Dave Angell Idaho Power 21st Annual Hands-On Relay School

Symmetrical Components I

An Introduction to Power System Fault Analysis Using Symmetrical Components

Dave Angell

Idaho Power

21st Annual

Hands-On Relay School

What Type of Fault?What Type of Fault?

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VA VB VC IA IB IC


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Basic Course TopicsBasic Course Topics

TerminologyTerminology PhasorsPhasors EquationsEquations Fault Analysis ExamplesFault Analysis Examples CalculationsCalculations

Unbalanced FaultUnbalanced Fault

Ia

Ib

Ic

Ia

Ib

Ic

Ia

Ib

Ic

Symmetrical Component Symmetrical Component PhasorsPhasors

The unbalanced three phase system The unbalanced three phase system can be transformed into three can be transformed into three balanced phasors.balanced phasors.– Positive SequencePositive Sequence– Negative SequenceNegative Sequence– Zero SequenceZero Sequence

Positive Phase Sequence (ABC)Positive Phase Sequence (ABC)

-1.0

-0.5

0.0

0.5

1.0

0.000 0.017 0.033 0.050

Time

Ma

gn

itu

de

Va Vb Vc

Positive Phase SequencePositive Phase Sequence

Each have the Each have the same magnitude.same magnitude.

Each positive Each positive sequence voltage sequence voltage or current quantity or current quantity is displaced 120° is displaced 120° from one another. from one another.

Positive Phase SequencePositive Phase Sequence

The positive The positive sequence sequence quantities have a-quantities have a-b-c, counter clock-b-c, counter clock-wise, phase wise, phase rotation.rotation.

Reverse Phase Sequence (ACB)Reverse Phase Sequence (ACB)

-1.0

-0.5

0.0

0.5

1.0

0.000 0.017 0.033 0.050

Time

Ma

gn

itu

de

Va Vb Vc

Negative Phase SequenceNegative Phase Sequence

Each have the Each have the same magnitude.same magnitude.

Each negative Each negative sequence voltage sequence voltage or current quantity or current quantity is displaced 120° is displaced 120° from one another.from one another.

Negative Phase SequenceNegative Phase Sequence

The negative The negative sequence sequence quantities have a-quantities have a-c-b, counter clock-c-b, counter clock-wise, phase wise, phase rotation.rotation.

Zero Zero PhasePhase Sequence Sequence

-1.0

-0.5

0.0

0.5

1.0

0.000 0.017 0.033 0.050

Time

Ma

gn

itu

de

Va Vb Vc

Zero Phase SequenceZero Phase Sequence

Each zero Each zero sequence quantity sequence quantity has the same has the same magnitude.magnitude.

All three phasors All three phasors with no angular with no angular displacement displacement between them, all between them, all in phase. in phase.

Va0

Vb0

Vc0

Symmetrical Components Symmetrical Components Equations Equations

Each phase quantity is equal to the Each phase quantity is equal to the sum of its symmetrical phasors.sum of its symmetrical phasors.

Va = VaVa = Va00 + Va + Va11 +Va+Va22

Vb = VbVb = Vb00 + Vb + Vb11 +Vb +Vb22

Vc = VcVc = Vc00 + Vc + Vc11 +Vc +Vc22

The common form of the equations The common form of the equations are written in a-phase terms.are written in a-phase terms.

The The aa Operator Operator

Used to shift the a-phase terms to Used to shift the a-phase terms to coincide with the b and c-phasecoincide with the b and c-phase

Shorthand to indicate 120° rotation. Shorthand to indicate 120° rotation. Similar to the Similar to the jj operator of 90°. operator of 90°.

Va

Rotation of the Rotation of the aa Operator Operator

120° counter clock-wise rotation. 120° counter clock-wise rotation. A vector multiplied by 1 A vector multiplied by 1 /120°/120° results in results in

the same magnitude rotated 120°.the same magnitude rotated 120°.

Va

aVa

Rotation of the Rotation of the aa22 Operator Operator

240° counter clock-wise rotation. 240° counter clock-wise rotation. A vector multiplied by 1 A vector multiplied by 1 /240°/240° results in results in

the same magnitude rotated 240°.the same magnitude rotated 240°.

Va

a2Va

B-Phase Zero SequenceB-Phase Zero Sequence

We replace the We replace the Vb sequence Vb sequence terms by Va terms by Va sequence terms sequence terms shifted by the shifted by the aa operator.operator.

