Joule heating

linerate.equations.joule_heating.compute_resistance(conductor_temperature, temperature1, temperature2, resistance_at_temperature1, resistance_at_temperature2)

Compute the (possibly AC-)resistance of the conductor at a given temperature.

The resistance is linearly interpolated/extrapolated based on the two temperature-resistance measurement pairs provided as arguments.

Parameters:

• conductor_temperature (Annotated[float | float64 | ndarray[Any, dtype[float64]], '°C']) – \(T~\left[^\circ\text{C}\right]\). The average conductor temperature.

• temperature1 (Annotated[float | float64 | ndarray[Any, dtype[float64]], '°C']) – \(T_1~\left[^\circ\text{C}\right]\). The first temperature measurement.

• temperature2 (Annotated[float | float64 | ndarray[Any, dtype[float64]], '°C']) – \(T_2~\left[^\circ\text{C}\right]\). The second temperature measurement.

• resistance_at_temperature1 (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'Ω/m']) – \(R_1~\left[\Omega~\text{m}^{-1}\right]\). The resistance at temperature \(T=T_1\).

• resistance_at_temperature2 (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'Ω/m']) – \(R_2~\left[\Omega~\text{m}^{-1}\right]\). The resistance at temperature \(T=T_2\).

Returns:

\(R~\left[\Omega\right]\). The resistance at the given temperature.

Return type:

Union[float, float64, ndarray[Any, dtype[float64]]]

linerate.equations.joule_heating.correct_resistance_acsr_magnetic_core_loss(ac_resistance, current, aluminium_cross_section_area, constant_magnetic_effect, current_density_proportional_magnetic_effect, max_relative_increase)

Correct for extra resistance in ACSR conductors due to magnetic effects in the steel core.

Aluminium-conductor steel-reinforced (ACSR) conductors have an additional resistance due to magnetic effects in the steel core. Particularly conductors with an odd number of layers with aluminium wires.

According to [5], we can assume a linear relationship between the current density and the relative increase in resistance due to the steel core of three-layer ACSR. However, the task force also says that we can assume that the increase saturates at 6%. In [6], it is stated that the maximum increase for three-layer ACSR is 5%.

For ACSR with an even number of layers, the effect is negligible since we get cancelling magnetic fields, and for mono-layer ACSR, the effect behaves differently. Still, some software providers use the same correction scheme for mono-layer ACSR, but with a higher saturation point (typically 20%, since that is the maximum resistance increase in mono-layer ACSR [6]).

The linear but saturating increase in resistance leads to the following correction scheme

\[R_\text{corrected} = R \min(c_\text{max}, b + mJ),\]

where \(R_\text{corrected}\) is the corrected AC resistance for ACSR conductors, \(R\) is the uncorrected value for the AC resistance at the given temperature, \(c_\text{max}\) is the maximum relative increase (e.g. 6% or \(1.06`\) for three-layer ACSR conductors), \(J\) is the current density (current divided by the aluminium cross-section area, \(I/A\)) and \(b\) and \(m\) are the constant and current-density proprtional magnetic effects, respectively (for example obtained from linear regression curve).

Parameters:

• ac_resistance (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'Ω/m']) – \(R~\left[\Omega\right]\). The AC resistance of the conductor.

• current (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'A']) – \(I~\left[\text{A}\right]\). The current going through the conductor.

• aluminium_cross_section_area (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'm²']) – \(A_{\text{Al}}~\left[\text{m}^2\right]\). The cross sectional area of the aluminium strands in the conductor.

• constant_magnetic_effect (Annotated[float | float64 | ndarray[Any, dtype[float64]], ''] | None) – \(b\). The constant magnetic effect, most likely equal to 1. If None, then no correction is used (useful for non-ACSR cables).

• current_density_proportional_magnetic_effect (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'm²/A'] | None) – \(m\). The current density proportional magnetic effect. If None, then it is assumed equal to 0.

• max_relative_increase (Annotated[float | float64 | ndarray[Any, dtype[float64]], '']) – \(c_\text{max}\). Saturation point of the relative increase in conductor resistance.

Returns:

\(R_\text{corrected}~\left[\Omega\right]\). The resistance of the conductor after taking steel core magnetization effects into account.

Return type:

Union[float, float64, ndarray[Any, dtype[float64]]]

linerate.equations.joule_heating.compute_joule_heating(current, resistance)

Compute the Joule heating, assuming AC-resistance for AC lines.

Parameters:

• current (Annotated[float | float64 | ndarray[Any, dtype[float64]], 'A']) – \(I~\left[\text{A}\right]\). The current going through the conductor.

• Resistance – \(R~\left[\Omega\right]\). The (possibly AC-)resistance of the conductor, correcting for all possible magnetisation effects.

Returns:

\(P_J~\left[\text{W}~\text{m}^{-1}\right]\). The Joule heating of the conductor

Return type:

Union[float, float64, ndarray[Any, dtype[float64]]]