Center-tapped transformer¶
Center-tapped transformers allow to convert two phases primary connection into a split-phase secondary connection, with the neutral at the center secondary winding. It is modelled as follows:
Non-ideal 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 \(\mathrm {b}\). Nevertheless, center-tapped transformers can be connected between any two phases as long as the center phase at the secondary is always \(\mathrm{n}\).
Equations¶
The following equations are used:
\[\begin{split}\begin{equation}
\left\{
\begin{aligned}
\underline{U_{2,\mathrm{a}}^0} &= -\underline{U_{2,\mathrm{b}}^0} \\
k \cdot \underline{U_{1,\mathrm{ab}}} &= \underline{U_{2,\mathrm{a}}^0} - \underline{U_{2,\mathrm
{b}}^0} \\
\underline{I_{1,\mathrm{a}}} - Y_{\mathrm{m}} \cdot \underline{U_{1,\mathrm{ab}}} &=
-k \cdot \frac{\underline{I_{2,\mathrm{a}}} + \underline{I_{2,\mathrm{b}}}}{2} \\
\underline{I_{1,\mathrm{a}}} &= -\underline{I_{1,\mathrm{n}}} \\
\underline{I_{2,\mathrm{a}}} + \underline{I_{2,\mathrm{b}}} + \underline{I_{2,\mathrm{n}}} &= 0 \\
\end{aligned}
\right.
\end{equation}\end{split}\]
Where \(\underline{Z_2}\) is the series impedance, \(\underline{Y_{\mathrm{m}}}\) is the magnetizing admittance of the transformer, \(k\) the transformation ratio, and:
\[\begin{split}\begin{equation}
\left\{
\begin{aligned}
\underline{U_{2,\mathrm{a}}^0} &= \underline{U_{2,\mathrm{a}}} - \frac{Z_2}{2} \underline{I_{2,\mathrm{a}}} \\
\underline{U_{2,\mathrm{b}}^0} &= \underline{U_{2,\mathrm{b}}} - \frac{Z_2}{2} \underline{I_{2,\mathrm{b}}}
\end{aligned}
\right.
\end{equation}\end{split}\]
Example¶
import functools as ft
import numpy as np
import roseau.load_flow as rlf
# Create a ground and set it as the reference potential
ground = rlf.Ground("ground")
pref = rlf.PotentialRef("pref", ground)
# Create a source bus and voltage source (MV)
source_bus = rlf.Bus("source_bus", phases="abcn")
ground.connect(source_bus)
vs = rlf.VoltageSource(id="vs", bus=source_bus, voltages=20e3 / np.sqrt(3))
# Create a load bus and a load (MV)
load_bus = rlf.Bus(id="load_bus", phases="abc")
mv_load = rlf.PowerLoad("mv_load", load_bus, powers=[10000, 10000, 10000])
# Connect the two MV buses with a line
lp = rlf.LineParameters.from_catalogue(
name="U_AL_150", model="iec"
) # Underground, ALuminium, 150mm²
line = rlf.Line("line", source_bus, load_bus, parameters=lp, length=1.0, ground=ground)
# Create a low-voltage bus and a load
lv_bus = rlf.Bus(id="lv_bus", phases="abn")
ground.connect(lv_bus)
lv_load = rlf.PowerLoad("lv_load", lv_bus, powers=[-2000, 0])
# Create a transformer
tp = rlf.TransformerParameters.from_open_and_short_circuit_tests(
"t",
"center", # <--- Center-tapped transformer
sn=630e3,
uhv=20000.0,
ulv=230.0,
i0=0.018,
p0=1300.0,
psc=6500.0,
vsc=0.04,
)
transformer = rlf.Transformer("transfo", load_bus, lv_bus, parameters=tp)
# Create the network and solve the load flow
en = rlf.ElectricalNetwork.from_element(source_bus)
en.solve_load_flow()
# The current flowing into the line from the source side
en.res_lines[["current1"]].dropna().transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('current1', 'absolute') | ('current1', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('line', 'a') | 1.58451 | 45.1554 |
# | ('line', 'b') | 1.28415 | -55.5618 |
# | ('line', 'c') | 1.84471 | -178 |
# The current flowing into the transformer from the source side
en.res_transformers[["current1"]].dropna().transform(
[np.abs, ft.partial(np.angle, deg=True)]
)
# | | ('current1', 'absolute') | ('current1', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') | 0.564366 | -63.5557 |
# | ('transfo', 'b') | 0.564366 | 116.444 |
# The current flowing into the line from the load side
en.res_lines[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('current2', 'absolute') | ('current2', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('line', 'a') | 1.22632 | 155.665 |
# | ('line', 'b') | 0.726784 | 19.6741 |
# | ('line', 'c') | 0.866034 | -60.0009 |
# The current flowing into the transformer from the load side
en.res_transformers[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# | | ('current2', 'absolute') | ('current2', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') | 17.3904 | 30.0135 |
# | ('transfo', 'b') | 0 | 0 |
# | ('transfo', 'n') | 17.3904 | -149.987 |
# We can see the secondary phase "b" of the transformer does not carry any current as
# the load has 0VA on this phase.
# 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') |
# |:---------------------|--------------------------:|-----------------------:|
# | ('source_bus', 'an') | 11547 | 9.20565e-25 |
# | ('source_bus', 'bn') | 11547 | -120 |
# | ('source_bus', 'cn') | 11547 | 120 |
# | ('load_bus', 'ab') | 19999.8 | 29.9994 |
# | ('load_bus', 'bc') | 19999.9 | -90.0009 |
# | ('load_bus', 'ca') | 19999.7 | 149.999 |
# | ('lv_bus', 'an') | 115.006 | 30.0135 |
# | ('lv_bus', 'bn') | 114.999 | -150.001 |