Page 137 - Power Electronic Control in Electrical Systems
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Power electronic control in electrical systems 125
This equation may be partitioned as follows
2 3
V 2
6 7
6 V 3 7
(4:33)
. 7
I 1 Y 11 V 1 Y 12 Y 13 Y 1n 6
. . 5
4
V n
2 3 2 3 2 32 3
I 2 Y 21 Y 22 Y 23 Y 2n V 2
I 3 Y 31 Y 32 Y 33 Y 3n V 3
6 7 6 7 6 76 7
6 7 6 7 6 76 7
. . . (4:34)
6 . 7 6 . 7 V 1 6 . . . . 76 . 7
. . 5 . . 5 . . . . . 54 . . 5
4 4 4
I n Y n1 Y n2 Y n3 Y nn V n
Equation (4.34), in rearranged form, is used to find the solution for the unknown
n 1 nodal complex voltages by iteration
(r 1) 8 (r) 9
2 3 2 3 2 3 2 3
V 2 Z 22 Z 23 Z 2n > I 2 Y 21 >
> >
> >
Z 32 Z 33
6 7 6 7 6 7 6 7 =
<
. . . . 7 V 1 (4:35)
6 V 3 7 6 Z 3n 7 6 I 3 7 6 Y 31 7
6 . 7 6 . . . . 7 6 . 7 6
. . 5 . . . . . 5 4 . . 5 . . 5
4 4 > 4 >
> >
> >
: ;
V n Z n2 Z n3 Z nn I n Y n1
where (r) is an iteration counter and V 1 is the complex voltage at the Slack node. The
voltage phase angle at this node is normally specified to be 0 and provides a reference
for the remaining n 1 voltage phase angles in the network. The voltage magnitude
is also specified at this node.
The vector of complex current injections is obtained from the active and reactive
powers consumed by the n 1 loads and voltage information calculated at the
previous iteration
2 3 (r) 2 load load (r) 3
I 2 (P 2 jQ 2 )=V 2
6 load load (r) 7
6 7 (P jQ )=V
6 3 3 3 7
. 7 6 7
I 3 7
6 (4:36)
6 .
. 6 . 7
4 . 5 4 . 5
I n (P load jQ load )=V (r)
n n n
The voltages V 2 , ... V n are not known at the beginning and initial estimates are given
to these nodes to start the iterative process. It should be noted that the current
expressions in equation (4.36) are derived from the complex power equation, i.e.
S VI , and that the negative sign outside the parentheses is due to the fact that
the loads consume power. The symbol is used to denote the conjugate complex
operation.
The convergence characteristic of equation (4.35) is reasonably good for small and
medium size systems. Although limited to networks containing only one generator,
the method has found application in industrial systems where the Slack node is the
electricity company's infeed point to the plant. Equation (4.35) may also be modified
to include voltage-controlled nodes (Stagg and El-Abiad, 1968) but it has been found
that its convergence characteristic deteriorates rapidly with the number of voltage-
controlled nodes in the network. In spite of its simplicity and intuitive appeal, this
has prevented its use in most practical applications. To overcome this limitation,
