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.

bersambung ......

untuk lebih COMPLETE ~download~