EEE415-Week3-4 1 PowerTrans-PerUnit · í ì l í ì l î ì í ò ó µ l µ } À h v ] À ] Ç ,...

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EEE 415Power System Analysis I

Department of Electrical and Electronics Engineering

Power Transformers & Per Unit System

Çukurova UniversityDepartment of Electrical and Electronics Engineering


• For an ideal transformer, the following are assumed:

• There is no leakage flux so, the entire flux φc is confined to the core and links both windings.

• The windings have zero resistance; therefore, the I2Rlosses in the windings are zero.

• The core permeability µc is infinite, which corresponds to zero core reluctance.

• B-H characteristic of the magnetic material is single valued, and linear.

• There are no core losses.

Ideal Transformer Model

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• If there is no leakage flux,


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1 1

2 2

c

c

N

N

Using Faraday’s Law,

1 1 2 2

1 1

2 2

c cd dv N v N

dt dtv N

v N



• Using Ampere’ s Law,


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1 1 2 2

1 1 2 2 1 1 2 2or

c c enclosed

c c

c cc c

c

H l i

H l i N i N

B li N i N i N i N

If the core permeability µc is infinite,

1 1 2 2

1 2

2 1

0c c

i N i N

i N

i N

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• An equivalent circuit of ideal transformer is represeneted as;

Ideal Transformer

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1 1

2 2

v Na turn ratio

v N



• The complex power at primary side is as the complex power at secondary side.

Ideal Transformer

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• This means, the ideal transformer has no real and reactive power loss.



• Impedances can be reflected from one side of a transformer to the other side of a transformer.

• Using the impedance reflection, two separate primary and secondary circuits can be combined to make calculations easily.

Ideal Transformer

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2

' 12 1

2

' 22 1

NZ Z

N

Z a Z

Secondary side impedancereferred to primary side:



• As a result, for an ideal transformer;• The voltages behave according to the turns ratio.• The currents act inversly to the turn ratio.• The impedances perform according to the square of the turns ratio.

Ideal Transformer

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• An equivalent circuit for a practical transformer differs from the ideal transformer as follows:

• The windings have resistance.• The magnetic flux is not entirely confined to

the core, in other words the primary andseconary windings have leakage flux.

• The core permeability µc is finite.• There are real and reactive power losses in

the core.

Practical Transformer Model

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• The resistances of primary and secondary windings are represented in theequivalent circuit model as;


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• The leakage flux flows outside the core (in the air).• The leakage flux linkages of primary and secondary windings are denoted as λl1

and λl2 , and are equal to;


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1 1 1

2 2 2

l

l

L i

L i

• L1 is the leakage inductance of the primary side winding and L2 is the leakageinductance of the secondary side winding. L1 and L2 are represented in theequivalent circuit model as;



• If the core permeability µc has a finite value, recall from Ampere’ s Law that;


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1 1 2 2

1 1 2 2

21 2

1 1

c c enclosed

c c

c c

c

c cm

c

H l i

H l i N i N

B li N i N

B lNi i i

N N

ఓேభ

is called as magnetizing current, im.

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• From Faraday’ s Law, the voltage is induced in the primary winding is;


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1 1

1 1

c

cc c

dv N

dtdH

v N Adt

• Hc is eliminated by using Ampere’ s Law and the primaty side induced voltage is equal to;

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c c m

c

N A div

l dt



•ேభఓ

is called as a magnetizing inductance, Lm of the transformer and it is

represented in the equivalent circuit model of transformer as;


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• The real losses of the transformer core is caused by the eddy and hysteresislosses. The real losses of transformer is represented in the equivalent circuitmodel of transformer as;


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• By refering the secondary side impedance to primary side, the equivalent circuitmodel of practical transformer are shown as;


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• Typically the core resistance and magnetizing inductance have very large values. Thus, the currents flowing through them have very small values and the currentflowing through primary and secondary winding impedances are almost same. Because of these, the equivalent circuit model of practical transfomer can be organized as;


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Transformer Tests


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• For medium power transformers, the shunt branch can be omitted, which corresponds to neglecting the exciting current. Since the exciting current is usually less than 5% of rated current, neglecting it in power system studies is often valid unless transformer efficiency or exciting current phenomena are of particular concern.


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• For large power transformers rated more than 500 kVA , the winding resistances, which are small compared to the leakage reactances, can often be neglected;


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• The following are not represented by the steady-state equivalent circuit of practical transformer:

• Saturation• Inrush current• Nonsinusoidal exciting current• Surge phenomena


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• Motivations for Per Unit System• Transformer introduces various voltage levels.• So far we can only reflect the load from one side of the transformer to

another. Still we need to consider different voltage level at each side of the transformer when we try to find voltage and current.

• It is difficult to calculate voltage and current of the system at various points.• It is even more difficult for the system operator to observe the current

situation of the system.

Per Unit System

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• In per unit system, the voltage and current are normalized at each location of thepower system.

• The per unit quantity of voltage, current, power and impedance is found from dividing the actual quantity by a base value of that quantity.

Per Unit System

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• The base value has the same units as the actual quantity, this making the per-unit quantity unitless. Per unit value is denoted by ‘p.u.’

• The base value is always a real number. Therefore, the angle of the per-unit quantity is the same as the angle of the actual quantity.



Base Value Selection• Only single base value is selected for the power quantities in the whole power

system. The base values of real power, reactive power, and apparent power are all the same real number.

• Generally, voltage base values are selected as transformer voltage ratings.• For single phase calculations, the base voltage VbaseLN is selected for either a single-phase

circuit or for one phase of a three phase circuit. • For three phase calculations, the base voltage, VbaseLL is selected as line to line voltage of

three phase circuit.

