carsons


Namecarsons JSON
Version 1.0.1 PyPI version JSON
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SummaryA python library computing carson's equations.
upload_time2023-01-20 21:18:16
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authorOpus One Solutions
requires_python
licensecopyright 2018 Opus One Solutions Energy Corporation Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
keywords carsons cables lines power systems energy
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            carsons
=======

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This is an implementation of Carson's Equations, a mathematical model
for deriving the equivalent impedance of an AC transmission or
distribution line.

Implementation
--------------

`carsons` is developed using python 3.6 support for unicode characters
like π, ƒ, ρ, μ, ω etc. This feature allows us to avoid translating the
problem into a more typical programming syntax, so the code is dense and
can easily be compared to published formulations of the problem.

For example, we implement the kron reduction, a matrix decomposition
step, using unicode notation to indicate the slightly different meaning
of impedance values before and after a kron reduction:

```python
def perform_kron_reduction(z_primitive):
     Ẑpp, Ẑpn = z_primitive[0:3, 0:3], z_primitive[0:3, 3:]
     Ẑnp, Ẑnn = z_primitive[3:,  0:3], z_primitive[3:,  3:]
     Z_abc = Ẑpp - Ẑpn @ inv(Ẑnn) @ Ẑnp
     return Z_abc
```

Take a look at the [source
code](https://github.com/opusonesolutions/carsons/blob/add-documentation/carsons/carsons.py)
to see more cool unicode tricks!

Installation
------------

```bash
~/$ pip install carsons
```

Usage
-----

Carsons model requires a line model object that maps each phase to
properties of the conductor for that phase.

```python
from carsons import CarsonsEquations, calculate_impedance

class Line:
   geometric_mean_radius: {
       'A': geometric_mean_radius_A in meters
       ...
   }
   resistance: {
        'A': per-length resistance of conductor A in ohms/meters
        ...
   }
   wire_positions: {
        'A': (x, y) cross-sectional position of the conductor in meters
        ...
   }
   phases: {'A', ... }
     # map of phases 'A', 'B', 'C' and 'N<>' which are described in the
     # gmr, r and phase_positions attributes

line_impedance = calculate_impedance(CarsonsEquations(Line()))
```

The model supports any combination of ABC phasings (for example BC, BCN
etc...) including systems with multiple neutral cables; any phases that
are not present in the model will have zeros in the columns and rows
corresponding to that phase.

Multiple neutrals are supported, as long as they have unique labels
starting with `N` (e.g. `Neutral1`, `Neutral2`).

Intermediate results such as primitive impedance matrix are also
available.

```python
z_primitive = CarsonsEquations(Line()).build_z_primitive()
```

For examples of how to use the model, see the [overhead wire
tests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_overhead_line.py).

`carsons` is tested against several cable configurations from the [IEEE
test feeders](http://sites.ieee.org/pes-testfeeders/resources/), as well as
examples from  EPRI's [OpenDSS documentation](http://svn.code.sf.net/p/electricdss/code/trunk/Distrib/Doc/TechNote%20CableModelling.pdf).

### Concentric Neutral Cable

`carsons` also supports modelling of concentric neutral cables of any
phasings. Its usage is very similar to the example above, only requiring
a few more parameters about the neutral conductors in the line model
object.

```python
from carsons import (ConcentricNeutralCarsonsEquations,
                     calculate_impedance)

class Cable:
   resistance: {
       'A': per-length resistance of conductor A in ohm/meters
       ...
   }
   geometric_mean_radius: {
       'A': geometric mean radius of conductor A in meters
       ...
   }
   wire_positions: {
        'A': (x, y) cross-sectional position of conductor A in meters
        ...
   }
   phases: {'A', 'NA', ... }
   neutral_strand_gmr: {
       'NA': neutral strand gmr of phase A in meters
       ...
   }
   neutral_strand_resistance: {
       'NA': neutral strand resistance of phase A in ohm/meters
       ...
   }
   neutral_strand_diameter: {
       'NA': neutral strand diameter of phase A in meters
       ...
   }
   diameter_over_neutral: {
       'NA': diameter over neutral of phase A in meters
       ...
   }
   neutral_strand_count: {
       'NA': neutral strand count of phase A
       ...
   }

cable_impedance = calculate_impedance(ConcentricNeutralCarsonsEquations(Cable()))
```

For examples of how to use the model, see the [concentric cable
tests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_concentric_neutral_cable.py).