VbVb00 = Va = Va00Va0

Vb0

Vc0

B-Phase Positive SequenceB-Phase Positive Sequence

We replace the Vb We replace the Vb sequence terms by sequence terms by Va sequence terms Va sequence terms shifted by the shifted by the aa operatoroperator

VbVb11 = a = a22VaVa11Va1

Vb1

Vc1

B-Phase Negative SequenceB-Phase Negative Sequence

We replace the Vb We replace the Vb sequence terms by sequence terms by Va sequence terms Va sequence terms shifted by the shifted by the aa operatoroperator

VbVb22 = aVa = aVa22 Va2

Vc2

Vb2

C-Phase Zero SequenceC-Phase Zero Sequence

We replace the We replace the Vc sequence Vc sequence terms by Va terms by Va sequence terms sequence terms shifted by the shifted by the aa operator.operator.

VcVc00 = Va = Va00

Va0

Vb0

Vc0

C-Phase Positive SequenceC-Phase Positive Sequence

We replace the Vc We replace the Vc sequence terms by sequence terms by Va sequence terms Va sequence terms shifted by the shifted by the aa operatoroperator

VcVc11 = aVa = aVa11 Va1

Vb1

Vc1

C-Phase Negative SequenceC-Phase Negative Sequence

We replace the We replace the Vc sequence Vc sequence terms by Va terms by Va sequence terms sequence terms shifted by the shifted by the aa operatoroperator

VcVc22 = a = a22VaVa22

Va2

Vc2

Vb2

What have we produced?What have we produced?

Va = VaVa = Va00 + Va+ Va11 + Va + Va22

Vb = VaVb = Va00 + a + a22VaVa11 + aVa + aVa22

Vc = VaVc = Va00 + aVa + aVa11 + a + a22VaVa22

Symmetrical Components Symmetrical Components EquationsEquations

AnalysisAnalysis– To find out of the amount of the To find out of the amount of the

componentscomponents SynthesisSynthesis

– The combining of the component The combining of the component elements into a single, unified entityelements into a single, unified entity

Symmetrical Components Symmetrical Components Synthesis EquationsSynthesis Equations

Va = VaVa = Va00 + Va+ Va11 + Va + Va22

Vb = VaVb = Va00 + a + a22VaVa11 + aVa + aVa22

Vc = VaVc = Va00 + aVa + aVa11 + a + a22VaVa22

Symmetrical Components Symmetrical Components Analysis Equations Analysis Equations

VaVa00 = = 11//33 ( Va + Vb + Vc) ( Va + Vb + Vc)

VaVa11= = 11//33 (Va + aVb + a(Va + aVb + a22Vc)Vc)

VaVa22= = 11//33 (Va + a (Va + a22Vb + aVc)Vb + aVc)

Symmetrical Components Symmetrical Components Analysis Equations - 1/3 ??Analysis Equations - 1/3 ??

Where does the 1/3 come from?Where does the 1/3 come from?


Va = VaVa = Va0 + 0 + VaVa1 + 1 + VaVa22 When balancedWhen balanced0 0



Adding the phasesAdding the phases

V a


VaVa11= = 11//33 (Va + aVb + (Va + aVb +

aa22Vc)Vc) Adding the phases yieldsAdding the phases yields

V a aV b

V c

V a

V b



Adding the phases yields 3 Va.Adding the phases yields 3 Va. Divide by the 3 and now Va Divide by the 3 and now Va

= Va= Va11

a2V cV a aV b

V c

V a

V b

Example VectorsExample VectorsAn Unbalanced VoltageAn Unbalanced Voltage

Va

Vc

Vb

Va = 13.4 Va = 13.4 /0°/0° Vb = 59.6 Vb = 59.6 /-104°/-104° Vc = 59.6 Vc = 59.6 /104°/104°

Analysis Results in These Analysis Results in These Sequence QuantitiesSequence Quantities

Va 0Vb 0Vc 0

Va2Vc2

Vb2

Va 1

Vb 1

Vc 1

Va0 = -5.4Va0 = -5.4 Va1 = 42.9 Va2 = -24.1

Synthesize by Summing the Synthesize by Summing the Positive, Negative and …Positive, Negative and …

