Per-unit transmission-line constants

Abstract

F OR MANY YEARS, the per-unit or per cent method has been used to define the constants of such equipment as generators, reactors, and transformers. However, the constants of transmission lines still are expressed in terms of ohmic impedances. This fact is not important for short lines where the line impedance is generally small in relation to terminal impedances. However, for long lines, the line constants are one of the principal factors which determine the steady-state and transient power limits of the system. Therefore, it is desirable to use a per-unit system for trans mission-line constants which is based upon the power limits of the transmission line.A second difficulty which is encountered in the solution of long-line problems is the determination of the exact line constants by hyperbolic functions. This is a time-consum-2.5, (9 Z o 5 3 go-.,* DELIVERABLE F SURGE-IMPE SI_L»2.5(KV o 1-3 a. \( tt \ \ it ίο 3( 1 \ö M )o il » ύ X) TRANSMISSION DISTANCE IN MILES Figure 1. Permissible loading of straightaway transmission lines as a function of line length in miles for voltages from 69kv to 500kving task which is subject to mechanical errors. The pur pose of this article is to illustrate that transmission-line con stants can be generalized and expressed in per-unit terms so that the constants of any practical line can be read from a group of curves.For any 4-terminal network composed of linear, bilateral elements, the sending-and receiving-end voltages and cur rents are related by the expressions:The ABCD constants of a transmission line are complex quantities which can be expressed in terms of line length and the per-mile values of series resistance, series reactance, and shunt reactance, as follows:3) B» y/(r+jx)(-jx') sinh Is J^J -l +j -\ ohms (4) ^V( f J)(-;V) sinh [ 5 V^V-1+^] mhoS D = A (5) Shunt and Series Reactances of a TransmissionLine. An ex amination of the quantity Λ/Χ/Χ' for a large number of typi cal transmission lines shows that the average value of this quantity is 2.06 X 1 0 -3 (1/miles) with a maximum deviation from this value of about 0.5 per cent. Therefore, this value can be substituted in equations 3 to 6, reducing the hyper bolic quantities to functions of length and r/x ratio. Surge Impedance and Surge Impedance Loading. The surge impedance loading (SIL) of a transmission line is defined by the equation kv* SIL --megawatts Zo where (7) Zo/ T = V(r+>)(-/*') r = -V 2 t a n -i r / *Neglecting line resistance, when a line is loaded with a resistance equal to its surge impedance, the sending-and receiving-end voltages are equal and the sending-and re ceiving-end power factors are unity. On this basis, SIL is a desirable form of loading. However, it does not take into account the factors of line length, terminal impedances, and stability. The results of an investigation by S. B. Griscom 1 of the effect of these factors on allowable line loadings is reproduced in Figure 1. This figure shows that for long lines, allowable loadings are in the neighborhood of one SIL. This suggests that the SIL in ...