### Multi-Conductor Cable

`carsons` also supports modelling of phased duplex, triplex, quadruplex cables and triplex secondary.
It only requires a few more parameters to describe cable's geometry.

```python
from carsons import (MultiConductorCarsonsEquations,
                     calculate_impedance)

class Cable:
    resistance: {
        'A': per-length resistance of conductor A in ohm/meters
        ...
    }
    geometric_mean_radius: {
        'A': geometric mean radius of conductor A in meters
        ...
    }
    wire_positions: {
        'A': (x, y) cross-sectional position of conductor A in meters
        ...
    }
    outside_radius: {
        'A': outside radius of conductor A, including insulation and jacket thickness
        ...
    }
    insulation_thickness: {
        'A': insulation thickness of conductor A
        ...
    }
    phases: {'A', ... }

cable_impedance = calculate_impedance(MultiConductorCarsonsEquations(Cable()))
```

To model a triplex secondary cable, the inputs should be keyed on secondary conductors `S1` and `S2`. The impedance result
is a 2 x 2 matrix.

```python
class Cable:
    resistance: {
        'S1': per-length resistance of conductor S1 in ohm/meters
        ...
    }
    geometric_mean_radius: {
        'S1': geometric mean radius of conductor S1 in meters
        ...
    }
    wire_positions: {
        'S1': (x, y) cross-sectional position of conductor S1 in meters
        ...
    }
    outside_radius: {
        'S1': outside radius of conductor S1, including insulation and jacket thickness
        ...
    }
    insulation_thickness: {
        'S1': insulation thickness of conductor S1
        ...
    }
    phases: {'S1', ... }
```

For examples of how to use the model, see the [multi-conductor cable
tests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_multi_conductor.py).


### Tape Shield Cable

`carsons` also supports modelling of tape shield cables of any
phasings. Its usage is very similar to the example above, only requiring
a few more parameters about the tape shield conductors in the line model
object.

```python
from carsons import (TapeShieldedCableCarsonsEquations,
                     calculate_impedance)

class Cable:
   resistance: {
       'A': per-length resistance of conductor A in ohm/meters
       ...
   }
   geometric_mean_radius: {
       'A': geometric mean radius of conductor A in meters
       ...
   }
   wire_positions: {
        'A': (x, y) cross-sectional position of conductor A in meters
        ...
   }
   phases: {'A', ... }
   tape_shield_thickness: {
       'A': thickness of tape shield conductor on phase A cable in meters
       ...
   }
   tape_shield_outer_diameter: {
       'A': outer diameter of tape shield conductor on phase A cable in meters
       ...
   }
   

cable_impedance = calculate_impedance(TapeShieldedCableCarsonsEquations(Cable()))
```

For examples of how to use the model, see the [tape shielded cable
tests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_tape_shielded_cables.py).



## Problem Description
-------------------

Carsons equations model an AC transmission or distribution line into an
equivalent set of phase-phase impedances, which can be used to model the
line in a power flow analysis.

For example, say we have a 4-wire system on a utility pole, with `A`,
`B`, `C` phase conductors as well as a neutral cable N. We know that
when conductors carry electrical current, they exhibit a magnetic field
--- so its pretty easy to imagine that, e.g., the magnetic field
produced by `A` would interact with the `B`, `C`, and `N` conductors.

                            B
                              O
                              |
                              |
                  A        N  |       C
                    O        O|         O
                    ----------|-----------
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
                              |
        ==============[Ground]============================
        /     /     /     /     /     /     /     /     /
             /     /     /     /     /     /     /
                  /     /     /     /     /
     
     
     
     
     
     
     
     
     
     
                     A*       N*          C*
                       0        0           0
     
                               B*
                                 0

    Figure: Cross-section of a 4-wire distribution line, with
            ground return.

However, each conductor also has a ground return path (or 'image') ---
shown as `A*`, `B*`, `C*`, and `N*` in the figure above --- which is a
magnetically induced current path in the ground. When A produces a
magnetic field, that field *also* interacts with `B*`, `C*`, `N*`, *and*
`A*`. Carsons equations model all these interactions and reduce them to
an equivalent impedance matrix that makes it much easier to model this
system.

In addition `carsons` implements the kron reduction, a conversion that
approximates the impedances caused by neutral cables by incorporating
them into the impedances for phase `A`, `B`, and `C`. Since most AC and
DC powerflow formulations don't model the neutral cable, this is a
valuable simplification.

