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{{Short description|N x N matrix describing a linear power system with N buses}}
{{about|the admittance matrix in electrical engineering|the graph theoretic admittance matrix|Laplacian matrix}}

In [[power engineering]], '''nodal admittance matrix''' (or just '''admittance matrix''') is an ''N x N'' [[matrix (mathematics)|matrix]] describing a linear power system with ''N'' [[Bus (power engineering)|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 [[electric power transmission|transmission line]]s.<ref>
{{cite book
    | first1=John
    | last1=Grainger
    | year=1994
    | title=Power System Analysis
    | publisher=McGraw-Hill Science/Engineering/Math
    | isbn=978-0070612938}}
</ref>  The nodal admittance matrix is used in the formulation of the [[power-flow study|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 circuit laws|Kirchhoff's laws]] to the admittance diagram of the power system. Starting from the [[One-line diagram|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 <math>N</math> buses. The ''vector of bus voltages'', <math>V</math>, is an <math>N \times 1</math> vector where <math>V_{k}</math> is the voltage of bus <math>k</math>, and ''vector of bus current injections'', <math>I</math>, is an <math>N \times 1</math> vector where <math>I_{k}</math> is the cumulative current injected at bus <math>k</math> by all loads and sources connected to the bus. The admittance between buses <math>k</math> and <math>i</math> is a complex number <math>y_{ki}</math>, and is the sum of the admittance of all lines connecting busses <math>k</math> and <math>i</math>. The admittance between the bus <math>i</math> and ground is <math>y_{k}</math>, and is the sum of the admittance of all the loads connected to bus <math>k</math>.

Consider the ''current injection'', <math>I_{k}</math>, into bus <math>k</math>. Applying [[Kirchhoff's circuit laws|Kirchhoff's current law]]
:<math>
  I_{k} = \sum_{i=1, 2, \ldots, N} I_{ki}
</math>
where <math>I_{ki}</math> is the current from bus <math>k</math> to bus <math>i</math> for <math>k \neq i</math> and <math>I_{kk}</math> is the current from bus <math>k</math> to ground through the bus load. Applying [[Ohm's law|Ohm's]] law to the admittance diagram, the ''bus voltages'' and the line and load currents are linked by the relation
:<math>
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}
</math>
Therefore,
:<math>
  I_{k}
= \sum_{i=1, 2, \ldots, N \atop i \neq k} { ( V_{k} - V_{i} ) y_{ki} } + V_{k} y_{k}
= V_{k} \left( y_{k} + \sum_{i=1, 2, \ldots, N \atop i \neq k} y_{ki} \right) - \sum_{i=1, 2, \ldots, N \atop i \neq k} V_{i} y_{ki}
</math>
This relation can be written succinctly in matrix form using the admittance matrix. The nodal admittance matrix <math>Y</math> is a <math>N \times N</math> matrix such that bus voltage and current injection satisfy Ohm's law
:<math>
  Y V = I
</math>
in vector format. The entries of <math>Y</math> are then determined by the equations for the ''current injections'' into buses, resulting in
:<math>
Y_{kj} = \begin{cases}
  y_{k} + \sum_{i=1, 2, \ldots, N \atop i \neq k} {y_{ki}}, & \mbox{if} \quad k = j \\
  -y_{kj},  & \mbox{if} \quad k \neq j.
\end{cases}
</math>

[[File:3 bus network.jpg|400px|thumbnail|'''Figure 1''': The admittance diagram of a three bus network.]]

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:
:<math>
Y = \begin{pmatrix}
  y_{1} + y_{12} + y_{13} & -y_{12} & -y_{13} \\
  -y_{12} & y_{2} + y_{12} + y_{23} & -y_{23} \\
  -y_{13} & -y_{23} & y_{3} + y_{13} + y_{23} \\
\end{pmatrix}
</math>

The diagonal entries <math>Y_{11}, Y_{22}, ..., Y_{nn}</math> are called the ''self-admittances'' of the network nodes. The non-diagonal entries are the ''mutual admittances'' of the nodes corresponding to the subscripts of the entry. The admittance matrix <math>Y</math> is typically a [[symmetric matrix]] as <math>Y_{ki} = Y_{ik}</math>. However, extensions of the line model may make <math>Y</math> asymmetrical. For instance, modeling phase-shifting transformers, results in a [[Hermitian matrix|Hermitian]] admittance matrix.<ref>
{{cite book
    | first1=Hadi
    | last1=Saadat
    | title=Power System Analysis
    | publisher=WCB/McGraw-Hill
    | location=United Kingdom
    | year=1999
    | isbn=978-0075616344
    | chapter=6.7 Tap changing transformers
}}
</ref>

== Applications ==

The admittance matrix is most often used in the formulation of the [[Power-flow study|power flow problem]].<ref>
{{cite web
    | last1=McCalley
    | first1=James
    | title=The Power Flow Equations
    |url=http://home.eng.iastate.edu/~jdm/ee458_2011/PowerFlowEquations.pdf
    |website=Iowa State Engineering
}}</ref><ref>
{{cite book
    | first1=Hadi
    | last1=Saadat
    | title=Power System Analysis
    | publisher=WCB/McGraw-Hill
    | location=United Kingdom
    | year=1999
    | isbn=978-0075616344
}}
</ref>

== See also ==
* [[Admittance parameters]]
* [[Nodal analysis]]
* [[Zbus]]

== References ==
{{Reflist}}

== External links ==
*[https://web.archive.org/web/20060206073952/http://www.jcmiras.net/jcm/item/73/ A C/C++ Program and Source Code for Computing Ybus and Zbus Matrices]

[[Category:Electrical engineering]]
[[Category:Electric power]]