Fault Analysis: Introduction To

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Introduction to


Louelle Lucio A. Sanchez, PEE Metering Services Senior Engineer, NGCP 2019 Board of Director, IIEE Cebu Chapter


• Per Unit Calculations • Symmetrical Components • Sequence Impedances of Power System Elements and Sequence Networks

Per Unit Calculations



Per Unit Quantities • Analyses of power systems employing actual values such as electrical units of voltampere(VA), voltage, current, and impedance do not adapt themselves easily to computations. • Impedances, currents, voltages, and power are preferably expressed in per unit (pu) values rather than their actual units.

Per Unit Quantities The per-unit value of any quantity is defined as the ratio of the actual value to its base value expressed as a decimal.

Advantages of Per Unit Values • Per unit values of equipment (generators, transformers, etc.) that have widely varying ratings normally fall within a narrow range while their actual ohmic values differ from equipment to equipment having different ratings. Therefore, pu values can also be selected, when ohmic values of impedances are not available, from tables which provide average values of various categories of equipment.



Advantages of Per Unit Values •Per unit values of quantities, when the base values are appropriately selected, are independent of the side of the transformer to which they are connected. On the other hand, ohmic values have to be referred to one side of the transformer. •Per unit values are independent of the type of the power system, that is, whether the power system is single phase or three phase.

Advantages of Per Unit Values •Although the voltage bases on the two sides of a transformer in a three-phase circuit must have a definite relationship, the pu values of impedances are independent of the way in which transformers are connected •The equipment, supplied by manufacturers, provides the equipment parameters in pu on the name plate rating. •Analyses of power systems are simplified considerably by using pu values.

Formulas • Single Phase Systems

• Three Phase Systems



Conversion of pu Quantities from One Base to Another Base

Example 1. A system operates at 220 KVA and 11 KV. Using these quantities as base values, find the base impedance and base current for the system.

Example 2. A three phase wye connected 6.25 kVA, 220 V synchronous generator has a reactance of 8.4 ohms per phase. Using the rated kVA and voltage as base values, determine the per-unit reactance. Then refer this per unit value to a 230 V, 7.5 kVA base.





Symmetrical Components



Majority of


• Unsymmetrical nature • Unsymmetrical currents • Magnitude have unequal phase displacement

Unsymmetrical Faults Faults on the power system, which give rise to unsymmetrical fault

currents - unequal fault currents with unequal phase displacement

Solution 1. Kirchhoff’s laws 2. Symmetrical components method •

a simple method and gives more generality to be given to fault performance studies. provides a useful tool for the protection engineers, particularly in connection with tracing out of fault currents.



Symmetrical Components Method

Symmetrical Components Method A system of n unbalanced set of phasors can be resolved into sets of n - 1 balanced phasor systems of different phase sequences plus one zero-phase sequence system

Symmetrical Components Method

Three unbalanced phasors of a three-phase system can be resolved into three balanced systems of phasors as follows: 1. Positive-sequence components 2. Negative-sequence components 3. Zero-sequence components



Positive-sequence components consist of three phasors equal in magnitude, displaced in phase from each other by 120°, and having the same phase sequence as the original phasors

Negative-sequence components consist of three phasors equal in magnitude, displaced in phase from each other by 120°, and having phase sequence opposite to the original phasors

Zero-sequence components consist of three phasors equal in magnitude, and with zero phase displacement from each other



Symmetrical Components Va = Va1 + Va2 + Va0 Vb = Vb1 + Vb2 + Vb0 Vc = Vc1 + Vc2 + Vc0

Operator “a” • commonly used to designate the operator that causes a rotation of 120° in the counterclockwise direction • a complex number of unit magnitude with an angle of 120°

a = 1∟120° = -0.5+j0.866 a2 = 1∟240° = -0.5-j0.866 a3 = 1∟360° = 1∟0° = 1 a4 = a3 . a = a

Symmetrical Components of Unsymmetrical Phasors Va = Va1 + Va2 + Va0 -> Eq. 1 Vb = Vb1 + Vb2 + Vb0 -> Eq. 2 Vc = Vc1 + Vc2 + Vc0 -> Eq. 3 Vb1 = a2Va1 Vc1 = aVa1 Vb2 = aVa2 Vc2 = a2Va2 Vb0 = Vao Vc0 = Vao

