Kron reduction Method in power system

KRON REDUCTION METHOD

By Tariq Kamal UET Peshawar

Overview

Introduction to Kron Reduction Method Mathmatically form with Explanation

Introduction to Kron Reduction Method

The

size of a real Ybus, admittance matrix, is very large. Computational time can be a problem, therefore, we needed to come up of reduce the size of such matrix. The selection of the buses to be eliminated (in order to reduce the size of the matrix) is usually determined by the fact that there is no current being injected and /or the bus is of no importance to the analysis. „‟As a rule if there is no external load and/or there are not generating sources connected then we can eliminate such bus‟‟. When we want to eliminate a bus, we use the method of Kron.s reduction.

Mathmatically form

Kron’s reduction method is given as:

Y jk ( new)



Y jk 



Y jpY pk  Y  pp

“p” is the bus number to be eliminated. “j” row “k” column  j and k take values from 1 to n Therefore Ybus (new) has a new dimension (n-1)(n-1).

Mathmatically form Cond’t

Let consider a three-bus system in which I3=0 , So we may write admittance equation in the form

 I1  Y11  I   Y  2   21 0  Y31

Y12 Y22 Y32

Y 13  V 1 

   Y23 V 2   Y 33  V 3 

And eliminating node (3) we obtain 2 *2 system

Mathmatically form Cond’t The current injection into node 3 is zero and may be eliminated as follow: 1.Using the third row of the matrix,express V3 in terms of V1 & V3 2.The dependence of I1 and I2 on V1,V2,and V3 may now be expressed in terms of V1 and V2 3.Collect the terms into a new 2*2 admittance matrix Carrying out this procedure,we have

1. 0  Y31V1  Y32V3  Y33V3  or V3 

Y32 Y33

V1 

Y 32 Y 33

V2  

Mathmatically form Cond’t

 Y31 Y 32  2.  I1  Y11V1  Y12V2  Y13   V1  V 2  Y 33   Y33  Y31 Y 32   I 2  Y21V1  Y22V2  Y23   V1  V 2  Y 33   Y33 Y13Y31  Y13Y32    V 1   I1      Y11  Y  Y 12  Y      33  33         3.          Y Y Y Y   23 31 23 32   Y 22     Y21    Y33  Y 33   V 2   I 2  

Mathmatically form Cond’t

Regardless of which node has to zero current injection a system can be kron reduced without having to rearrange the equations.for  example if Ip=0 in the nodal equation of the N-bus system we may directly calculate the elements of the new,reduced bus admittance matrix by chossing Ypp as the pivot and p using the general formula

Y jk ( new)  Y jk  

Y jpY pk  Y  pp

i, j  1, 2,....., n