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4 Symmetrical Components: A Review

4.1 INTRODUCTION AND BACKGROUND

The method of symmetrical components provides a practical technology for understanding and analyzing the operation of a system during power unbalanced conditions, such as those caused by faults between phases and ground, open phases, unbalanced impedances, and so on. In addition, many protective relays operate from symmetrical component quantities. Thus, a good under- standing of this subject is of great value and another very important tool in protection.

In a sense, symmetrical components can be called the language of the relay engineer or technician. Its value is both in thinking or visualizing unbalances, and it is a means of detailed analysis of them from the system parameters. In this, it is like a language in that it requires experience and practice for each access and application. Faults and unbalances occur infre- quently and many do not require detailed analysis, so it becomes difficult to practice the language. This has increased with the ready availability of fault studies by computers. These provide rapid access to voluminous data, often with little understanding of the background or method that provides the data. Hence, this review of the method is intended to provide the fundamentals, basic circuits and calculations, and an overview directed at clear understanding and visualization. The method of symmetrical components was discovered by Charles L.

Fortescue, who was mathematically investigating the operation of induction motors under unbalanced conditions, late in 1913. At the 34th Annual Convention of the AIEE—on June 28, 1918, in Atlantic City—he presented an 89-page paper entitled ‘‘Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks.’’ The six discussants, including Charles Proteus Steinmetz, added 25 pages. Practical application for system fault analysis was developed by C.F. Wagner and R.D. Evans in the later part of 1920s and early 1930s, with W.A. Lewis adding valuable simplifications in 1933. Tables of fault and unbalance connections were provided by E.L. Harder in 1937. At the same time Edith Clarke was also developing notes and lecturing in this area, but formal publication of her work did not occur until 1943. Additional material and many examples for further study are found in Blackburn (1993).

Only symmetrical components for three-phase systems are reviewed in this chapter. For these systems there are three distinct sets of components: positive, negative, and zero for both current and voltage. Throughout this discussion, the sequence quantities are always line-to-neutral or line-to-ground and appropriate to the situation. This is an exception for voltage connections, whereas while in the power system line-to-line voltages are commonly indicated, in symmetrical components they are always given as line-to-neutral (or possibly line-to-ground).

4.2 POSITIVE-SEQUENCE SET

The positive-sequence set consists of balanced three-phase currents and line- to-neutral voltages supplied by the system generators. Thus, they are always equal in magnitude and are phase-displaced by 1208C. Figure 4.1 shows a positive-sequence set of phase currents, with the power system phase sequence in the order of a, b, c. A voltage set is similar, except for line- to-neutral voltage of the three phases, with equal magnitude and which displaces at 1208C. These are phasors that rotate in the counterclockwise direction at the system frequency.

To document the angle displacement, it is convenient to use a unit phasor with an angle displacement of 120o. This is designated as a so that

FIGURE 4.1 Positive-sequence current phasors. Phasor rotation is counterclockwise.

It is most important to emphasize that the set of sequence currents or sequence voltages always exists as defined. The phasors Ia1 or Ib1 or Ic1 can never exist alone or in pairs, but always as a set of three. Thus, it is necessary to define only one of the phasors (any one) from which the other two will be as documented in Equation 4.2.

4.3 NOMENCLATURE CONVENIENCE

It will be noted that the designation subscript for phase a was dropped in the second expression for the currents and voltages in Equation 4.2 (and also in the following equations). This is a common shorthand notation used for convenience. When the phase subscript is not mentioned, it can be assumed that the reference is to phase a. If phase b or phase c quantities are intended, the phase subscript must be correctly designated; otherwise, it is assumed as phase a. This shortcut will be used throughout the book and is common in practice.

4.4 NEGATIVE-SEQUENCE SET

The negative-sequence set is also balanced with three equal magnitude quantities at 1208 separately, but only when the phase rotation or sequence is reversed as illustrated in Figure 4.2. Thus, if positive sequence is a, b, c; negative will be a, c, b. When positive sequence is a, c, b, as in some power systems; negative sequence is a, b, c.

4.5 ZERO-SEQUENCE SET

The members of this set of rotating phasors are always equal in magnitude and exist in phase (Figure 4.3).

