SYMMETRICAL COMPONENTS CALCULATION FOR TRANSFORMERS

As a passive device, the positive- and negative-sequence impedance magnitudes for transformers are identical and are equal to the nameplate leakage reactance provided by the manufacturer. However, in modeling transformers in symmetrical components, recognizing that an inherent phase shift is associated with delta-connected windings is sometimes necessary.

Wye-delta and delta-wye transformers built under ANSI standards are designed so that high-voltage quantities always lead the corresponding low-voltage quantities by 30°. The complete positive-sequence model for a delta-wye or wye-delta transformer, therefore, should include a 30° phase shift.

Negative-sequence quantities, however, are shifted in the opposite direction, and so the negative-sequence representation should include a phase shift opposite to the shift considered in positive sequence. These relationships are illustrated in Figure 2-7a.

Figure 2-7a—Positive- and negative-sequence equivalent circuits for delta-wye or wye-delta transformer


NOTES
1—The phase shift in positive sequence is in the same direction as in the physical transformer: high voltage leads low voltage by 30° for ANSI standard transformers.
2—The phase shift in the negative-sequence circuit is opposite in direction.



Inclusion of these phase shifts is important only if a rigorous calculation is needed to determine exact phase currents and voltages on both sides of the transformer, including phase angles. Analysts often take the shortcut of neglecting phase shifts if the calculations are restricted to determining information on only one side of the transformer.

No inherent phase shift occurs in wye-wye transformers; therefore, the positive- and negativesequence equivalent circuits for these transformers also do not require phase shifts.

ZERO SEQUENCE IMPEDANCE OF POWER TRANSFORMER

The zero-sequence impedance of a transformer is controlled by a number of factors. The best way to determine a magnitude of this impedance is by an actual test, but the following comments, supplemented by information in some of the references, may be used to predict a value that is close enough for many applications.

First, the zero-sequence impedance seen looking into a transformer depends upon the configuration of the winding. The zero-sequence impedance of a delta winding is infinite (an open circuit), whereas the zero-sequence impedance of a wye-connected winding is a series composite of the zero-sequence impedance of the transformer and the impedance of any neutral grounding devices that might be present.

Thus, an ungrounded wye winding would present an infinite zero-sequence impedance because the absence of a neutral grounding connection appears as an open circuit in series with the zero-sequence impedance of the transformer winding itself (see Figure 2-7b).

Figure 2-7b—Zero-sequence equivalent circuit for delta-wye-grounded transformer


NOTE—The circuit is open on the side corresponding to the delta winding on the physical transformer


The impedance of the transformer itself depends upon several factors in the construction of the transformer. Three-phase transformers, which are constructed so that a closed, low-impedance path exists for the flow of zero-sequence flux within the transformer, have a lower zerosequence impedance than transformers without such a path.

One such path is the transformer core. Transformers with core-form construction have lower zero-sequence impedances than units with shell-form cores.

Three-phase transformers with delta windings have the lowest zero-sequence impedance, and in the absence of actual test data, it is often assumed that the zero-sequence impedance of core-form transformers with delta windings is about 0.85 times the positive-sequence leakage reactance of such transformers.

The zero-sequence impedance of shell-form transformers has about the same magnitude as the positive sequence leakage reactance of such transformers. Conversely, a three-phase transformer bank consisting of three, single-phase transformers connected wye-wye has a very high zero-sequence impedance.

SYMMETRICAL COMPONENTS CALCULATION FOR TRANSMISSION LINES

Determining impedances for transmission lines is more challenging and generally involves making calculations from the physical parameters of the line and its conductors. The algorithm and equations given in this subclause describe the procedure, and experienced protection engineers find understanding the theoretical basis for this procedure helpful.

All the equations given in this subclause are for 60 Hz systems; impedances for systems at other frequencies can be determined by ratio or by modifying the formulae. Alternatively, computer programs are available to calculate line impedances.

The first consideration is that the positive- and negative-sequence impedances of transmission lines are equal. A transmission line is a passive component that responds in the same way to positive- and negative-sequence excitation.

Because sequence impedances are the relationships between respective sequence voltages and currents, calculation of one impedance suffices for both needs.

