A Solution for Faults at Two Locations in Three-Phase 



Power Systems 



By E. F. VAAGE 



This paper is an outgrowth of studies of double faults to ground 

 in three-phase power systems made by the author in connection 

 with work of the Joint Subcommittee on Development and Re- 

 search, Edison Electric Institute and Bell Telephone System. The 

 paper provides a systematic solution, based on the method of 

 symmetrical components, by means of which currents and voltages 

 can be determined at times of fault involving any combination of- 

 phases at one or two locations on three-phase power systems. 



1. Introduction 



A KNOWLEDGE of the magnitude and phase relation of power 

 system voltages and currents for various types of faults in three- 

 phase systems is of importance in the study of various problems, among 

 which are relaying studies, the efficacy of current limiting devices and 

 their reaction on the power network, and estimates of induction in 

 paralleling communication circuits. 



The method of symmetrical components as developed by Fortescue ^ 

 and others is now extensively used in the solution for currents and 

 voltages in three-phase power systems under fault (short-circuit) 

 conditions. Formulas for special cases of faults, such as single and 

 double line-to-ground faults, can be found in various text books on this 

 subject. The solution for simultaneous faults at two locations has 

 been treated by Miss Clark,^ in a form particularly adaptable to the 

 use of a calculating board. 



The present development provides a complete and systematic solu- 

 tion for currents and voltages at times of fault on any number of 

 phases at one or two locations in a three-phase system, in which gener- 

 ators may be assumed in phase and of the same internal voltage, and 

 where load currents can be neglected. These are the usual assumptions 

 made in computing fault currents, except for certain special problems, 

 such as that of power system stability. The methods employed herein 

 could be extended to cases where generators of dififerent phase angles 

 and voltages of more than two points of fault are involved. Formally 

 such cases can be treated in a manner similar to that given in the 

 paper. The number of impedances to which an w-terminal network 



' Reference numbers refer to references appearing at the end of the article. 



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