References
----------

The following works were used to produce this formulation:

-   [Leonard L. Grigsby -- Electrical Power Generation, Transmission and
    Distribution](https://books.google.ca/books?id=XMl8OU4wIEQC&lpg=SA21-PA4&dq=kron%20reduction%20carson%27s%20equation&pg=SA21-PA4#v=onepage&q=kron%20reduction%20carson's%20equation&f=true)
-   [William H. Kersting -- Distribution System Modelling and Analysis
    2e](https://books.google.ca/books?id=1R2OsUGSw_8C&lpg=PA84&dq=carson%27s%20equations&pg=PA85#v=onepage&q=carson's%20equations&f=false)
-   [William H. Kersting, Distribution System Analysis Subcommittee --
    Radial Distribution Test
    Feeders](http://sites.ieee.org/pes-testfeeders/files/2017/08/testfeeders.pdf)
-   [Timothy Vismore -- The Vismor
    Milieu](https://vismor.com/documents/power_systems/transmission_lines/S2.SS1.php)
-   [Daniel Van Dommelen, Albert Van Ranst, Robert Poncelet -- GIC
    Influence on Power Systems calculated by Carson's
    method](https://core.ac.uk/download/pdf/34634673.pdf)
-   [Andrea Ballanti, Roger Dugan -- Cable Modelling in OpenDSS](http://svn.code.sf.net/p/electricdss/code/trunk/Distrib/Doc/TechNote%20CableModelling.pdf)

            

Raw data

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    "_id": null,
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    "requires_python": "",
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    "keywords": "carsons,cables,lines,power systems,energy",
    "author": "Opus One Solutions",
    "author_email": "rnd@opusonesolutions.com",
    "download_url": "https://files.pythonhosted.org/packages/28/95/e96722d50938c472788956c9333c13c2809edb5e849269b51be22abfffce/carsons-1.0.1.tar.gz",
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    "description": "carsons\n=======\n\n[![latest release on pypi](https://badge.fury.io/py/carsons.svg)](https://badge.fury.io/py/carsons)\n[![versons of python supported by carsons](https://img.shields.io/pypi/pyversions/carsons.svg)](https://pypi.python.org/pypi/carsons)\n[![GitHub license](https://img.shields.io/github/license/opusonesolutions/carsons.svg)](https://github.com/opusonesolutions/carsons/blob/master/LICENSE.txt)\n[![build passing or failing](https://github.com/opusonesolutions/carsons/blob/master/.github/workflows/python-package.yaml/badge.svg)](https://travis-ci.org/opusonesolutions/carsons)\n[![test coverage](https://coveralls.io/repos/github/opusonesolutions/carsons/badge.svg?branch=master)](https://coveralls.io/github/opusonesolutions/carsons?branch=master)\n[![Maintainability](https://api.codeclimate.com/v1/badges/22cfed180fd6032fe29b/maintainability)](https://codeclimate.com/github/opusonesolutions/carsons/maintainability)\n\nThis is an implementation of Carson's Equations, a mathematical model\nfor deriving the equivalent impedance of an AC transmission or\ndistribution line.\n\nImplementation\n--------------\n\n`carsons` is developed using python 3.6 support for unicode characters\nlike \u03c0, \u0192, \u03c1, \u03bc, \u03c9 etc. This feature allows us to avoid translating the\nproblem into a more typical programming syntax, so the code is dense and\ncan easily be compared to published formulations of the problem.\n\nFor example, we implement the kron reduction, a matrix decomposition\nstep, using unicode notation to indicate the slightly different meaning\nof impedance values before and after a kron reduction:\n\n```python\ndef perform_kron_reduction(z_primitive):\n     \u1e90pp, \u1e90pn = z_primitive[0:3, 0:3], z_primitive[0:3, 3:]\n     \u1e90np, \u1e90nn = z_primitive[3:,  0:3], z_primitive[3:,  3:]\n     Z_abc = \u1e90pp - \u1e90pn @ inv(\u1e90nn) @ \u1e90np\n     return Z_abc\n```\n\nTake a look at the [source\ncode](https://github.com/opusonesolutions/carsons/blob/add-documentation/carsons/carsons.py)\nto see more cool unicode tricks!\n\nInstallation\n------------\n\n```bash\n~/$ pip install carsons\n```\n\nUsage\n-----\n\nCarsons model requires a line model object that maps each phase to\nproperties of the conductor for that phase.\n\n```python\nfrom carsons import CarsonsEquations, calculate_impedance\n\nclass Line:\n   geometric_mean_radius: {\n       'A': geometric_mean_radius_A in meters\n       ...\n   }\n   resistance: {\n        'A': per-length resistance of conductor A in ohms/meters\n        ...\n   }\n   wire_positions: {\n        'A': (x, y) cross-sectional position of the conductor in meters\n        ...\n   }\n   phases: {'A', ... }\n     # map of phases 'A', 'B', 'C' and 'N<>' which are described in the\n     # gmr, r and phase_positions attributes\n\nline_impedance = calculate_impedance(CarsonsEquations(Line()))\n```\n\nThe model supports any combination of ABC phasings (for example BC, BCN\netc...) including systems with multiple neutral cables; any phases that\nare not present in the model will have zeros in the columns and rows\ncorresponding to that phase.\n\nMultiple neutrals are supported, as long as they have unique labels\nstarting with `N` (e.g. `Neutral1`, `Neutral2`).\n\nIntermediate results such as primitive impedance matrix are also\navailable.\n\n```python\nz_primitive = CarsonsEquations(Line()).build_z_primitive()\n```\n\nFor examples of how to use the model, see the [overhead wire\ntests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_overhead_line.py).\n\n`carsons` is tested against several cable configurations from the [IEEE\ntest feeders](http://sites.ieee.org/pes-testfeeders/resources/), as well as\nexamples from  EPRI's [OpenDSS documentation](http://svn.code.sf.net/p/electricdss/code/trunk/Distrib/Doc/TechNote%20CableModelling.pdf).\n\n### Concentric Neutral Cable\n\n`carsons` also supports modelling of concentric neutral cables of any\nphasings. Its usage is very similar to the example above, only requiring\na few more parameters about the neutral conductors in the line model\nobject.\n\n```python\nfrom carsons import (ConcentricNeutralCarsonsEquations,\n                     calculate_impedance)\n\nclass Cable:\n   resistance: {\n       'A': per-length resistance of conductor A in ohm/meters\n       ...\n   }\n   geometric_mean_radius: {\n       'A': geometric mean radius of conductor A in meters\n       ...\n   }\n   wire_positions: {\n        'A': (x, y) cross-sectional position of conductor A in meters\n        ...\n   }\n   phases: {'A', 'NA', ... }\n   neutral_strand_gmr: {\n       'NA': neutral strand gmr of phase A in meters\n       ...\n   }\n   neutral_strand_resistance: {\n       'NA': neutral strand resistance of phase A in ohm/meters\n       ...\n   }\n   neutral_strand_diameter: {\n       'NA': neutral strand diameter of phase A in meters\n       ...\n   }\n   diameter_over_neutral: {\n       'NA': diameter over neutral of phase A in meters\n       ...\n   }\n   neutral_strand_count: {\n       'NA': neutral strand count of phase A\n       ...\n   }\n\ncable_impedance = calculate_impedance(ConcentricNeutralCarsonsEquations(Cable()))\n```\n\nFor examples of how to use the model, see the [concentric cable\ntests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_concentric_neutral_cable.py).\n\n### Multi-Conductor Cable\n\n`carsons` also supports modelling of phased duplex, triplex, quadruplex cables and triplex secondary.\nIt only requires a few more parameters to describe cable's geometry.\n\n```python\nfrom carsons import (MultiConductorCarsonsEquations,\n                     calculate_impedance)\n\nclass Cable:\n    resistance: {\n        'A': per-length resistance of conductor A in ohm/meters\n        ...