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ECE 5984: Power Distribution System Analysis

Lecture 9: Three-Phase Transformer Models

Reference: Textbook, Chapter 8

Instructor: V. Kekatos

Distribution system transformers

• Found at the substation and in-line

substation transformer pole-mounted transformer^ pad-mounted single-phase

transformer

Conventions

4

• ANSI/IEEE Std. C57.12.00 for Delta-Wye transformer connections

Voltages and currents on the high-voltage side lead by 30 degrees

• Distribution transformers of <200 kVA and HV<8.66 kV have additive polarity

subtractive additive

[source: electrical-engineering-portal.com]

Three-phase connections

1. Grounded Wye – grounded Wye

2. Delta – grounded Wye (step-up)

3. Delta – grounded Wye (step-down)

4. Ungrounded Wye – Delta (step-down)

5. Ungrounded Wye – Delta (step-up)

6. Grounded Wye – Delta (step-down)

7. Delta-Delta

8. Open Wye – open Delta

9. Open Delta – open Delta

• Transformer connections
• Multi-phase transformers usually implemented by connecting 1ϕ transformers

Current conversions (review)

• Phase to line conversion
• Line currents phase currents
• If line currents exit triangle (delta source), vector i gets negative sign or
• Given line currents, recover equivalent delta currents

I

A

I

C

I

B

i :=

I

a

I

b

I

c

i :=

I

ab

I

bc

I

ca

• Singularity (shift-invariance) can be waived by fixing the sum of delta currents

i = W

i

• Textbook follows a different derivation and finds matrix L with a zero column

i =

I

ab

I

ca

I

bc

I

ab

I

ca

I

bc

i = D

f

i

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

i :=

I

ba

I

cb

I

ac

• Backward model
  • Forward model

diagonal matrix with vector

z on its main diagonal

• Turns ratios are typically identical across all phases
  • Impedances may not be equal; transformers of different power ratings

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

V

AG

V

ta

= n t and

I

A

I

a

n t

AAACGHicbZBNS8NAEIY39avWr6pHL8EieKqJKHoRCl48SQX7AW0Jm+2kXbrZhN2JWEJ/hhf/ihcPinjtzX/jNi2irS8svDwzw+y8fiy4Rsf5snJLyyura/n1wsbm1vZOcXevrqNEMaixSESq6VMNgkuoIUcBzVgBDX0BDX9wPak3HkBpHsl7HMbQCWlP8oAzigZ5xRPp4VU7UJSlt14b4RHTWPGQquFo9EM0sEh2M+YVS07ZyWQvGndmSmSmqlcct7sRS0KQyATVuuU6MXZSqpAzAaNCO9EQUzagPWgZK2kIupNmh43sI0O6dhAp8yTaGf09kdJQ62Hom86QYl/P1ybwv1orweCyk3IZJwiSTRcFibAxsicp2V2ugKEYGkOZ4uavNutTkxKaLAsmBHf+5EVTPy2752Xn7qxUcWZx5MkBOSTHxCUXpEJuSJXUCCNP5IW8kXfr2Xq1PqzPaWvOms3skz+yxt/6LKIx

n t

N

primary

N

secondary

• Transformations

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

v

m

= v

t

dg(z)i

m

n

t

v

n

dg(z)i

m

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

i n

n t

i m

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

1) Y

G

Y

G

connection

• –

V ab V bc

V ag

I c I b I a

I C

I B

I A

I n

X 1 a

n t

X 2 b

X 3 c

H 1 A

H 2 B

H 3 C

g

Zt a

Zt b

Zt c

V CA

I AB

• – + –

V BC

I AB

I BC

I c

I a I b

I CA

I CA

Vt b

Vt c

Vt a

V ab

I A

V AB

I BC

V AB

V BC

Vt a

• – + – + Vt – c Vt b

• Backward model
• Forward model
• Sanity check (for ideal transformer)
• Voltage and current transformations

phasor diagrams

assume balanced

conditions

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

n t =

V LL,primary

V LN,secondary

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

i n

= D

f

i n

n t

D

f

i m

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

v m = v t dg(z)i m

n t

D

f v n dg(z)i m

• How can we determine it is a step-up without knowing the turns ratio?

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

˜ i

H

n

v˜ n

=

✓

1

n t

i

H

m

◆

(n t

v t

) = i

H

m

v m

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

2) Step-up Y

G

connection

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

v ˜ n = n t v t and

i n

I

AB

I

BC

I

CA

n t

i m

11

V ab

V bc

V ag

I c

I b I a

I n

X 1 a

n t

X 2 b X 3 c

H 1 A

H 2 B

H 3 C

g

Zt a Zt b Zt c

V CA

V CA

• – + – + –

V BC

V BC

Vt b

Vt c

Vt a

V ab

V AN

V AB

V AB

Vt a

• – + – + Vt – c Vt b

• Voltage transform where

permutation

matrix

• Key properties

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

A

v

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

p1) A

1

v

= A

v

p2) A

v

D

f

= D

f

why step-down?

