] = [
Ȳm Ȳd V̄d
Id
(2.8)
Where,
Ȳc =
Z̄d
;
2
Z̄c Z̄d − Z̄m
Ȳd =
Z̄c
;
2
Z̄c Z̄d − Z̄m
and Ȳm = −
Z̄m
2
Z̄c Z̄d − Z̄m
⎡ ⎤
⎢V̄u ⎥
⎢ ⎥
V̄c
V̄u − V̄v
1 −1 0 0 ⎢⎢V̄v ⎥⎥
Now from Fig. 2.9, [ ] = [
]=[
]⎢ ⎥
V̄d
V̄x − V̄y
0 0 1 −1 ⎢⎢V̄x ⎥⎥
⎢ ⎥
⎢V̄y ⎥
⎣ ⎦
Or,
⎡ ⎤
⎢V̄u ⎥
⎢ ⎥
⎢V̄ ⎥
V̄c
1 −1 0 0
⎢ v⎥
[ ] = [C] ⎢ ⎥ where, C = [
]
⎢
⎥
V̄d
0 0 1 −1
⎢V̄x ⎥
⎢ ⎥
⎢V̄y ⎥
⎣ ⎦
19
(2.9)
(2.10)
Again, from Fig. 2.9,
⎡¯ ⎤ ⎡ ¯ ⎤ ⎡
⎢Iu ⎥ ⎢ Ic ⎥ ⎢
⎥ ⎢
⎢ ⎥ ⎢
⎢I¯ ⎥ ⎢ −I¯ ⎥ ⎢
⎢ v⎥ ⎢ c ⎥ ⎢
⎢¯ ⎥ = ⎢ ¯ ⎥ = ⎢
⎢Ix ⎥ ⎢ Id ⎥ ⎢
⎥ ⎢
⎢ ⎥ ⎢
⎢¯ ⎥ ⎢ ¯ ⎥ ⎢
⎢Iy ⎥ ⎢ −Id ⎥ ⎢
⎦ ⎣
⎣ ⎦ ⎣
⎤
1 0⎥
⎥
¯c
T I
−1 0 ⎥⎥ I¯c
⎥ [ ¯ ] = [C] [ ¯ ]
Id
0 1 ⎥⎥ Id
⎥
0 −1 ⎥⎦
(2.11)
From equations (2.8) and (2.10),
I¯c
Ȳc Ȳm V̄c
Ȳc
[¯ ] = [
][ ] = [
Id
Ȳm Ȳd V̄d
Ȳm
Or,
⎡ ⎤
⎢V̄u ⎥
⎢ ⎥
⎢V̄ ⎥
Ȳm
⎢ v⎥
] [C] ⎢ ⎥
⎢V̄x ⎥
Ȳd
⎢ ⎥
⎢ ⎥
⎢V̄y ⎥
⎣ ⎦
⎡ ⎤
⎡¯ ⎤
⎢V̄u ⎥
⎢Iu ⎥
⎢ ⎥
⎢ ⎥
⎢V̄ ⎥
⎢I¯ ⎥
¯c
T I
Ȳc Ȳm
T
⎢ v⎥
⎢ v⎥
[C] [ ¯ ] = ⎢ ¯ ⎥ = [[C] ] [
] [C] ⎢ ⎥
⎢V̄x ⎥
⎢Ix ⎥
Ȳm Ȳd
Id
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢¯ ⎥
⎢V̄y ⎥
⎢Iy ⎥
⎣ ⎦
⎣ ⎦
(2.12)
Now,
T Ȳc
[C] [
Ȳm
⎡
⎢
⎢
⎢
Ȳm
⎢
] [C] = ⎢
⎢
Ȳd
⎢
⎢
⎢
⎣
⎤
1 0⎥
⎥
−1 0 ⎥⎥ Ȳc Ȳm 1 −1 0 0
⎥[
][
]
0 1 ⎥⎥ Ȳm Ȳd
0 0 1 −1
⎥
0 −1 ⎥⎦
(2.13)
Or,
T Ȳc
[C] [
Ȳm
⎡
⎢
⎢
⎢
Ȳm
⎢
] [C] = ⎢
⎢
Ȳd
⎢
⎢
⎢
⎣
⎤
1 0⎥
⎥
−1 0 ⎥⎥ Ȳc −Ȳc Ȳm −Ȳm
⎥[
]
0 1 ⎥⎥ Ȳm −Ȳm Ȳd −Ȳd
⎥
0 −1 ⎥⎦
(2.14)
Or,
T Ȳc
[C] [
Ȳm
⎡
⎤
⎢ Ȳc −Ȳc Ȳm −Ȳm ⎥
⎢
⎥
⎢ −Ȳ
Ȳm
Ȳc −Ȳm Ȳm ⎥⎥
⎢
c
⎥
] [C] = ⎢
⎢ Ȳm −Ȳm
Ȳd
Ȳd −Ȳd ⎥⎥
⎢
⎢
⎥
⎢ −Ȳm Ȳm −Ȳd
Ȳd ⎥⎦
⎣
20
(2.15)
Hence, from equations (2.12) and (2.15)
⎤⎡ ⎤
⎡¯ ⎤ ⎡
⎢Iu ⎥ ⎢ Ȳc −Ȳc Ȳm −Ȳm ⎥ ⎢V̄u ⎥
⎥⎢ ⎥
⎢ ⎥ ⎢
⎢I¯ ⎥ ⎢ −Ȳ
Ȳc −Ȳm Ȳm ⎥⎥ ⎢⎢V̄v ⎥⎥
⎢ v⎥ ⎢
c
⎥⎢ ⎥
⎢¯ ⎥ = ⎢
⎢Ix ⎥ ⎢ Ȳm −Ȳm
Ȳd −Ȳd ⎥⎥ ⎢⎢V̄x ⎥⎥
⎢ ⎥ ⎢
⎥⎢ ⎥
⎢¯ ⎥ ⎢
⎢Iy ⎥ ⎢ −Ȳm Ȳm −Ȳd
Ȳd ⎥⎦ ⎢⎣V̄y ⎥⎦
⎣ ⎦ ⎣
(2.16)
From equation (2.16),
I¯u = Ȳc V̄u − Ȳc V̄v + Ȳm V̄x − Ȳm V̄y
= Ȳc V̄u − Ȳc V̄v + Ȳm V̄x − Ȳm V̄y + Ȳm V̄u − Ȳm V̄u
= Ȳc (V̄u − V̄v ) + (−Ȳm )(V̄u − V̄x ) + Ȳm (V̄u − V̄y )
(2.17)
Or,
I¯u = I¯uv + I¯ux + I¯uy
(2.18)
Similarly,
I¯v = −Ȳc V̄u + Ȳc V̄v − Ȳm V̄x + Ȳm V̄y
= −Ȳc V̄u + Ȳc V̄v − Ȳm V̄x + Ȳm V̄y + Ȳm V̄v − Ȳm V̄v
= Ȳc (V̄u − V̄v ) + (−Ȳm )(V̄v − V̄y ) + Ȳm (V̄v − V̄x )
(2.19)
Or,
I¯v = I¯vu + I¯vy + I¯vx
