Implement three-phase transmission line section with lumped parameters

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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}*,

See the PI Section Line block reference page for an explanation on how to compute RLC parameters taking into account hyperbolic corrections.

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 `power_lineparam`

reference page to learn how to
use this tool.

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). Default is

`60`

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

`[ 0.01273 0.3864]`

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

`[ 0.9337e-3 4.1264e-3]`

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

`[12.74e-9 7.751e-9]`

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

`100`

.

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.