### Chapter 1 History of Power Systems

```Review of AC Circuits
Smith College, EGR 325
March 27, 2006
1
Objectives
• Power calculations and terminology
• Expand understanding of electrical
power
– from simple linear circuits to
– a high voltage power system
2
Overview
• Basic Circuits
•
•
•
•
•
•
Sinusoidal waveform representation
Root mean square
Phase shift
Phasors
Complex numbers
Complex impedance
• Electric Power
• Complex: real & reactive power
• Power factor and power factor correction
3
ac Waveform
v
Vmax
v  Vmax sin  t
t
 2  f
f is thefrequencyof the waveform
4
How AC is Generated
Stator
N
S
Rotor
Windings
5
How AC is Generated
f
v
2700
900
N
Angle
S
1800
3600
X
6
AC Phasor Representation
v
t
v  Vmax sin t
Vmax
Vrms  V 
2
  2  f  377 rad / sec
7
V1
Reference

V2
v1
v2
V1  V10
V2  V2    

t
v1  V1max sin t
v2  V2 max sin (t  )
8
V1
Reference

V2
v1
v2
V1  V10
V2  V2    

t
v1  V1max cost
v2  V2 max cos(t  )
9
Phasors
j
V  Vm e  Vm 

v1 (t )  Re Ve
jt

10
Representing Power
11
Power Calculations
• P = VI
• P = I2R
• P = V2/R
• S = VI
• S = I2Z
• S = V2/Z
12
Resistance  Impedance
•
•
•
•
Resistance in 
Capacitance in F
Inductance in H
Z = R + jX
13
Instantaneous Electric Power [p(t)]
v(t )  Vmax sin(t )
v
i
i(t )  I max sin(t   )
V
t


I
 (t )  v(t ) * i(t )  Vmax I max sin(t ) sin(t  )
Vmax I max
 (t ) 
[cos(  )  cos( 2t   ) ]
2
Fixed average
Zero average
14
Instantaneous vs. Average Power
1
1
p(t )  Vm I m cos( v   i )  Vm I m cos( 2t   v   i )
2
2
15
Instantaneous vs. Average Power
• Instantaneous power is written as
1
1
p(t )  Vm I m cos( v   i )  Vm I m cos( 2t   v   i )
2
2
• The average of this expression is
1
P  Vm I m cos(  v   i )
2
16
Real & Reactive Power – Time Domain
Vmax I max
p(t ) 
[cos(  vi )  cos( 2t   vi ) ]
2
p(t )  P  Q(t )

p
Q(t)
t

t
17
Complex Power
IMPORTANT
 is the power factor angle
S V I
V

*
I
I  I  
S  V I V0 I VI 
*
S  V I cos  j V I sin 
S  P j Q
Real Power
Reactive Power
18
Example: Current Flow
19
Example: Power Flow
20
Power System
Operations
21
Operating Challenges
• Load is stochastic and is not
controlled
• Power flows cannot be directed or
controlled
• Electricity cannot be stored
• Everything happens in real-time
• Generation can be controlled
22
Power System Variables
• Generators produce complex power
– S = P + jQ
– Real power, P, able to perform useful
work
– Reactive power, Q, supports the system
electromagnetically
• Single system frequency, f
• Voltage profile, V
23
Real Power Flow – Voltage Relation
Voltage (pu)
• In normal system operation, frequency/real-power
dynamics are decoupled from voltage/reactive-power
Power (pu)
24
Real Power and Frequency
• P and f dynamics are coupled
– Demand > Supply: frequency will decrease
(more energy drained from system than produced,
acts like brakes on the turbines)
– Supply > Demand: frequency will increase
(more energy in the power system than consumed,
acts like an accelerator so turbines spin faster)
• Generation-based frequency regulation
– Generator inertia
– Generator governors
25
Frequency Problems
• Imbalances in supply and demand beyond
the capabilities of these generator controls
– Load may be dropped, or “shed” by operators
– Equipment protection may disconnect
generators
– Operators may disconnect regional tie lines
26
Reactive Power Analogy
• Voltage and reactive power allow real
power to flow
• Reactive power
– Energy stored in capacitance and inductance
– Supports the electromagnetic fields along
transmission lines
– Cannot be transmitted long distances
• Analogy
– Inflatable water pipes
27
Voltage Collapse
• The real power demanded is above the
transfer capability of a transmission line
• Return to the water pipe analogy
– Load draws too much power – dips into the
stored reactive power – “collapses” the pipe
• Equations: P = V*I, I = V/Z
– Load wants more power: Decrease apparent
impedance (Z), to increase current draw (I),
which allows increased P
– But, if P at limit, result is to decrease V
28
Voltage (pu)
Real Power Flow – Voltage Relation
Power (pu)
29
Power System Response to Outages
• Power flows on the paths of least
impedance
• As elements are removed (fail), the
impedance changes and so power flows
change  Instantaneously
• Human and computer monitoring of and
reaction to problems is on a much slower
timescale
30
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