Nodal admittance matrix explained
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
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
}, & \mbox \quad i = k \\ (V_ - V_) y_, & \mbox \quad i \neq k.\endTherefore,
Ik=\sumi=1,{(Vk-Vi)yki}+Vkyk=Vk\left(yk+\sumi=1,yki\right)-\sumi=1,Viyki
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
Ykj=\begin{cases}
yk+\sumi=1,{yki
}, & \mbox \quad k = j \\ -y_, & \mbox \quad k \neq j.\end
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:
Y=\begin{pmatrix}
y1+y12+y13&-y12&-y13\\
-y12&y2+y12+y23&-y23\\
-y13&-y23&y3+y13+y23\\
\end{pmatrix}
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
The admittance matrix is most often used in the formulation of the power flow problem.[3] [4]
See also
External links
Notes and References
- Book: John . Grainger . 1994 . Power System Analysis . McGraw-Hill Science/Engineering/Math . 978-0070612938.
- Book: Hadi . Saadat . Power System Analysis . WCB/McGraw-Hill . United Kingdom . 1999 . 978-0075616344 . 6.7 Tap changing transformers.
- Web site: McCalley . James . The Power Flow Equations . Iowa State Engineering.
- Book: Hadi . Saadat . Power System Analysis . WCB/McGraw-Hill . United Kingdom . 1999 . 978-0075616344.