Transcript EE 372 Fundamentals of Power Systems

EE 372
Fundamentals of Power Systems
Textbook:
John J. Grainger, William D. Stevenson, Jr., “Power System
Analysis”, McGraw-Hill, Inc., 1994.
Objective:
To teach the fundamental concepts of electric power system engineering.
Basics
 Power: Instantaneous consumption of energy
 Power Units
Watts = voltage x current for dc (W)
kW
–
1 x 103 Watt
MW
–
1 x 106 Watt
GW
–
1 x 109 Watt
Basics
 Energy: Amount of Work
 Energy Units (for electrical power)
Wh
-1 x 100 Watthour
kWh
–
1 x 103 Watthour
MWh –
1 x 106 Watthour
GWh –
1 x 109 Watthour
 Relationship of power and energy
Energy Consumed
Average Power
W
P
t
Duration
Sinusoidal Signals
Circular rotation of a magnetized rotor in Synchronous Generator
produces sinusoidal voltage in stator windings due to FARADAY LAW.
(Look at EE 471 Notes)
Sinusoidal Signals
THREE-PHASE SYNCHRONOUS GENERATOR
Sinusoidal Signals
How do you write the mathematical
equation for this periodic function?
?
?
Sinusoidal Signals
Period
: 0.01 s.
Frequency : 100 Hz.
 (t )  200cos(2100t )
Sinusoidal Signals
 (t )  200sin(2 50t )  100
Period
: 0.02 s.
Frequency : 50 Hz.
OR

 (t )  200 cos( 2 50 t  )  100
2
Sinusoidal Signals
 (t )  100cos(2 50t )
 (t )  100sin(2 50t )

 (t )  100 cos( 2 50 t  )
2

sin( )  cos(   )
2

 (t )  100 sin( 2 50 t  )
2

cos(  )  sin(  )
2
Sinusoidal Signals
400
Voltage
Current
200
0.0200
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
-100
0.0025
0
0.0000
-0.0025
-0.0050
-0.0075
-0.0100
-0.0125
-0.0150
-0.0175
100
-0.0200
Volts, Amperes
300
-200
?


-300
-400
Time (seconds)
?
Peak voltage : 310 V.
Period
: 0.02 s.
Frequency : 50 Hz.
 (t )  Vm sin(2 f t )  310sin(314t )
Peak current : 150 A.
Period
: 0.02 s.
Frequency : 50 Hz.
i(t )  I m sin(2 f t   )  150sin(314t  2.355)
1350
radian
Complex Numbers
Euler’s Formula : Relates exponential and sinusoidal functions
Rectangular
Notation
Polar
Notation
R
Im

R
j
z  x  j y  R e  R cos( )  j R sin( )
R  z  x2  y 2
 y
  arctan 
x
Re
Attention:
z  1 j
  arctan( 1)  
z  1  j

4
  450
 45 0
  450  1800  1350
Complex Numbers
Addition and subtraction of complex numbers are easier with
the rectangular notation.
(a  j b)  (c  j d )  (a  c)  j (b  d )
Multiplication and division of complex numbers are easier with
the polar notation.
( A ).(B )  ( A.B)(   )
A A
 (   )
B B
Attention:
Rectangular
Polar
Phasors
Phasors are complex numbers used to represent sinusoids.
Phasor representation of a sinusoidal function:
 (t )  Vm cos(t   )
V  Vm e j  Vm 
Phasor
If we multiply phasor
V
by
e j t
and apply Euler’s formula
Ve j t  Vm e j e j  t  Vme j ( t  )  Vm cos(t   )  j Vm sin(t   )
 (t )  e Ve j  t 
Phasors
Consider the derivative of sinusoidal signal represented as a
phasor
Derivative:





d
d
j t
Ve

Vm e j e j  t  j Vm e j e j  t  j Ve j  t
dt
dt
d
dt

j

Phasors
Examples:
Inductor
Capacitor
d i (t )
L
  (t )
dt
d  (t )
C
 i (t )
dt
d I e j t
L
 Ve j t
dt
d V e j t
C
 I e j t
dt




j L I e j t  Ve j t
j L I  V




j C V e j t  I e j t
j C V  I
Phasors
 (t )
I
i (t )
360
Ref.
V
 (t )  311cos(2 50t )
Vrms
311

 220 V
2
V  22000 V

i (t )  141 cos( 2 50 t  )
5
141
I rms 
 100 A
2
I  100360 A
Important: In power systems, RMS values are used for the magnitudes.
Phasors
20
i(t)
v(t)
10
second
0.2
0.1
0.1
10
20
0.2