SYMMETRICAL COMPONENTS CALCULATION FOR TRANSMISSION LINES
Determining impedances for transmission lines is more challenging and generally involves making calculations from the physical parameters of the line and its conductors. The algorithm and equations given in this subclause describe the procedure, and experienced protection engineers find understanding the theoretical basis for this procedure helpful.
All the equations given in this subclause are for 60 Hz systems; impedances for systems at other frequencies can be determined by ratio or by modifying the formulae. Alternatively, computer programs are available to calculate line impedances.
The first consideration is that the positive- and negative-sequence impedances of transmission lines are equal. A transmission line is a passive component that responds in the same way to positive- and negative-sequence excitation.
Because sequence impedances are the relationships between respective sequence voltages and currents, calculation of one impedance suffices for both needs.
The positive-sequence reactance of a transmission line can be thought of as the impedance that would relate voltage and current when the three conductors or a transmission line are shorted together at one end, while excited by a positive-sequence source of voltages at the other end. This impedance can be calculated using the following equation:
GMD is the geometric mean spacing between phase conductors (e.g., the cube root of the product of the three-phase spacings) (m),
GMR is the geometric mean radius of the phase conductor (m).
GMD should be calculated for the specific spacings of the array of conductors making up the transmission line, while GMR is a parameter for the conductor that is available from the conductor manufacturer.
Positive-sequence resistance can be read directly from conductor tables. Calculating the zero-sequence impedance of a transmission line is more challenging. The concept can be viewed as follows:
All three phases of a transmission line are shorted together to ground at the source end, while all three conductors are shorted together and to both ground and the overhead ground wire (OHGW) at the other end.
When a single phase source of voltage is then applied at the source end, a current flows. The ratio of the single-phase driving voltage to the resulting current flow is the zero-sequence impedance of the line.
Physically, current flows from the faulted conductor into both ground and the OHGW as depicted in Figure 2-6a; the current flows from the source out through the phase conductors and returns through a complex path consisting of the OHGW and the earth.
Figure 2-6a—Illustration of insulation flashover on open wire line showing return current flowing through OHGW of transmission line and through earth
From this physical picture, it is apparent that the zero-sequence impedance should, therefore, consist of three branches as indicated in Figure 2-6b: the zero-sequence impedance of the phase conductors, the zero-sequence impedance of the static wire return (OHGW), and the zero-sequence impedance of the earth return.
Figure 2-6b—Zero-sequence equivalent circuit that accounts for self impedance of transmission line and the impedances of earth and OHGW return paths
Values can be calculated for the various branches of Figure 2-6b using the following equations:
Ra is the resistance of the phase conductor (Ω/km),
GMD2 is the geometric mean spacing of all conductors—phase and static (OHGW) wires (m),
GMR2 is the geometric mean radius of k static (OHGW) wires (m),
k is the number of static (OHGW) wires,
r is the earth resistivity (typically 100) (Ω⋅m),
Rgw is the resistance of one ground wire (Ω/km),
f is the system frequency.