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Lesson 8 Symmetrical Components. Symmetrical Components. Due to C. L. Fortescue (1918): a set of n unbalanced phasors in an n-phase system can be resolved into n balanced phasors by a linear transformation The n sets are called symmetrical components

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Lesson 8 symmetrical components

Lesson 8Symmetrical Components

Notes on Power System Analysis


Symmetrical components
Symmetrical Components

  • Due to C. L. Fortescue (1918): a set of n unbalanced phasors in an n-phase system can be resolved into n balanced phasors by a linear transformation

    • The n sets are called symmetrical components

    • One of the n sets is a single-phase set and the others are n-phase balanced sets

    • Here n = 3 which gives the following case:

Notes on Power System Analysis


Symmetrical component definition
Symmetrical component definition

  • Three-phase voltages Va, Vb, and Vc (not necessarily balanced, with phase sequence a-b-c) can be resolved into three sets of sequence components:

    Zero sequence Va0=Vb0=Vc0

    Positive sequence Va1, Vb1, Vc1 balancedwith phase sequence a-b-c

    Negative sequence Va2, Vb2, Vc2 balanced with phase sequence c-b-a

Notes on Power System Analysis



where

a = 1/120°= (-1 + j 3)/2

a2 = 1/240°= 1/-120°

a3 = 1/360°= 1/0 °

Notes on Power System Analysis


Vp = A VsVs= A-1Vp

Notes on Power System Analysis


Vp = A VsVs= A-1Vp

Ip = A Is Is= A-1Ip

Notes on Power System Analysis


Va = V0 + V1 + V2

Vb = V0 + a2V1 + aV2

Vc = V0 + aV1 + a2V2

V0 = (Va + Vb + Vc)/3

V1 = (Va + aVb + a2Vc)/3

V2 = (Va + a2Vb + aVc)/3

These are the phase a symmetrical (or sequence) components. The other phases follow since the sequences are balanced.

Notes on Power System Analysis


Sequence networks

a

Zy

b

Zy

c

Zn

g

Sequence networks

  • A balanced Y-connected load has

    three impedances Zy connected line to neutral

    and one impedance Zn connected neutral to ground

Notes on Power System Analysis


Sequence networks1
Sequence networks

or in more compact notation Vp= ZpIp

Notes on Power System Analysis


a

Zy

b

Zy

c

Zn

g

Zy

Vp = ZpIp

Vp = AVs = ZpIp= ZpAIs

AVs = ZpAIs

Vs= (A-1ZpA) Is

Vs = Zs Is where

Zs = A-1ZpA

n

Notes on Power System Analysis


V0 = (Zy + 3Zn) I0 = Z0 I0

V1 = ZyI1 = Z1 I1

V2 = ZyI2 = Z2 I2

Notes on Power System Analysis


I2

I0

I1

Zy

Zy

Zy

n

a

a

a

V2

V0

V1

3 Zn

n

g

n

Zero-

sequence

network

Positive-

sequence

network

Negative-

sequence

network

Sequence networks for Y-connected load impedances

Notes on Power System Analysis


I2

I0

I1

ZD/3

ZD/3

ZD/3

n

a

a

a

V2

V0

V1

n

g

n

Positive-

sequence

network

Negative-

sequence

network

Zero-

sequence

network

Sequence networks for D-connected load impedances.

Note that these are equivalent Y circuits.

Notes on Power System Analysis


Remarks
Remarks

  • Positive-sequence impedance is equal to negative-sequence impedance for symmetrical impedance loads and lines

  • Rotating machines can have different positive and negative sequence impedances

  • Zero-sequence impedance is usually different than the other two sequence impedances

  • Zero-sequence current can circulate in a delta but the line current (at the terminals of the delta) is zero in that sequence

Notes on Power System Analysis


Notes on Power System Analysis


Z0 = (Zaa+Zbb+Zcc+2Zab+2Zbc+2Zca)/3

Z1 = Z2 = (Zaa+Zbb+Zcc–Zab–Zbc–Zca)/3

Z01 = Z20 = (Zaa+a2Zbb+aZcc–aZab–Zbc–a2Zca)/3

Z02 = Z10 = (Zaa+aZbb+a2Zcc–a2Zab–Zbc–aZca)/3

Z12 = (Zaa+a2Zbb+aZcc+2aZab+2Zbc+2a2Zca)/3

Z21 = (Zaa+aZbb+a2Zcc+2a2Zab+2Zbc+2aZca)/3

Notes on Power System Analysis


Notes on Power System Analysis


Z0 = Zaa+ 2Zab

Z1 = Z2 = Zaa– Zab

Z01=Z20=Z02=Z10=Z12=Z21= 0

Vp = ZpIpVs = ZsIs

  • This applies to impedance loads and to series impedances (the voltage is the drop across the series impedances)

