First Course on
POWER SYSTEMS
Ned Mohan
Oscar A. Schott Professor of Power Electronics and Systems
Department of Electrical and Computer Engineering
University of Minnesota
Minneapolis, MN 55455
USA
© Copyright Ned Mohan 2006
1
A 345-kV Example System
Bus-1
Bus-3
200km
P + jQ
Pm1
Pe1
150km
150km
Bus-2
Pe 2
Pm 2
© Copyright Ned Mohan 2006
2
TOPICS IN POWER SYSTEMS
Week Book Chapters
1
Chapter 1: Overview
Chapter 2: Fundamentals
Laboratory
Lab 1: Visit to a local substation; otherwise a
virtual substation
2
Chapter 3: Energy Sources
Lab 2: Introduction to PSCAD/EMTDC; 3phase circuits, vars, power-factor correction
3
Chapter 4: Transmission Lines
Lab 3: Transmission Lines using PSCADEMTDC
4
Chapter 5: Power Flow
Lab 4: Power Flow using MATLAB and
PowerWorld
5
Chapter 6: Transformers
Lab 5: Including Transformers in Power Flow
using PowerWorld and MATLAB
6
Chapter 7: HVDC, FACTS
Lab 6: Power Converters and HVDC using
PSCAD-EMTDC, HVDC in PowerWorld
7
Chapter 8: Distribution Systems
Lab 7: Power Quality using PSCAD-EMTDC
8
Chapter 9: Synchronous
Generators
Lab 8: Synchronous Generators and AVR
using PSCAD-EMTDC.
9
Chapter 10: Voltage Stability
Lab 9: Voltage Regulation using PowerWorld
10
Chapter 11: Transient Stability
Lab 10: Transient Stability using MATLAB
11
Chapter 12: Interconnected
Systems, Economic Dispatch
Lab 11: AGC using Simulink, and Economic
Dispatch using PowerWorld
12
Chapter 13: Short-Circuit
Faults, Relays, Circuit Breakers
Lab 12: Transmission Line Faults using
PowerWorld and MATLAB
13
Chapter 14: Transient OverVoltages, Surge Arrestors,
Insulation Coordination
Lab 13: Over-voltages and Surge Arrestors
using PSCAD-EMTDC
© Copyright Ned Mohan 2006
3
Chapter 1
POWER SYSTEMS: A CHANGING
LANDSCAPE
© Copyright Ned Mohan 2006
4
NATURE OF POWER SYSTEMS
Fig. 1-1 Interconnected North American Power Grid [2].
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5
Control Areas
Fig. 1-2 NERC Interconnections [3]. Source: NERC.
© Copyright Ned Mohan 2006
6
One-line Diagram
Step up
Transformer
Generator
Transmission
line
13.8 kV
Feeder
Load
Fig. 1-3 One-line diagram as an example.
© Copyright Ned Mohan 2006
7
CHANGING LANDSCAPE OF POWER SYSTEMS AND UTILITY
DEREGULATION
(a)
( b)
Fig. 1-4 Changing landscape [4]. Source: ABB.
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8
CHAPTER 2
REVIEW OF BASIC
ELECTRIC CIRCUITS AND
ELECTROMAGNETIC
CONCEPTS
© Copyright Ned Mohan 2006
9
Symbols and
Conventions
+
a
+
vab
−
i
b
+
va
vb
−
−
Fig. 2-1 Convention for voltages and currents.
© Copyright Ned Mohan 2006
10
Phasors
Imaginary
positive
angles
V = V ∠0 Real
−φ
I = I∠ − φ
Fig. 2-2 Phasor diagram.
© Copyright Ned Mohan 2006
11
Phasor Analysis
i( t )
v( t )
= 2V cos( ω t )
I
+
L
+
V = V ∠0
−
R
jω L = j X L
−
R
(b)
− jX c
jX L
Z
⎛ 1 ⎞
− j⎜
⎟ = − j XC
⎝ω C ⎠
C
(a )
Im
R
Re
0
(c)
Fig. 2-3 A circuit (a) in time-domain and (b) in phasor-domain; (c) impedance triangle.
© Copyright Ned Mohan 2006
12
Example of Impedance
Calculation
j 0.1 Ω
− j5 Ω
2Ω
Fig. 2-4 Impedance network of Example 2-1.
© Copyright Ned Mohan 2006
13
Example of Impedance
Calculation
0.3Ω
j 0.5 Ω
j 0.2 Ω
+
I1
V1
−
7.0Ω
j15 Ω
Im
I2
Fig. 2-5 Circuit of Example 2-2.