Similarly, I0 or V0 exists equally in all three phases, but never alone in a phase.

4.6 GENERAL EQUATIONS

Any unbalanced current or voltage can be determined from the sequence components given in the following fundamental equations:

where Ia, Ib, and Ic or Va, Vb, and Vc are general unbalanced line to neutral phasors.

From these, equations defining the sequence quantities from a three-phase unbalanced set can be determined:

FIGURE 4.3 Zero-sequence current phasors. Phasor rotation is counterclockwise.

FIGURE 4.4 Zero-sequence current and voltage networks used for ground-fault protection. See Figure 3.9 and Figure 3.10 for typical fault operations.

These three fundamental equations are the basis for determining if the sequence quantities exist in any given set of unbalanced three-phase currents or voltages. They are used for protective-relaying operations from the sequence quantities. For example, Figure 4.4 shows the physical application of current transformers (CTs) and voltage transformers (VTs) to measure zero sequence as required in Equation 4.8 and as used in ground-fault relaying.

Networks operating from CTs or VTs are used to provide an output proportional to I2 or V2 and are based on physical solutions (Equation 4.10). This can be accomplished with resistors, transformers, or reactors, by digital solutions of Equation 4.8 through Equation 4.10.

4.7 SEQUENCE INDEPENDENCE

The factor that makes the concept of dividing the unbalanced three-phase quantities into the sequence components practical is the independence of the components in a balanced system network. For all practical purposes, electric power systems are balanced or symmetrical from the generators to the point of single-phase loading, except in an area of a fault or unbalance, such as an open conductor. In this effectively balanced area, the following conditions

exist:

Positive-sequence currents flowing in the symmetrical or balanced network produce only positive-sequence voltage drops, no negative- or zero-sequence drops.

Negative-sequence currents flowing in the balanced network produce only negative-sequence voltage drops, no positive- or zero-sequence voltage drops.

Zero-sequence currents flowing in the balanced network produce only zero-sequence voltage drops, no positive- or negative-sequence voltage drops.

This is not true for any unbalanced or nonsymmetrical point or area, such as an unsymmetrical fault, open phase, and so on.

Positive-sequence current flowing in an unbalanced system produces positive-, negative-, and possibly zero-sequence voltage drops.

Negative-sequence currents flowing in an unbalanced system produces positive-, negative-, and possibly zero-sequence voltage drops.

Zero-sequence current flowing in an unbalanced system produces all three: positive-, negative-, and zero-sequence voltage drops.

This important fundamental condition permits setting up three independent networks, one for each of the three sequences, which can be interconnected only at the point or area of unbalance. Before continuing with the sequence networks, a review of the source of fault current is useful.

4.8 POSITIVE-SEQUENCE SOURCES

A single-line diagram of the power system or area under study is the starting point for setting up the sequence networks. A typical diagram for a section of a power system is shown in Figure 4.5. In these diagrams, circles are used to designate the positive-sequence sources, which are the rotating machines in the system; generators, synchronous motors, synchronous condensers, and probably induction motors. The symmetrical current supplied by these to the power-system faults decreases exponentially with time from a relatively high initial value to a low steady-state value. During this transient period three reactance values are possible for use in the positive-sequence network and for the

calculation of fault currents. These are the direct-axis subtransient reactance

the direct-axis transient reactance

, and the unsaturated direct-axis

The values of these reactances vary with the designs of the machines and the specific values are supplied by the manufacturer. In their absence, typical

FIGURE 4.5 Single-line diagram of a section of a power system.

values are shown in Blackburn (1993, p. 279) and in many other references. Generally, typical values at the machines rated MVA (kVA) and kV are:

Xd= 0.1 to 0.3 pu, with time constants in the order of 0.6–1.5 sec; Xd=1.2-2.0 time Xd, with time constants in the order of 0.6–1.5 sec; Xd for faults is the unsaturated value that can range from 6 to 14 timesXd.

