Chapter 3: Network Admittance and Impedance Matrices

Direct Determination of Zbus Matrix

We shall now use the methods given in Sections 3.3.1 to 3.3.4 for the direct determination of the Zbus matrix without forming the Ybus matrix first. To accomplish this we shall consider the system of Fig. 3.2 and shall use the system data given in Example 3.1. Note that for the construction of the Z bus matrix we first eliminate all the voltage sources from the system.

Step-1 : Start with bus-1. Assume that no other buses or lines exist in the system. We add this bus to the reference bus with the impedance of j 0.25 per unit. Then the Zbus matrix is

 (3.43)

Step-2 : We now add bus-2 to the reference bus using (3.31). The system impedance diagram is shown in Fig. 3.10. We then can modify (3.43) as

 (3.44)

Fig. 3.10 Network of step-2.

Step-3 : We now add an impedance of j 0.2 per unit between buses 1 and 2 as shown in Fig. 3.11. The interim Z bus matrix is then obtained by applying (3.41) on (3.44) as

Eliminating the last row and last column using the Kron's reduction of (3.31) we get

 (3.45)

Step-4 : We now add bus-3 to bus-1 through an impedance of j 0.25 per unit as shown in Fig. 3.12. The application of (3.35) on (3.45) will then result in the following matrix

Fig. 3.11 Network of step-3.

 (3.46)

Fig. 3.12 Network of step-4.

Step-5 : Connect buses 2 and 3 through an impedance of j 0.4 per unit as shown in Fig. 3.13. The interim Zbus matrix is then formed from (3.41) and (3.46) as

Fig. 3.13 Network of step-5.

Using the Kron's reduction we get the following matrix

 (3.47)

Step-6 : We now add a new bus-4 to bus-2 through an impedance of j 0.5 as shown in Fig. 3.14. Then the application of (3.35) on (3.47) results in the following matrix

 (3.48)

Fig. 3.14 Network of step-6.

Step-7 : Finally we add buses 3 and 4 through an impedance of j 0.4 to obtain the network of Fig. 3.3 minus the voltage sources. The application of (3.41) on (3.48) results in the interim Zbus matrix of

Eliminating the 5th row and column through Kron's reduction we get the final Zbus as

 (3.49)

The Zbus matrix given in (3.49) is the as that given in Example 3.1 which is obtained by inverting the Ybus matrix.