© Copyright Ned Mohan 2006
14
Power Flow
+
Subcircuit 1
v (t )
Subcircuit 2
−
p (t ) = v (t ) i (t )
Figure 2-6 A generic circuit divided into two sub-circuits.
© Copyright Ned Mohan 2006
15
Real and Reactive
Power
average
power
p (t )
0
t
v(t )
φ /ω
p (t )
average
power
0
t
v (t )
(a )
i (t )
( b)
i (t )
Figure 2-7 Instantaneous power with sinusoidal currents and voltages.
© Copyright Ned Mohan 2006
16
P, Q and VA by Phasors
I
+
Subcircuit 1
Subcircuit 2
V
−
S = P + jQ
(a)
Im
V = V ∠φv
φ
Re
Im
S
Q
φ
I = I ∠φi
(b)
P
Re
(c)
Fig. 2-8 (a) Circuit in phasor-domain; (b) phasor diagram; (c) power triangle.
© Copyright Ned Mohan 2006
17
Example of Power Factor
Correction
P = PL
+
− jQC
V1
−
− j13.963 Ω
PL + jQL
Fig. 2-9 Power factor correction in Example 2-5.
© Copyright Ned Mohan 2006
18
One-line Diagram
Step up
Transformer
Generator
Transmission
line
13.8 kV
Feeder
Load
Fig. 2-10 One-line diagram of a three-phase transmission and distribution system.
© Copyright Ned Mohan 2006
19
Three-Phase Voltages
van (t ) vbn (t ) vcn (t )
0
ωt
Vcn
a −b−c
positive
sequence
120°
120°
120°
2π
3
2π
Vbn
Van
( b)
3
(a )
Fig. 2-11 Three-phase voltages in time and phasor domain.
© Copyright Ned Mohan 2006
20
Balanced Three-Phase
Circuit Analysis
a
Ia
Ia
+
ZL
V an −
V cn
+
−
n
− V bn
+
V an
N
c
Ic
(a)
Ib
b
a
+
−
V cn − n − V bn
+
+
Ic
ZL
In
N
c
Ib
b
(b)
Fig. 2-12 Balanced wye-connected, three-phase circuit.
© Copyright Ned Mohan 2006
21
Per-Phase Analysis
Ia
a
+
a
V cn
V an
−
n
Ib
(Hypothetical)
N
Ic
φ
V an
Ia
V bn
(a)
(b)
Fig. 2-13 Per-phase circuit and the corresponding phasor diagram.
© Copyright Ned Mohan 2006
22
Balanced Mutual
Coupling
Ia
Z self
a
A
Ib
Z self
Z mutual
b
a
Z aA
A
Z mutual
B
Ic
Z self
c
Z mutual
(Hypothetical)
C
(a)
( b)
Fig. 2-14 Balanced three-phase network with mutual couplings.
© Copyright Ned Mohan 2006
23
Line-Line Voltages
Vcn
Vca
−Vb
Vab
30 o
Van
Vbn
Vbc
Fig. 2-15 Line-to-line voltages in a three-phase circuit.
© Copyright Ned Mohan 2006
24
Wye-Delta
Transformation
Ia
Ia
a
ZΔ
a
ZΥ
I ab Z Δ
I ca
Ibc
c
b
c
ZΥ
ZΥ
b
ZΔ
(a)
(b)
Fig. 2-16 Delta-wye transformation.
© Copyright Ned Mohan 2006
25
Power Flow in AC
Systems
I
jX
+
Vs
+
Vs
jXI
δ
φ
VR
−
−
VR
I
( b)
(a )
Fig. 2-17 Power transfer between two ac systems.
© Copyright Ned Mohan 2006
26
Power-Angle Diagram
P / Pmax
1.0
0.5
0
0
0
δ
180
90
Fig. 2-18 Power as a function of δ .
© Copyright Ned Mohan 2006
27
Per Unit Quantities
Vbase
Rbase , Xbase , Zbase =
Ibase
(in Ω)
(2-48)
Ibase
Gbase , Bbase ,Ybase =
Vbase
(in
(2-49)
Pbase ,Qbase ,(VA)base =Vbase Ibase
(in Watt, VAR, or VA)
)
(2-50)
In terms of these base quantities, the per-unit quantities can be specified as
actual value
Per-UnitValue =
base value
© Copyright Ned Mohan 2006
(2-51)
28
Energy Efficiency of
Apparatus
Pin
Power
System
Apparatus
Po
Ploss
Fig. 2-19 Energy Efficiency η = Po / Pin .
© Copyright Ned Mohan 2006
29
Electro-Magnetic
Concepts:
Ampere’s Law
dl
H
i3
i1
i2
(a)
(b)
(c)
Fig. 2-20 Ampere’s Law.
© