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Implement three-phase transmission line section with lumped parameters
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}, k_{l1}, k_{c1}, k_{r0}, k_{l0}, k_{c0} — positive-sequence and zero-sequence hyperbolic correction factors
Note 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:
$$\begin{array}{l}Rs=\left(2{R}_{1}+{R}_{0}\right)/3\\ Ls=\left(2{L}_{1}+{L}_{0}\right)/3\\ {R}_{m}=\left({R}_{0}-{R}_{1}\right)/3\\ {L}_{m}=\left({L}_{0}-{L}_{1}\right)/3\\ {C}_{p}={C}_{1}\\ {C}_{g}=3{C}_{1}{C}_{0}/\left({C}_{1}-{C}_{0}\right)\end{array}$$
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).
The positive- and zero-sequence resistances in ohms/kilometer (Ω/km).
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.
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.
The line section length in kilometers (km).
The power_triphaselinepower_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.