Three-phase transformer¶

Three-phase transformers are modeled with three separate single-phase non-ideal transformers. The windings of the individual transformers are connected with different configurations to the primary side (generally the high voltage side) and to the secondary side (generally the low voltage side). The non-ideal transformer losses are represented by \(\underline{Z_2}\) the series impedances and \(\underline{Y_{\mathrm{m}}}\) the magnetizing admittances.

European standards

American standards

For example, the windings with a \(Dyn11\) configuration are represented by the following diagram:

European standards

American standards

Windings¶

There are several ways to connect the windings of the individual internal transformers in Roseau Load Flow. They are represented in the following windings diagrams:

Wye secondary¶

Delta secondary¶

Zigzag secondary¶

Equations¶

The following equations are used to model 3-phase transformers:

\[\begin{split}\begin{equation} \left\{ \begin{aligned} K_{\mathrm{UXYZ}} \cdot \underline{U_{\mathrm{XYZ}}} &= K_{\mathrm{VABC}} \cdot \underline{V_{\mathrm{ABC}}} - K_{\mathrm{N}} \cdot \underline{V_{\mathrm{N}}} \\ K_{\mathrm{Uxyz}} \cdot M_{\mathrm{TV}}\cdot \underline{U_{\mathrm{XYZ}}} + o_r \cdot \underline{Z_2} \cdot \underline{I_{\mathrm{xyz}}} &= K_{\mathrm{Vabc}} \cdot \underline{V_{\mathrm{abc}}} - K_{\mathrm{n}} \cdot \underline{V_{\mathrm{n}}} \\ K_{\mathrm{IABC}} \cdot \underline{I_{\mathrm{ABC}}} &= K_{\mathrm{IXYZ}} \cdot \left( \underline{Y_{\mathrm{m}}} \cdot \underline{U_{\mathrm{XYZ}}} + M_{\mathrm{TI}} \cdot \underline{I_{\mathrm{xyz}}} \right)\\ K_{\mathrm{Iabc}} \cdot \underline{I_{\mathrm{abc}}} &= K_{\mathrm{Ixyz}} \cdot \underline{I_{\mathrm{xyz}}} \\ \underline{I_{\mathrm{N}}} &= - K_{\mathrm{N}}^\top \cdot \underline{I_{\mathrm{ABC}}} \\ \underline{I_{\mathrm{n}}} &= - K_{\mathrm{n}}^\top \cdot \underline{I_{\mathrm{abc}}} \end{aligned} \right. \end{equation}\end{split}\]

Where \(\underline{Z_2}\) is the series impedance and \(\underline{Y_{\mathrm{m}}}\) is the magnetizing admittance of the transformer. \(o_r\) is the orientation variable, equal to \(1\) for direct windings and \(-1\) for inverse windings. The other quantities are the matrices defined below.

Matrices¶

The following matrices are used to model the windings configurations described above:

Transformation matrices¶

Winding	\(M_{\mathrm{TV}}\)	\(M_{\mathrm{TI}}\)
Dd, Yy, Dy and Yd	\(k\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(k\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)
Yz	\(k\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(k\begin{pmatrix} -1 & 0 & 1\\ 1 & -1 & 0\\ 0 & 1 & -1 \end{pmatrix}\)
Dz	\(k\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(k\begin{pmatrix} -1 & 1 & 0\\ 0 & -1 & 1\\ 1 & 0 & -1 \end{pmatrix}\)

Where \(k\) is the transformation ratio of the internal transformers defined as:

Winding	\(k\)
Dy	\(\dfrac{U_{\mathrm{LV}}}{\sqrt{3} \cdot U_{\mathrm{HV}}}\)
Yy	\(\dfrac{U_{\mathrm{LV}}}{U_{\mathrm{HV}}}\)
Dd	\(\dfrac{U_{\mathrm{LV}}}{U_{\mathrm{HV}}}\)
Yd	\(\dfrac{\sqrt{3} \cdot U_{\mathrm{LV}}}{U_{\mathrm{HV}}}\)
Dz	\(\dfrac{U_{\mathrm{LV}}}{3 \cdot U_{\mathrm{HV}}}\)
Yz	\(\dfrac{U_{\mathrm{LV}}}{\sqrt{3} \cdot U_{\mathrm{HV}}}\)

Primary winding matrices¶

Winding	\(K_{\mathrm{VABC}}\)	\(K_{\mathrm{UXYZ}}\)	\(K_{\mathrm{IABC}}\)	\(K_{\mathrm{IXYZ}}\)	\(K_{\mathrm{N}}\)
Dx	\(\begin{pmatrix} 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}\)
Yx	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)
Zx	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)

