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.

Construction from a single line diagram

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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   buses. The vector of bus voltages,  , is an   vector where   is the voltage of bus  , and vector of bus current injections,  , is an   vector where   is the cumulative current injected at bus   by all loads and sources connected to the bus. The admittance between buses   and   is a complex number  , and is the sum of the admittance of all lines connecting busses   and  . The admittance between the bus   and ground is  , and is the sum of the admittance of all the loads connected to bus  .

Consider the current injection,  , into bus  . Applying Kirchhoff's current law

 

where   is the current from bus   to bus   for   and   is the current from bus   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   is a   matrix such that bus voltage and current injection satisfy Ohm's law

 

in vector format. The entries of   are then determined by the equations for the current injections into buses, resulting in

 
 
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:

 

The diagonal entries   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   is typically a symmetric matrix as  . However, extensions of the line model may make   asymmetrical. For instance, modeling phase-shifting transformers, results in a Hermitian admittance matrix.[2]

Applications

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The admittance matrix is most often used in the formulation of the power flow problem.[3][4]

See also

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References

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  1. ^ Grainger, John (1994). Power System Analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0070612938.
  2. ^ Saadat, Hadi (1999). "6.7 Tap changing transformers". Power System Analysis. United Kingdom: WCB/McGraw-Hill. ISBN 978-0075616344.
  3. ^ McCalley, James. "The Power Flow Equations" (PDF). Iowa State Engineering.
  4. ^ Saadat, Hadi (1999). Power System Analysis. United Kingdom: WCB/McGraw-Hill. ISBN 978-0075616344.
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