### Analysis of Power Swings in a Multi – Machine System

```Module 7 : Out of Step Protection
Lecture 25 : Analysis of Power Swings in a Multi – Machine System
Objectives
In this lecture we will
Analyse a given transmission network.
Reduce it to a two source equivalent.
Determine whether the power swing in R-X plane cuts through any transmission line.
Now let the current in the transmission line AB,
when it is connected to the network be I ij from
the i th end and I ji at the j th end respectively.
Then, under the assumption of linearity, the
effect of the transmission line connection can be
evaluated by superimposing injected currents
and
on the network in
which
line
i-j
was
disconnected.
Thus,
. Since, we
are only interested in characterizing behavior at
busses i and j, we can ignore the remaining rows
in (2).
Thus,
(3)
Now if we assure a simple series model of the line, then,
where Y ij is the primitive admittance of the transmission line
Equivalently, (3) can be expressed as follows.
(4)
where
matrix is the inverse of corresponding
symmetry assumption,
restricted Z bus matrix. Under usual
.
The equivalent circuit of fig 25.1 can be reduced as shown in fig 25.2. Now from the equivalent circuit of
fig 25.2 we get that,
Therefore, the incremental model is given by,
(5)
Equating (5) to (4) we get,
;
;
;
Thus the equivalent circuit is as
shown in fig 25.3.
Determination of Power Swing Locus for Multi Machine System
Impedance seen by relay on line L,
Determination of Power Swing Locus for Multi Machine System
where
Thus,
(6)
So, it has to be checked whether
is less than
or not. If it is less than
then
electrical center is formed on line L. If it is greater than Z L , then electrical center lies outside the
transmission line L.
Example 1
For the system shown in fig
25.3, determine the two port
equivalent and find out whether
the power swing locus passes
through
(a) transmission line ‘c'
(b) transmission line ‘b'
Solution:
(a)
In order to analyze whether the power swing will pass through the transmission line ‘c', we need to
develop a two
machine equivalent across the line. For that, the transmission line ‘c' is disconnected from the network
and Z bus matrix is formed.
Let us form the nodal admittance matrix or Y bus for the given system with transmission line ‘c'
disconnected.
=
Now Z bus is obtained by inverting Y bus.
Example 1 (contd..)
Solution:
Since we are considering the transmission line ‘c' which is connected between buses 2 and 3, we can
ignore the first row and column of the Z bus matrix and the reduced model will be,
Therefore,
i.e.
Example 1 (contd..)
Solution:
Hence the network will be as shown in fig 25.4, with transmission line ‘c' connected across it.
The total impedance between the two sources is given by,
,
The swing impedance
impedance line at
trajectory
intersects the
, which is the location of electrical center
on R-X plane. From fig 25.5 it can be seen that
electrical center of the swing lies on the transmission
line ‘c'.
Example 1 (contd..)
Solution:
(b)
Now consider the transmission line ‘b'. For forming the two machine equivalent, transmission line ‘b' is
disconnected
from the system and Z bus is formed by inverting Y bus.
Since we are considering only the buses 1 and 2 across which line ‘b' is connected, we can ignore the third
row and third column of the Z bus matrix. Thus, the reduced system model is given by,
and
=
Since,
Example 1 (contd..)
Solution:
Solving the above, we get
. Now, connect the transmission
line ‘b' across the network as shown in fig 25.6 and then the total impedance across the sources will be,
,
The swing impedance
impedance line at
trajectory
intersects
the
. Since the line impedance is j0.2, we can
see from fig 25.7 we can see that the intersecting
point of swing impedance trajectory with impedance
line lies outside the transmission line ‘b'.
Review Questions
1.
For the system shown in fig 25.3, determine whether swing locus passes through transmission line 'f'.
2.
For the system shown in fig 25.8, find out whether power swing passes through any of the transmission
lines?
Recap
In this lecture we have learnt the following:
Developed a two source equivalent of the power system.
Analysed the system to determine whether swing locus passes through any transmission line.
Found that swing locus passes through one of the transmission line.
```