Secondary windings matrices¶

Direct¶

Winding	\(K_{\mathrm{Vabc}}\)	\(K_{\mathrm{Uxyz}}\)	\(K_{\mathrm{Iabc}}\)	\(K_{\mathrm{Ixyz}}\)	\(K_{\mathrm{n}}\)
Dd0	\(\begin{pmatrix} 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}\)
Yd11	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}\)
Yy0 and Dy11	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)
Dz0	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)
Yz11	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)

Inverse¶

Winding	\(K_{\mathrm{Vabc}}\)	\(K_{\mathrm{Uxyz}}\)	\(K_{\mathrm{Iabc}}\)	\(K_{\mathrm{Ixyz}}\)	\(K_{\mathrm{n}}\)
Dd6	\(\begin{pmatrix} 1 & -1 & 0\\ 0 & 1 & -1\\ -1 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 1\\ 1 & -1 & 0\\ 0 & 1 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}\)
Yd5	\(\begin{pmatrix} 1 & 0 & -1\\ -1 & 1 & 0\\ 0 & -1 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 1 & 0\\ 0 & -1 & 1\\ 1 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}\)
Yy6 and Dy5	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)
Dz6	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 1\\ 1 & -1 & 0\\ 0 & 1 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)
Yz5	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 1 & 0\\ 0 & -1 & 1\\ 1 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\)	\(\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\)	\(\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}\)

Example¶

The following example shows a 160kVA MV/LV transformer with a \(Dyn11\) configuration that connects a voltage source on the MV network to a load on the LV network.

import functools as ft
import numpy as np
import roseau.load_flow as rlf

# Create a MV bus
bus_mv = rlf.Bus(id="bus_mv", phases="abc")

# Create a LV bus
bus_lv = rlf.Bus(id="bus_lv", phases="abcn")

# Set the potential references of the MV and LV networks
pref_mv = rlf.PotentialRef(id="pref_mv", element=bus_mv)
pref_lv = rlf.PotentialRef(id="pref_lv", element=bus_lv)

# Create a voltage source and connect it to the MV bus
vs = rlf.VoltageSource(id="vs", bus=bus_mv, voltages=20e3)

# Create a MV/LV transformer
tp = rlf.TransformerParameters.from_open_and_short_circuit_tests(
    id="SE_Minera_A0Ak_100_kVA",
    type="Dyn11",
    sn=100.0 * 1e3,
    uhv=20e3,
    ulv=400.0,
    i0=0.5 / 100,
    p0=145.0,
    psc=1250.0,
    vsc=4.0 / 100,
)
transformer = rlf.Transformer(
    id="transfo",
    bus1=bus_mv,
    bus2=bus_lv,
    phases1="abc",
    phases2="abcn",
    parameters=tp,
    tap=1.025,
)

# Create a balanced constant-power 9kW LV load (3kW per phase)
load = rlf.PowerLoad(id="load", bus=bus_lv, phases="abcn", powers=3e3)

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

# The current flowing into the transformer from the MV bus
en.res_transformers[["current1"]].dropna().transform(
    [np.abs, ft.partial(np.angle, deg=True)]
)
# |                  |   ('current1', 'absolute') |   ('current1', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') |                   0.275904 |                -38.8165 |
# | ('transfo', 'b') |                   0.275904 |               -158.817  |
# | ('transfo', 'c') |                   0.275904 |                 81.1835 |

# The current flowing into the transformer from the LV bus
en.res_transformers[["current2"]].transform([np.abs, ft.partial(np.angle, deg=True)])
# |                  |   ('current2', 'absolute') |   ('current2', 'angle') |
# |:-----------------|---------------------------:|------------------------:|
# | ('transfo', 'a') |               12.6872      |                179.813  |
# | ('transfo', 'b') |               12.6872      |                 59.8133 |
# | ('transfo', 'c') |               12.6872      |                -60.1867 |
# | ('transfo', 'n') |                2.25156e-13 |                -80.4634 |

# 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') |
# |:-----------------|--------------------------:|-----------------------:|
# | ('bus_mv', 'ab') |                 20000     |               0        |
# | ('bus_mv', 'bc') |                 20000     |            -120        |
# | ('bus_mv', 'ca') |                 20000     |             120        |
# | ('bus_lv', 'an') |                   236.459 |              -0.186695 |
# | ('bus_lv', 'bn') |                   236.459 |            -120.187    |
# | ('bus_lv', 'cn') |                   236.459 |             119.813    |