Lines¶

Definition¶

Lines are modeled using passive components lumped in a PI section. The lumped parameters are defined using the series impedance matrix \(\underline{Z}\) and the shunt admittance matrix \(\underline{Y}\).

Matrices definition¶

Before diving into the different line models, lets define the series impedance matrix \(\underline{Z}\), and the shunt admittance matrix \(\underline{Y}\) used to model the lines.

Series impedance matrix¶

The series impedance matrix \(\underline{Z}\), in \(\Omega\), consists of the series resistance of the conductors (\(R\in{\mathbb{R}^+}^4\)), the self-inductances (\(L\in\mathbb{R}^4\)) and the mutual inductances (\(M\in\mathbb{R}^{12}\)).

\[\begin{split}\begin{aligned} \underline{Z} &= R + j \cdot X \\ \underline{Z} &= \begin{pmatrix} \underline{Z_{\mathrm{aa}}} & \underline{Z_{\mathrm{ab}}} & \underline{Z_{\mathrm{ac}}} & \underline{Z_{\mathrm{an}}}\\ \underline{Z_{\mathrm{ba}}} & \underline{Z_{\mathrm{bb}}} & \underline{Z_{\mathrm{bc}}} & \underline{Z_{\mathrm{bn}}}\\ \underline{Z_{\mathrm{ca}}} & \underline{Z_{\mathrm{cb}}} & \underline{Z_{\mathrm{cc}}} & \underline{Z_{\mathrm{cn}}}\\ \underline{Z_{\mathrm{na}}} & \underline{Z_{\mathrm{nb}}} & \underline{Z_{\mathrm{nc}}} & \underline{Z_{\mathrm{nn}}}\\ \end{pmatrix}\\ \underline{Z} &= \underbrace{ \begin{pmatrix} R_{\mathrm{a}} & 0 & 0 & 0\\ 0 & R_{\mathrm{b}} & 0 & 0\\ 0 & 0 & R_{\mathrm{c}} & 0\\ 0 & 0 & 0 & R_{\mathrm{n}}\\ \end{pmatrix} }_{R} + j \cdot \underbrace{ \omega \cdot \begin{pmatrix} L_{\mathrm{a}} & M_{\mathrm{ab}} & M_{\mathrm{ac}} & M_{\mathrm{an}}\\ M_{\mathrm{ba}} & L_{\mathrm{b}} & M_{\mathrm{bc}} & M_{\mathrm{bn}}\\ M_{\mathrm{ca}} & M_{\mathrm{cb}} & L_{\mathrm{c}} & M_{\mathrm{cn}}\\ M_{\mathrm{na}} & M_{\mathrm{nb}} & M_{\mathrm{nc}} & L_{\mathrm{n}}\\ \end{pmatrix} }_{X} \end{aligned}\end{split}\]

Admittance matrix¶

Warning

The admittance matrix \(\underline{y}\) shouldn’t be confused with the shunt admittance matrix \(\underline{Y}\) defined below.

\(\underline{y}\) represents the admittances between each node, while \(\underline{Y}\) is used to compute the currents and voltages.

\[\begin{split}\begin{aligned} \underline{y} &= G + j \cdot B \\ \underline{y} &= \begin{pmatrix} \underline{y_{\mathrm{ag}}} & \underline{y_{\mathrm{ab}}} & \underline{y_{\mathrm{ac}}} & \underline{y_{\mathrm{an}}}\\ \underline{y_{\mathrm{ab}}} & \underline{y_{\mathrm{bg}}} & \underline{y_{\mathrm{bc}}} & \underline{y_{\mathrm{bn}}}\\ \underline{y_{\mathrm{ac}}} & \underline{y_{\mathrm{bc}}} & \underline{y_{\mathrm{cg}}} & \underline{y_{\mathrm{cn}}}\\ \underline{y_{\mathrm{an}}} & \underline{y_{\mathrm{bn}}} & \underline{y_{\mathrm{cn}}} & \underline{y_{\mathrm{ng}}} \end{pmatrix}\\ \underline{y} &= \underbrace{ \begin{pmatrix} G_{\mathrm{a}} & 0 & 0 & 0\\ 0 & G_{\mathrm{b}} & 0 & 0\\ 0 & 0 & G_{\mathrm{c}} & 0\\ 0 & 0 & 0 & G_{\mathrm{n}} \end{pmatrix} }_{G} + j \cdot \underbrace{ \omega \cdot \begin{pmatrix} C_{\mathrm{a}} & C_{\mathrm{ab}} & C_{\mathrm{ac}} & C_{\mathrm{an}}\\ C_{\mathrm{ab}} & C_{\mathrm{b}} & C_{\mathrm{bc}} & C_{\mathrm{bn}}\\ C_{\mathrm{ac}} & C_{\mathrm{bc}} & C_{\mathrm{c}} & C_{\mathrm{cn}}\\ C_{\mathrm{an}} & C_{\mathrm{bn}} & C_{\mathrm{cn}} & C_{\mathrm{n}} \end{pmatrix} }_{B} \end{aligned}\end{split}\]

with \(G\in\mathbb{R}^4\) the conductance of the line, \(B\in\mathbb{R}^4\) the susceptance of the line and \(C\in\mathbb{R}^{16}\) the transverse susceptances of the line.

Shunt admittance matrix¶

The shunt admittance matrix \(\underline{Y}\) is defined from the admittance matrix \(\underline{y}\) as:

\[\begin{split}\underline{Y} = \begin{pmatrix} \underline{Y_{\mathrm{aa}}} & \underline{Y_{\mathrm{ab}}} & \underline{Y_{\mathrm{ac}}} & \underline{Y_{\mathrm{an}}}\\ \underline{Y_{\mathrm{ba}}} & \underline{Y_{\mathrm{bb}}} & \underline{Y_{\mathrm{bc}}} & \underline{Y_{\mathrm{bn}}}\\ \underline{Y_{\mathrm{ca}}} & \underline{Y_{\mathrm{cb}}} & \underline{Y_{\mathrm{cc}}} & \underline{Y_{\mathrm{cn}}}\\ \underline{Y_{\mathrm{na}}} & \underline{Y_{\mathrm{nb}}} & \underline{Y_{\mathrm{nc}}} & \underline{Y_{\mathrm{nn}}}\\ \end{pmatrix} \quad \text{with} \quad \left\{ \begin{aligned} \underline{Y_{ii}} &= \sum_{k\in\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{n},\mathrm{g}\}}{\underline{y_{ik}}}\\ \underline{Y_{ij}} &= -\underline{y_{ij}}\\ \end{aligned} \right.\text{, }\forall(i,j)\in\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{n}\}^2\end{split}\]

Line parameters¶

The parameters of the lines are defined using the LineParameters class. It takes the series impedance matrix \(\underline{Z}\) and optionally, the shunt admittance matrix \(\underline{Y}\). The first one must be given in \(\Omega\)/km (or an equivalent unit) and the second must be given in \(S/km\) (or an equivalent unit).

import numpy as np
import roseau.load_flow as rlf

# An impedance matrix
z_line = rlf.Q_(
    np.array(
        [
            [0.3 + 0.35j, 0.25j, 0.25j, 0.25j],
            [0.25j, 0.3 + 0.35j, 0.25j, 0.25j],
            [0.25j, 0.25j, 0.3 + 0.35j, 0.25j],
            [0.25j, 0.25j, 0.25j, 0.3 + 0.35j],
        ]
    ),
    "ohm/km",
)

# A shunt admittance matrix
y_shunt = rlf.Q_(
    np.array(
        [
            [20 + 475j, -68j, -10j, -68j],
            [-68j, 20 + 475j, -68j, -10j],
            [-10j, -68j, 20 + 475j, -68j],
            [-68j, -10j, -68j, 20 + 475j],
        ]
    ),
    "uS/km",  # micro Siemens per kilometer
)

# The line parameter for a simple line (no shunt)
simple_line_parameters = rlf.LineParameters(id="simple_line_parameters", z_line=z_line)

# The line parameter for a line with a shunt
shunt_line_parameters = rlf.LineParameters(
    id="shunt_line_parameters", z_line=z_line, y_shunt=y_shunt
)

Tip

The Line instance itself has the z_line and y_shunt properties. They retrieve the line impedance in \(\Omega\) and the line shunt admittance in Siemens (taking into account the length of the line).

There are several alternative constructors for LineParameters objects. The description of them can be found in the dedicated Line parameters page.

Available models¶

The following line models are available in Roseau Load Flow. Please also have a look at the parameters page to define the parameters of lines.

Lines

API Reference¶

Bases: Identifiable, JsonMixin, CatalogueMixin[DataFrame]

Parameters that define electrical models of lines.

LineParameters constructor.

Parameters:

id – A unique ID of the line parameters, typically its canonical name.
z_line – The Z matrix of the line (Ohm/km).
y_shunt – The Y matrix of the line (Siemens/km). This field is optional if the line has no shunt part.
max_current – The maximum current loading of the line (A). The maximum current is optional, it is not used in the load flow but can be used to check for overloading. See also Line.res_violated.
line_type – The type of the line (overhead, underground, twisted). The line type is optional, it is informative only and is not used in the load flow. This field gets automatically filled when the line parameters are created from a geometric model or from the catalogue.
conductor_type – The type of the conductor material (Aluminum, Copper, …). The conductor type is optional, it is informative only and is not used in the load flow. This field gets automatically filled when the line parameters are created from a geometric model or from the catalogue.
insulator_type – The type of the cable insulator (PVC, XLPE, …). The insulator type is optional, it is informative only and is not used in the load flow. This field gets automatically filled when the line parameters are created from a geometric model or from the catalogue.

property line_type: LineType | None: The type of the line. Informative only, it has no impact on the load flow.

property conductor_type: ConductorType | None: The type of the conductor material. Informative only, it has no impact on the load flow.

property insulator_type: InsulatorType | None: The type of the cable insulator. Informative only, it has no impact on the load flow.

property section: Quantity[float] | None: The cross-section area of the cable (in mm²). Informative only, it has no impact on the load flow.

property max_current: Quantity[float] | None: The maximum current loading of the line (A) if it is set.

Create line parameters from a symmetric model.

Parameters:

id – A unique ID of the line parameters, typically its canonical name.
z0 – Impedance - zero sequence - \(r_0+x_0\cdot j\) (ohms/km)
z1 – Impedance - direct sequence - \(r_1+x_1\cdot j\) (ohms/km)
y0 – Admittance - zero sequence - \(g_0+b_0\cdot j\) (Siemens/km)
y1 – Conductance - direct sequence - \(g_1+b_1\cdot j\) (Siemens/km)
zn – Neutral impedance - \(r_{\mathrm{n}}+x_{\mathrm{n}}\cdot j\) (ohms/km)
xpn – Phase to neutral reactance (ohms/km)
bn – Neutral susceptance (siemens/km)
bpn – Phase to neutral susceptance (siemens/km)
max_current – An optional maximum current loading of the line (A). It is not used in the load flow.

Returns:

The created line parameters.

Notes

As explained in the Line parameters alternative constructor documentation, the model may be “degraded” if the computed impedance matrix is not invertible.

Create line parameters from its geometry.

Parameters:

id – The id of the line parameters type.
line_type – Overhead or underground. See also LineType.
conductor_type – Type of the conductor. If None, ACSR is used for overhead lines and AL for underground or twisted lines. See also ConductorType.
insulator_type – Type of insulator. If None, XLPE is used for twisted lines and PVC for underground lines. See also InsulatorType.
section – Cross-section surface area of the phases (mm²).
section_neutral – Cross-section surface area of the neutral (mm²). If None it will be the same as the section of the other phases.
height – Height of the line (m). It must be positive for overhead lines and negative for underground lines.
external_diameter – External diameter of the cable (m).
max_current – An optional maximum current loading of the line (A). It is not used in the load flow.

Returns:

The created line parameters.

Method to get the electrical parameters of a LV line from its canonical name. Some hypothesis will be made: the section of the neutral is the same as the other sections, the height and external diameter are pre-defined, and the insulator is PVC.

Parameters:

name – The name of the line the parameters must be computed. E.g. “U_AL_150”.
section_neutral – Surface of the neutral (mm²). If None it will be the same as the section of the other phases.
height – Height of the line (m). If None a default value will be used.
external_diameter – External diameter of the wire (mm). If None a default value will be used.
max_current – An optional maximum current loading of the line (A). It is not used in the load flow.

Returns:

The corresponding line parameters.

Deprecated since version 0.6.0: Use LineParameters.from_geometry() instead.

classmethod from_name_mv(name: str, max_current: float | Quantity[float] | None = None) → Self

Get the electrical parameters of a MV line from its canonical name (France specific model)

Parameters:

name – The canonical name of the line parameters. It must be in the format lineType_conductorType_crossSection. E.g. “U_AL_150”.
max_current – An optional maximum current loading of the line (A). It is not used in the load flow.

Returns:

The corresponding line parameters.

Create a line parameters object from PowerFactory “TypLne” data.

Parameters:

id – A unique ID of the line parameters.
r0 – PwF parameter rline0 (AC-Resistance R0’). Zero sequence resistance in (ohms/km).
r1 – PwF parameter rline (AC-Resistance R1’). Direct sequence resistance in (ohms/km).
x0 – PwF parameter xline0 (Reactance X0’). Zero sequence reactance in (ohms/km).
x1 – PwF parameter xline (Reactance X1’). Direct sequence reactance in (ohms/km).
b0 – PwF parameter bline0 (Susceptance B0’). Zero sequence susceptance in (µS/km).
b1 – PwF parameter bline (Susceptance B’). Direct sequence susceptance in (µS/km).
rn – PwF parameter rnline (AC-Resistance Rn’). Neutral resistance in (ohms/km).
xn – PwF parameter xnline (Reactance Xn’). Neutral reactance in (ohms/km).
bn – PwF parameter bnline (Susceptance Bn’). Neutral susceptance in (µS/km).
rpn – PwF parameter rnpline (AC-Resistance Rpn’). Phase-Neutral coupling resistance in (ohms/km).
xpn – PwF parameter xnpline (Reactance Xpn’). Phase-Neutral coupling reactance in (ohms/km).
bpn – PwF parameter bnpline (Susceptance Bpn’). Phase-Neutral coupling susceptance in (µS/km).
nphase – PwF parameter nlnph (Phases). The number of phases of the line between 1 and 3. This should not count the neutral conductor.
nneutral – PwF parameter nneutral (Number of Neutrals). The number of neutrals of the line. It can be either 0 or 1.
cohl – PwF parameter cohl_ (Cable/OHL). The type of the line; ‘Cable’ or 0 mean an underground cable and ‘OHL’ or 1 mean an overhead line.
inom – PwF parameter sline or InomAir (Rated Current in ground or in air). The rated current in (kA) of the line. It is used as the maximum current for analysis of network constraint violations. Pass the sline parameter if the line is an underground cable (cohl=’Cable’) or the InomAir parameter if the line is an overhead line (cohl=’OHL’).
conductor – PwF parameter mlei (Conductor Material). The material used for the conductors. It can be one of: ‘Al’ (Aluminium), ‘Cu’ (Copper), ‘Ad’ (Aldrey AlMgSi), ‘As’ (Aluminium-Steel), ‘Ds’ (Aldrey-Steel).
insulation – PwF parameter imiso (Insulation Material). The material used for the conductor’s insulation. It can be one of ‘PVC’ (0), ‘XLPE’ (1), ‘Mineral’ (2), ‘Paper’ (3) or ‘EPR’ (4).
section – PwF parameter qurs (Nominal Cross Section). The nominal cross-sectional area of the conductors in (mm²).

Returns:

The created line parameters.

Create a line parameters object from OpenDSS “LineCode” data.

Parameters:

id – The unique ID of the line parameters.
nphases – OpenDSS parameter: NPhases. Number of phases in the line this line code represents. To create single-phase lines with a neutral pass nphases=2, for three-phase lines with a neutral nphases=4, etc.
r1 – OpenDSS parameter: R1. Positive-sequence resistance in (ohm/km).
r0 – OpenDSS parameter: R0. Positive-sequence resistance in (ohm/km).
x1 – OpenDSS parameter: X1. Positive-sequence reactance in (ohm/km).
x0 – OpenDSS parameter: X0. Positive-sequence reactance in (ohm/km).
c1 – OpenDSS parameter: C1. Positive-sequence capacitance in (nF/km).
c0 – OpenDSS parameter: C0. Positive-sequence capacitance in (nF/km).
basefreq – OpenDSS parameter: BaseFreq. Frequency at which impedances are specified (Hz). Defaults to 50 Hz.
normamps – OpenDSS parameter: NormAmps. Normal ampere limit on line (A). This is the so-called Planning Limit. It may also be the value above which load will have to be dropped in a contingency. Usually about 75% - 80% of the emergency (one-hour) rating. This value is passed to max_current and used for violation checks.
linetype – OpenDSS parameter: LineType. Code designating the type of line. Only "OH" (overhead) and "UG" (underground) are currently supported.

Returns:

The corresponding line parameters object.

Example usage:

# DSS command: `New linecode.240sq nphases=3 R1=0.127 X1=0.072 R0=0.342 X0=0.089 units=km`
lp = LineParameters.from_open_dss(
    id="linecode-240sq",
    nphases=3,  #  creates 3x3 Z,Y matrices
    r1=Q_(0.127, "ohm/km"),
    x1=Q_(0.072, "ohm/km"),
    r0=Q_(0.342, "ohm/km"),
    x0=Q_(0.089, "ohm/km"),
    c1=Q_(3.4, "nF/km"),  # default value used in OpenDSS code
    c0=Q_(1.6, "nF/km"),  # default value used in OpenDSS code
)

# DSS command: `New LineCode.16sq NPhases=1 R1=0.350, X1=0.025, R0=0.366, X0=0.025, C1=1.036, C0=0.488 Units=kft NormAmps=400`
lp = LineParameters.from_open_dss(
    id="linecode-16sq",
    nphases=1,  # creates 1x1 Z,Y matrices
    r1=Q_(0.350, "ohm/kft"),
    x1=Q_(0.025, "ohm/kft"),
    r0=Q_(0.366, "ohm/kft"),
    x0=Q_(0.025, "ohm/kft"),
    c1=Q_(1.036, "nF/kft"),
    c0=Q_(0.488, "nF/kft"),
    normamps=Q_(400, "A"),
)

classmethod catalogue_path() → Path: Get the path to the catalogue.

classmethod catalogue_data() → DataFrame: Get the catalogue data.