Va 2

Vb 2

Vc 2

Va 1

Vb 1

Vc 1

Zero SequencesZero Sequences

Va 2

Vb 2

Vc 2

Va 0

Vb 0

Vc 0

Va 1

Vb 1

Vc 1

The Synthesis Equation Results The Synthesis Equation Results in the Original Unbalanced in the Original Unbalanced

VoltageVoltage

Va 2

Vb 2

Vc 2

Va 0

Vb 0

Vc 0

Va 1

Vb 1

Vc 1

Va

Vc

Vb

Symmetrical Components Symmetrical Components Present During Shunt FaultsPresent During Shunt Faults

Three phase faultThree phase fault– PositivePositive

Phase to phase Phase to phase faultfault

– PositivePositive– NegativeNegative

Phase to Phase to ground faultground fault

– PositivePositive– NegativeNegative

– ZeroZero

Symmetrical Component Symmetrical Component Review of Faults TypesReview of Faults Types

Let’s return to the example fault Let’s return to the example fault reports and view the sequence reports and view the sequence quantities present quantities present

Three Phase Fault, Right?Three Phase Fault, Right?

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A Symmetrical Component View A Symmetrical Component View of an Three-Phase Faultof an Three-Phase Fault

Component Magnitude Angle

Ia0 7.6 175

Ia1 2790 -64

Ia2 110 75.8

Va0 0 0

Va1 18.8 0

Va2 0.7 337

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315I1

V1

A to Ground Fault, Okay?A to Ground Fault, Okay?

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A Symmetrical Component View A Symmetrical Component View of an A-Phase to Ground Faultof an A-Phase to Ground Fault


Ia0 7340 -79

Ia1 6447 -79

Ia2 6539 -79

Va0 46 204

Va1 123 0

Va2 79 178

0

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315I0I1I2

V0 V1V2

Single Line to Ground FaultSingle Line to Ground Fault

VoltageVoltage– Negative and zero sequence 180Negative and zero sequence 180 out of out of

phase with positive sequencephase with positive sequence CurrentCurrent

– All sequence are in phaseAll sequence are in phase

A to B Fault, Easy?A to B Fault, Easy?

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A Phase Symmetrical Component A Phase Symmetrical Component View of an A to B Phase FaultView of an A to B Phase Fault


Ia0 3 -102

Ia1 5993 -81

Ia2 5961 -16

Va0 1 45

Va1 99 0

Va2 95 -117

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I1

I2

V1

V2

C Phase Symmetrical Component C Phase Symmetrical Component View of an A to B Phase FaultView of an A to B Phase Fault


Ic0 3 138

Ic1 5993 279

Ic2 5961 104

Vc0 1 -75

Vc1 99 0

Vc2 95 2.5

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I1

I2

V1V2

Line to Line FaultLine to Line Fault

VoltageVoltage– Negative in phase with positive Negative in phase with positive

sequencesequence CurrentCurrent

– Negative sequence 180Negative sequence 180 out of phase out of phase with positive sequencewith positive sequence

B to C to GroundB to C to Ground

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A Symmetrical Component View A Symmetrical Component View of a B to C to Ground Faultof a B to C to Ground Fault


Ia0 748 97

Ia1 2925 -75

Ia2 1754 101

Va0 8 351

Va1 101 0

Va2 18 348

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I0

I1

I2

V0V1

V2

Line to Line to Ground FaultLine to Line to Ground Fault

VoltageVoltage– Negative and zero in phase with positive Negative and zero in phase with positive


– Negative and zero sequence 180Negative and zero sequence 180 out of out of phase with positive sequencephase with positive sequence

Again, What Type of Fault?Again, What Type of Fault?

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C Symmetrical Component View C Symmetrical Component View of a C-Phase Open Faultof a C-Phase Open Fault


Ic0 69 184

Ic1 101 4

Ic2 32 183

Vc0 0 162

Vc1 79 0

Vc2 5 90

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I0I1

I2V1

V2

One Phase Open (Series) One Phase Open (Series) FaultsFaults

VoltageVoltage– No zero sequence voltageNo zero sequence voltage– Negative 90Negative 90 out of phase with positive out of phase with positive


– Negative and zero sequence 180Negative and zero sequence 180 out of phase out of phase with positive sequencewith positive sequence

What About This One?What About This One?

