Unsymmetrical Fault on Three Power System

Unsymmetrical faults on three power systems are those faults in the power system that result in unsymmetrical fault currents, or unequal fault currents in the lines with unequal phase displacement.

The currents in the three lines become uneven as well as their phase displacement when an unsymmetrical fault occurs on a three-power system. It should be noted that only the fault itself and the resulting line currents fall under the definition of “unsymmetry.” However, the generators, transmission lines, synchoronous reactors, and other major system components ensure that the impedances and source voltages are always symmetrical. Unsymmetrical faults can happen in a power system in three different ways (see Fig. 18.1).

1. Single line-to-ground fault (L — G)
2. Line-to-line fault (L — L)
3. Double line-to-ground fault (L — L— G)

So Kirchhoff’s laws or the method of symmetrical components can both be used to solve problems involving asymmetrical faults on a three-power system. The latter approach is preferred for the following explanations:

• It is an easy method that gives fault performance studies more generality.
• It gives protection engineers a useful tool, especially when it comes to tracing out fault currents.

Symmetrical Components Method

A scientist from the United States named Dr. C.L. Fortescue demonstrated in 1918 that any unbalanced system of three-phase currents (or voltages) can be viewed as being composed of three distinct sets of balanced vectors, namely.

• Positive (or normal) phase order in a balanced system of three-phase currents. Positive phase sequence components are what these are known as.
• A balanced set of three-phase currents with the reverse phase sequence. The negative phase sequence components are those.
• A set of three equal-amplitude currents with zero phase displacement. The components of a zero phase sequence are known as these.

So The positive, negative, and zero phase sequence components are referred to as the original unbalanced system’s symmetrical components. The unbalanced three-phase system has been reduced to three sets of balanced (or symmetrical) components, so the term “symmetrical” is appropriate. So Positive, negative, and zero phase sequence components are typically denoted by the subscripts 1, 2, and 0, respectively. For instance, IR0 denotes the current in the red phase’s zero phase sequence component. IY1 also denotes the positive phase sequence component of the yellow phase current.

So Let’s now use a three-phase unbalanced system to apply the symmetrical components theory. Suppose a 3-phase system with the phase sequence RYB experiences an Unsymmetrical Faults on Three Power System. The resulting unbalanced currents IR, IY, and IB (see Fig. 18.2) can be resolved into the following according to symmetrical components theory:

1. a balanced system of 3-phase currents, I_R1,I_Y1,I_B1 having positive phase sequence (i.e. RYB) as shown in Fig. 18.3 (i). These are the positive phase sequence components.
2. a balanced system of 3-phase currents I_R2,I_Y2,I_B2 having negative phase sequence (I e. RBY) as shown in_: 18.3 (ii). These are the negative phase sequence components.
3. a system of three currents I_R0,I_Y0,I_B0 equal in magnitude with zero phase displacement from each other as shown in Fig. 18.3 (iii). These are the zero phase sequence components.

So The current in any phase is equal to the vector sum of positive, negative and zero phase sequence currents in that phase as shown in Fig. 18.4.

The following points may be noted :

• The balanced system of currents is made up of the positive phase sequence currents (IR1, IY1, IB1), the negative phase sequence currents (IR2, IY2, IB2), and the zero phase sequence currents (IR0, IY0, IB0). They are referred to as symmetrical components of the unbalanced system as a result.
• 3-phase currents and voltages, both in terms of line and phase values, are both covered by the symmetrical component theory.
• The symmetrical parts do not exist independently. They are merely mathematical elements of the system’s actual unbalanced currents (or voltages).
• Negative and zero phase sequence currents are zero in a three-phase system that is balanced.

Operator ‘a’

It is desirable to develop an operator that should cause a 120° rotation because the symmetrical component theory uses the idea of 120° displacement in both the positive and negative sequence sets. Operator ‘a’ is used for this purpose (occasionally, ‘h’ or ” are used in place of ‘a’). It is described as follows:

The operator ‘a’ is one, which when multiplied to a vector rotates the vector through 120° in the anticlockwise direction.

Think about a vector I represented by the OA in Figure 18.5 as an example. So This vector rotates through 120 degrees anticlockwise and assumes the position OB if it is multiplied by operator “a”.

The vector that is assuming position OB is further rotated through 120 degrees in an anticlockwise direction and assumes position OC if it is multiplied by operator “a.”

Thus, the vector will be rotated 240 degrees anticlockwise by the operator ‘a2. So This is equivalent to rotating the vector by 120 degrees in a clockwise direction.

Properties or Operator ‘a’

1. Adding exps. (i) and (ii), we get,

2. Subtracting exp. (ii) from exp. (i), we get,