09-05-2014, 04:55 PM
Power System Matrices and Matrix Operations
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Admittance Matrix
Most power system networks are analyzed by first forming the admittance matrix. The
admittance matrix is based upon Kirchhoff's current law (KCL), and it is easily formed and very
sparse.
Consider the three-bus network shown in Figure that has five branch impedances and one current
source.
Voltage sources, if present, can be converted to current sources using the usual network rules. If
a bus has a zero-impedance voltage source attached to it, then the bus voltage is already known,
and the dimension of the problem is reduced by one.
A simple observation of the structure of the above admittance matrix leads to the following rule
for building Y:
1. The diagonal terms of Y contain the sum of all branch admittances connected directly to the
corresponding bus.
2. The off-diagonal elements of Y contain the negative sum of all branch admittances connected
directly between the corresponding busses.
These rules make Y very simple to build using a computer program. For example, assume that
the impedance data for the above network has the following form
Kron Reduction
Gaussian elimination can be made more computationally efficient by simply not performing
operations whose results are already known. For example, instead of arithmetically forcing
elements below the diagonal to zero, simply set them to zero at the appropriate times. Similarly,
instead of dividing all elements below and to the right of a diagonal element by the diagonal
element, divide only the elements in the diagonal row by the diagonal element, make the
diagonal element unity, and the same effect will be achieved. This technique, which is actually a
form of Gaussian elimination, is known as Kron reduction.