Voltage source¶

Definition¶

It represents an ideal voltage source that maintains a fixed voltage independently of the load resistance or the output current.

Connections¶

A voltage source can be either star-connected or delta-connected depending on whether its phases include a neutral or not.

Star (wye) connection¶

The diagram of the star voltage source is:

European standards

American standards

The equations that model a star voltage source are:

\[\begin{split}\left\{ \begin{split} \underline{V_{\mathrm{a}}}-\underline{V_{\mathrm{n}}} &= \underline{U_{\mathrm{an}}} \\ \underline{V_{\mathrm{b}}}-\underline{V_{\mathrm{n}}} &= \underline{U_{\mathrm{bn}}} \\ \underline{V_{\mathrm{c}}}-\underline{V_{\mathrm{n}}} &= \underline{U_{\mathrm{cn}}} \end{split} \right.\end{split}\]

Where \(\underline{U}\in\mathbb{C}^3\) is the voltage vector (user defined parameter) and \(\underline{V}\in\mathbb{C}^4\) is the node potentials vector (variable).

Note

You can create star connected sources even on buses that don’t have a neutral. In this case, the source’s neutral will be floating and its potential can be accessed similar to normal star sources.

Delta connection¶

The diagram of the delta voltage source is:

European standards

American standards

The equations that model a delta voltage source are:

\[\begin{split}\left\{ \begin{split} \underline{V_{\mathrm{a}}}-\underline{V_{\mathrm{b}}} &= \underline{U_{\mathrm{ab}}} \\ \underline{V_{\mathrm{b}}}-\underline{V_{\mathrm{c}}} &= \underline{U_{\mathrm{bc}}} \\ \underline{V_{\mathrm{c}}}-\underline{V_{\mathrm{a}}} &= \underline{U_{\mathrm{ca}}} \end{split} \right.\end{split}\]

Where \(\underline{U}\in\mathbb{C}^3\) is the voltage vector (user defined parameter) and \(\underline{V}\in\mathbb{C}^3\) is the node potentials vector (variable).

Usage¶

A voltage source defined with a neutral phase is a star-connected voltage source, otherwise it is a delta-connected voltage source. The phases of the source must be a subset of the phases of the bus it is connected to. A voltage source takes the same phases as the bus by default.

import numpy as np
import roseau.load_flow as rlf

bus = rlf.Bus(id="bus", phases="abcn")
# The phases of the source are the same as the bus by default
vs1 = rlf.VoltageSource("vs1", bus=bus, voltages=230)  # phases="abcn" implied
vs1.phases  # "abcn"
vs1.voltage_phases  # ["an", "bn", "cn"]

# Explicitly define the phases of the source (star connection)
vs2 = rlf.VoltageSource("vs2", bus=bus, phases="abcn", voltages=230)  # Same as vs1
vs2.phases  # "abcn"
vs2.voltage_phases  # ["an", "bn", "cn"]

# Explicitly define the phases of the source (delta connection)
vs3 = rlf.VoltageSource("vs3", bus=bus, phases="abc", voltages=400)
vs3.phases  # "abc"
vs3.voltage_phases  # ["ab", "bc", "ca"]

# Incorrect phases: the source's phases must be a subset of the bus's phases
bus2 = rlf.Bus(id="bus2", phases="an")
rlf.VoltageSource("vs4", bus=bus2, phases="bn", voltages=230)  # Error

A scalar (potentially complex) voltage value can be used to define the source’s balanced voltages. For a single-phase source, the scalar value is used as the voltage of the source’s phase. For a two-phase source, the second voltage value is the negative of the first value (180° phase shift). For a three-phase source, the second and third values are calculated by rotating the first value by -120° and 120°, respectively (120° phase shift clockwise).

bus = rlf.Bus(id="bus", phases="abcn")

# Three-phase connection (star)
# -----------------------------
rlf.VoltageSource("vs1", bus=bus, phases="abcn", voltages=230)
# {'an': (230+0j), 'bn': (-115-199.18584287042083j), 'cn': (-115+199.1858428704209j)}

# Three-phase connection (delta)
# ------------------------------
rlf.VoltageSource("vs2", bus=bus, phases="abc", voltages=400)
# {'ab': (400+0j), 'bc': (-200-346.41016151377534j), 'ca': (-200+346.4101615137755j)}

# Two-phase connection
# --------------------
rlf.VoltageSource("vs3", bus=bus, phases="abn", voltages=230)
# {'an': (230+0j), 'bn': (-230+0j)}

# Single-phase connection
# -----------------------
rlf.VoltageSource("vs4", bus=bus, phases="an", voltages=230)
# {'an': (230+0j)}

# Unbalanced source, explicit voltage vector
# ------------------------------------------
rlf.VoltageSource(
    "vs5",
    bus=bus,
    phases="abcn",
    voltages=[230, 115 * np.exp(1j * np.pi / 2), 115 * np.exp(-1j * np.pi / 2)],
)
# {'an': (230+0j), 'bn': (115j), 'cn': (-115j)}

# Incorrect voltage vector: only two elements!!
rlf.VoltageSource(
    id="vs6", bus=bus, phases="abc", voltages=400 * np.exp([0, -2j * np.pi / 3])
)  # Error

A voltage vector (list or numpy array) can be used to create an unbalanced voltage source if needed. The voltage vector must have the same size as the number of the phase-to-phase or phase-to-neutral connections of the source.

bus = rlf.Bus(id="bus", phases="abcn")

# Unbalanced source, explicit voltage vector
# ------------------------------------------
rlf.VoltageSource(
    "vs1",
    bus=bus,
    phases="abcn",
    voltages=[230, 115 * np.exp(1j * np.pi / 2), 115 * np.exp(-1j * np.pi / 2)],
)
# {'an': (230+0j), 'bn': (115j), 'cn': (-115j)}

# Incorrect voltage vector: only two voltage values!!
rlf.VoltageSource(
    id="vs2", bus=bus, phases="abc", voltages=400 * np.exp([0, -2j * np.pi / 3])
)  # Error

API Reference¶

class VoltageSource(id: Id, bus: roseau.load_flow.models.buses.Bus, *, voltages: roseau.load_flow.typing.ComplexScalarOrArrayLike1D, phases: str | None = None, connect_neutral: bool | None = None)

Bases: roseau.load_flow.models.core.Element

A voltage source fixes the voltages on the phases of the bus it is connected to.

The source can be connected in a wye or star configuration (i.e with a neutral) or in a delta configuration (i.e without a neutral).