Three phase Bergeron transmission line

This section describes Bergeron three phase transmission line based on a distributed parameter traveling wave model with lumped resistance.

Properties for this model are length of the line in kilometers, per length resistance and ground resistance in ohms/km, per length core and shield capacitance in farads/km, per length inductance and mutual inductance between phases in henries/km and execution rate in seconds.

Description

In case of balanced line

mathematical model of the circuit can be written in the matrix form:

$-\left(\frac{\partial }{\partial x}\right)\left[e\right]=\left[L\right]\left(\frac{\partial }{\partial t}\right)\left[i\right]$

$-\left(\frac{\partial }{\partial x}\right)\left[i\right]=\left[C\right]\left(\frac{\partial }{\partial t}\right)\left[e\right]$

where matrices [ L ] and [ C ] are: $L\left[\begin{array}{ccc}Ls& M& M\\ M& Ls& M\\ M& M& Ls\end{array}\right]$

$C\left[\begin{array}{ccc}2{C}_{c}+{C}_{e}& -{C}_{e}& -{C}_{e}\\ -{C}_{e}& 2{C}_{c}+{C}_{e}& -{C}_{e}\\ -{C}_{e}& -{C}_{e}& 2{C}_{c}+{C}_{e}\end{array}\right]$

Model of the three phase system is transformed into three decoupled single phase line models, so called "modal" lines, where transform matrix is: $T\left[\begin{array}{ccc}1& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$ and inverse transform matrix is ${T}^{-1}\left[\begin{array}{ccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& -\frac{1}{3}& 0\\ \frac{1}{3}& 0& -\frac{1}{3}\end{array}\right]$

To transform matrices [ L ] and [ C ] into modal domain we use transform matrix:

$\left[{L}_{mode}\right]=\left[{T}^{-1}\right]\left[L\right]\left[T\right]$

$\left[{C}_{mode}\right]=\left[{T}^{-1}\right]\left[C\right]\left[T\right]$

For each of single phase modal lines, travel time can be found and simulation time step, T , selected according to them (Note: in discrete system simulation adjusted time steps can only be integer multiples of T).

The calculations previously done in Single phase Bergeron line are made in the modal domain before being converted back to phase values .

Representation of the lossless three phase power line can be expanded to model losses by adding series conductors' resistances and ground resistance.

$R\left[\begin{array}{ccc}R+{R}_{g}& {R}_{g}& {R}_{g}\\ {R}_{g}& R+{R}_{g}& {R}_{g}\\ {R}_{g}& {R}_{g}& R+{R}_{g}\end{array}\right]$

To transform [ R ] into modal domain:

$\left[{R}_{mode}\right]=\left[{T}^{-1}\right]\left[R\right]\left[T\right]$

References

[1] Dommel, H., “Digital Computer Solution of Electromagnetic Transients in Single and Multiple Networks,” IEEE® Transactions on Power Apparatus and Systems, Vol. PAS-88, No. 4, April, 1969.