Shunt line¶
The first model of line which can be used is a PI model with series impedance and shunt admittance. The corresponding diagram is:
Equations¶
The corresponding equations are:
\[\begin{split}\left\{
\begin{aligned}
\underline{V_1} &= \underline{a} \cdot \underline{V_2} - \underline{b} \cdot \underline{I_2} + \underline{g}
\cdot \underline{V_{\mathrm{g}}} \\
\underline{I_1} &= \underline{c} \cdot \underline{V_2} - \underline{d} \cdot \underline{I_2} + \underline{h}
\cdot \underline{V_{\mathrm{g}}} \\
\underline{I_{\mathrm{g}}} &= \underline{f}^\top \cdot \left(\underline{V_1} + \underline{V_2} - 2\cdot
\underline{V_{\mathrm{g}}}\right)
\end{aligned}
\right.\end{split}\]
where
\[\begin{split}\left\{
\begin{aligned}
\underline{a} &= \mathcal{I}_4 + \dfrac{1}{2} \cdot \underline{Z} \cdot \underline{Y} \\
\underline{b} &= \underline{Z} \\
\underline{c} &= \underline{Y} + \dfrac{1}{4}\cdot \underline{Y} \cdot \underline{Z} \cdot \underline{Y} \\
\underline{d} &= \mathcal{I}_4 + \dfrac{1}{2} \cdot \underline{Y} \cdot \underline{Z} \\
\underline{f} &= -\dfrac{1}{2} \cdot \begin{pmatrix} \underline{y_{\mathrm{ag}}} & \underline{y_{\mathrm{bg}}
} & \underline{y_{\mathrm{cg}}} & \underline{y_{\mathrm{ng}}} \end{pmatrix} ^\top \\
\underline{g} &= \underline{Z} \cdot \underline{f} \\
\underline{h} &= 2 \cdot \underline{f} + \frac{1}{2}\cdot \underline{Y} \cdot \underline{Z} \cdot \underline{f} \\
\end{aligned}
\right.\end{split}\]
with \(\underline{Z}\) the series impedance matrix and \(\underline{Y}\) the shunt admittance matrix.
Usage¶
To create a shunt line, create an instance of LineParameter with the y_shunt argument. The
ground argument of the Line constructor is mandatory for shunt lines. Here is a line that
connects a constant power load to a voltage source.
import functools as ft
import numpy as np
import roseau.load_flow as rlf
# Two buses
bus1 = rlf.Bus(id="bus1", phases="abcn")
bus2 = rlf.Bus(id="bus2", phases="abcn")
# Create a ground element and set its potential to zero
ground = rlf.Ground(id="ground")
pref = rlf.PotentialRef(id="pref", element=ground)
# A shunt line
z_line = rlf.Q_(
np.array(
[
[0.3 + 0.35j, 0.25j, 0.25j, 0.25j],
[0.25j, 0.3 + 0.35j, 0.25j, 0.25j],
[0.25j, 0.25j, 0.3 + 0.35j, 0.25j],
[0.25j, 0.25j, 0.25j, 0.3 + 0.35j],
]
),
"ohm/km",
)
y_shunt = rlf.Q_(
np.array(
[
[20 + 475j, -68j, -10j, -68j],
[-68j, 20 + 475j, -68j, -10j],
[-10j, -68j, 20 + 475j, -68j],
[-68j, -10j, -68j, 20 + 475j],
]
),
"uS/km", # micro Siemens per kilometer
)
line_parameters = rlf.LineParameters(id="line_parameters", z_line=z_line, y_shunt=y_shunt)
line = rlf.Line(
id="line",
bus1=bus1,
bus2=bus2,
parameters=line_parameters,
length=rlf.Q_(1, "km"),
ground=ground,
)
# A voltage source on the first bus
un = 400 / np.sqrt(3)
voltages = rlf.Q_(un * np.exp([0, -2j * np.pi / 3, 2j * np.pi / 3]), "V")
vs = rlf.VoltageSource(id="source", bus=bus1, voltages=voltages)
# A power load on the second bus
load = rlf.PowerLoad(
id="load", bus=bus2, powers=rlf.Q_(np.array([5.0, 2.5, 0]) * (1 - 0.3j), "kVA")
)
# The impedance matrix (in Ohm) can be accessed from the line instance
line.z_line
