JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III POWER SYSTEMS ANALYSIS Previous Question Paper (January/February - 2021 - R16)

The bus admittance matrix Y_BUS is formed using the singular transformation method by relating the primitive admittance matrix y to the bus incidence matrix A. The procedure is as follows:

1. Identify all nodes (buses) and branches in the network. Assign a reference bus (usually the slack bus).
2. Construct the primitive admittance matrix y, which is a diagonal matrix of size (b × b) where b is the number of branches. Each element y_ii is the admittance of the i-th branch.
3. Define the bus incidence matrix A of size (n × b), where n is the number of independent buses (excluding the reference bus). An element A_ij is 1 if branch j leaves bus i, -1 if it enters bus i, and 0 if the branch is not connected to bus i.
4. Incorporate generator buses by treating them as nodes with shunt admittances or by including their internal impedances in the branch list if they are represented as voltage sources with series impedance.
5. Calculate the Y_BUS matrix using the transformation formula:

The resulting Y_BUS is an (n × n) symmetric matrix. For a system with n buses, the diagonal element Y_ii is the sum of all admittances connected to bus i, and the off-diagonal element Y_ij is the negative of the sum of admittances connected directly between buses i and j.

When a new element with impedance Z_b is added between two existing buses (bus p and bus q) in a system where the Z_BUS matrix is already known, the new Z_BUS matrix is calculated by augmenting the existing matrix and applying Kron's reduction. Let the original Z_BUS be an n x n matrix. Adding an element between bus p and bus q creates a new loop.

First, we define the current I_b flowing through the new element. The voltage drop across the new element is given by:

The voltages at any bus k in the original network are expressed as:

This can be written in matrix form by adding an extra row and column representing the new branch b:

Where Z_p and Z_q are the p-th and q-th columns of the original Z_BUS. Let the new augmented matrix be Z_aug of size (n+1) x (n+1). The final Z_BUS is obtained by eliminating the branch current I_b using Kron's reduction formula:

Where Z_ib is the (n+1)-th column, Z_bj is the (n+1)-th row, and Z_bb is the element at the (n+1, n+1) position of the augmented matrix. This reduction effectively incorporates the constraint of the new loop into the existing bus impedance network.