POWER SYSTEM ANALYSIS

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Chapter 2: Basic Principles (A review)
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Power in single-phase AC circuits
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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
dtts
T 0
)(
1
T: period
Let’s first find the average value of
 
T
VIdttpR
T 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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Secondly let’s find the average value of
 
T
dttpX
T 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 with
active 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 ?
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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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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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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 Bus2
Transmission 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 ?
Realpowerflow
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 ?
Reactivepowerflow
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;
Where
Vp: rms value of phase voltage
Ip: rms value of phase current
θ: phase angle between
voltage and current (power factor angle)
Average (real, active) power
in Watts (W), or kW, MW
Reactive power
in Vars, or kVar, MVar
Complex power
in VA, or kVA, MVA
or
or
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Solution:
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Solution:
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Solution:
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End of Chapter 2