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:

Consider an admittance graph with

N

buses. The vector of bus voltages,

V

, is an

N x 1

vector where

Vk

is the voltage of bus

k

, and vector of bus current injections,

I

, is an

N x 1

vector where

Ik

is the cumulative current injected at bus

k

by all loads and sources connected to the bus. The admittance between buses

k

and

i

is a complex number

yki

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

k

and

i

. The admittance between the bus

i

and ground is

yk

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

k

.

Consider the current injection,

Ik

, into bus

k

. Applying Kirchhoff's current law

Ik=\sumi=1,Iki

where

Iki

is the current from bus

k

to bus

i

for

ki

and

Ikk

is the current from bus

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

Iki=\begin{cases} Vk{yk

}, & \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

Y

is a

N x N

matrix such that bus voltage and current injection satisfy Ohm's law

YV=I

in vector format. The entries of

Y

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

Y11,Y22,...,Ynn

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

Y

is typically a symmetric matrix as

Yki=Yik

. However, extensions of the line model may make

Y

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

  1. Book: John . Grainger . 1994 . Power System Analysis . McGraw-Hill Science/Engineering/Math . 978-0070612938.
  2. Book: Hadi . Saadat . Power System Analysis . WCB/McGraw-Hill . United Kingdom . 1999 . 978-0075616344 . 6.7 Tap changing transformers.
  3. Web site: McCalley . James . The Power Flow Equations . Iowa State Engineering.
  4. Book: Hadi . Saadat . Power System Analysis . WCB/McGraw-Hill . United Kingdom . 1999 . 978-0075616344.