Mathematical model of the thermal mode of overhead power lines considering temperature variations along the line length

A mathematical model representing the temperature mode of an overhead power line wire and taking into account the axial heat transfer was developed. Processes in overhead power lines were analyzed using analytical and numerical methods for solving differential equations, including the finite difference method. The equation of thermal conductivity for AS-240/32 and SIP-2 3х95+1х95 wires was solved for the case of current variations along the line length. An analytical solution to the equation of thermal conductivity was proposed for the steady-state operation of an overhead wire under the same current in all sections of the line, taking into account the temperature dependence of active resistance. The results obtained by the analytical method agree well with those obtained by the method of finite differences. The boundary conditions at the beginning and at the end of the line were established to affect the line temperature only within a few meters. At the same time, despite the slight increase in the degree of this effect at an increase in the current due to the temperature dependence of heat emission, it remains small up to emergency level currents. Therefore, the calculations of the line thermal mode require no high accuracy in setting boundary conditions. A line with a uniformly distributed load demonstrates differing results at large current variations along the wire length. Thus, the absolute error of the analytical solution (compared to the finite difference method) for the maximum temperature equals 77.9°C, while the relative error for losses equals 10%. The same errors in temperature calculations for an infinitely long wire in terms of the length function comprise 2.5°C and 0.1%, respectively. Therefore, despite the high thermal conductivity of a metal, a model with a zero thermal conductivity along the wire axis gives more accurate results as compared to a model with an infinitely high thermal conductivity. The obtained results are applicable when clarifying the total loss of active power, as well as for estimating the line capacity according to the maximum permissible temperature, which depends on the type of wires and comprises 70 and 90°C for uninsulated steel-aluminum and self-supporting insulated wires, respectively.

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