Skip to content

User Guide

Introduction

The temperature of overhead line conductors is determined by an equilibrium between heating and cooling phenomena. Several factors affect this equilibrium. Here we will have a word about the most common sources of heating and cooling.

All these phenomena correspond to an object PowerTerm in the code. If you want to use one which is not included or modify an existing one, you can define your own power term.

Joule Heating

Joule heating is the result of the flowing through an imperfect conductor. The dissipated power is proportional to the conductor's electric resistance and to the square of the current ($ P=RI^2 $). The conductor's electric resistance usually increases with the conductor temperature. Depending on the conductor composition, some magnetic effects can occurs and may be taken into account.

Solar Heating

Sunlight directly heats the conductor. The absorbed power depends on the relative position of the sun regarding the conductor, the solar irradiance, and the conductor surface properties through an absorption coefficient (darker conductors absorb more heat).

Convection Cooling

Convection (or convective heat transfer) is the transfer of heat from one place to another due to the movement of a fluid (the air). In the case of conductor temperature estimation, two types of convection are actually considered:

  • Natural convection : air movement due to temperature difference between the air close to and the air far from the conductor (hence a density difference and the movement).
  • Forced convection : when the cooling mechanism is driven by the wind.

Convection cooling is often the most influential cooling mechanism. Usually both types or convection are computed and the maximum value is taken.

Radiative Cooling

When heated above ambient temperature, a conductor emits thermal radiation to the cooler surroundings. The energy dissipated is given by the Stefan–Boltzmann Law ($ P=\sigma\varepsilon T^4 $).

Evaporative Cooling

This phenomemon occurs when the conductor is wet (rain or snow). If the cable is hot enough, the evaporation of water draws heat away. It is not included in standard (CIGRE & IEEE) models but may be introduced in further developments. Some work about precipitation cooling can be found here.

Heat Equations

General case

The heat conduction equation is a partial differential equation, that describes the evolution of temperature within time and space. In the case of an overhead line conductor, we use it in cylindrical coordinates: $ \rho c_{\text{p}} \frac{\partial T}{\partial t} = \lambda \left[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 T}{\partial \theta^2} + \frac{\partial^2 T}{\partial z^2} \right] $ .

Simplification for overhead conductors

Overhead line conductors are typically modeled with simplifying assumptions:

  • No angular dependency ($ \partial/\partial\theta=0 $) ;
  • No axial dependency ($ \partial/\partial z=0 $) .

The previous equation is reduced to a radial, one-dimensional heat equation. When adding the sources (joule heating), boundary and initial conditions, we eventually have :

  • heat equation : $ \rho c_{\text{p}} \frac{\partial T}{\partial t} = \lambda \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + p_{\text{joule}} $ ;
  • boundary conditions : $ \frac{\partial T}{\partial r}=0 $ for $ r=0 $ and $ -\frac{\partial T}{\partial r}= \frac{q_{\sigma}}{\lambda} $ for $ r=R $ ;
  • $ T(r, t=0) = T^0(r) $ a given initial condition.

In this problem:

  • $ T $ is the temperature (in Kelvin);
  • $ t $ is the time (in seconds);
  • $ \rho $ the conductor volumic mass (in kg.m-3);
  • $ c_{\text{p}} $ the conductor specific heat capacity (in J.K-1.kg-1);
  • $ \lambda $ the conductor heat conductivity (in W.m-1.K-1);
  • $ p_{\text{joule}} $ the volumic heat source term (in W.m-3);
  • $ q_{\sigma} $ the heat exchange terms (usually the sum of solar heating, convective and radiative cooling, in W.m-1).

This equation can be solved with numerical methods (eg FEM), but in order to apply it to a full electrical network, some simplifications must be carried.

Heat Equations in ThermOHL

Single-temperature model

If we multiply the above equation by $ r $, integrate it before multiplying by two over the squared conductor radius, we have the formula of the average temperature for the conductor. If we assume that the heat terms $ p_{\text{joule}} $ and $ q_{\sigma} $ do not depend on $ r $, we have the simplified problem

$ \rho c_{\text{p}} \frac{\partial \bar{T}}{\partial t} = p_{\text{joule}} - \frac{1}{\pi R^2} q_{\sigma} $ ,

which can be rewritten under the more classic form (we multiply by $ \pi R^2 $, then the volumic mass becomes a lineic mass $ m $, the volumic joule term become the classic joule term $ q_{\text{joule}} $, and we expanded $ q_{\sigma} = -q_{\text{solar}} + q_{\text{convection}} + q_{\text{radiation}} $) :

$ m c_{\text{p}} \frac{\partial T}{\partial t} = q_{\text{joule}} + q_{\text{solar}} - q_{\text{convection}} - q_{\text{radiation}} $ .

In steady mode, we have the even simpler power balance equation (power terms may depend on temperature $ T $) : $ q_{\text{joule}} + q_{\text{solar}} - q_{\text{convection}} - q_{\text{radiation}} $.

Three-temperatures model

In order to have a more accurate resolution than the single-temperature model, but with faster solving time than the full resolution, RTE developed a specific model with three temperatures for the conductor :

  • the surface temperature;
  • the average temperature;
  • the core temperature.

Uncertainty in cable temperature computation

For the conductor temperature computation with the three-temperatures "legacy" solver, we provide a standard uncertainty value estimated using following formula:

$ u_T^2 = (\frac{\partial T}{\partial I}*u_I)^2 $

$ + (\frac{\partial T}{\partial T_{amb}}*u_{T_{amb}})^2 $

$ + (\frac{\partial T}{\partial W} u_W)^2 $

$ + (\frac{\partial T}{\partial Azm}*u_{Azm})^2 $

$ + (\frac {\partial T}{\partial Q_s}*u_{Q_s})^2 $

where

$ u_I = 0.05 * I $

with following notations:

  • $ u_x $ = standard uncertainty about the variable x (see below)
  • $ I $ = transit
  • $ T_{amb} $ = ambient temperature
  • $ W $ = wind speed
  • $ Azm $ = wind azimuth
  • $ Q_s $ = solar irradiance

The uncertainties about ambient temperature, wind speed, wind azimuth and solar irradiance are constants set as follows:

Variable Standard uncertainty
Ambient temperature 1 °C
Wind speed 1 m/s
Wind azimuth 10°
Solar irradiance 100 $ W/m^2 $