Per Unit System

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, , ,

ForSinglePhaseCircuits;

base base baseS P Q ,3 ,3 ,3

ForThreePhaseCircuits;

base base baseS P Q

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Base Value Selection• Current base values are calculated from the base power and the base voltage.

• Impedance base values (same value for impedance, resistance, or reactance) are calculated from voltage and current.

Per Unit System

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,,

,


basebase

base l n

SI

V

,,

,


3base

base

base l l

SI

V

2,

,


base

base base basebase l n

VZ R X

S

2,

,3


base l l

base base basebase

VZ R X

S



Per Unit Equivalent Circuit of Single Phase Transformer

Per Unit System

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For the ideal transformer,

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Conversion of Per Unit Values from Old to New Base Values• Manufacturers usually specify equipment impedances in per unit values together

with voltage ratings (V) and apparent power rating (VA).• The actual impedance base values can be found from the ratings of the

equipment.• Different equipments have different ratings and the selected voltage and power

base values for the power system may differ from the ratings of equipments. Thus, we may need to calculate per unit values on the new basis.

• To convert a per-unit impedance from ‘‘old’’ to ‘‘new’’ base values, use;

Per Unit System

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Steps for Per-Unit Analysis1) Choose a base power value for the entire system.2) Select base voltage values for different zones (usually follows transformer

voltage ratings).3) Calculate for different zones.4) Express all quantities in p.u.5) Draw impedance diagram and solve for p.u. quantities.6) Convert back to actual quantities if needed.

Per Unit System

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Advantages• Per unit system simplifies the calculations by eliminating the turn ratios of

transformers.• If the equipment ratings are used as base values, per-unit quantities in power

system usually lie within a narrow numerical range by means of this, per unitsystem;

• helps to spot errors in data.• helps to detect abnormality in the system.

Per Unit System

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Three zones of a single-phase circuit are identified in Figure. The zones are connected by transformers T1 and T2, whose ratings are also shown. Using base values of 30 kVA and 240 volts in zone 1, draw the per-unit circuit and determine the per-unit impedances and the per-unit source voltage. Then calculate the load current both in per-unit and in amperes. Transformer winding resistances and shunt admittance branches are neglected.

Example 1

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A balanced Y-connected voltage source with Eab=480∟0° volts is applied to a balanced-Δ load with ZΔ=3∟40°Ω. The line impedance between the source and load is ZL=1∟85°Ω for each phase. Calculate the per-unit and actual current in phase a of the line using Sbase3φ=10 kVA and VbaseLL=480 volts.

Example 2

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• The three phase transformers can be constructed either as a 3-phase bank of independent identical transformers (can be replaced independently) or as a single transformer wound on a single 3-legged core (lighter, smaller, cheaper and slightly more efficient).

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Three Phase Transformers

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• There are four basic three phase transformer connections as;• Y-Y• Y-Δ • Δ-Y• Δ-Δ

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Y-Y Connection

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• 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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• The per unit equivalent circuit of Y-Y connected transformer

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Y-Δ Connection

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• Y-Δ connection is generally used in stepping down from a high voltage to a medium.

• The secondary voltage is shifted by -30°with respect to the primary voltage. This can cause problems when paralleling 3-phase transformers since transformerssecondary voltages must be in-phase to be paralleled. Therefore, we must pay attention to these shifts.

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• The per unit equivalent circuit of Y-Δ connected transformer

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Δ-Y Connection

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• Δ-Y connection is generally used for thegenerator step-up transformers and thedistribution from a high voltage to a lowvoltage levels.

• The secondary voltage is shifted by 30°with respect to the primary voltage.

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• The per unit equivalent circuit of Δ-Y connected transformer

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Δ- Δ Connection

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• The Δ-Δ connection has no phase shift. Per phase analysis similar to Y-Y except impedances are decreased by a factor of 3.

• Key disadvantage of Δ-Δ connectionis that can not be grounded; not commonly used.

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• The per unit equivalent circuit of Δ-Δ connected transformer

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Power Transformers & Per Unit SystemExample 3

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Three Winding Transformers



• The shunt admittance branch, a core loss resistor in parallel with a magnetizing inductor, can be evaluated from an open-circuit test.

• When one winding is left open, the three-winding transformer behaves as a two-winding transformer, and standard short-circuit tests can be used to evaluate per-unit leakage impedances, which are defined as follows:

• Z12 = per-unit leakage impedance measured from winding 1 ; with winding 2 shorted and winding 3 open



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• Note that each of the windings on a three-winding transformer may have a different kVA rating. If the leakage impedances from short-circuit tests areexpressed in per-unit based on winding ratings, they must first be converted toper-unit on a common Sbasebefore the calculations.

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• Autotransformers are transformers in which the primary and secondary windings are coupled magnetically and electrically.

• This results in lower cost, and smaller size and weight.• The key disadvantage is the loss of electrical isolation between voltage levels.

Hence auto-transformers are not used when the turn ratio is large. For example in stepping down 36kV/400V, we do not ever want 36kV on the low voltage side!

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Autotransformers

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• Phase shifting transformers are used to control the phase angle across the transformer

• Also called phase angle regulators (PARs) or quadrature booster transformers

• Since power flow through the transformer depends upon phase angle, this allows the transformer to regulate the power flow through the transformer

• Phase shifters can be used to prevent inadvertent "loop flow" and to prevent line overloads.

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Phase Shifting Transformers



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EEE415-Week3-4 1 PowerTrans-PerUnit · í ì l í ì l î ì í ò ó µ l µ } À h v ] À ] Ç ,...

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