• Forward model

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

v m

= v t

dg(z)i m

n t

A

v

v˜ n dg(z)i m

n t

A

v

D

f

v n

dg(z)i m

n t

D

f

v n

dg(z)i m

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

3) Step-down Y

G

connection

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

v ˜ n

V

AB

V

BC

V

CA

= n t

A

v

v t

[Gonen]

• Typical distribution substation connection
• Easier balancing of (large) 1ϕ loads across all three transformers
• Cannot operate with two transformers (no open delta – open Y)

Discussion 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 Y connection

14

• Voltage and current transformations

V ab V bc

V ca

• –

I b I c I a

I B

I C

I A

X1-a

X2-b X3-c

n t

H1-A H2-B H3-C

Zt ab Zt bc Zt ca

V CN 1

V AN

V AB V BC

• – +
• • – +

V BN 1

V CN

V ca 1

V ab 1

V an 1

V bc 1

V AN 1

V AB 1

V BN

Vt ab

I ba I cb I ac

• – (^) + + Vt – ca

Vt bc

ungrounded Y

• Key point: phase currents in delta are zero-sum
• Backward model
• Forward model

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

4) Step-down Y connection

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

i n

n t

Li m

matrix L can be either W

T or

the matrix in (8.63) of book

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

i m

= D

f

i m

i m

= W

i m

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

v m = W

⇣

v ˜ t dg(z)

˜ i m

⌘

=

1

n t

Wv n Wdg(z)Li m

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

v n = n t v˜ t and i n =

1

n t

˜ i m =

1

n t

2

4

I ba

I cb

I ac

3

5

[Gonen]

Discussion on Y connection

AAACPXicdVA7SwNBEN7zbXxFLW0Wo2Chcic+O0ELSwXjg9wR9jaTuGRv79idE8Nxf8zG/2BnZ2OhiK2te0mEKDrswsc3M9/MfGEihUHXfXKGhkdGx8YnJktT0zOzc+X5hQsTp5pDlccy1lchMyCFgioKlHCVaGBRKOEybB8V+ctb0EbE6hw7CQQRaynRFJyhperl88zvitR0Kwxo5m663VgfADuud7Dr5T7CHWbHwvDUFHLUvpXrDf8YJLIVymOlgBeieV4vV7776QDoCVGvz1RIP07r5Ue/EfM0AoVcMmNqnptgkDGNgkvIS35qIGG8zVpQs1CxCEyQdRfP6aplGrQZa/sV0i472JGxyJhOFNrKiOGN+Z0ryL9ytRSb+0EmVJIiKN4b1EwlxZgWVtKG0PZi2bGAcS3srpTfMM04WsNL1oTvS+n/4GJr07P4bLtyuN+3Y4IskWWyRjyyRw7JCTklVcLJPXkmr+TNeXBenHfno1c65PR7FsmPcD6/AClcqyM=AAACPXicdVA7SwNBEN7zbXxFLW0Wo2Chcic+O0ELSwXjg9wR9jaTuGRv79idE8Nxf8zG/2BnZ2OhiK2te0mEKDrswsc3M9/MfGEihUHXfXKGhkdGx8YnJktT0zOzc+X5hQsTp5pDlccy1lchMyCFgioKlHCVaGBRKOEybB8V+ctb0EbE6hw7CQQRaynRFJyhperl88zvitR0Kwxo5m663VgfADuud7Dr5T7CHWbHwvDUFHLUvpXrDf8YJLIVymOlgBeieV4vV7776QDoCVGvz1RIP07r5Ue/EfM0AoVcMmNqnptgkDGNgkvIS35qIGG8zVpQs1CxCEyQdRfP6aplGrQZa/sV0i472JGxyJhOFNrKiOGN+Z0ryL9ytRSb+0EmVJIiKN4b1EwlxZgWVtKG0PZi2bGAcS3srpTfMM04WsNL1oTvS+n/4GJr07P4bLtyuN+3Y4IskWWyRjyyRw7JCTklVcLJPXkmr+TNeXBenHfno1c65PR7FsmPcD6/AClcqyM=AAACPXicdVA7SwNBEN7zbXxFLW0Wo2Chcic+O0ELSwXjg9wR9jaTuGRv79idE8Nxf8zG/2BnZ2OhiK2te0mEKDrswsc3M9/MfGEihUHXfXKGhkdGx8YnJktT0zOzc+X5hQsTp5pDlccy1lchMyCFgioKlHCVaGBRKOEybB8V+ctb0EbE6hw7CQQRaynRFJyhperl88zvitR0Kwxo5m663VgfADuud7Dr5T7CHWbHwvDUFHLUvpXrDf8YJLIVymOlgBeieV4vV7776QDoCVGvz1RIP07r5Ue/EfM0AoVcMmNqnptgkDGNgkvIS35qIGG8zVpQs1CxCEyQdRfP6aplGrQZa/sV0i472JGxyJhOFNrKiOGN+Z0ryL9ytRSb+0EmVJIiKN4b1EwlxZgWVtKG0PZi2bGAcS3srpTfMM04WsNL1oTvS+n/4GJr07P4bLtyuN+3Y4IskWWyRjyyRw7JCTklVcLJPXkmr+TNeXBenHfno1c65PR7FsmPcD6/AClcqyM=AAACPXicdVA7SwNBEN7zbXxFLW0Wo2Chcic+O0ELSwXjg9wR9jaTuGRv79idE8Nxf8zG/2BnZ2OhiK2te0mEKDrswsc3M9/MfGEihUHXfXKGhkdGx8YnJktT0zOzc+X5hQsTp5pDlccy1lchMyCFgioKlHCVaGBRKOEybB8V+ctb0EbE6hw7CQQRaynRFJyhperl88zvitR0Kwxo5m663VgfADuud7Dr5T7CHWbHwvDUFHLUvpXrDf8YJLIVymOlgBeieV4vV7776QDoCVGvz1RIP07r5Ue/EfM0AoVcMmNqnptgkDGNgkvIS35qIGG8zVpQs1CxCEyQdRfP6aplGrQZa/sV0i472JGxyJhOFNrKiOGN+Z0ryL9ytRSb+0EmVJIiKN4b1EwlxZgWVtKG0PZi2bGAcS3srpTfMM04WsNL1oTvS+n/4GJr07P4bLtyuN+3Y4IskWWyRjyyRw7JCTklVcLJPXkmr+TNeXBenHfno1c65PR7FsmPcD6/AClcqyM=

