Electric Power Generation, Transmission, and Distribution

Performing the matrix operations inEq. (21.195):

Ifa¼IPaðÞYaaVaxþYabVbxþYacVcx YaVxg

Ifb¼IPbðÞYbaVaxþYbbVbxþYbcVcx YbVxg

If¼IPcðÞYcaVaxþYcbVbxþYccVcx YcVxg (21:208)

where

Ya¼YaaþYabþYac

Yb¼YbaþYbbþYbc

Yc¼YcaþYcbþYcc (21:209)

Equations (21.208) become the general equations that are used to simulate all types of short circuits.
Basically there are three equations and seven unknowns (Ifa,Ifb,Ifc,Vax,Vbx,Vcx, andVxg). The other
three variables in the equations (IPa,IPb, andIPc) are functions of the total impedance and the Thevenin
voltages and are therefore known. In order to solve Eq. (21.208), it will be necessary to specify four of the
seven unknowns. These specifications are functions of the type of fault being simulated. The additional
required four knowns for various types of faults are given below:
Three-phase faults

Vax¼Vbx¼Vcx¼ 0

IaþIbþIc¼ (^0) (21:210)
Three-phase-to-ground faults
Vax¼Vbx¼Vcx¼Vxg¼ 0 (21:211)
Line-to-line faults (assume i–j fault with phase k unfaulted)
Vix¼Vjx¼ 0
Ifk¼ 0
IfiþIfj¼ 0 (21:212)
Line-to-line-to-ground faults (assume i–j to ground fault with k unfaulted)
Vix¼Vjx¼Vxg¼ 0
Vkx¼
IPk
Ykk
(21:213)
Line-to-ground faults (assume phase k fault with phases i and j unfaulted)
Vkx¼Vxg¼ 0
Ifi¼Ifj¼ 0 (21:214)
Notice that Eqs. (21.212) through (21.214) will allow the simulation of line-to-line, line-to-line-to-
ground, and line-to-ground faults for all phases. There is no limitation to b–c faults for line-to-line and
a–g for line-to-ground as is the case when the method of symmetrical components is employed.