Branches

Both line models below are composite components: they do not stamp the system matrix directly but are built from resistor, inductor and capacitor subcomponents, each of which contributes its own stamp. See subcomponents for how that composition works, and RLC elements for the stamps of the individual elements.

The transformer is documented separately under transformer.

RX-Line

The RX line represents a line by its series resistance and series inductance only, ignoring the shunt admittance. It is the appropriate choice for short lines, where the charging current is negligible, and it is what the CIM reader produces for an ACLineSegment when no shunt data is present.

The model is composed of a series resistor and a series inductor between the two terminals:

An additional resistor from the inductor terminal to ground is present to make initialisation well posed. It is not part of the physical model.

RxLine exists in DP::Ph1, EMT::Ph3, SP::Ph1 and SP::Ph3.

PI-Line

The PI line adds the shunt admittance of the line, split evenly between the two terminals, which matters once the line is long enough for the charging current to affect the result. The name comes from the shape of the equivalent circuit: a series branch with one shunt branch at each end.

The series branch carries the resistance and inductance as above. Each terminal additionally carries half of the total shunt capacitance and half of the total shunt conductance:

The shunt capacitance and conductance are specified as totals for the line, and the halving between the two ends is part of the model rather than something the user does.

Decoupling Line

The decoupling line is a distributed parameter line based on the Bergeron travelling wave method. Unlike the two models above it is not primarily a fidelity improvement: its purpose is to remove the direct coupling between the two terminals so that the network on either side can be solved as an independent system, which is what makes splitting a network across solvers or across simulators possible.

The method rests on the behaviour of a lossless line. For a line with distributed inductance and capacitance, the quantity $v + Z_c, i$ observed at one end reappears unchanged at the other end one travel time later, and likewise in the opposite direction. Nothing propagates faster than that travel time, so the two ends cannot influence each other within it. The surge impedance and the travel time follow from the line’s total inductance and capacitance,

Each terminal is then represented by a resistance to ground in parallel with a current source. The resistance is $Z_c + R/4$, and the current source carries the history term, whose value depends on the voltage and current recorded at the other terminal one travel time ago. Because that value is already known when the step begins, it enters the system as a constant injection rather than as a coupling into the admittance matrix, and the matrix separates into two blocks that can be factorised and solved independently.

The series resistance is not distributed along the line. It is lumped, with $R/4$ placed at each end and the remainder in the middle of the equivalent, which is why the terminating resistance and the history coefficients carry $R/4$ terms rather than the full $R$.

The travel time is not required to be a whole number of time steps. The recorded quantities are held in a buffer of $\lceil \tau / \Delta t \rceil$ samples and the value one travel time ago is recovered by linear interpolation between the two nearest entries. The one hard requirement is that the travel time exceed the time step; a line whose $\tau$ is shorter than $\Delta t$ cannot decouple anything, and setting one up is rejected rather than silently approximated.

In the dynamic phasor domain the history terms carry an additional rotation $e^{-j \omega_s \tau}$. This is a direct consequence of working with envelopes: a delay of $\tau$ applied to an instantaneous waveform becomes, for the envelope, the same delay together with a phase rotation of the carrier over that interval, as described under dynamic phasors. Note that this rotation is currently evaluated at a fixed 50 Hz rather than at the system frequency in use.

The decoupling is exact for the lossless travelling wave line it is derived from. The error introduced in practice comes from the lumped treatment of the series resistance and from the interpolation of the delayed quantities, and it grows as the time step approaches the travel time.

Choosing between them

Use the RX line when the shunt admittance can be neglected and you want the smaller system matrix, since the PI line introduces additional nodes for its shunt branches. Use the PI line when the line is long enough that its charging current matters, or when you are comparing against a reference tool that models the shunt branch.

Both are lumped parameter models and therefore do not reproduce travelling wave behaviour. Use the decoupling line when you need that behaviour, or when the reason for reaching for a line model is to split the network in the first place. For the domains each model is available in, see model availability.