mid-tap of one secondary is

grounded and connected to

secondary neutral conductor

• –

V AG

V BG

V CG

I B

I C

I A

V AN

V CN

V BN

I A

• I B

• I C

I N

Zt ab

I ba

I ac

I cb

Vt ab

Vt bc

Vt ca

Zt ca

Zt bc

I a

I c

I b

V ca

V ab

V bc

Z g

• Challenging part is to relate delta currents to secondary line currents
• Exploit the fact that LL voltages across the delta loop are zero-sum

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

6) Step-down Y

G

connection

Step-down Y

G

connection

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

• Get one more equation from zero-sum LL voltages across delta
• Goal : express delta currents in terms of line currents
• Solve for delta currents if
• Put the three equations together

delta currents depend on line currents and primary voltages!

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

i m

= D

f

i m

• There are two independent linear equations from

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

v˜ m

(5)

v¯ n

n t z +

3 z g

n t

i m

2 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

4

I a

I b

1

v¯ n

3

5 = K

1 ˜ i m where K

1

2

4

1 0 1

1 1 0

n t

z ab

3 zg

n t

n t

z bc

3 zg

n t

n t

z ca

3 zg

n t

3

5

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

K = [k 1

k 2

k 3

]

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

i m = [k 1 k 2 0 ] i m

• k 3

v¯ n

Step-down Y

G

connection

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

• Substitute (7) into (1) to get the backward update with
• Substitute (7) into (6) to get the forward update with
• This is the only connection with a C matrix!

AAADa3icrVJdb9MwFHUaPkb52scb8BBRDXViq5IBghekSggEb0Oi66QmRI7jpFYdJ7JvKrrI/B1+Da8gfgT/AadJUbeON64S5fice5zr6xsVnClw3V9Wx752/cbNrVvd23fu3ru/vbN7qvJSEjoiOc/lWYQV5UzQETDg9KyQFGcRp+No9qbWx3MqFcvFJ1gUNMhwKljCCAZDhTvW0M8wTKOkequfvF7hsfY5TaDvJxKTytOVCEGvxA/aOWrlmpFZFae6v1LP9cHTxnYepheN3hr67ENe+JKlUzhYsTMdPvtHht+tn1Z7Zyo92ij1P9VSbzZZq8j7urY4/rtwdWMIuuF2zx24y3A2gdeCHmrjxPR8149zUmZUAOFYqYnnFhBUWAIjnOquXypaYDLDKZ0YKHBGVVAt71o7+4aJnSSX5hXgLNkrHYfxnBWqgWqeNgCwGYyg+tJstu6rcKbUIosOUyrqr/lTfVR1Oacmr9ImJSSvgoqJogQqSFNoUnIHcqcePCdmkhLgCwMwkcyc1SFTbC4HzHjWXfQu92wTnB4PvBcD9+Pz3tBt+7mFHqLHqI889BIN0Xt0gkaIWN+s79YP62fnt71nP7AfNakdq/XsoQth7/8BbBAiQA==

E = W

n t

I

dg(z) +

z g

n t

k 3

F = W

dg(z) +

z g

n t

[k 1 k 2

0 ]

• Matrix C usually multiplies secondary voltages v m . However, this equation can still be

used during backward sweep as primary voltages are known and fixed during this update.

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

i n

= C

0 v¯ n

• Fi m

where C

0

n t

k 3

and D =

n t

[k 1

k 2

0 ]