Pi Line (3-phase)

Three-phase nominal-pi transmission line — series R+L between the two ports plus a C/2 shunt to ground at each end (per phase, no neutral coupling). EMT companion combines the series-RL trapezoidal Norton with two trapezoidal shunt-capacitor companions; load-flow row carries R, X (= ωL), and B (= ωC, total line-charging susceptance, the solver splits half/half between the two ends). Use for medium-length overhead lines or cables where the line-charging current cannot be neglected. For a short line with negligible charging, RLC Branch (3-phase) is simpler. For long cables that require travelling-wave behaviour, use Transmission Line (PI / Bergeron).

Category: Three-Phase / Transmission

Model

The Pi Line represents a three-phase transmission line by its nominal-pi equivalent: a lumped series impedance carrying the through-current, with half of the total line-charging capacitance shunted to ground at each end. The three phases are modeled independently — there is no mutual (neutral) coupling between phases in v1.

Per phase, the series impedance and total shunt admittance are

$$Z=R+j\omega L,Y=j\omega C_{\text{total}},\omega =2\pi f.$$

Each end carries half the charging admittance, $Y/2=j\omega C_{\text{total}}/2$, so the model is symmetric end-to-end.

EMT companion

In the electromagnetic-transient (EMT) solve the line is discretized with the trapezoidal rule. The series R+L branch becomes a Norton companion whose equivalent conductance and history current at step $n$ are

$$G_{\text{series}}={(R+\frac{2L}{\mathrm{\Delta }t})}^{-1},i_{\text{hist}}^n=G_{\text{series}}(v^{n-1}+\frac{2L}{\mathrm{\Delta }t}{i}^{n-1})-i^{n-1},$$

and each shunt half-capacitor contributes its own trapezoidal companion

$$G_C=\frac{2(C_{\text{total}}/2)}{\mathrm{\Delta }t}=\frac{C_{\text{total}}}{\mathrm{\Delta }t},i_{C,\text{hist}}^n=G_C{v}_C^{n-1}+i_C^{n-1}.$$

The published per-phase observable branchCurrent_x is the series current $i^n$ flowing from i to j; it excludes the shunt-charging current drawn at either end.

Load-flow row

For power-flow the line is emitted as a branch carrying the per-unit $R$, $X$, and $B$ values

$$R_{\text{pu}}=\frac{R}{Z_{\text{base}}},X_{\text{pu}}=\frac{\omega L}{Z_{\text{base}}},B_{\text{pu}}=\omega C_{\text{total}}{Z}_{\text{base}},Z_{\text{base}}=\frac{V_{\text{base}}^2}{S_{\text{base}}}.$$

The solver places half of $B_{\text{pu}}$ at each terminal bus, matching the EMT $C/2$ split.

Worked example

A 100 km, 230 kV overhead line with $R=0.5\mathrm{\Omega }$, $L=10\text{mH}$, $C_{\text{total}}=1.2\mu \text{F}$ at $f=60\text{Hz}$ gives

$$X=\omega L=2\pi (60)(0.01)\approx 3.77\mathrm{\Omega },B=\omega C_{\text{total}}=2\pi (60)(1.2\times {10}^{-6})\approx 4.52\times {10}^{-4}\text{S}.$$

On a $100\text{MVA}$, $230\text{kV}$ base ($Z_{\text{base}}={230}^2/100=529\mathrm{\Omega }$):

$$R_{\text{pu}}\approx 9.5\times {10}^{-4},X_{\text{pu}}\approx 7.1\times {10}^{-3},B_{\text{pu}}\approx 0.239.$$

When to use something else

• Negligible charging (short line): use RLC Branch (3-phase) — a pure series R+L with no shunt, which is cheaper and avoids the two extra companion solves.
• Long cables needing travelling-wave behaviour: use the distributed-parameter Transmission Line (PI / Bergeron) model, which captures wave propagation delay that the lumped nominal-pi cannot.

Ports

Name	Direction	Value type	Notes
i	electrical_3ph	double
j	electrical_3ph	double

Parameters

Config

Name	Label	Type	Default	Units	Description
r_ohms	R	double	0.5	Ohms (Ω, mΩ, kΩ, MΩ, pu)	Per-phase total series resistance. Drives the EMT series-RL Norton and the load-flow R column (R_pu = R / Z_base, from-bus side). Pick a unit from the dropdown; pu reads the bases from the Units tab.
l_henries	L	double	0.01	H (H, mH, μH, Ω, pu, X.pu)	Per-phase total series inductance. Drives the EMT companion and load-flow X (X_pu = ωL / Z_base). Pick a unit from the dropdown; X (Ω reactance), pu, and X.pu conversions read the bases from the Units tab.
c_total_farads	C_total	double	0	F (F, μF, nF, pF, S, pu, B.pu)	Per-phase TOTAL line-charging C. EMT splits C/2 at each port; load-flow B column receives the full ωC·Z_base (pu), solver applies B/2 per bus. 0 disables charging (= pure series-RL). Pick a unit from the dropdown; B (S susceptance), pu, and B.pu conversions read the bases from the Units tab.
measure_current	Measure current	enum (Off / On)	0	—	Emit per-phase series-branch currents (positive = `i` → `j`). Names on Current Monitoring; blank skips a phase. Shunt-capacitor currents are not published in v1.

Current Monitoring

Name	Label	Type	Default	Units	Description
i_name_a	Current name A	string	Ia	—	Signal name for Phase A series-branch current. Blank skips this phase.
i_name_b	Current name B	string	Ib	—	Signal name for Phase B series-branch current.
i_name_c	Current name C	string	Ic	—	Signal name for Phase C series-branch current.

Units

Name	Label	Type	Default	Units	Description
S_base	S_base (MVA)	double	100	—	System base apparent power in MVA. Used by `pu` impedance / admittance conversions in this component's `computations` block.
V_base	V_base (kV)	double	230	—	Base RMS line-to-line voltage in kV. Used by `pu` impedance / admittance conversions in this component's `computations` block.
Freq	Frequency (Hz)	double	60	—	Frequency in Hz. Used by `B` (susceptance) / `X` (reactance) conversions in this component's `computations` block.

Observables

Signal	Type	Default name	Enable	Description
branchCurrent_a	signal	from i_name_a	measure_current	Phase A series-branch current (A). Positive = current flowing from `i` to `j` through the series R+L (excludes the shunt-capacitor charging current at either end).
branchCurrent_b	signal	from i_name_b	measure_current	Phase B series-branch current (A).
branchCurrent_c	signal	from i_name_c	measure_current	Phase C series-branch current (A).