Kron reduction

In power engineering, Kron reduction is a method used to reduce or eliminate the desired node without need of repeating the steps like in Gaussian elimination.[1]

It is named after American electrical engineer Gabriel Kron.

Kron reduction is a useful tool to eliminate unused nodes in a Y-parameter matrix[2]. For example, three linear elements linked in series with a port at each end may be easily modeled as a 4X4 nodal admittance matrix of Y-parameters, but only the two port nodes normally need to be considered for modeling and simulation. Kron reduction may be used to eliminate the internal nodes, and thereby reducing the 4th order Y-parameter matrix to a 2nd order Y-parameter matrix. The 2nd order Y-parameter matrix is then more easily converted to a Z-parameter matrix or S-parameter matrix when needed.

Consider a general Y-parameter matrix that may be created from a combination of linear elements constructed such that two internal nodes exist.

${\displaystyle Y{_{4}}{_{X}}{_{4}}={\begin{bmatrix}Y{_{1}}{_{1}}&Y{_{1}}{_{2}}&Y{_{1}}{_{3}}&Y{_{1}}{_{4}}\\Y{_{2}}{_{1}}&Y{_{2}}{_{2}}&Y{_{2}}{_{3}}&Y{_{2}}{_{4}}\\Y{_{3}}{_{1}}&Y{_{3}}{_{2}}&Y{_{3}}{_{3}}&Y{_{3}}{_{4}}\\Y{_{4}}{_{1}}&Y{_{4}}{_{2}}&Y{_{4}}{_{3}}&Y{_{4}}{_{4}}\\\end{bmatrix}}}$

While it is possible to use the 4X4 matrix in simulations or to construct a 4X4 S-parameter matrix, is may be simpler to reduce the Y-parameter matrix to a 2X2 by eliminating the two internal nodes through Kron Reduction, and then simulating with a 2X2 matrix and/or converting to a 2X2 S-parameter or Z-Parameter matrix.

${\displaystyle Y'{_{2}}{_{X}}{_{2}}={\begin{bmatrix}Y'{_{1}}{_{1}}&Y'{_{1}}{_{2}}\\Y'{_{2}}{_{1}}&Y'{_{2}}{_{2}}\end{bmatrix}}\qquad \mathrm {Where\ Y'\ is\ the\ Kron\ Reduced\ Matrix} }$

The process for executing a Kron reduction is as follows[3]:

Select the Kth row/column used to model the undesired internal nodes to be eliminated. Apply the below formula to all other matrix entries that do not reside on the Kth row and column. Then simply remove the Kth row and column of the matrix, which reduces the size of the matrix by one.

Kron Reduction for the Kth row/column of an NXN matrix:

${\displaystyle {\begin{aligned}\sum _{i=1}^{N}\sum _{j=1}^{N}Y'{_{i}}{_{j}}=Y{_{i}}{_{j}}-{\frac {Y{_{i}}{_{k}}Y{_{j}}{_{k}}}{Y{_{k}}{_{k}}}},&\qquad for\ i\neq k,,j\neq k\\&\qquad \mathrm {Where\ Y'\ is\ The\ Replacement\ Matrix\ Entry} \\\end{aligned}}}$

Linear elements that are also passive always form a symmetric Y-parameter matrix, that is, ${\displaystyle Y{_{i}}{_{J}}=Y{_{j}}{_{i}}}$ in all cases. The number of computations of a Kron reduction may be redcued by taking advantage of this symmetry, as shown ion the equation below.

Kron Reduction for symmetric NXN matrices:

${\displaystyle {\begin{aligned}&\sum _{i=1}^{N}\sum _{j=i}^{N}Y{_{i}}{_{j}}=Y{_{i}}{_{j}}-{\frac {Y{_{i}}{_{k}}Y{_{j}}{_{k}}}{Y{_{k}}{_{k}}}},\qquad for\ i\neq k,j\neq k\\&Y{_{j}}{_{i}}=Y{_{i}}{_{j}},\qquad for\ i\neq j\\\end{aligned}}}$

Once all the matrix entries have been modified by the Kron Reduction equation, the Kth row/column me be eliminated, and the matrix order is reduced by one. Repeat for all internal nodes desired to be eliminated