Symmetrical Components & Fault Calculation | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) PDF Download

Symmetrical Components and Fault Calculation

INTRODUCTION

• When the insulation of the system fails at one or more points or an conducting object comes in contact with a live point, a short circuit or fault occurs. 
• While one involving only one or two phases is known as asymmetrical fault. 
• Fault calculations involve finding the voltage and current distribution throughout the system during the fault. 
• When fault occurs at a point in a power system, the corresponding MVA is referred to as the fault level at that point.

SYMMETRICAL FAULT

A fault involving all the three phases is known symmetrical fault.

Remember : 

• The circuit breaker rated MVA breaking capacity is based on 3-phase fault MVA.

Assumptions: 

• The emfs of all generators are 1Ð0 per unit.

This means that the system voltage is at its nominal value and the system is operating on no load at the time of fault. The effect of this is that all generators can be replaced by a single generator since all emfs are equal and in phase. 

• Shunt elements in the transformer model that account for magnetizing current and core loss are neglected. 
• Shunt capacitances of the transmission line are neglected. 
• System res istance is neglected and only inductive reactance of the system is taken into account.

Steps in Calculation 

• Draw a single line diagram of the system. 
• Select a common base and find out the per unit reactances of all generators, transformers, lines etc., as referred to common base. 
• From the single line diagram draw a single line reactance diagram showing one phase and neutral.
  Indicate all the reactances, etc., on the reactance diagram. 
• Reduce the reactance diagram (by series parallel and star delta transformation) keeping the identify of the fault point intact. Find the reactance of the system as seen from the fault point. 
• Find the fault current and fault MVA in per unit Convert these per unit values to actual values.

Consideration of Pre-Fault Load Current 

• Generally the fault currents are much larger than load currents and, therefore, the load currents can be neglected during fault calculations. 
• In some situations it may be necessary to consider the effect of load current. This can be done through the use of superposition theorem.

Steps 

• Calculate the pre-fault voltage at the fault point.
• The fault current can then be determined using this value of the pre-fault voltage.
• The total current is the phasor sum of the load current and fault current.

Remember: 

• The directions of the load current and fault current are the same for the generator (both the currents flow out of generator terminals.) 
• The directions of the two currents are opposite to each other for the motor because of the load current flows into the motor terminals while the fault current flows out of the motor terminals.

SYMMETRICAL COMPONENTS

•  A balanced 3-f system is one in which all the three phasor are equal in magnitude and are equalspaced in the space.

• To analyze an asymmetrical fault, an unbalanced 3-phase circuit has to be solved. The direct solution of such a circuit is very difficult. 
• The solution can be more easily obtained by suing symmetrical components since this yields three (frictious) single phase networks.

General Principles 

Any set of unbalanced 3-phase voltages (or current) can be transformed into 3 balanced into 3 balanced sets these are : 

• A positive sequence set of three symmetrical voltages (i.e., all numerically equal and all displaced from each other by 120°) having the same phase sequence abc as the original set and denoted by V_a1, V_b1 and V_c1 as shown in figure. 
• A negative sequence set of three symmetrical voltages having the phase sequence opposite to that of the original set and denoted by V_a2, V_b2 and V_c2 as shown in figure. 
• A zero sequence set of three voltages, all equal in magnitude and in phase with each other and denoted by V_a0, V_b0 and V_c0.
• The positive, negative and zero sequence sets are known as symmetrical components. 
• The three-phase voltage obtained by adding the symmetrical components.

Operator ‘a’ 

• The operator ‘a’ is defined as 1 Ð120°, an operator which when multiplied with a phasor rotates it through 120° in the positive (naticlockwise) direction without changing the magnitude.

Properties of Operator ‘a’ 1 + a + a² = 0

Remember: 
1 + a² = 1Ж60° = 0.5 – j0.866
1 + a = 1Ð60° 0.5+ j0.866

Relation Between Phase Voltages and Symmetrical Component Voltages

Where [A] is known as operator matrix 

• In matrix form 
  
  

Where

Note: 

• All the above equation are valid for current also
  

Effect on 3-f Power 

• The total complex power in a 3-phase circuit

• In matrix form

Observations:  

• Sum of powers of the three symmetrical components equals the three-phase power. 
• Symmetrical component transformation is power invariant.

Effect on Impedance Matrix 

• The relation between phase voltages and currents

Conclusions:

• v_a1 = v_a2 = v_a0 
• I_a1 + I_a2 + I_a0 = 0
• This open conductor condtion leads to equal series voltage drops in all the three sequence networks in the direction of current flow from P to Q. 
• One open conductor condition leads to equal series voltage rises, in the three sequence networks, from Q to P tending to send currents from Q to P. These sequence current flow, due to these voltage rises, are opposed by series impedances of the equipments.

Two Conductors Open 

• Figure shows the 3-phase line with break in conductors b and c.

• The terminal conditions

v_a1 = v_a2 - V_a0'=0

Conclusions: