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. The nodal admittance matrix is used in the formulation of the power flow problem.
Construction from a single line diagram 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:
replacing each line in the diagram with its equivalent admittance, and converting all voltage sources to their equivalent current source. Consider an admittance graph with N {\displaystyle N} buses. The vector of bus voltages, V {\displaystyle V} , is an N × 1 {\displaystyle N\times 1} vector where V k {\displaystyle V_{k}} is the voltage of bus k {\displaystyle k} , and vector of bus current injections, I {\displaystyle I} , is an N × 1 {\displaystyle N\times 1} vector where I k {\displaystyle I_{k}} is the cumulative current injected at bus k {\displaystyle k} by all loads and sources connected to the bus. The admittance between buses k {\displaystyle k} and i {\displaystyle i} is a complex number y k i {\displaystyle y_{ki}} , and is the sum of the admittance of all lines connecting busses k {\displaystyle k} and i {\displaystyle i} . The admittance between the bus k {\displaystyle k} and ground is y k {\displaystyle y_{k}} , and is the sum of the admittance of all the loads connected to bus k {\displaystyle k} . Consider the current injection, I k {\displaystyle I_{k}} , into bus k {\displaystyle k} . Applying Kirchhoff's current law
I k = ∑ i = 1 , 2 , … , N I k i {\displaystyle I_{k}=\sum _{i=1,2,\ldots ,N}I_{ki}}
where I k i {\displaystyle I_{ki}} is the current from bus k {\displaystyle k} to bus i {\displaystyle i} for k ≠ i {\displaystyle k\neq i} and I k k {\displaystyle I_{kk}} is the current from bus k {\displaystyle k} 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
I k i = { V k y k , if i = k ( V k − V i ) y k i , if i ≠ k . {\displaystyle I_{ki}={\begin{cases}V_{k}{y_{k}},&{\mbox{if}}\quad i=k\\(V_{k}-V_{i})y_{ki},&{\mbox{if}}\quad i\neq k.\end{cases}}}
Therefore,
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