Implement three-phase transmission line section with lumped parameters

Fundamental Blocks/Elements

The Three-Phase PI Section Line block implements a balanced three-phase transmission line model with parameters lumped in a PI section.

Contrary to the Distributed Parameter Line model where the resistance, inductance, and capacitance are uniformly distributed along the line, the Three-Phase PI Section Line block lumps the line parameters in a single PI section as shown in the figure below.

The line parameters R, L, and C are specified as positive- and zero-sequence parameters that take into account the inductive and capacitive couplings between the three phase conductors, as well as the ground parameters. This method of specifying line parameters assumes that the three phases are balanced.

The self and mutual resistances (*Rs*, *Rm*), self
and mutual inductances (*Ls*, *Lm*)
of the three coupled inductors, as well as phase capacitances *Cp* and
ground capacitances *Cg*, are deduced from the positive-
and zero-sequence RLC parameters as follows.

Let us assume the following line parameters:

r, _{1}r_{0} | Positive- and zero-sequence resistances per unit length (Ω/km) |

l, _{1}l_{0} | Positive- and zero-sequence inductances per unit length (H/km) |

c, _{1}c_{0} | Positive- and zero-sequence capacitances per unit length (F/km) |

f | Frequency (Hz) |

lsec | Line section length (km) |

The total positive- and zero-sequence RLC parameters including hyperbolic corrections are first evaluated:

$$\begin{array}{l}{R}_{1}={r}_{1}\cdot l\mathrm{sec}\cdot {k}_{r1}\\ {L}_{1}={l}_{1}\cdot l\mathrm{sec}\cdot {k}_{l1}\\ {C}_{1}={c}_{1}\cdot l\mathrm{sec}\cdot {k}_{c1}\\ {R}_{0}={r}_{0}\cdot l\mathrm{sec}\cdot {k}_{r0}\\ {L}_{0}={l}_{0}\cdot l\mathrm{sec}\cdot {k}_{l0}\\ {C}_{0}={c}_{0}\cdot l\mathrm{sec}\cdot {k}_{c0}\end{array}$$

where

*k _{r1}*,

The Powergui block provides a graphical tool for the calculation
of resistance, inductance, and capacitance per unit length based on
the line geometry and the conductor characteristics. See the |

For a short line section (approximately *lsec* <
50 km), these correction factors are negligible (close to unity).
However, for long lines, these hyperbolic corrections must be taken
into account in order to get an exact line model at the specified
frequency.

The RLC line section parameters are then computed as follows:

$$\begin{array}{l}Rs=\left(2{R}_{1}+{R}_{0}\right)/3\\ Ls=\left(2{L}_{1}+{L}_{0}\right)/3\\ {R}_{m}=\left({R}_{0}-{R}_{1}\right)/3\\ {L}_{m}=\left({L}_{0}-{L}_{1}\right)/3\\ {C}_{p}={C}_{1}\\ {C}_{g}=3{C}_{1}{C}_{0}/\left({C}_{1}-{C}_{0}\right)\end{array}$$

**Frequency used for rlc specification**The frequency used for specification of per unit length rlc line parameters, in hertz (Hz). This is usually the nominal system frequency (50 Hz or 60 Hz).

**Positive- and zero-sequence resistances**The positive- and zero-sequence resistances in ohms/kilometer (Ω/km).

**Positive- and zero-sequence inductances**The positive- and zero-sequence inductances in henries/kilometer (H/km). The zero-sequence inductance can not be zero, because it would result in an invalid propagation speed computation.

**Positive- and zero-sequence capacitances**The positive- and zero-sequence capacitances in farads/kilometer (F/km). The zero-sequence capacitance can not be zero, because it would result in an invalid propagation speed computation.

**Line section length**The line section length in kilometers (km).

The `power_triphaseline`

example
illustrates voltage transients at the receiving end of a 200 km line
when only phase A is energized. Voltages obtained with two line models
are compared: 1) the Distributed Parameters Line block and 2) a PI
line model using two Three-Phase PI Section Line blocks.

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