**Z**_{BUS} Formulation:

_{BUS}Formulation:

Z_{BUS} Formulation is given by

**By Inventing Y _{BUS}**

The sparsity of Y_{BUS} may be retained by using an efficient inversion technique and nodal impedance matrix can then be calculated directly from the factorized admittance matrix.

**Current Injection Method:**

Equation (9.33) can be written in the expanded form

It immediately follows from Eq. (9.34) that

Also Z_{ij }= Z_{ji}; (Z_{BUS} Formulation_{ }is a symmetrical matrix).

As per Eq. (9.35) if a unit current is injected at bus (node) j, while the other buses are kept open circuited, the bus voltages yield the values of the jth column of Z_{BUS}. However, no organized computerizable techniques are possible for finding the bus voltages. The technique had utility in AC Network Analyzers where the bus voltages could be read by a voltmeter.

**Z**_{BUS} Building Algorithm:

_{BUS}Building Algorithm:

It is a step-by-step programmable technique which proceeds branch by branch. It has the advantage that any modification of the network does not require complete rebuilding of Z_{BUS} Formulation.

Consider that Z_{BUS} Formulation has been formulated up to a certain stage and another branch is now added. Then

Upon adding a new branch, one of the following situations is presented.

- Z
_{b}is added from a new bus to the reference bus (i.e. a new branch is added and the dimension of Z_{BUS}goes up by one). This is type-I modification. - Z
_{b}is added from a new bus to an old bus (i.e., a new branch is added and the dimension of Z_{BUS}goes up by one). This is type-2 modification. - Z
_{b}connects an old bus to the reference branch (i.e., a new loop is formed but the dimension of Z_{BUS}does not change). This is type-3 modification. - Z
_{b}connects two old buses (i.e., new loop is formed but the dimension of Z_{BUS}does not change). This is type-4 modification. - Z
_{b}connects two new buses (Z_{BUS}remains unaffected in this case). This situation can be avoided by suitable numbering of buses and from now on wards will be ignored.

Notation: i, j—old buses; r—reference bus; k—new bus.

**Type-1 Modification:**

Figure 9.24 shows a passive (linear) n-bus network in which branch with impedance Z_{b} is added to the new bus k and the reference bus r. Now

Hence

**Type-2 Modification:**

Z_{b} is added from new bus k to the old bus j as in Fig. 9.25. It follows from this figure that

Rearranging,

Consequently

**Type****–****3 Modification:**

Z_{b} connects an old bus (j) to the reference bus (r) as in Fig. 9.26. This case follows from Fig. 9.25 by connecting bus k to the reference bus r, i.e. by setting V_{k} = 0.

Thus

Eliminate I_{k} in the set of equations contained in the matrix operation (9.38),

or

Now

Substituting Eq. (9.40) in Eq. (9.39)

Equation (9.37) can be written in matrix form as

**Type-4 Modification:**

Z_{b} connects two old buses as in Fig. 9.27. Equations can be written as follows for all the network buses.

Similar equations follow for other buses.

The voltages of the buses i and j are, however, constrained by the equation (Fig. 9.27)

Rearranging

Collecting equations similar to Eq. (9.43) and Eq. (9.45) we can write

Eliminating I_{k} in Eq. (9.46) on lines similar to what was done in Type-2 modification, it follows that

With the use of four relationships Eqs (9.36), (9.37), (9.42) and (9.47) bus impedance matrix can be built by a step-by-step procedure

When the network undergoes changes, the modification procedures can be employed to revise the bus impedance matrix of the network. The opening of a line (Z_{ij}) is equivalent to adding a branch in parallel to it with impedance — Z_{ij.}