-> Eq. 4



Symmetrical Components of Unsymmetrical Phasors Repeating Eq. 1 and Substituting Eq. 4 to Eq. 2 and 3 Va = Va1 + Va2 + Va0

Vb = a2Va1 + aVa2 + Va0 Vc = aVa1 + a2Va2 + Va0

Resolving three unsymmetrical phasors into their symmetrical components, yields: Va0 = 1/3(Va + Vb + Vc) Va1 = 1/3(Va + aVb + a2Vc) Va2 = 1/3(Va + a2Vb + aVc)

Symmetrical Components of Unsymmetrical Phasors The same applies to currents Ia = Ia1 + Ia2 + Ia0 Ib = a2Ia1 + aIa2 + Ia0 Ic = aIa1 + a2Ia2 + Ia0

Ia0 = 1/3 (Ia + Ib + Ic) Ia1 = 1/3 (Ia + aIb + a2Ic) Ia2 = 1/3 (Ia + a2Ib + aIc)

Current In in the return through the neutral In = Ia + Ib + Ic

In = 3Ia0

In the absence of path through the neutral or a deltaconnected load of a three-phase system, In is zero and the line currents contain no zero-sequence components

Sequence Impedances of Power System Elements and Sequence Networks



Sequence Impedances • Each element of power system will offer impedance to different phase sequence components of current which may not be the same. • The impedance which any piece of equipment offers to positive sequence current will not necessarily be the same as offered to negative sequence current or zero sequence current.

Sequence Impedances • In unsymmetrical fault calculations, each piece of equipment will have three values of impedance 1. Positive sequence impedance (Z1) 2. Negative sequence impedance (Z2) 3. Zero sequence impedance (Z0)

Sequence Impedances of Power System Elements •The concept of impedances of various elements of power system to positive, negative and zero sequence currents is of considerable importance in determining the fault currents in a 3-phase unbalanced system. •The following three main power system elements are considered: 1. Synchronous machines 2. Transformers 3. Transmission lines



Synchronous Machines Positive Sequence Impedances The positive sequence impedance of synchronous machines varies depending on the time period after the machine experiences fault, and these are: • Subtransient reactance Xd” (at an instant of fault) • Transient reactance Xd’ (1/2 to 4 cycles after) • Synchronous reactance Xd (more than 4 cycles)

Synchronous Machines Negative Sequence Impedances Generally somewhat less than the positive sequence impedance and varies with the type of winding, type of machine, # of poles, etc.

Synchronous Machines Zero Sequence Impedances • Usually less than the positive and negative sequence impedance • Any impedance Zg in the earth connection of a star-connected system has the effect to introduce an impedance of 3*Zg per phase. It is because the three equal zero-sequence currents, being in phase, do not sum to zero at the star point, but they flow back along the neutral earth connection.



Transformers Positive and Negative Sequence Impedances Since transformers have the same impedance with reversed phase rotation, their positive and negative sequence impedances are equal; this value being equal to the impedance of the transformer.

Transformers Zero Sequence Impedances The zero sequence networks of three-phase transformers is dependent on the type of transformer connections and whether a path is available to the flow of zero sequence currents.

Transformers Equivalent Sequence Circuits for Three Phase Two Winding Transformers



Transformers Equivalent Sequence Circuits for Three Phase Two Winding Transformers

Transmission Lines Positive and Negative Sequence Impedances The same model as the positive-sequence network is used for transmission lines in as much as the positive sequence and negative sequence impedances of transmission lines are the same

Transmission Lines Zero Sequence Impedances • The zero-sequence network model for a transmission line is the same as that of the positive- and negativesequence networks. • The sequence impedance of the model is the zerosequence impedance of the line. This is normally higher than the positive- and negative-sequence impedances because of the influence of the earth’s resistivity and the ground wire/s.



Sequence Networks • the path for the flow of that sequence current in the system • composed of impedances offered to that sequence current in the system.

Sequence Networks Positive-sequence network:

Va1 = Ea - I a1Z1

Sequence Networks Negative-sequence network:

Va2 = - I a2Z2



Sequence Networks Zero-sequence network:

Va0 = - I a0Z0

Positive Sequence Network Example

Negative Sequence Network Example



Zero Sequence Network Example

Are you ready now to calculate



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