UNSYMMETRICAL-FAULTS- - University of Florida ? ‚ Mostof-the-faults-thatoccur-on-power-systems-are-unsymmetrical-faults ... applied-to-the-analysis-of-unsymmetrical-faults.-

  • View
    243

  • Download
    5

Embed Size (px)

Text of UNSYMMETRICAL-FAULTS- - University of Florida ? ‚...

  • UNSYMMETRICAL FAULTS

    updated 11/11/13

    Unsymmetrical Faults (c) 2013 H. Zmuda 1 11/11/13

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 2

    Introductory Comments Most of the faults that occur on power systems are unsymmetrical faults, which may consist of unsymmetrical short circuits, unsymmetrical faults through impedances, or open conductors. Unsymmetrical faults occur as single line-to-ground faults, line-to-line faults, or double line-to-ground faults. The path of the fault current from line to line or line to ground may or may not contain impedance. One or two open conductors result in unsymmetrical faults, through either the breaking of one or two conductors or the acHon of fuses and other devices that may not open the three phases simultaneously.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 3

    Introductory Comments Since any unsymmetrical fault causes unbalanced currents to flow in the system, the method of symmetrical components is very used in the analysis to determine the currents and voltages in all parts of the system aLer the occurrence of the fault. We will consider faults on a power system by applying Thvenin's theorem, which allows us to find the current in the fault by replacing the enHre system by a single generator and series impedance, and we will show how the bus impedance matrix is applied to the analysis of unsymmetrical faults.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 4

    Unsymmetrical Faults In the derivaHon of equaHons for the symmetrical components of currents and voltages in a general network the currents flowing out of the original balanced system from phases a , b, and c at the fault point will be designated as Ia, lb, and lc, respecHvely. We can visualize these currents by as follows: This shows the three lines a, b, and c of the three-phase system at the part of the network where the fault occurs. The flow of current from each line into the fault is indicated by arrows shown beside hypotheHcal stubs connected to each line at the fault locaHon.

    I fa

    I fb

    I fc

    a

    b

    c

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 5

    Unsymmetrical Faults Appropriate connecHons of the stubs represent the various types of fault. For instance, direct connecHon of stubs b and c produces a line-to-line fault through zero impedance. The current in stub a is then zero, and lb equals - lc. The line-to-ground voltages at any bus j of the system during the fault will be designated Vja, Vjb and Vjc and we shall conHnue to use superscripts 1, 2, and 0, respecHvely, to denote posiHve-, negaHve-, and zero-sequence quanHHes. Thus, for example, V(1)ja, V(2)jb and V(0)jc will denote, respecHvely, the posiHve-, negaHve-, and zero-sequence components of the line-to-ground voltage Vja at bus j during the fault.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 6

    Unsymmetrical Faults The line-to-neutral voltage of phase a at the fault point before the fault occurs will be designated simply by Vf, which is a posiHve-sequence voltage since the system is balanced. We considered the prefault voltage Vf previously when calculaHng the currents in a power system with a symmetrical three-phase fault applied.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 7

    Unsymmetrical Faults Consider a single-line diagram of a power system containing two synchronous machines. This simple system is sufficiently general that the equaHons derived are applicable to any balanced system regardless of the complexity. The point where a fault is assumed to occur is marked P, and in this example it is called bus k on the single-line diagram and in the sequence networks.

    Single line diagram of a balanced three-phase system

    P

    k

  • Unsymmetrical Faults

    Thvenin Equivalent of PosiKve-Sequence Network PosiKve-Sequence Network

    11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 8

    P

    k

    Reference

    P k

    +

    Vf

    I fa

    1( )

    +

    Vf

    P I fa

    1( )

    k

    Zkk1( )

    +

    Vka1( )

  • Unsymmetrical Faults

    Thvenin Equivalent of NegaKve-Sequence Network NegaKve-Sequence Network

    11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 9

    P

    k

    Reference

    P k

    I fa

    2( )P

    I fa2( )

    k

    Zkk2( )

    +

    Vka2( )

  • Unsymmetrical Faults

    Thvenin Equivalent of Zero-Sequence Network Zero-Sequence Network

    11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 10

    P

    k

    Reference

    P k

    I fa

    0( )P

    I fa0( )

    k

    Zkk0( )

    +

    Vka0( )

