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In power engineering, Y Matrix or Ybus is an n x n symmetric matrix describing a power system with n buses. It represents the nodal admittance of the buses in a power system. In realistic systems which contain thousands of buses, the Y matrix is quite sparse. Each bus in a real power system is usually connected to only a few other buses through the transmission lines. The Y Matrix is also one of the data requirements needed to formulate a power flow study.

Ybus matrix form: $\begin{bmatrix} Y_{11} & Y_{12} & \cdots & Y_{1n} \\ Y_{21} & Y_{22} & \cdots & Y_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ Y_{n1} & Y_{n2} & \cdots & Y_{nn} \end{bmatrix}$

## Context

Electric power transmission needs optimization. Only computer simulation allows the complex handling required. The Ybus matrix is a tool in that domain.

## Design

Starting from the single line diagram of a power system, there are four main steps in creating the Y Matrix. First, the single line diagram is converted to an impedance diagram. Next, all voltage sources are converted to their equivalent current source representations. From here, the impedance diagram is then converted to an admittance diagram. Finally, the Y Matrix itself is created.

The admittances Y11, Y22,... Ynn are called the self-admittances at the nodes, and each equals the sum of all the admittances terminating on the node identified by the repeated subscripts. The other admittances are the mutual admittances of the nodes, and each equals the negative of the sum of all admittances connected directly between the nodes identified by the double subscripts.

For small transmission systems of about less than 10 nodes or buses, Y matrix can be calculated manually. But for a realistic system with relatively large number of nodes or buses, say 1000 nodes, a computer program for computing Ybus is more practical to use.

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