Single-phase transformer#

Single-phase transformers are modelled as follows:

Single-phase transformer diagram
Single-phase transformer diagram

Non-ideal transformer models are used in Roseau Load Flow. The series impedances \(\underline{Z_2}\) and the magnetizing admittances \(\underline{Y_{\mathrm{m}}}\) are included in the model.

Note

Figures and equations on this page are related to a transformer connected between the phases \(\mathrm{a}\) and \(\mathrm {n}\). Nevertheless, single-phase transformers can be connected between any two phases.

Equations#

The following equations are used:

\[\begin{split}\begin{equation} \left\{ \begin{aligned} k \cdot \underline{U_{1,\mathrm{a}}} &= \underline{U_{2,\mathrm{a}}} - \underline{Z_2} \cdot \underline{I_{2, \mathrm{a}}} \\ \underline{I_{1,\mathrm{a}}} - \underline{Y_{\mathrm{m}}} \cdot \underline{U_{1,\mathrm{a}}} &= -k \cdot \underline{I_{2,\mathrm{a}}} \\ \underline{I_{1,\mathrm{a}}} &= -\underline{I_{1,\mathrm{n}}} \\ \underline{I_{2,\mathrm{a}}} &= -\underline{I_{2,\mathrm{n}}} \\ \end{aligned} \right. \end{equation}\end{split}\]

Where \(\underline{Z_2}\) is the series impedance, \(\underline{Y_{\mathrm{m}}}\) is the magnetizing admittance of the transformer, and \(k\) the transformation ratio.

Example#

The following examples shows a single-phase load connected via an isolating single-phase transformer to a three-phase voltage source.

import functools as ft
import numpy as np
from roseau.load_flow import (
    Bus,
    ElectricalNetwork,
    PotentialRef,
    PowerLoad,
    Transformer,
    TransformerParameters,
    VoltageSource,
)

# Create the source bus and the voltage source
bus1 = Bus(id="bus1", phases="abcn")
pref1 = PotentialRef(id="pref1", element=bus1)

voltages = 400 / np.sqrt(3) * np.exp([0, -2j * np.pi / 3, 2j * np.pi / 3])
vs = VoltageSource(id="vs", bus=bus1, voltages=voltages)

# Create the load bus and the load
bus2 = Bus(id="bus2", phases="an")
pref2 = PotentialRef(id="pref2", element=bus2)

load = PowerLoad(id="load", bus=bus2, powers=[100], phases="an")

# Create the transformer
tp = TransformerParameters(
    id="Example_TP",
    type="single",  # <--- Single-phase transformer
    sn=800,
    uhv=400,
    ulv=400,
    i0=0.022,
    p0=17,
    psc=25,
    vsc=0.032,
)
transformer = Transformer(
    id="transfo",
    bus1=bus1,
    bus2=bus2,
    phases1="an",
    phases2="an",
    parameters=tp,
)

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

# The current flowing into the transformer from the source side
en.res_branches[["current1"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# |                  |   ('current1', 'absolute') |   ('current1', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') |                   0.462811 |               -0.956008 |
# | ('transfo', 'n') |                   0.462811 |              179.044    |

# The current flowing into the transformer from the load side
en.res_branches[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# |                  |   ('current2', 'absolute') |   ('current2', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') |                   0.438211 |              179.85     |
# | ('transfo', 'n') |                   0.438211 |               -0.149761 |

# The power flow in the transformer
en.res_branches[["power1", "power2"]].abs()
# |                  |   power1 |   power2 |
# |:-----------------|---------:|---------:|
# | ('transfo', 'a') |  106.882 |      100 |
# | ('transfo', 'n') |    0     |        0 |

# The voltages at the buses of the network
en.res_buses_voltages.transform([np.abs, ft.partial(np.angle, deg=True)])
# |                |   ('voltage', 'absolute') |   ('voltage', 'angle') |
# |:---------------|--------------------------:|-----------------------:|
# | ('bus1', 'an') |                    230.94 |               0        |
# | ('bus1', 'bn') |                    230.94 |            -120        |
# | ('bus1', 'cn') |                    230.94 |             120        |
# | ('bus2', 'an') |                    228.2  |              -0.149761 |