In electrical engineering, the method of symmetrical components is used to
simplify analysis of unbalanced three-phase power systems under both
normal and abnormal conditions. The basic idea is that an asymmetrical set
of N phasors can be expressed as a linear combination of N symmetrical sets
of phasors by means of a complex linear transformation. In the most common case
of three-phase system, the resulting "symmetrical" components are referred to
as direct, inverse and zero. The analysis of power system is much simpler
in the domain of symmetrical components, because the resulting equations are
mutually linearly independent if the circuit itself is balanced.
Description In 1918 Charles Legeyt Fortescue
presented a paper which demonstrated that any set of N unbalanced phasors
could be expressed as the sum of N symmetrical sets of balanced phasors,
for values of N that are prime. Only a single frequency component is
represented by the phasors. In a three-phase system, one set of
phasors has the same phase sequence as the system under study, the second set
has the reverse phase sequence, and in the third set the phasors A, B and C are
in phase with each other. Essentially, this method converts three unbalanced
phases into three independent sources, which makes asymmetric fault analysis
more tractable. By expanding a one-line diagram to show
the positive sequence, negative sequence and zero sequence impedances of
generators, transformers and other devices including overhead lines and
cables, analysis of such unbalanced conditions as a single line to ground
short-circuit fault is greatly simplified. The technique can also be
extended to higher order phase systems. Physically, in a three phase winding a
positive sequence set of currents produces a normal rotating field, a
negative sequence set produces a field with the opposite rotation, and the zero
sequence set produces a field that oscillates but does not rotate between
phase windings. Since these effects can be detected physically with sequence
filters, the mathematical tool became the basis for the design of protective
relays, which used negative-sequence voltages and currents as a reliable
indicator of fault conditions. Such relays may be used to trip circuit
breakers or take other steps to protect electrical systems.
The analytical technique was adopted and advanced by engineers at General
Electric and Westinghouse and after World War II it was an accepted method
for asymmetric fault analysis. As shown in the figure to the right, the
three sets of symmetrical components add up to create the system of three
unbalanced phases as pictured in the bottom of the diagram. The imbalance
between phases arises because of the difference in magnitude and phase shift
between the sets of vectors. Notice that the colors of the separate sequence
vectors correspond to three different phases. To arrive at the final plot, the
sum of vectors of each phase is calculated. This resulting vector is the
effective phasor representation of that particular phase. This process,
repeated, produces the phasor for each of the three phases.
The three-phase case Symmetrical components are most commonly
used for analysis of three-phase electrical power systems. In a perfectly
balanced three-phase power system, the voltage phasors have equal magnitudes
but are 120 degrees apart. In an unbalanced system, the magnitudes and
phases of the phasors are different. Decomposing the voltage phasors into
their symmetrical components can help in analyzing the system as well as
visualizing any imbalances. If the phase quantities are expressed in phasor
notation using complex numbers, a vector can be formed for the three phase
quantities. For example, a vector for three phase voltages could be written as
where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and
negative sequence components. The sequence components differ only by their
phase angles, which are symmetrical and so are radians or 120°. Define the
operator phasor vector forward by that angle.
Note that α3 = 1 so that α−1 = α2. The zero sequence components are in
phase; denote them as: and the other phase sequences as:
Thus, where
Conversely, the sequence components are generated from the analysis equations
where An intuitive feeling
The phasors form a closed triangle. To find the synchronous and inverse
components of the phases, take any side of the outer triangle and draw the two
possible equilateral triangles sharing the selected side as base. These two
equilateral triangles represent a synchronous and inverse system. If the
phasors V were a perfectly synchronous system, the vertex of the outer triangle
not on the base line would be at the same position as the corresponding
vertex of the equilateral triangle representing the synchronous system. Any
amount of inverse component would mean a deviation from this position. The
deviation is exactly 3 times the inverse phase component. The synchronous
component is in the same manner 3 times the deviation from the "inverse
equilateral triangle". The directions of these components are correct for the
relevant phase. It seems counter intuitive that this works for all three
phases regardless of the side chosen but that is the beauty of this illustration.
For an illustration see Napoleon's Theorem.
Poly-phase case It can be seen that the transformation
matrix above is a discrete Fourier transform, and as such, symmetrical
components can be calculated for any poly-phase system. However, by
Pontryagin duality, only certain groups have a unique inverse, which is
necessary for use in fault analysis. See also
Symmetry Dqo transformation
Alpha–beta transformation References
Notes Bibliography
J. Lewis Blackburn Symmetrical Components for Power Systems
Engineering, Marcel Dekker, New York. ISBN 0-8247-8767-6
William D. Stevenson, Jr. Elements of Power System Analysis Third Edition,
McGraw-Hill, New York. ISBN 0-07-061285-4.
History article from IEEE on early development of symmetrical components,
retrieved May 12, 2005. Westinghouse Corporation, Applied
Protective Relaying, 1976, Westinghouse Corporation, no ISBN, Library of
Congress card no. 76-8060 - a standard reference on electromechanical
protective relays