# array(
# [[0.3+0.35j, 0. +0.25j, 0. +0.25j, 0. +0.25j],
# [0. +0.25j, 0.3+0.35j, 0. +0.25j, 0. +0.25j],
# [0. +0.25j, 0. +0.25j, 0.3+0.35j, 0. +0.25j],
# [0. +0.25j, 0. +0.25j, 0. +0.25j, 0.3+0.35j]]
# ) <Unit('ohm')>
# The shunt admittance matrix (in Siemens) can be accessed from the line instance
line.y_shunt
# array(
# [[2.e-05+4.75e-04j, 0.e+00-6.80e-05j, 0.e+00-1.00e-05j, 0.e+00-6.80e-05j],
# [0.e+00-6.80e-05j, 2.e-05+4.75e-04j, 0.e+00-6.80e-05j, 0.e+00-1.00e-05j],
# [0.e+00-1.00e-05j, 0.e+00-6.80e-05j, 2.e-05+4.75e-04j, 0.e+00-6.80e-05j],
# [0.e+00-6.80e-05j, 0.e+00-1.00e-05j, 0.e+00-6.80e-05j, 2.e-05+4.75e-04j]]
# ) <Unit('siemens')>
# For a shunt line, the property `with_shunt` is True
line.with_shunt
# True
# Create a network and solve a load flow
en = rlf.ElectricalNetwork.from_element(bus1)
en.solve_load_flow()
# The current "entering" into the line from the bus1
en.res_branches[["current1"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('current1', 'absolute') | ('current1', 'angle') |
# |:--------------|---------------------------:|------------------------:|
# | ('line', 'a') | 23.9459 | 15.6886 |
# | ('line', 'b') | 11.2926 | -104.492 |
# | ('line', 'c') | 0.119763 | -157.68 |
# | ('line', 'n') | 20.6151 | 167.381 |
# The current "entering" into the line from the bus2
en.res_branches[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('current2', 'absolute') | ('current2', 'angle') |
# |:--------------|---------------------------:|------------------------:|
# | ('line', 'a') | 23.9202 | -164.585 |
# | ('line', 'b') | 11.2551 | 74.9044 |
# | ('line', 'c') | 5.68434e-14 | 0 |
# | ('line', 'n') | 20.6273 | -12.625 |
# The currents in the series components of the line
en.res_lines[["series_current"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('series_current', 'absolute') | ('series_current', 'angle') |
# |:--------------|---------------------------------:|------------------------------:|
# | ('line', 'a') | 23.9333 | 15.5496 |
# | ('line', 'b') | 11.2737 | -104.796 |
# | ('line', 'c') | 0.0598601 | -157.579 |
# | ('line', 'n') | 20.6215 | 167.376 |
# The losses of the series components of line can also be accessed
en.res_lines[["series_losses"]].transform([np.real, np.imag])
# | | ('series_losses', 'real') | ('series_losses', 'imag') |
# |:--------------|----------------------------:|----------------------------:|
# | ('line', 'a') | 171.841 | 57.2802 |
# | ('line', 'b') | 38.1291 | 12.7097 |
# | ('line', 'c') | 0.00107497 | 0.000358324 |
# | ('line', 'n') | 127.574 | 42.5246 |
# The shunt losses can be computed. Notice that the shunt losses are not null for the shunt line.
res_lines = en.res_lines
total_losses = res_lines["power1"] + res_lines["power2"] # total = series + shunt
shunt_losses = total_losses - res_lines["series_losses"]
shunt_losses.to_frame("shunt_losses").transform([np.real, np.imag])
# | | ('shunt_losses', 'real') | ('shunt_losses', 'imag') |
# |:--------------|---------------------------:|---------------------------:|
# | ('line', 'a') | -1.59017 | -26.6385 |
# | ('line', 'b') | 1.01834 | -28.5657 |
# | ('line', 'c') | 3.69511 | -27.4104 |
# | ('line', 'n') | 0.0351686 | 0.0139828 |