\n    }\n    geometric_mean_radius: {\n        'A': geometric mean radius of conductor A in meters\n        ...\n    }\n    wire_positions: {\n        'A': (x, y) cross-sectional position of conductor A in meters\n        ...\n    }\n    outside_radius: {\n        'A': outside radius of conductor A, including insulation and jacket thickness\n        ...\n    }\n    insulation_thickness: {\n        'A': insulation thickness of conductor A\n        ...\n    }\n    phases: {'A', ... }\n\ncable_impedance = calculate_impedance(MultiConductorCarsonsEquations(Cable()))\n```\n\nTo model a triplex secondary cable, the inputs should be keyed on secondary conductors `S1` and `S2`. The impedance result\nis a 2 x 2 matrix.\n\n```python\nclass Cable:\n    resistance: {\n        'S1': per-length resistance of conductor S1 in ohm/meters\n        ...\n    }\n    geometric_mean_radius: {\n        'S1': geometric mean radius of conductor S1 in meters\n        ...\n    }\n    wire_positions: {\n        'S1': (x, y) cross-sectional position of conductor S1 in meters\n        ...\n    }\n    outside_radius: {\n        'S1': outside radius of conductor S1, including insulation and jacket thickness\n        ...\n    }\n    insulation_thickness: {\n        'S1': insulation thickness of conductor S1\n        ...\n    }\n    phases: {'S1', ... }\n```\n\nFor examples of how to use the model, see the [multi-conductor cable\ntests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_multi_conductor.py).\n\n\n### Tape Shield Cable\n\n`carsons` also supports modelling of tape shield cables of any\nphasings. Its usage is very similar to the example above, only requiring\na few more parameters about the tape shield conductors in the line model\nobject.\n\n```python\nfrom carsons import (TapeShieldedCableCarsonsEquations,\n                     calculate_impedance)\n\nclass Cable:\n   resistance: {\n       'A': per-length resistance of conductor A in ohm/meters\n       ...\n   }\n   geometric_mean_radius: {\n       'A': geometric mean radius of conductor A in meters\n       ...\n   }\n   wire_positions: {\n        'A': (x, y) cross-sectional position of conductor A in meters\n        ...\n   }\n   phases: {'A', ... }\n   tape_shield_thickness: {\n       'A': thickness of tape shield conductor on phase A cable in meters\n       ...\n   }\n   tape_shield_outer_diameter: {\n       'A': outer diameter of tape shield conductor on phase A cable in meters\n       ...\n   }\n   \n\ncable_impedance = calculate_impedance(TapeShieldedCableCarsonsEquations(Cable()))\n```\n\nFor examples of how to use the model, see the [tape shielded cable\ntests](https://github.com/opusonesolutions/carsons/blob/master/tests/test_tape_shielded_cables.py).\n\n\n\n## Problem Description\n-------------------\n\nCarsons equations model an AC transmission or distribution line into an\nequivalent set of phase-phase impedances, which can be used to model the\nline in a power flow analysis.\n\nFor example, say we have a 4-wire system on a utility pole, with `A`,\n`B`, `C` phase conductors as well as a neutral cable N. We know that\nwhen conductors carry electrical current, they exhibit a magnetic field\n--- so its pretty easy to imagine that, e.g., the magnetic field\nproduced by `A` would interact with the `B`, `C`, and `N` conductors.\n\n    \u2001                       B\n    \u2001                         O\n    \u2001                         |\n    \u2001                         |\n    \u2001             A        N  |       C\n    \u2001               O        O|         O\n    \u2001               ----------|-----------\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001                         |\n    \u2001   ==============[Ground]============================\n    \u2001   /     /     /     /     /     /     /     /     /\n    \u2001        /     /     /     /     /     /     /\n                  /     /     /     /     /\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001\n    \u2001                A*       N*          C*\n    \u2001                  0        0           0\n    \u2001\n    \u2001                          B*\n    \u2001                            0\n\n    Figure: Cross-section of a 4-wire distribution line, with\n            ground return.\n\nHowever, each conductor also has a ground return path (or 'image') ---\nshown as `A*`, `B*`, `C*`, and `N*` in the figure above --- which is a\nmagnetically induced current path in the ground. When A produces a\nmagnetic field, that field *also* interacts with `B*`, `C*`, `N*`, *and*\n`A*`. Carsons equations model all these interactions and reduce them to\nan equivalent impedance matrix that makes it much easier to model this\nsystem.\n\nIn addition `carsons` implements the kron reduction, a conversion that\napproximates the impedances caused by neutral cables by incorporating\nthem into the impedances for phase `A`, `B`, and `C`. Since most AC and\nDC powerflow formulations don't model the neutral cable, this is a\nvaluable simplification.\n\nReferences\n----------\n\nThe following works were used to produce this formulation:\n\n-   [Leonard L. Grigsby -- Electrical Power Generation, Transmission and\n    Distribution](https://books.google.ca/books?id=XMl8OU4wIEQC&lpg=SA21-PA4&dq=kron%20reduction%20carson%27s%20equation&pg=SA21-PA4#v=onepage&q=kron%20reduction%20carson's%20equation&f=true)\n-   [William H. Kersting -- Distribution System Modelling and Analysis\n    2e](https://books.google.ca/books?id=1R2OsUGSw_8C&lpg=PA84&dq=carson%27s%20equations&pg=PA85#v=onepage&q=carson's%20equations&f=false)\n-   [William H. Kersting, Distribution System Analysis Subcommittee --\n    Radial Distribution Test\n    Feeders](http://sites.ieee.org/pes-testfeeders/files/2017/08/testfeeders.pdf)\n-   [Timothy Vismore -- The Vismor\n    Milieu](https://vismor.com/documents/power_systems/transmission_lines/S2.SS1.php)\n-   [Daniel Van Dommelen, Albert Van Ranst, Robert Poncelet -- GIC\n    Influence on Power Systems calculated by Carson's\n    method](https://core.ac.uk/download/pdf/34634673.pdf)\n-   [Andrea Ballanti, Roger Dugan -- Cable Modelling in OpenDSS](http://svn.code.sf.net/p/electricdss/code/trunk/Distrib/Doc/TechNote%20CableModelling.pdf)\n",
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