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Ground Fault with Reverse LoadGround Fault with Reverse Load

Ic0 164 -22

Ic1 89 -113

Ic2 41 -6

Vc0 4 -123

Vc1 38 0

Vc2 6 -130

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I0

I1

I2

V0

V1

V2

Finally, The Last One!Finally, The Last One!

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Ic0 45 40

Ic1 153 -4

Ic2 132 180

Vc0 0.5 331

Vc1 40 0

Vc2 0.5 93

Fault on Distribution System Fault on Distribution System with Delta – Wye Transformerwith Delta – Wye Transformer

0

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I0

I1I2

V0V1

V2

Use of Sequence Quantities in Use of Sequence Quantities in RelaysRelays

Zero Sequence filtersZero Sequence filters– Current Current – VoltageVoltage

Relay operating quantityRelay operating quantity Relay polarizing quantityRelay polarizing quantity

Zero Sequence CurrentZero Sequence Current

IaIbIc

Direction of theprotected line

Ia+Ib+IcIa+Ib+Ic3I0

Zero Sequence VoltageZero Sequence Voltage(Broken Delta)(Broken Delta)

Va

Vb

Vc

3V 0

Zero Sequence VoltageZero Sequence Voltage

VaVaVc

Vb

Va

Vb

3Vo

Va

Vb

Sequence Operating QuantitiesSequence Operating Quantities

Zero and negative sequence currents Zero and negative sequence currents are not present during balanced are not present during balanced conditions.conditions.

Good indicators of unbalanced faultsGood indicators of unbalanced faults

Sequence Polarizing QuantitiesSequence Polarizing Quantities

Polarizing quantities are used to Polarizing quantities are used to determine direction.determine direction.

The quantities used must provide a The quantities used must provide a consistent phase relationship.consistent phase relationship.

Zero Sequence Voltage Zero Sequence Voltage PolarizingPolarizing

3Vo is out of phase with Va3Vo is out of phase with Va -3Vo is used to polarize for ground faults-3Vo is used to polarize for ground faults

Va

Vb

3Vo

Learning CheckLearning Check

Given three current sourcesGiven three current sources How can zero sequence be produced How can zero sequence be produced

to test a relay?to test a relay?

How can negative sequence How can negative sequence produced?produced?

How can zero sequence be How can zero sequence be produced to test a relay?produced to test a relay?

A single source provides positive, A single source provides positive, negative and zero sequencenegative and zero sequence– Note that each sequence quantity will Note that each sequence quantity will

be 1/3 of the total currentbe 1/3 of the total current Connect the three sources in parallel Connect the three sources in parallel

and set their amplitude and the and set their amplitude and the phase angle equal to one anotherphase angle equal to one another– The sequence quantities will be equal to The sequence quantities will be equal to

each source outputeach source output

How can negative sequence How can negative sequence produced?produced?

A single source provides positive, negative A single source provides positive, negative and zero sequenceand zero sequence– Each sequence quantity will be 1/3 of the total Each sequence quantity will be 1/3 of the total

currentcurrent Set the three source’s amplitude equal to Set the three source’s amplitude equal to

one another and the phase angles to one another and the phase angles to produce a reverse phase sequence (Ia produce a reverse phase sequence (Ia at at /0/0oo, Ib at , Ib at /120/120oo and Ic at and Ic at /-120/-120oo))– Only negative sequence will be producedOnly negative sequence will be produced

Advanced Course TopicsAdvanced Course Topics

Sequence NetworksSequence Networks Connection of Networks for FaultsConnection of Networks for Faults Per Unit SystemPer Unit System Power System Element ModelsPower System Element Models

ReferencesReferences

Symmetrical Components for Power Symmetrical Components for Power Systems Engineering, J Lewis Systems Engineering, J Lewis BlackburnBlackburn

Protective Relaying, J Lewis BlackburnProtective Relaying, J Lewis Blackburn Power System Analysis, StevensonPower System Analysis, Stevenson Analysis of Faulted Power System, Paul Analysis of Faulted Power System, Paul

AndersonAnderson

ConclusionConclusion

Symmetrical components provide:Symmetrical components provide:– balanced analysis of an unbalanced balanced analysis of an unbalanced

system.system.– a measure of system unbalancea measure of system unbalance– methods to detect faultsmethods to detect faults– an ability to distinguish fault directionan ability to distinguish fault direction

Documents

Dave Angell Idaho Power 21st Annual Hands-On Relay School