Q7Power System Analysis

(a) Fault Current for Line-to-Line Fault by Symmetrical Component Method

For a line-to-line (LL) fault between phases b and c at a bus with pre-fault positive-sequence EMF E and sequence reactances X1 and X2 (X0 does not appear in the line-to-line fault analysis at all, since no ground or neutral return path is involved in a line-to-line fault), the boundary conditions at the fault point are: Ia=0 (the unfaulted phase carries no fault current), Ib=-Ic (the fault current flows out through one faulted phase and back through the other, so they are equal in magnitude and opposite in sign), and Vb=Vc (the two faulted phases are shorted together, so their voltages are equal) at the fault, through fault impedance Zf if present.

Applying the symmetrical component transformation to the boundary condition Ia=0 gives I0+I1+I2=0; since a line-to-line fault by definition does not involve the ground or neutral (only two of the three phase conductors are shorted to each other), no zero-sequence current path exists in this fault type at all, so I0=0 identically, which combined with I0+I1+I2=0 immediately gives I1=-I2 - this equal-and-opposite positive/negative sequence current condition (with I0 always exactly zero) is the single most distinguishing characteristic of the line-to-line fault, in contrast to the single line-to-ground fault's equal-sequence-current condition (I0=I1=I2) examined elsewhere in this examination.

This I1=-I2 condition (with I0=0) dictates that the positive- and negative-sequence networks must be connected in parallel with each other (rather than in series, as for the LG fault), and applying the corresponding voltage boundary condition Vb=Vc leads, after working through the algebra of the symmetrical component voltage transformation, to the standard result for the line-to-line fault current magnitude:

The magnitude of the line-to-line fault current is therefore |Ib|=|Ic|=sqrt(3)*E/(X1+X2+Zf) (for a solid fault, Zf=0: |Ib|=sqrt(3)E/(X1+X2)) - comparing this to the three-phase fault current E/X1, and noting that X2 is typically reasonably close to X1 in magnitude for most synchronous machines, the line-to-line fault current is generally somewhat smaller than the three-phase fault current (by a factor of roughly sqrt(3)/2, if X1 approximately equals X2), which is why the line-to-line fault is generally, though not universally, considered less severe than the three-phase fault, in contrast to the single line-to-ground fault which can, under certain zero-sequence-impedance conditions, actually exceed the three-phase fault current as discussed in relation to other questions in this examination.

(b) Why Single Line-to-Ground Fault Can Be More Severe Than a 3-Phase Fault at Alternator Terminals

The three-phase fault current at an alternator's terminals is E/X1 (limited purely by the positive-sequence, i.e., subtransient or transient, reactance), whereas the single line-to-ground fault current is 3E/(X1+X2+X0). Whether the LG fault current exceeds the three-phase fault current depends entirely on whether (X1+X2+X0)/3 is smaller than X1, i.e., whether X2+X0 is smaller than 2*X1.

At an alternator's own terminals with a solidly grounded neutral, the zero-sequence reactance X0 is often considerably smaller than the positive-sequence subtransient reactance X1 (a typical characteristic of many synchronous generator designs, where the zero-sequence reactance, being associated with a simpler, single-phase-equivalent zero-sequence flux path rather than the full rotating-field positive-sequence flux path, tends to be a comparatively low fraction of X1, sometimes as low as 0.15 to 0.6 times X1 depending on machine design) - and the negative-sequence reactance X2 is typically reasonably close to X1 in magnitude (often only slightly smaller). When X0 is sufficiently small relative to X1 in this way, the average of the three sequence reactances (X1+X2+X0)/3 can fall below X1 itself, making the resulting single line-to-ground fault current 3E/(X1+X2+X0) larger than the three-phase fault current E/X1.

This phenomenon is specifically associated with a solidly-grounded (zero or very low grounding impedance) alternator neutral - if the neutral were instead grounded through a significant grounding impedance (a neutral grounding resistor or reactor), the effective zero-sequence impedance (X0 plus 3 times the grounding impedance, per the general LG fault formula) would be correspondingly increased, generally suppressing the LG fault current back down below the three-phase fault current level. This is precisely why alternator neutral grounding practice (solid grounding versus resistance or reactance grounding, and the specific value chosen if impedance grounding is used) is an important power system protection design decision, directly affecting both the magnitude of ground-fault current that must be interrupted by protective equipment and the relative severity ranking of ground faults compared to three-phase faults at that particular generator.

It is also worth explicitly cross-referencing the line-to-line fault current derivation here against the single line-to-ground fault derivation presented elsewhere in this examination, since comparing the two fault types' distinct sequence-network connection patterns (series for the LG fault, parallel for the LL fault) and their correspondingly different resulting fault current expressions (3E/(X1+X2+X0) for LG, versus sqrt(3)E/(X1+X2) for LL) together illustrate the general symmetrical component analysis principle that each distinct unsymmetrical fault type is characterized by its own specific set of boundary conditions at the fault point, which in turn dictates a specific, fault-type-unique interconnection pattern among the three otherwise-independent sequence networks.

It is also worth noting that the same underlying zero-sequence-reactance-dependent severity condition explained here for the single line-to-ground fault at a solidly-grounded alternator's terminals is precisely why grounding-transformer and neutral-grounding-impedance selection is such a carefully engineered aspect of power system design: system designers deliberately select neutral grounding impedance (where impedance grounding rather than solid grounding is used) specifically to keep ground-fault current within a desired range - large enough to reliably operate ground-fault protective relaying, but not so large (in the solidly-grounded, low-X0 case) as to exceed the three-phase fault current and potentially the interrupting or thermal withstand rating of equipment sized primarily around the expected three-phase fault level.

In summary, the line-to-line fault current derivation and the explanation of why single line-to-ground faults can exceed three-phase fault severity together demonstrate how the symmetrical component method not only computes fault current magnitudes correctly for each distinct fault type but also provides genuine engineering insight into why different fault types behave so differently, insight that is essential for sound protective relay coordination and equipment rating decisions in real power system design.

This combined derivation-and-explanation approach, addressing both the mathematics of fault current calculation and the physical reasoning behind observed severity differences between fault types, reflects the depth of understanding expected of a power system engineer responsible for protective relay setting and coordination decisions in practice.