Q8Power System Analysis

(a)(i) Symmetrical vs Unsymmetrical Faults

A symmetrical fault is a fault condition in which the three-phase system remains balanced after the fault occurs - the only fault type meeting this description is the three-phase (and three-phase-to-ground) fault, in which all three phases are affected identically and simultaneously, so the resulting fault currents and voltages remain balanced (equal magnitude, 120-degree phase-displaced) three-phase quantities, exactly as under normal balanced operation, just at a different (typically much larger current, much lower voltage) operating point. Because the system remains balanced, a symmetrical fault can be analyzed using only the positive-sequence network alone (equivalently, an ordinary single-phase per-phase circuit), without needing the symmetrical component decomposition into positive, negative, and zero sequence networks at all.

An unsymmetrical fault, by contrast, is any fault condition that disturbs the three-phase balance of the system - this includes the single line-to-ground fault, line-to-line fault, and double line-to-ground (line-to-line-to-ground) fault, all of which affect the three phases differently (asymmetrically) and therefore require the full symmetrical component method (decomposing the fault condition into positive-, negative-, and zero-sequence networks connected together according to the specific fault type's boundary conditions, as examined in detail elsewhere in this examination) for correct analysis, since a simple single-phase equivalent circuit alone cannot represent the resulting unbalanced three-phase currents and voltages. Unsymmetrical faults are, in practice, considerably more common than symmetrical (three-phase) faults on real power systems - single line-to-ground faults alone account for the large majority (typically 70-80%) of all faults occurring on overhead transmission and distribution lines, reflecting the greater likelihood of a single conductor experiencing an insulation failure or being struck by lightning/a falling object compared to all three phases being affected simultaneously and symmetrically.

(a)(ii) Short Circuit Fault vs Open Circuit Fault

A short circuit fault is a fault condition in which two or more conductors (or a conductor and ground) come into direct or low-impedance electrical contact that should not normally exist, creating an abnormally low-impedance path that causes current to rise well above its normal rated value - short circuit faults include all the fault types discussed elsewhere in this examination (three-phase, line-to-line, single line-to-ground, double line-to-ground faults), and their analysis centers on determining the resulting excessive fault current magnitude, which must be safely interrupted by protective devices (circuit breakers) before thermal or mechanical damage occurs to system equipment.

An open circuit fault, by contrast, is a fault condition in which one or more conductors that should normally be carrying current become interrupted (broken) - common causes include a conductor breaking due to mechanical failure, a fuse operating in only one phase, or a single-phase-open condition in a circuit breaker that fails to close or open all three of its phases together (a 'stuck pole' or single-phase-open condition). Open circuit faults, unlike short circuit faults, do not typically produce a dangerously large current; instead, their principal concern is the resulting unbalanced (or, in a complete three-phase open circuit, entirely interrupted) power flow and the potentially damaging unbalanced currents that can flow in connected rotating machinery (motors), since a partial (one- or two-phase) open circuit forces the remaining connected phases to carry an unbalanced current distribution that can cause excessive machine heating and vibration even without the current magnitude itself necessarily exceeding a normal short-circuit protective threshold. Open circuit faults are also analyzed using symmetrical components, but with different boundary conditions (typically, a zero-current condition imposed on the open phase(s) rather than a zero-impedance condition as in short circuit fault analysis) than those used for short circuit fault analysis.

(b) Sequence Reactances from Three Different Fault Tests

Given: 50 MVA, 11 kV, 3-phase alternator, neutral solidly grounded. Fault currents measured: 3-phase fault = 1870 A, line-to-line fault = 2590 A, single line-to-ground fault = 4130 A.

First, compute the machine's rated (base) current:

From the three-phase fault, using I3ph = E/X1 = 1.0/X1 (taking E=1.0 pu):

From the line-to-line fault, using ILL = sqrt(3)*E/(X1+X2) = sqrt(3)/(X1+X2):

From the single line-to-ground fault (solidly grounded, Zf=0), using ILG = 3E/(X1+X2+X0) = 3/(X1+X2+X0):

So the three per unit sequence reactances of the alternator are: X1 approximately 1.403 p.u. (positive sequence, corresponding here to the machine's effective subtransient reactance during the fault test), X2 approximately 0.352 p.u. (negative sequence), and X0 approximately 0.151 p.u. (zero sequence). Notably, X0 (0.151 pu) is considerably smaller than X1 (1.403 pu) in this machine, and indeed the measured LG fault current (4130A) is larger than the three-phase fault current (1870A) - directly consistent with, and a good numerical illustration of, the general condition explained elsewhere in this examination for why a single line-to-ground fault at an alternator's solidly-grounded terminals can be more severe than a three-phase fault, precisely because this machine's small zero-sequence reactance relative to its positive-sequence reactance satisfies that severity condition.

It is also worth noting that the symmetrical/unsymmetrical and short-circuit/open-circuit fault classifications discussed in part (a) are independent of each other, meaning a complete fault-type taxonomy actually spans both dimensions simultaneously: the three-phase fault is a symmetrical short-circuit fault, while single line-to-ground, line-to-line, and double line-to-ground faults are all unsymmetrical short-circuit faults, and separately, a single-phase-open condition is an unsymmetrical open-circuit fault, illustrating that 'symmetrical versus unsymmetrical' and 'short-circuit versus open-circuit' are two genuinely distinct classification criteria that combine to describe the complete space of possible power system fault conditions requiring protection and analysis.

It is also worth noting the broader diagnostic value of computing all three sequence reactances from field or factory fault-test data, as performed in part (b): rather than relying solely on manufacturer-nameplate reactance values (which represent design or type-test values that may not perfectly reflect an individual installed machine's actual characteristics due to manufacturing tolerances or in-service degradation), directly measuring the three-phase, line-to-line, and single line-to-ground fault currents at a specific installed machine and back-calculating its effective sequence reactances, exactly as done in this problem, provides commissioning and protection engineers with verified, machine-specific reactance values that can be used to properly set and coordinate that specific generator's protective relaying with confidence.

In summary, the symmetrical/unsymmetrical and short-circuit/open-circuit fault classification framework, combined with the worked sequence-reactance extraction from measured fault-test data, together illustrate both the conceptual taxonomy and the practical measurement-based determination of the machine parameters that underlie all power system fault analysis examined throughout this paper.