Nodal admittance matrix

In power engineering, nodal admittance matrix (or just admittance matrix) is an N x N matrix describing a linear 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 admittance matrix is quite sparse.

Each bus in a real power system is usually connected to only a few other buses through the transmission lines.

[1] The nodal admittance matrix is used in the formulation of the power flow problem.

The nodal admittance matrix of a power system is a form of Laplacian matrix of the nodal admittance diagram of the power system, which is derived by the application of Kirchhoff's laws to the admittance diagram of the power system.

Starting from the single line diagram of a power system, the nodal admittance diagram is derived by: Consider an admittance graph with

The vector of bus voltages,

, and vector of bus current injections,

is the cumulative current injected at bus

by all loads and sources connected to the bus.

, and is the sum of the admittance of all lines connecting busses

, and is the sum of the admittance of all the loads connected to bus

Applying Kirchhoff's current law where

to ground through the bus load.

Applying Ohm's law to the admittance diagram, the bus voltages and the line and load currents are linked by the relation Therefore, This relation can be written succinctly in matrix form using the admittance matrix.

The nodal admittance matrix

matrix such that bus voltage and current injection satisfy Ohm's law in vector format.

are then determined by the equations for the current injections into buses, resulting in As an example, consider the admittance diagram of a fully connected three bus network of figure 1.

The admittance matrix derived from the three bus network in the figure is: The diagonal entries

are called the self-admittances of the network nodes.

However, extensions of the line model may make

For instance, modeling phase-shifting transformers, results in a Hermitian admittance matrix.

[2] The admittance matrix is most often used in the formulation of the power flow problem.