Simplified line#

If the line is modeled with only series impedance, the model is simplified as there is no coupling with the ground. This is a common model of short lines, typically in distribution networks.

The corresponding diagram is:

Simplified line diagram
Simplified line diagram

Equations#

With \(\underline{Y} = 0\), the equations become:

\[\begin{split}\left\{ \begin{aligned} \underline{V_1} - \underline{V_2} &= \underline{Z} \cdot \underline{I_1} \\ \underline{I_2} &= -\underline{I_1} \end{aligned} \right.\end{split}\]

Usage#

To create a simplified line, create an instance of LineParameter without providing y_shunt. Here is a simplified line connecting a constant power load to a voltage source.

import functools as ft
import numpy as np
from roseau.load_flow import (
    Bus,
    ElectricalNetwork,
    Line,
    LineParameters,
    PotentialRef,
    PowerLoad,
    Q_,
    VoltageSource,
)

# Two buses
bus1 = Bus(id="bus1", phases="abcn")
bus2 = Bus(id="bus2", phases="abcn")

# A line
lp = LineParameters(id="lp", z_line=Q_(0.35 * np.eye(4), "ohm/km"))
line = Line(id="line", bus1=bus1, bus2=bus2, parameters=lp, length=Q_(1, "km"))

# A voltage source on the first bus
un = 400 / np.sqrt(3)
voltages = Q_(un * np.exp([0, -2j * np.pi / 3, 2j * np.pi / 3]), "V")
vs = VoltageSource(id="source", bus=bus1, voltages=voltages)

# The neutral of the voltage source is fixed at potential 0
pref = PotentialRef(id="pref", element=bus1, phase="n")

# A power load on the second bus
load = PowerLoad(
    id="load", bus=bus2, powers=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(
#     [[0.35+0.j, 0.  +0.j, 0.  +0.j, 0.  +0.j],
#      [0.  +0.j, 0.35+0.j, 0.  +0.j, 0.  +0.j],
#      [0.  +0.j, 0.  +0.j, 0.35+0.j, 0.  +0.j],
#      [0.  +0.j, 0.  +0.j, 0.  +0.j, 0.35+0.j]]
# ) <Unit('ohm')>

# For a simplified line, the property `with_shunt` is False and the `y_shunt` matrix is zero
line.with_shunt
# False

line.y_shunt
# array(
#     [[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
#      [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
#      [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
#      [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]]
# ) <Unit('siemens')>

# Create a network and solve a load flow
en = ElectricalNetwork.from_element(bus1)
en.solve_load_flow()

# The current flowing into the line from bus1
en.res_branches[["current1"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# |               |   ('current1', 'absolute') |   ('current1', 'angle') |
# |:--------------|---------------------------:|------------------------:|
# | ('line', 'a') |                    24.1958 |                 16.4456 |
# | ('line', 'b') |                    11.3722 |               -105.263  |
# | ('line', 'c') |                     0      |                  0      |
# | ('line', 'n') |                    20.628  |                168.476  |

# The current flowing into the line from bus2
en.res_branches[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# |               |   ('current2', 'absolute') |   ('current2', 'angle') |
# |:--------------|---------------------------:|------------------------:|
# | ('line', 'a') |                    24.1958 |               -163.554  |
# | ('line', 'b') |                    11.3722 |                 74.7366 |
# | ('line', 'c') |                     0      |                  0      |
# | ('line', 'n') |                    20.628  |                -11.5242 |

# 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') |                          24.1958 |                       16.4456 |
# | ('line', 'b') |                          11.3722 |                     -105.263  |
# | ('line', 'c') |                           0      |                        0      |
# | ('line', 'n') |                          20.628  |                      168.476  |

# All currents are equal in magnitude for a simplified lines as no current escapes to the ground.
# The current from bus 2 has an opposite direction to the current from bus 1 as expected.

# The losses of the line can also be accessed. One can remark that there are no shunt losses
en.res_lines[["series_losses"]].transform([np.real, np.imag])
# |               |   ('series_losses', 'real') |   ('series_losses', 'imag') |
# |:--------------|----------------------------:|----------------------------:|
# | ('line', 'a') |                    204.904  |                -2.66329e-15 |
# | ('line', 'b') |                     45.2646 |                -8.96306e-16 |
# | ('line', 'c') |                      0      |                 0           |
# | ('line', 'n') |                    148.93   |                 7.62657e-16 |

# With a simplified model, all the losses are caused by the series impedance of the line
res_lines = en.res_lines
total_losses = res_lines["power1"] + res_lines["power2"]  # total = series + shunt
np.allclose(total_losses, res_lines["series_losses"])
# True