Notes on Power System Analysis


Power in sequence networks
Power in sequence networks

Sp = VagIa* + VbgIb* + VcgIc*

Sp = [VagVbgVcg] [Ia*Ib*Ic*]T

Sp = VpTIp*

= (AVs)T(AIs)*

= VsTATA* Is*

Notes on Power System Analysis


Power in sequence networks1
Power in sequence networks

Sp = VpTIp* = VsT ATA* Is*

Sp = 3 VsT Is*

Notes on Power System Analysis


Sp= 3 (V0 I0* + V1 I1* +V2 I2*) = 3 Ss

In words, the sum of the power calculated in the three sequence networks must be multiplied by 3 to obtain the total power.

This is an artifact of the constants in the transformation. Some authors divide A by 3 to produce a power-invariant transformation. Most of the industry uses the form that we do.

Notes on Power System Analysis


Sequence networks for power apparatus
Sequence networks for power apparatus

  • Slides that follow show sequence networks for generators, loads, and transformers

  • Pay attention to zero-sequence networks, as all three phase currents are equal in magnitude and phase angle

Notes on Power System Analysis


N

E

V1

Positive

I1

Z1

N

V2

I2

Negative

Zn

Z2

G

Y generator

V0

3Zn

Zero

I0

Z0

N

Notes on Power System Analysis


N

V1

Positive

I1

Z

N

V2

I2

Negative

Ungrounded Y load

Z

G

V0

Zero

I0

Z

N

Notes on Power System Analysis


Zero-sequence networks for loads

G

Y-connected load grounded through Zn

V0

3Zn

I0

Z

N

G

D-connected load ungrounded

V0

Z

Notes on Power System Analysis


Y y transformer
Y-Y transformer

H1

X1

A

a

Zeq+3(ZN+Zn)

A

a

B

I0

Va0

VA0

b

g

c

C

Zero-sequence

network (per unit)

N

n

ZN

Zn

Notes on Power System Analysis


Y y transformer1
Y-Y transformer

Zeq

H1

X1

A

a

A

a

I1

Va1

VA1

B

b

n

c

C

Positive-sequence

network (per unit)

Negative sequence

is same network

N

n

ZN

Zn

Notes on Power System Analysis


D y transformer
D-Y transformer

H1

X1

A

a

Zeq+3Zn

A

a

B

VA0

I0

Va0

b

g

c

C

Zero-sequence

network (per unit)

n

Zn

Notes on Power System Analysis


D y transformer1
D-Y transformer

H1

X1

A

a

Zeq

A

a

B

I1

Va1

VA1

b

n

c

C

Positive-sequence

network (per unit)

Delta side leads wye

side by 30 degrees

n

Zn

Notes on Power System Analysis


D y transformer2
D-Y transformer

H1

X1

A

a

Zeq

A

a

B

I2

Va2

VA2

b

n

c

C

Negative-sequence

network (per unit)

Delta side lags wye

side by 30 degrees

n

Zn

Notes on Power System Analysis


Three-winding (three-phase)

transformers Y-Y-D

H and X in grounded Y and T in delta

Zero sequence

Positive and negative

Neutral

Ground

ZT

X

ZX

ZH

ZH

H

H

X

ZX

ZT

T

T

Notes on Power System Analysis


Three-winding transformer data:

Windings Z Base MVA

H-X 5.39% 150

H-T 6.44% 56.6

X-T 4.00% 56.6

Convert all Z's to the system base of 100 MVA:

Zhx = 5.39% (100/150) = 3.59%

ZhT= 6.44% (100/56.6) = 11.38%

ZxT= 4.00% (100/56.6) = 7.07%

Notes on Power System Analysis


Calculate the equivalent circuit parameters:

Solving:

ZHX= ZH+ ZX

ZHT= ZH+ ZT

ZXT= ZX+ZT

Gives:

ZH= (ZHX+ ZHT- ZXT)/2 = 3.95%

ZX= (ZHX+ ZXT- ZHT)/2 = -0.359%

ZT= (ZHT+ ZXT- ZHX)/2 = 7.43%

Notes on Power System Analysis


Typical relative sizes of sequence impedance values
Typical relative sizes of sequence impedance values

  • Balanced three-phase lines:

    Z0 > Z1 = Z2

  • Balanced three-phase transformers (usually):

    Z1 = Z2 = Z0

  • Rotating machines: Z1 Z2 >Z0

Notes on Power System Analysis


Unbalanced short circuits
Unbalanced Short Circuits

  • Procedure:

    • Set up all three sequence networks

    • Interconnect networks at point of the fault to simulate a short circuit

    • Calculate the sequence I and V

    • Transform to ABC currents and voltages

Notes on Power System Analysis


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