**ZBUS Formulation:**

ZBUS Formulation is given by

**By Inventing Y _{BUS}**

The sparsity of Y_{BUS} may be retained by using an efficient inversion technique [1] and nodal impedance matrix can then be calculated directly from the factorized admittance matrix. This is beyond the scope of this book.

**Current Injection Technique**

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 }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} •

Consider that Z_{BUS} has been formulated upto 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 onwards will be ignored.

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

*Type-1 Mollification*

*Type-1 Mollification*

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*

*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*

*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),

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

Equation (9.37) can be written in matrix form as

**T***ype-4 Modification*

*ype-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 (bringing in one branch at a time) as illustrated in Example 9.8. This procedure being a mechanical one can be easily computerized.

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}