Create line parameters from a catalogue.

Parameters:

name – The name of the line parameters to get from the catalogue. It can be a regular expression.
line_type – The type of the line parameters to get. It can be "overhead", "twisted", or "underground". See also LineType.
conductor_type – The type of the conductor material (Al, Cu, …). See also ConductorType.
insulator_type – The type of insulator. See also InsulatorType.
section – The cross-section surface area of the phases (mm²).
id – A unique ID for the created line parameters object (optional). If None (default), the id of the created object will be its name in the catalogue. Note that this parameter is not used in the data filtering.
nb_phases – The number of phases of the line between 1 and 4, defaults to 3. It represents the size of the z_line and y_shunt matrices.

Returns:

The created line parameters.

Get the catalogue of available lines.

You can use the parameters below to filter the catalogue. If you do not specify any parameter, all the catalogue will be returned.

Parameters:

name – The name of the line parameters to get from the catalogue. It can be a regular expression.
line_type – The type of the line parameters to get. It can be "overhead", "twisted", or "underground". See also LineType.
conductor_type – The type of the conductor material (Al, Cu, …). See also ConductorType.
insulator_type – The type of insulator. See also InsulatorType.
section – The cross-section surface area of the phases (mm²).

Returns:

The catalogue data as a dataframe.

classmethod from_dict(data: dict[str, Any], *, include_results: bool = True) → Self

Line parameters constructor from dict.

Parameters:: data – The dictionary data of the line parameters.
Returns:: The created line parameters.

Bases: AbstractBranch

An electrical line PI model with series impedance and optional shunt admittance.

Line constructor.

Parameters:

id – A unique ID of the line in the network branches.
bus1 – The first bus (aka “from_bus”) to connect to the line.
bus2 – The second bus (aka “to_bus”) to connect to the line.
parameters – Parameters defining the electric model of the line using its impedance and shunt admittance matrices. This is an instance of the LineParameters class and can be used by multiple lines.
length – The length of the line (in km).
phases – The phases of the line. A string like "abc" or "an" etc. The order of the phases is important. For a full list of supported phases, see the class attribute allowed_phases. All phases of the line must be present in the phases of both connected buses. By default, the phases common to both buses are used.
ground – The ground element attached to the line if it has shunt admittance.
geometry – The geometry of the line i.e. the linestring.

allowed_phases: Final = frozenset({'a', 'ab', 'abc', 'abcn', 'abn', 'an', 'b', 'bc', 'bcn', 'bn', 'c', 'ca', 'can', 'cn', 'n'})

The allowed phases for a line are:

P-P-P or P-P-P-N: "abc", "abcn"
P-P or P-P-N: "ab", "bc", "ca", "abn", "bcn", "can"
P or P-N: "a", "b", "c", "an", "bn", "cn"
N: "n"

property phases: str: The phases of the line. This is an alias for phases1 and phases2.

property length: Quantity[float]: The length of the line (in km).

property parameters: LineParameters: The parameters defining the impedance and shunt admittance matrices of line model.

property z_line: Quantity[ndarray[Any, dtype[complex128]]]: Impedance of the line (in Ohm).

property y_shunt: Quantity[ndarray[Any, dtype[complex128]]]: Shunt admittance of the line (in Siemens).

property max_current: Quantity[float] | None: The maximum current loading of the line (in A).

property res_series_currents: Quantity[ndarray[Any, dtype[complex128]]]: Get the current in the series elements of the line (in A).

property res_series_power_losses: Quantity[ndarray[Any, dtype[complex128]]]: Get the power losses in the series elements of the line (in VA).

property res_shunt_currents: tuple[Quantity[ndarray[Any, dtype[complex128]]], Quantity[ndarray[Any, dtype[complex128]]]]: Get the currents in the shunt elements of the line (in A).

property res_shunt_power_losses: Quantity[ndarray[Any, dtype[complex128]]]: Get the power losses in the shunt elements of the line (in VA).

property res_power_losses: Quantity[ndarray[Any, dtype[complex128]]]: Get the power losses in the line (in VA).

property res_violated: bool | None

Whether the line current exceeds the maximum current (loading > 100%).

Returns None if the maximum current is not set.