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:
![{\displaystyle \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]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b280e62c6e35d5e720912a082d5ded773f37dd)
... Equ. (1)
Where 
, 
and 
are the phase voltages across the section of line (V)
, 
and 
are the phase currents through the section of line (A)
, 
and 
are the phase self-impedances (
)
, 
and 
are the mutual coupling impedances between phase conductors ()
and 
are the voltage and current across the earth wire
is the earth wire self-impedance ()
, 
and 
are the mutual coupling impedances between phase conductors and the earth wire ()
We can rewrite Equ. (1) in a partitioned form as follows:
![{\displaystyle \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]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99f6c6210e81e2c77cab9adaf4e95910fa9797ff)
Where ![{\displaystyle V_{abc}=\left[{\begin{matrix}V_{a}\\V_{b}\\V_{c}\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9cc71e165630efd879b1c5943cf9fc1b77b2d5f)
and ![{\displaystyle I_{abc}=\left[{\begin{matrix}I_{a}\\I_{b}\\I_{c}\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298586fc75d04393eb9da39e4df9db549c378c22)
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/419082fa3e21122e11a5dd21bd863bd582470c3b)
, ![{\displaystyle Z_{E1}=\left[{\begin{matrix}Z_{ae}\\Z_{be}\\Z_{ce}\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5858460b548baaaead1b784a80cd5e6de74d7c)
and ![{\displaystyle Z_{E2}'=\left[{\begin{matrix}Z_{ea}\\Z_{eb}\\Z_{eb}\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a08c6d8642a937625a7b7d2390d3460d786ed9)
If we assume that the voltage in the earth wire is zero, i.e. 
and thus:
We can solve for the current in the earth wire:
Plugging this back into the top part of the partitioned matrix:
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