13-03-2014, 02:20 PM
A Derivation of Symmetrical Component Theory and Symmetrical Component Networks
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INTRODUCTION
This paper provides a review of some of the theory behind symmetrical component analysis and
derives some of the basic calculations utilized in power system analysis for short circuit and
open phase conditions. The paper starts with a review of the concepts of system impedances in
the phase (ABC) domain and develops the three phase, two port, voltage drop equation,
VS - VR = Z·I, in substantial detail. Then, the paper reviews symmetrical component (012)
domain theory. The paper shows the conversion of the ABC domain voltage drop equation to
the equivalent 012 domain voltage drop equation and, in the process, correlates the terms ZS,
ZM, Z0, Z1, Z2, and other impedances and presents the relationship between the ZABC and Z012
impedance matrices. Next, the paper shows how the ZABC and Z012 matrices have a simplified
form for certain symmetrical impedance networks. Then, the paper applies the ABC and 012
forms of the voltage drop equation to some of the more common basic fault types in order to
show the development of the associated symmetrical component networks used for fault
analysis. The analysis process that is developed allows for the development of the equations for
faults not easily shown in classical symmetrical component networks, and some of these
equations are shown in an appendix.
Basic Impedance Concepts
Many papers and texts on power system analysis use lumped impedance concepts and models
and start from models such as figure 3. Let us back up, however, with a development of figure
in order to have a better foundation for our analysis. Going back to basic electromagnetic
theory, recall that around a conductive loop,
Ground vs. Neutral vs. Remote Equipotential Ground
The terms “neutral” and “ground” (and earth, in some literature) are very similar, and it leads to
some confusion. Though in some applications each has a specific meaning, in the overall body
of literature the terms are used somewhat interchangeably. In this paper, the term “ground” is
utilized for the source voltage reference, rather than “neutral”. We will proceed in the analysis
assuming that any power system can be reduced to the diagram shown in figure 2
Review of Symmetrical Components Concepts
A basic tenet of symmetrical component analysis of a power system is that any set of ABC
phase domain voltages or currents (called the ABC domain in the balance of the paper) can be
restated as a sum of three balanced symmetrical component domain voltages or currents
(called the 012 domain in the balance of this paper). An example set of equivalent ABC and 012
domain voltages is shown in figure 4.
The terms “symmetrical coordinates” (Fortescue’s original 1918 term), “sequence components,”
“symmetrical components,” “sequence component circuit,” and “symmetrical component
network” are very similar and used almost interchangeably in papers and texts on this topic. In
this paper, the term “symmetrical components” generally refers to the entire concept of this type
of analysis, and a “sequence component” generally refers to the individual elements (positive,
negative, and zero sequence) within the larger symmetrical component analysis approach. Also,
the term “symmetrical component network” will be used rather than “sequence component
circuit.”
Symmetrical Component Networks
With this groundwork on symmetrical component analysis, particularly the nature of the basic
three phase ABC and 012 domain voltage drop equation, let us proceed with development of
the symmetrical component networks and the general equations for short circuits and open
circuits. The process to develop these networks will follow these steps:
1. State the voltage drop equation in ABC domain for the known system values during the
fault. Also state the equation in the 012 domain.
2. Solve for the ABC and the 012 domain voltages and currents.
3. Devise an electric circuit (i.e., an symmetrical component network) consisting of 012 domain
voltages and impedances that has the same solution for currents as the mathematically
derived solution for 012 domain currents found in step 2.
Shunt Faults
Shunt faults to ground are modeled by specifying that in the voltage drop equation (16), VR on
one or more phases is 0 (due to a phase to ground fault), that a yet to be determined current IF
flows on the faulted phases, and that 0 current flows on the unfaulted phases. For a phase to
phase fault, we assume two voltages at the receiving end are identical relative to ground. On
the unfaulted phases, we assume that there is a high impedance or open circuit at the receiving
end on those phases (i.e., there is no load flow). We will start with the unbalanced phase to
ground and phase to phase faults and discuss the simple three phase fault last.