1. Introduction

The Zbus is the inverse of the Ybus, i.e.,

[pic](1)

Since we know that

[pic](2)

and therefore

[pic](3)

then

[pic](4)

So Zbus relates the nodal current injections to the nodal voltages, as seen in (4).

In developing the power flow problem, we choose to work with Ybus. The reason for this is that the power flow problem requires an iterative solution that can be made very efficient when we use Ybus, due to the sparsity (lots of zeros) in the matrix used in performing the iteration (the Jacobian matrix – we will discuss this more later). However, in developing fault analysis methods (done in EE 457), we will choose to work with Zbus. The main reason for choosing to work with Zbus in fault analysis is that, as we will see, Zbus quantities characterize conditions when all current injections are zero except one, corresponding to the faulted bus. We can use some creative thinking to express that one current injection (the fault current). Once we have that one current injection, eq. (4) is very easy to evaluate to obtain all bus voltages in the network, and once we have bus voltages, we can get all currents everywhere. These currents are the currents under the fault conditions and are used to design protection systems.

The Zbus is not sparse (no zeros). But fortunately, fault analysis does not require iterative solutions, and so computational benefit of sparsity is not significant in fault analysis.

2. The meaning of Zbus elements

We can write the Z-bus relation for the same network as

[pic](11)

We understand by eq. (11) that the independent sources are all current sources, and eq. (11) allows us to compute the voltages resulting from those current sources being injected into the network. These current sources are the equivalent representation of the generator voltage sources.

Let’s inspect more closely one of the equations in (11). Arbitrarily choose the second equation. [pic](12)

Now solve for driving point impedance of bus 2, Z22.

[pic](13)

But what if we set I1 and I3 to 0, i.e., what if we open-circuit buses 1 and 3? In other words, let’s idle all sources.

This means that we will open any current sources at nodes 1 and 3 so that there are no sources there (but there may be load impedances). Then eq. (13) is: [pic](14)

Equation (14) says that Z22 is the ratio of bus 2 voltage to the bus 2 current injection when all sources are idled. This is the definition of the Thevenin impedance!!!!

Conclusion: The diagonal elements of the Z-bus are the Thevenin equivalent impedances seen looking into the network at that bus. The remainder of these notes comes from chapter 9 of Bergen & Vittal. I cover it in EE 456. They show you how to construct the Z-bus for a large network without performing matrix inversion. You are not responsible for this material for this course. If you have not had EE 456 and you are not going to take it next fall, then I suggest you review this material together with the corresponding material in the Bergen & Vittal textbook.

One important attribute to building the Z-bus for fault analysis is that the generator subtransient reactances should be included. This means that it is necessary to include an additional bus for every generator in order to enable distinguishing between the high side of the generator internal voltage from the network side of the generator subtransient reactance. The figure below illustrates the difference between generator representation for power flow analysis (the focus of EE 456) and generator representation for fault analysis (the focus of EE 457). [pic]

3. Self admittance and driving point impedance

You should recall that it is easy to develop the Y-bus. From that, one can invert it to obtain the Z-bus. However, in spite of the fact that Matlab is quite capable of matrix inversion for small dimension, you are NOT ALLOWED to think about just inverting Ybus since we...