For system-protection fault studies, the almost universal practice is to use the subtransient Xd for the rotating machines in the positive-sequence networks. This provides a maximum value of fault current that is useful for

high-speed relaying. Although slower-speed protection may operate after the subtransient reactance has decayed into the transient reactance period, the general practice is to use Xd, except possibly for special cases where Xd would be used. There are special programs to account for the decremental decay in fault current with time in setting the slower-speed protective relays, but these tend to be difficult and tedious, and may not provide any substantial advantages. A guide to aid in the understanding of the need for special considerations is outlined in Figure 4.6. The criteria are very general and approximate.

Cases A and B (see Figure 4.6) are the most common situations, so that the use of Xd has a negligible effect on the protection. Here the higher system Zs tends to negate the source decrement effects.

Case C (see Figure 4.6) can affect the overall operation time of a slower-speed protection, but generally the decrease in fault current level with time will not cause coordination problems unless the time–current characteristics of various devices that are used are significantly different.

When ZM predominates, the fault levels tend to be high and well above the maximum-load current. The practice of setting the protection as sensitive as possible, but not operating on maximum load (phase devices) should provide good protection sensitivity in the transient reactance period. If protection operating times are very long, such that the current decays into the synchron- ous reactance period, special phase relays are required, as discussed in Chapter 8.

a) Utility systems outside generating station areas, industrial plants with utility tie and no or small local generation

b & c) industrial plants with utility tie and significant local generation. Near generating stations

d) generating stations, industrial plants with all local generation, no utility tie.

FIGURE 4.6 Guide illustrating the effects of rotating machine decrements on the symmetrical fault current.

Usually, induction motors are not considered as sources of fault current for protection purposes (see Figure 4.6, case D). However, it must be emphasized that these motors must be considered in circuit breakers’ applications under the ANSI=IEEE standards. Without a field source, the voltage that is developed by induction motors decays rapidly, within a few cycles; thus, they generally have a negligible effect on the protection. The DC offset that can result from sudden changes in current in the ac networks is neglected in symmetrical components. It is an important consideration in all protection.

An equivalent source, such as that shown in Figure 4.5, represents the equivalent of all the systems that are not shown up to the point of connection to that part of the system under study. This includes one or many rotating machines that may be interconnected together with anynetwork oftransformers, lines, and so on. In general, a network system can be reduced to two equivalent sources at each end of an area to be studied, with an equivalent interconnecting tie between these two equivalent sources. When the equivalent tie is large or infinite, indicating that little or no power is exchanged between the two source systems,it is convenientto expressthe equivalent source systemupto a specified bus or point in short-circuit MVA (or kVA). Appendix 4.1 outlines this and the conversion to the impedance or the reactance values. In Figure 4.5, the network to the right has reduced to a single equivalent impedance to represent it up to the M terminal of the three-winding transformer bank.

4.9 SEQUENCE NETWORKS

The sequence networks represent one of the three-phase-to-neutral or to-ground circuits of the balanced three-phase power system and document how their sequence currents will flow if they can exist. These networks are best explained by an example: let us now consider the section of a power system in Figure 4.5.

Reactance values have been indicated only for the generator and the transformers. Theoretically, impedance values should be used, but the resistances of these units are small and negligible for fault studies. However, if loads are included, impedance values should be used unless their values are small in relation to the reactances.

It is important that all values should be specified with a base [ voltage if ohms are used, or MVA (kVA) and kV if per-unit or percent impedances are used]. Before applying these to the sequence networks, all values must be changed to one common base. Usually, per-unit (percent) values are used, and a common base in practice is 100 MVA at the particular system kV.

4.9.1 POSITIVE-SEQUENCE NETWORK

This is the usual line-to-neutral system diagram for one of the three symmet- rical phases modified for fault conditions. The positive-sequence networks for the system in Figure 4.5 are shown in Figure 4.7. The voltages VG and VS are the system line-to-neutral voltages. VG is the voltage behind

the generator subtransient direct-axis reactance Xd , and Vs is the voltage behind the system equivalent impedance Z1S.

FIGURE 4.7 Positive-sequence networks for the system in Figure 4.5: (a) network including loads; (b) simplified network with no load—all system voltages equal and in phase.

XTG is the transformer leakage impedance for the bank bus G, and XHM is the leakage impedance for the bank at H between the H and M windings. More details on these are given in Appendix 4.2. The delta-winding L of this three-winding bank is not involved in the positive-sequence network unless a generator or synchronous motor is connected to it or unless a fault is to be considered in the L delta system. The connection would be as in Figure A4.2-3.

