Line parameters

A computation of line parameters is composed of a problem, which defines what is computed, and a formulation, which selects the formulas. The computation is executed by a backend, and all computations are invoked by the same call, compute(problem, formulation; kw...).

Problem definition

The problem LineParametersProblem(system, earth; temperature, frequencies) is defined by the cables of a CableSystem, the EarthModel below the air, the temperature of all conductors and dielectrics in °C, equal to 20 by default, and the frequencies of interest in Hz. The frequencies must be positive, finite and sorted in strictly increasing order. All numbers of the system, earth, temperature and frequencies are promoted together with Float64, so that the number type T of the problem is at least Float64.

Coaxial formulation

The CoaxialFormulation selects one formula for each physical contribution. Each slot takes a formula of its family, and the corresponding keyword accepts :id, :id => (key = value, ...) with the options of the formula, or a formula object. The default formulas are summarized below.

When both earth slots take the same identifier, their formulas must be equal, since the unified formula obtains the impedance and the potential coefficients from the same circumferential currents. The remaining keywords are the options of the formulation:

The formulas are described in the theory pages Matrix formulation, Conductors, Dielectrics and Earth return.

Computation

The call compute(problem, formulation; kw...) runs the default backend of the formulation type, given by the trait backend: LCMBackend, the analytical computation of this package, for CoaxialFormulation, and GmshBackend for FEM. Another backend is selected by passing it as the first argument, compute(backend, problem, formulation; kw...). The keywords are the computation options of the backend, and LCMBackend has one option, trace, equal to false by default.

At each frequency f, with s = j2πf, the computation of LCMBackend consists of the following steps:

1. Evaluation of the material laws, which give the resistivity of each conductor at the operating temperature, the admittivity of each dielectric layer and the properties of the air and the earth.
2. Computation of the surface impedances of the conductors and of the magnetic terms of the dielectric intervals, which are assembled by the loop recursion into the impedance matrix inside the cables, Zin.
3. Computation of the potential coefficients inside the cables, Pin, by the series combination of their layers.
4. Computation of the earth-return impedance Zg and potential coefficients Pg of the cables, from their outer surfaces.
5. Addition of the entries of Zg and Pg for each pair of cables, including each cable with itself, to every pair of their conductors.
6. Bundle and Kron reductions of the impedance and potential matrices.
7. Computation of the admittance Y = s P⁻¹ from the reduced potential matrix P.
8. Ideal transposition of Z and Y, when transposition is true.

The same steps are given in the docstring of solve!, in the Matrix formulation page.

Results

The result of the computation is a LineParameters object, with the following fields:

Each row is labeled as cable:<i>:<terminal>, where i is the index of the cable in the system and terminal is the name of the terminal. The label of a bundle joins the labels of its terminals with +. With trace = true, the trace of LCMBackend contains Zin in Ω/m and Pin in m/F per conductor, and Zg in Ω/m and Pg in m/F per cable, each with one slice per frequency. Otherwise, the four arrays are empty.

Options

The options of each owner are declared by defaults, where the owner is a formula or a formulation type for FormulationOptions, and a backend type for ComputationOptions. The record of options is built once, from the defaults and the keywords given. An unknown key throws an ArgumentError, each value is converted to the type of its default, and the record is checked by validate. The defaults of the coaxial formulation are:

using LineCableModels
LineCableModels.defaults(CoaxialFormulation)

(Γ = 0.0 + 0.0im, bundle = true, kron = true, transposition = false)

Reference systems

The test suite includes six reference systems, whose line parameters computed by the analytical and finite-element formulations are stored in test/references/:

• two_bare_wires: two copper wires with a radius of 42.5 mm, buried at 1 m depth with a horizontal spacing of 1 m, in a homogeneous earth of 0.1 Ω·m.
• two_insulated_wires: the same wires with an insulation 1 mm thick.
• coaxial_circular_strands: one coaxial cable with a core of circular strands, buried at 1 m depth in a homogeneous earth of 100 Ω·m.
• coaxial_rectangular_strands: the same cable with a core of rectangular strands.
• overhead_bare_wires: the two bare wires at a height of 1 m above an earth of 0.1 Ω·m.
• mixed_bare_wires: the two bare wires with one placed above the earth and the other buried.

The validation consists of comparing the analytical and finite-element formulations on these systems. The analytical results of the package agree with the reference values of Z and Y within 1e-9 of the largest diagonal entry at each frequency, with Zin and Pin within 1e-12 and Zg and Pg within 1e-9. The finite-element results are compared with the analytical ones by means of the relative errors defined in Finite element solver, at an accuracy level of 1 %.