Question 3.4: Form an incidence and reduced incidence matrix from the grap......

The bus incidence matrix is formed from the graph of the network in Figure 3.8c:

This matrix is singular. The matrix without the last row pertaining to the reference node, which is shown shaded, is a reduced incidence matrix. It can be partitioned as follows:

We can therefore write:

A_{(n-1), e}=(A_{b})_{(n-1), (n-1)} × (A_{L})_{(n-1), I}      (3.35)

Figure 3.8b ignores the shunt capacitance of the lines. Referring to the graph in Figure 3.8c, we can write

\left|\begin{matrix} 1 & 0 & 0 & 0 &-1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 & -1 \end{matrix} \right| \left|\begin{matrix} I_{1} \\ I_{2} \\ I_{3} \\ I_{4} \\ I_{5} \\ I_{6} \\ I_{7} \\ I_{8} \\ I_{9} \end{matrix} \right| = \left|\begin{matrix} I_{1j} \\ I_{2j} \\ I_{3j} \\ I_{4j} \end{matrix} \right|       (3.36)

where I_{1j}, I_{2j} are the injected currents at nodes I, 2, . . .
In abbreviated form
\overline{A} \ \overline{I}_{pr} =\overline{I}             (3.37)
where
  \overline{I}_{pr} is the column vector of the branch currents
  \overline{I} is column vector of the injected currents

With respect to voltages,

\left|\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \\ V_{4} \\ V_{5} \\ V_{6} \\ V_{7} \\ V_{8} \\ V_{9} \end{matrix} \right| =\left|\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 1 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} \right| \left|\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \\ V_{4} \end{matrix} \right|           (3.38)

Referring to Figure 3.8b,
V_{5} = V_{2} – V_{1}
V_{6} = V_{1} – V_{3}
V_{7} = V_{1} – V_{4}                                            (3.39)
V_{8} = V_{4} – V_{2}
V_{9} = V_{3} – V_{4}
Equation 3.38 can be written as
\overline{V}_{pr} = \overline{A}^t V                                         (3.40)
where
\overline{V}_{pr} is the column vector of the primitive branch voltage drops
\overline{V} is the vector of the bus voltages measured to the common node
From these relations a reader can prove Equation 3.30. Using this equation, the bus admittance matrix is

This matrix could be formed by simple examination of the network.