Mathematical Explanations of a Paradox Observed in a HVAC (High Voltage Alternating Current) Untransposed Overhead Line

The constructive asymmetry of the untransposed overhead lines of a high voltage alternating current is the cause of a great number of difficulties in their operation and modeling. In order to model the operating regimes of such lines, the symmetrical component method, based on constructive symmetry and thus the symmetry of the equivalent phase parameters, is inappropriate, which is why many research papers have been dedicated to either setting up improved modeling methods or to returning to phase coordinate modeling. This paper intends to justify a paradox found on some untransposed overhead lines of a high voltage alternating current during the no-load operating conditions by performing phase coordinate modeling. In such a situation, the transmission or distribution operators measured significant negative values for the active powers on one or two phases at the beginning of the lines. Considering the case of a real untransposed overhead line operating under no-load conditions, the paper starts from presenting the recorded electrical values. Then, the paper moves on to outlining the Carson’s simplified computing relations for calculating the series and shunt primitive equivalent parameters and Kron’s transformation relationships for calculating the phase equivalent parameters. After applying them to the real line, the calculation of the power flow for the no-load operating conditions, which is applied to an equivalent scheme of the line consisting of nine identical octopoles, is performed. Both the untransposed line and its transposed variant are studied here. The values of the electrical amounts obtained by the calculation for the untransposed line are basically similar to those obtained by measuring on the real line, which gives a mathematical confirmation of the so-called paradox. Its occurrence represents the effect of the asymmetry of the equivalent phase capacities, which causes a redistribution of the active powers between the phases of the network to which the overhead line operating in no-load conditions is connected.