Q1Power System - I

(a) Positive, Negative and Zero Sequence Components

Fortescue's theorem states that any unbalanced set of three phasors (voltages or currents) can be resolved into three balanced sets of phasors, called symmetrical components:

Positive-sequence components (Va1, Vb1, Vc1) consist of three phasors of equal magnitude, mutually displaced by 120°, and having the same phase sequence (a-b-c) as the original unbalanced phasors: Vb1 = a²Va1, Vc1 = aVa1 (where a = 1∠120°). This represents the normal balanced operating condition of the system.

Negative-sequence components (Va2, Vb2, Vc2) also consist of three phasors of equal magnitude, mutually displaced by 120°, but having the opposite phase sequence (a-c-b) to the original set: Vb2 = aVa2, Vc2 = a²Va2. Negative-sequence quantities arise only under unbalanced conditions (e.g., unbalanced faults or loads) and, in rotating machines, create a magnetic field rotating in the opposite direction to the rotor, inducing double-frequency currents in rotor/damper circuits and causing additional heating.

Zero-sequence components (Va0, Vb0, Vc0) consist of three phasors that are equal in magnitude and have zero phase displacement between them (all three are exactly in phase): Va0 = Vb0 = Vc0. Zero-sequence currents can only flow in circuits that provide a return path through a neutral or ground connection, since they represent a net, non-cancelling flow in all three phases simultaneously (e.g., 3·Ia0 flows through the neutral wire).

Example: for an unbalanced line-to-ground fault at phase A of an unloaded system where Ib = Ic = 0 and Ia = If (fault current), applying the symmetrical component transform gives Ia0 = Ia1 = Ia2 = If/3, illustrating a practical case where all three sequence currents in the faulted phase are equal — this is the standard boundary condition used to analyze single line-to-ground faults.

The original phase quantities are recovered from the sequence components by superposition: Va = Va0+Va1+Va2, Vb = Va0+a²Va1+aVa2, Vc = Va0+aVa1+a²Va2 — i.e., each phase phasor is the vector sum of the corresponding zero-sequence phasor, plus the appropriately rotated positive- and negative-sequence phasors.

(b) Sequence Components in an Unloaded Generator

For an unloaded, 3-phase synchronous generator with EMFs Ea, Eb, Ec (a balanced set, purely positive-sequence, since the generator's rotating field produces only fundamental-frequency, balanced EMFs), the machine is modeled by three separate sequence networks, each carrying only its own sequence current, since the machine's sequence impedances (Z1, Z2, Z0) are decoupled from one another under the assumption of a symmetrical, static machine structure (the sequence networks interact with each other only through the external fault/load boundary conditions, not internally within the machine itself).

Positive-sequence network: contains the internal generated EMF Ea (since the machine only generates positive-sequence voltage) in series with the positive-sequence impedance Z1 (equal to the sub-transient/transient/synchronous reactance depending on the time frame of interest): Va1 = Ea - Ia1·Z1.

Negative-sequence network: contains no EMF source (the machine does not generate negative-sequence voltage under normal balanced excitation), only the negative-sequence impedance Z2: Va2 = -Ia2·Z2.

Zero-sequence network: likewise contains no EMF source, only the zero-sequence impedance Z0 in series with three times the neutral grounding impedance (3Zn), since zero-sequence line currents combine in the neutral: Va0 = -Ia0·(Z0+3Zn).

These three independent sequence-network equations, combined with the specific boundary conditions imposed by the type of fault or unbalance occurring at the machine terminals (e.g., line-to-ground, line-to-line, or double line-to-ground fault), are solved simultaneously (often by connecting the three sequence networks in series, parallel or other combination depending on the fault type) to determine the actual fault currents and voltages, forming the standard method of unbalanced fault analysis in power systems.

Worked illustration — single line-to-ground fault at phase A: the boundary conditions at the fault point are Ib = Ic = 0 and Va = 0 (phase A solidly grounded). Transforming the current condition into sequence components gives Ia1 = Ia2 = Ia0 = Ia/3, i.e., all three sequence currents are equal. Substituting the sequence network voltage equations derived above, Va = Va0+Va1+Va2 = 0 becomes (Ea-Ia1Z1) + (-Ia2Z2) + (-Ia0(Z0+3Zn)) = 0. Since Ia1=Ia2=Ia0=Ia1 (all equal, call this common value Ia1), this reduces to Ea = Ia1(Z1+Z2+Z0+3Zn), so the fault current can be solved directly as Ia1 = Ea/(Z1+Z2+Z0+3Zn), and the total fault current is Ia = 3Ia1 = 3Ea/(Z1+Z2+Z0+3Zn). This single equation is obtained precisely because the boundary condition (Ia1=Ia2=Ia0) corresponds physically to connecting the positive-, negative- and zero-sequence networks of the machine in series with each other, a connection that is a direct graphical consequence of the equal-sequence-current boundary condition — this is the standard technique by which any unbalanced fault type at a generator's terminals is analyzed using the decoupled sequence networks derived above, only the manner of interconnecting the three networks (series for LG fault, parallel combinations for LL and LLG faults) changes with fault type.

Practical significance: this sequence-network approach converts an inherently 3-phase, mutually-coupled unbalanced problem into three simple, independent single-phase network problems (solvable with ordinary Ohm's-law-type equations), which are then coupled together only at the single point of unbalance through the fault boundary conditions — this is precisely why symmetrical components remain the standard industry technique for unbalanced fault calculations, relay setting studies, and protection coordination in power systems, since it avoids the far greater complexity of solving the original coupled 3-phase network directly.

Extension to line-to-line and double line-to-ground faults: the same three sequence networks derived for the generator are reused, unchanged, for analyzing every other unbalanced fault type — only the manner in which the boundary conditions at the fault point couple the networks together changes. For a line-to-line fault (phases B and C shorted, phase A healthy), the boundary conditions Ia=0, Ib=-Ic and Vb=Vc translate to Ia0=0 and Ia1=-Ia2, which corresponds to connecting only the positive- and negative-sequence networks in parallel opposition (the zero-sequence network carries no current and can be omitted entirely). For a double line-to-ground fault (phases B and C both grounded), the boundary conditions Ia=0 and Vb=Vc=0 translate to Va1=Va2=Va0, corresponding to connecting all three sequence networks in parallel. In every case, it is the same set of machine sequence impedances Z1, Z2 and Z0 (with their governing EMF-source rules established above) that is reused, with only the external interconnection pattern changing according to the fault boundary condition — underscoring why establishing the correct sequence-network model of the generator, as derived here, is the essential first step for all subsequent unbalanced-fault studies on that machine.