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Chapter 2: Basic Principles (A review)

1

Power in single-phase AC circuits

2

instantaneous voltage

instantaneous current

instantaneous power

instantaneous power is not practical to use so let’s simplify it!

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Trigonometric identity

iv

Phase angle between voltage and current

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Average value of a signal s(t) => T

dttsT

0

)(1

T: period

Let’s first find the average value of

T

VIdttpRT

0

cos)(1

cosVIP

Average power

Active power

Realpower

Unit is Watts (W)

kW

MW

Since P is not zero, it is converted into other forms of energy; such as motion,heat,light,etc...

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5

Secondly let’s find the average value of

T

dttpXT

0

0)(1

Meaning of zero is that pX(t) does not do any work !

It is the oscillating component of instantaneous power and should be considered together withactive power P

We define reactive power Q as the amplitude of this oscillating power

Unit is volt-ampere-reactive (VAR)

kVAR

MVAR

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BOX

i(t)

+

v(t)

For RESISTIVE component:

0 iv No phase angle difference between voltage and current

VIVIVIP 0coscos Real power associated with R

00sinsin VIVIQ No reactive power associated with R

Proof ?

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BOX

i(t)

+

v(t)

For INDUCTIVE component:

90 iv Phase angle difference between voltage and current

090coscos VIVIP No Real power associated with L

VIVIVIQ 90sinsin Reactive power associated with L

Proof ?

8

BOX

i(t)

+

v(t)

For CAPACITIVE component:

90 iv Phase angle difference between voltage and current

0)90cos(cos VIVIP No Real power associated with C

VIVIVIQ )90sin(sin Reactive power associated with C

Proof ?

What is the meaning of «-» sign ?

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9

Let’s remember BOX theory !

BOX

I

+

V ?

+

I: rms BOX current

V: rms BOX voltage

For the configuration above ( If i(t) enters to the BOX at (+) terminal)

Consumed Produced

P>0 P<0

Q>0 Q<0

In short: negative P or Q means production or generation !

Real Power

Reactive Power

Summary

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11

Figure 2.1

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Solution:

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Complex power

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voltage and current phasors

Phasor Diagram

Complex power

Power triangle

Apparent power

Complex power balance

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source load

PG

+ losses

QG

PL

QL

PLoss

QLoss

PG=PD=PL+Ploss

QG=QD=QL+Qloss

SG=sqrt(PG^2+QG^2)

Demand Side

SD=sqrt(PD^2+QD^2)SG=SD

Generation Side

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15

Power factor correction

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Lagging pf Unity pf

Power factor correction

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Power factor correction

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Complex power flow

19Bus1 Bus2Transmission line

Sflow

Turkey power grid

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Let’s find complex power flow from Bus1 to Bus2

Real power flow from Bus1 to Bus2

Reactive power flow from Bus1 to Bus2

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Which parameter(s) more or less will effect real power flow ?

Rea

l po

wer

flo

w

Less effective

More effective

• Tight bus voltage control (around 1.0 pu)• Voltage stability concerns

Real power can be effectively controlled/governed by angle difference

What is the maximum possible real power flow on a line?

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Which parameter(s) will effect reactive power flow ?

Rea

ctiv

e p

ow

er f

low

Less effective

Voltage difference(more effective)

• Angle difference is generally small• Transient stability concerns

Reactive power can be effectively controlled/governed by voltage magnitude difference

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BALANCED THREE-PHASE CIRCUITS

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PHASE-SEQUENCE

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Hint: check the 2nd phase (ph-B)

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Y-connected Loads

• Line-to-neutral voltages• Ref. A is chosen arbitrarily as reference• Positive sequence

Calculation of (Line-to-Line) or (Line) Voltages

Phasor Diagram

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Delta-connected Loads

Calculation of Line currents

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DELTA-WYE TRANSFORMATION

It simplifies calculations• You get a neutral point• Per-phase analysis is possible

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(in balanced conditions)

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PER-PHASE ANALYSIS

Why per-phase analysis ?• If you have a balanced system• You need to transform into Y if there is a delta-connected system• Neutral current is zero because of balanced current (I1+I2+I3=0)

Y-connected system

• Balanced power systems are solved on a «per-phase basis». • The other two phases carry identical currents except for the phase shift

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BALANCED THREE-PHASE POWER:

Consider a balanced three-phase circuit has the following voltages;

And consider the following set of balanced load phase-currents;

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BALANCED THREE-PHASE POWER:

The instantaneous power of the three-phase load:

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After simplification;

WhereVp: rms value of phase voltageIp: rms value of phase currentθ: phase angle between voltage and current (power factor angle)

Average (real, active) powerin Watts (W), or kW, MW

Reactive powerin Vars, or kVar, MVar

Complex powerin VA, or kVA, MVA

or

or

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34

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Solution:

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Solution:

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Solution:

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End of Chapter 2