Kron Reduction

The Kron Reduction is a relatively simple technique for eliminating nodes from a network when the voltage or current at that node is zero. The technique is named after Gabriel Kron, who described the method in [1].

Reduction of a Multi-conductor Line with Earth Wires

Consider a section of three-phase overhead line with an earth wire. From Ohm's law for the multi-conductor system:

$\left[{\begin{matrix}V_{a}\\V_{b}\\V_{c}\\-\\V_{e}\end{matrix}}\right]=\left[{\begin{matrix}Z_{aa}&Z_{ab}&Z_{ac}&|&Z_{ae}\\Z_{ba}&Z_{bb}&Z_{bc}&|&Z_{be}\\Z_{ca}&Z_{cb}&Z_{cc}&|&Z_{ce}\\--&--&--&|&--\\Z_{ea}&Z_{eb}&Z_{ec}&|&Z_{ee}\end{matrix}}\right]\left[{\begin{matrix}I_{a}\\I_{b}\\I_{c}\\-\\I_{e}\end{matrix}}\right]\,$ ... Equ. (1)

Where $V_{a}\,$, $V_{b}\,$ and $V_{c}\,$ are the phase voltages across the section of line (V)

$I_{a}\,$, $I_{b}\,$ and $I_{c}\,$ are the phase currents through the section of line (A)

$Z_{aa}\,$, $Z_{bb}\,$ and $Z_{cc}\,$ are the phase self-impedances ($\Omega$)

$Z_{ab}=Z_{ba}\,$, $Z_{ac}=Z_{ca}\,$ and $Z_{bc}=Z_{cb}\,$ are the mutual coupling impedances between phase conductors ($\Omega$)

$V_{e}\,$ and $I_{e}\,$ are the voltage and current across the earth wire

$Z_{ee}\,$ is the earth wire self-impedance ($\Omega$)

$Z_{ea}=Z_{ae}\,$, $Z_{eb}=Z_{be}\,$ and $Z_{ec}=Z_{ce}\,$ are the mutual coupling impedances between phase conductors and the earth wire ($\Omega$)

We can rewrite Equ. (1) in a partitioned form as follows:

$\left[{\begin{matrix}V_{abc}\\-\\V_{e}\end{matrix}}\right]=\left[{\begin{matrix}Z_{A}&|&Z_{E1}\\--&|&--\\Z_{E2}&|&Z_{ee}\end{matrix}}\right]\left[{\begin{matrix}I_{abc}\\-\\I_{e}\end{matrix}}\right]\,$

Where $V_{abc}=\left[{\begin{matrix}V_{a}\\V_{b}\\V_{c}\end{matrix}}\right]$ and $I_{abc}=\left[{\begin{matrix}I_{a}\\I_{b}\\I_{c}\end{matrix}}\right]$

$Z_{A}=\left[{\begin{matrix}Z_{aa}&Z_{ab}&Z_{ac}\\Z_{ba}&Z_{bb}&Z_{bc}\\Z_{ca}&Z_{cb}&Z_{cc}\end{matrix}}\right]$, $Z_{E1}=\left[{\begin{matrix}Z_{ae}\\Z_{be}\\Z_{ce}\end{matrix}}\right]$ and $Z_{E2}'=\left[{\begin{matrix}Z_{ea}\\Z_{eb}\\Z_{eb}\end{matrix}}\right]$

If we assume that the voltage in the earth wire is zero, i.e. $V_{e}=0\,$ and thus:

$Z_{E2}I_{abc}+Z_{ee}I_{e}=0\,$

We can solve for the current in the earth wire:

$I_{e}=-Z_{ee}^{-1}Z_{E2}I_{abc}\,$

Plugging this back into the top part of the partitioned matrix:

$V_{abc}=Z_{A}I_{abc}-Z_{E1}Z_{ee}^{-1}Z_{E2}I_{abc}\,$

$=\left(Z_{A}-Z_{E1}Z_{ee}^{-1}Z_{E2}\right)I_{abc}\,$

$=Z_{abc}I_{abc}\,$

This is the reduced form of the multi-conductor system, where the earth wire node is eliminated leaving only the three phase conductor nodes. Using similar logic to that described above, the Kron reduction can be used on multi-conductor systems with an arbitrary number of earth wires and neutral conductors.

References

• [1] Kron, G., "Tensor Analysis of Networks", John Wiley and Sons, 1965