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 11

    Unsymmetrical Faults Machines are represented by their subtransient internal voltages in series with their subtransient reactances when subtransient fault condiHons are being studied. Previously we used the bus impedance matrix composed of posiHve-sequence impedances to determine currents and voltages upon the occurrence of a symmetrical three-phase fault. The method can be easily extended to apply to unsymmetrical faults by realizing that the negaHve- and zero-sequence networks also can be represented by bus impedance matrices. The bus impedance matrix will now be wriben symbolically for the posiHve-, negaHve-, and zero-sequence networks in the following form

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 12

    Unsymmetrical Faults

    bus0,1,2( ) =

    Z110,1,2( ) Z12

    0,1,2( ) Z1k0,1,2( ) Z1N

    0,1,2( )

    Z210,1,2( ) Z22

    0,1,2( ) Z2k0,1,2( ) Z2 N

    0,1,2( )

    Zk10,1,2( ) Zk 2

    0,1,2( ) Zkk0,1,2( ) ZkN

    0,1,2( )

    ZN10,1,2( ) ZN 2

    0,1,2( ) ZNk0,1,2( ) ZNN

    0,1,2( )

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 13

    Unsymmetrical Faults The Thvenin equivalent circuit between the fault point P and the reference node in each sequence network can be used for the analysis. As before, the voltage source in the posiHve-sequence network and its Thvenin equivalent circuit is Vf, the prefault voltage to neutral at the fault point P, which happens to be bus k in this illustraHon. The Thvenin impedance measured between point P and the reference node of the posiHve-sequence network is Z(1)kk, and its value depends on the values of the reactances used in the network. Recall that subtransient reactances of generators and 1.5 Hmes the subtransient reactances (or else the transient reactances) of synchronous motors are the values used in calculaHng the symmetrical current to be interrupted.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 14

    Unsymmetrical Faults There are no negaHve- or zero-sequence currents flowing before the fault occurs, and the prefault voltages are zero at all buses of the negaHve- and zero-sequence networks. Therefore, the prefault voltage between point P and the reference node is zero in the negaHve- and zero-sequence networks and no electromoHve forces (emfs) appear in their Thvenin equivalents. The negaHve- and zero-sequence impedances between point P at bus k and the reference node in the respecHve networks are represented by the the impedances Z(2)kk and Z(0)kk, the Diagonal elements of Z(2)bus and Z(0)bus, respecHvely.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 15

    Unsymmetrical Faults Since Ifa is the current flowing from the system into the fault, its symmetrical components flow out of the respecHve sequence networks and their equivalent circuits at point P, as shown. Thus, the currents I(1)fa, I(2)g and I(0)fc represent injected currents into the faulted bus k of the posiHve-, negaHve-, and zero-sequence networks due to the fault. These current injecHons cause voltage changes at the buses of the posiHve-, negaHve-, and zero-sequence networks, which can be calculated from the bus impedance matrices in the manner similare to what we have done before.

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 16

    Unsymmetrical Faults For instance, due to the injecHon I(1)fa into bus k, the voltage changes in the posiHve-sequence network of the N-bus system are given in general terms by:

    V1a1( )

    V2a1( )

    Vka1( )

    VNa1( )

    =

    Z111( ) Z12

    1( ) Z1k1( ) Z1N

    1( )

    Z211( ) Z22

    1( ) Z2k1( ) Z2 N

    1( )

    Zk11( ) Zk 2

    1( ) Zkk1( ) ZkN

    1( )

    ZN11( ) ZN 2

    1( ) ZNk1( ) ZNN

    1( )

    00

    I fa1( )

    0

    =

    Z1k1( )I fa

    1( )

    Z2k1( )I fa

    1( )

    Zkk1( )I fa

    1( )

    ZNk1( ) I fa

    1( )

  • 11/11/13 Unsymmetrical Faults (c) 2013 H. Zmuda 17

    Unsymmetrical Faults Once again, it is industry pracHce to regard all prefault currents as being zero and to designate the voltage Vf as the posiHve-sequence voltage at all buses of the system before the fault occurs. Using superposiHon, the total posiHve-sequence voltage of phase a at each bus during the fault is:

    V1a1( )

    V2a1( )

    Vka1( )

    VNa1( )

    =

    VfVf

    Vf

    Vf

    +

    V1a1( )

    V2a1( )

    Vka1( )

    VNa1( )

    =

    Vf Z1k1( )I fa

    1( )

    Vf Z2k1( )I fa

    1( )

    Vf Zkk1( )I fa

    1( )

    Vf ZNk1( ) I fa

    1( )

  • 11/11/13 Unsymmetrical