EE 369POWER SYSTEM ANALYSIS

Lecture 9Transformers, Per Unit Calculations

Tom Overbye and Ross Baldick

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Announcements• For lectures 8 to 10 read Chapter 3• Homework 7 is 5.8, 5.15, 5.17, 5.24, 5.27, 5.28,

5.29, 5.34, 5.37, 5.38, 5.43, 5.45; due 10/22. • Homework 8 is 3.1, 3.3, 3.4, 3.7, 3.8, 3.9, 3.10,

3.12, 3.13, 3.14, 3.16, 3.18; due 10/29.• Homework 9 is 3.20, 3.23, 3.25, 3.27, 3.28,

3.29, 3.35, 3.38, 3.39, 3.41, 3.44, 3.47; due 11/5.

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Per Unit Change of MVA BaseParameters for equipment are often given using

power rating of equipment as the MVA baseTo analyze a system all per unit data must be on

a common power base

2 2

Hence Z

Z

/base base

OriginalBase NewBasepu actual pu

OriginalBase NewBasepu puOriginalBase NewBase

BaseBaseNewBase

OriginalBase NewBaseBasepu puOriginalBase

Base

Z Z Z

V VZ

S S

S ZS

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Per Unit Change of Base Example•A 54 MVA transformer has a leakage reactance of 3.69% (on its own MVA base). •What is the reactance on a 100 MVA base?

1000.0369 0.0683 p.u.54eX

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Transformer ReactanceTransformer reactance is often specified as a

percentage, say 10%. This is a per unit value expressed as a percentage on the power base of the transformer.

Example: A 350 MVA, 230/20 kV transformer has leakage reactance of 10%. What is p.u. value on 100 MVA base? What is value in ohms (230 kV)?

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1000.10 0.0286 p.u.350

2300.0286 15.1100

eX

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Three Phase Transformers•There are 4 different ways to connect 3 transformers

Y-Y -

Usually 3 transformers are constructed so all windingsshare a common core 6

3 Transformer Interconnections-Y Y-

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Y-Y Connection

Magnetic coupling with An/an, Bn/bn & Cn/cn1, ,An AB A

an ab a

V V Ia aV V I a

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Y-Y Connection: 3 Detailed Model

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Y-Y Connection: Per Phase Model

Per phase analysis of Y-Y connections is exactly the same as analysis of a single phase transformer.

Y-Y connections are common in transmission systems.

Key advantages are the ability to ground each side and there is no phase shift is introduced.

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- Connection

Magnetic coupling with AB/ab, BC/bb & CA/ca1 1, ,AB AB A

ab ab a

V I IaV I a I a

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- Connection: 3 Detailed Model

To use the per phase equivalent we need to usethe delta-wye load transformation

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- Connection: Per Phase Model

Per phase analysis similar to Y-Y except impedances are decreased by a factor of 3.

Key disadvantage is - connections can not be grounded; not commonly used.

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-Y Connection

Magnetic coupling with AB/an, BC/bn & CA/cn

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-Y Connection V/I Relationships, also 3 30

3030Hence 3 and 3

For current we get1

13 30 303

1 303

AB ABan ab an

an

AnABab an

ABa AB

a

A AB AB A

a A

V Va V V VV a

VVV Va a

I I a II a

I I I I

I a I

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-Y Connection: Per Phase Model

Note: Connection introduces a 30 degree phase shift!

Common for transmission/distribution step-down sincethere is a neutral on the low voltage side.

Even if a = 1 there is a sqrt(3) step-up ratio

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Y- Connection: Per Phase Model

Exact opposite of the -Y connection, now with a phase shift of -30 degrees.

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