The positive-sequence reactance of a transmission line can be thought of as the impedance that would relate voltage and current when the three conductors or a transmission line are shorted together at one end, while excited by a positive-sequence source of voltages at the other end. This impedance can be calculated using the following equation:

where

GMD is the geometric mean spacing between phase conductors (e.g., the cube root of the product of the three-phase spacings) (m),

GMR is the geometric mean radius of the phase conductor (m).

GMD should be calculated for the specific spacings of the array of conductors making up the transmission line, while GMR is a parameter for the conductor that is available from the conductor manufacturer.

Positive-sequence resistance can be read directly from conductor tables. Calculating the zero-sequence impedance of a transmission line is more challenging. The concept can be viewed as follows:

All three phases of a transmission line are shorted together to ground at the source end, while all three conductors are shorted together and to both ground and the overhead ground wire (OHGW) at the other end.

When a single phase source of voltage is then applied at the source end, a current flows. The ratio of the single-phase driving voltage to the resulting current flow is the zero-sequence impedance of the line.

Physically, current flows from the faulted conductor into both ground and the OHGW as depicted in Figure 2-6a; the current flows from the source out through the phase conductors and returns through a complex path consisting of the OHGW and the earth.

Figure 2-6a—Illustration of insulation flashover on open wire line showing return current flowing through OHGW of transmission line and through earth


From this physical picture, it is apparent that the zero-sequence impedance should, therefore, consist of three branches as indicated in Figure 2-6b: the zero-sequence impedance of the phase conductors, the zero-sequence impedance of the static wire return (OHGW), and the zero-sequence impedance of the earth return.

Figure 2-6b—Zero-sequence equivalent circuit that accounts for self impedance of transmission line and the impedances of earth and OHGW return paths


Values can be calculated for the various branches of Figure 2-6b using the following equations:

where
Ra is the resistance of the phase conductor (Ω/km),
GMD2 is the geometric mean spacing of all conductors—phase and static (OHGW) wires (m),
GMR2 is the geometric mean radius of k static (OHGW) wires (m),

k is the number of static (OHGW) wires,
r is the earth resistivity (typically 100) (Ω⋅m),
Rgw is the resistance of one ground wire (Ω/km),
f is the system frequency.

SYMMETRICAL COMPONENTS TUTORIALS FOR PROTECTIVE RELAYING

Symmetrical components are applied to calculations of unbalanced fault currents and voltages and in rotating machine analysis. Theory of symmetrical components can be briefly stated thus: a coplanar vector is defined by the position of its terminal and length and has 2 degrees of freedom.

A three-phase balanced system has 2 degrees of freedom because the current or voltage vectors (phasors) are displaced from each other by equal angles of separation of 120◦ and are of equal length. A three-phase unbalanced system of currents or voltages has 6 degrees of freedom because the vectors are of varying length at varying displacement angles from each other.

Such an unbalanced system can be resolved into three symmetrical systems, each system having three vectors with 2 degrees of freedom. Positive-sequence system is a set of balanced three-phase components of the same phase sequence as the original unbalanced set.

Negative-sequence system is a set of three-phase components of opposite phase sequence to the positive sequence system but vectors (phasors) of the same magnitude. Zero-sequence system consists of three single-phase components of the same magnitude and cophasial.

These are related by the following equations:

 

and

 

Where Va, Vb, and Vc are the original unbalanced voltages; a is a unit vector operator that rotates 120◦ in the counterclockwise direction; and V+a, V −a, and V0a are the positive, negative, and zero sequence components of the original unbalanced set.

Characteristics of Sequence Components

In a three-phase wye connected and ungrounded system, no zero sequence current flows. If the wye point is grounded, neutral carries the out-of-balance current. In a delta connection, no zero sequence currents can appear in the line currents.

In a balanced three-phase system with balanced loads, only positive sequence currents can flow. Negative sequence currents are set up in circuits of unbalanced impedances and voltages.

In symmetrical circuits, currents and voltages of different sequence do not affect each other (i.e., the positive sequence currents produce only positive sequence voltage drops and the theorem of superposition applies). Sequence impedance networks must be constructed for unbalanced fault current calculations and data input to digital computers.

As an example, the single-line-to-ground fault is given by the expression:

 

 

Where Ig is the single-line-to-ground fault current; E is the line-to-neutral voltage; and Z+, Z−, and Z0 are the positive, negative, and zero sequence impedances to the fault point.