For the line between buses G and H, Z1GH is the line-to-neutral impedance of this three-phase circuit. For open-wire transmission lines, an approximate estimating value is 0.8 Ω/mi for single conductor and 0.6 Ω/mi for bundled conductors. Typical values for shunt capacitance of these lines are 0.2 MΩ/mi for single conductor and 0.14 MΩ/mi for bundled conductors. Normally, this capacitance is neglected, as it is very high in relation to all other impedances that are involved in fault calculations. These values should be used either for estimating or in the absence of specific line constants. The impedances of cables vary considerably, so specific data are necessary for these.

The impedance angle of lines can vary quite widely, depending on the voltage and type of cable or open wire that is used. In computer fault programs, the angles are considered and included, but for hand calculation,

it is often practical and convenient to simplify calculations by assuming that all the equipment involved in the fault calculation is at 90^o. Otherwise, it is better to use reactance values only. Sometimes it may be preferred to use the line impedance values and treat them as reactances. Unless the network consists of a large proportion of low-angle circuits, the error of using all values as 90^o will not be too significant.

Load is shown to be connected at buses G and H. Normally, this would be specified as kVA or MVA and can be converted into impedance.

Equation 4.11 is a line-to-neutral value and could be used for ZLG and ZLH, representing the loads at G and H as shown in Figure 4.7a. If load is represented, the voltages VG and VS will be different in magnitude and angle, varying according to the system load.

The value of load impedance is usually quite large compared with the system impedances, such that the load has a negligible effect on the faulted- phase current. Thus, it becomes practical and simplifies the calculations to neglect load for shunt faults. With no load, ZLG and ZLH are infinite. VG and VS are equal and in phase, and so they are replaced by a common voltage V as in Figure 4.7b. Normally, V is considered as 1 pu, the system-rated line-to-neutral voltages.

Conventional current flow is assumed to be from the neutral bus N1 to the area or point of unbalance. With this the voltage drop V1x at any point in the network is always

where V is the source voltage (VG or Vs in Figure 4.7a) and

is the sum of the drops along any path from the N1 neutral bus to the point of measurement.

4.9.2 NEGATIVE-SEQUENCE NETWORK

The negative-sequence network defines the flow of negative-sequence currents when they exist. The system generators do not generate negative sequence, but negative-sequence current can flow through their windings. Thus, these generators and sources are represented by an impedance without voltage, as shown in Figure 4.8. In transformers, lines, and so on, the phase

FIGURE 4.8 Negative-sequence networks for the system in Figure 4.5: (a) network including loads; (b) network neglecting loads.

sequence of the current does not change the impedance encountered; hence, the same values as in the positive-sequence network are used.

A rotating machine can be visualized as a transformer with one stationary and one rotating winding. Thus, DC in the field produces positive sequence in the stator. Similarly, the DC offset in the stator ac current produces an ac component in field. In this relative-motion model, with the single winding rotating at synchronous speed, negative sequence in the stator results in a double-frequency component in the field.Thus,the negative-sequence flux component inthe air gap alternates between and under the poles at this double frequency. One common expression for the negative-sequence impedance of a synchronous machine is

or the average of the direct and substransient reactance of quadrature axes.

For a round-rotor machine,

, so that

. For salient-pole machines, X2 will be different, but this is frequently neglected unless calculating a fault very near the machine terminals. Where normally negative-sequence network is equivalent to the positive-sequence network except for the omission of voltages.

Loads can be shown, as in Figure 4.8a, and will be the same impedance as that for positive sequence, provided they are static loads. Rotating loads, such as those of induction motors, have quite a different positive- and negative-sequence impedances when in operation. This is discussed further in Chapter 11.

Similarly, when the load is normally neglected, the network is as shown in Figure 4.8b and is the same as the positive-sequence network (see Figure 4.7b), except that there is no voltage.

Conventional current flow is assumed to be from the neutral bus N2 to the area or point of unbalance. With this the voltage drop V2x at any point in the network is always

Where

is the sum of the drops along any path from the N neutral bus to the point of measurement.

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