(2.20)
I¯x = Ȳm V̄u − Ȳm V̄v + Ȳd V̄x − Ȳd V̄y
= Ȳm V̄u − Ȳm V̄v + Ȳd V̄x − Ȳd V̄y + Ȳm V̄x − Ȳm V̄x
= Ȳd (V̄x − V̄y ) + (−Ȳm )(V̄x − V̄u ) + Ȳm (V̄x − V̄u )
(2.21)
Or,
I¯x = I¯xy + I¯xu + I¯xv
(2.22)
Equations (2.18), (2.20) and (2.22) can be represented by the partial networks shown in Figs.
2.10, 2.11 and 2.12 respectively. Combining Figs. 2.10, 2.11 and 2.12, Fig. 2.13 is obtained.
Again from the last row of equation (2.16),
I¯y = −Ȳm V̄u + Ȳm V̄v − Ȳd V̄x + Ȳd V̄y
= −Ȳm V̄u + Ȳm V̄v − Ȳd V̄x + Ȳd V̄y + Ȳm V̄y − Ȳm V̄y
= Ȳd (V̄y − V̄x ) + (−Ȳm )(Ȳy − V̄v ) + Ȳm (Ȳy − V̄u )
(2.23)
It can be observed that equation (2.23) is also represented by Fig. 2.13. Therefore, the voltage21
Figure 2.10: Partial network corresponding to equation (2.18)
current relationship of equation (2.16) is adequately represented by Fig. 2.13. Thus, Fig. 2.13
can be considered as an equivalent circuit of Fig. 2.9. As Fig. 2.13 does not contain any mutual
admittance, usual method for ȲBUS formulation can be adopted for this circuit also.
Fig. 2.13 shows the most general case in which all the four nodes are distinct from each other.
However, in many cases mutual coupling exists between two elements which have one common node
between them. The equivalent circuit for this case can also be derived from Fig. 2.13. For example,
in Fig. 2.13, if nodes ‘v’ and ‘y’ are common (say ‘w’), then the equivalent circuit becomes as shown
in Fig. 2.14. Moreover, if the nodes ‘u’ and ‘x’ are also common (say ‘s’), then the equivalent circuit
is shown in Fig. 2.15. Again, the usual method for ȲBUS formulation can be adopted for these two
circuits also.
We are now in a position to write down the basic power flow equation, which we will take up in
the next lecture.
22
Figure 2.11: Partial network corresponding to equation (2.20)
Figure 2.12: Partial network corresponding to equation (2.22)
23
Figure 2.13: Combined network of Figs. 2.10, 2.11 and 2.12
Figure 2.14: Equivalent circuit with one common node
24
Figure 2.15: Equivalent circuit with two common nodes
25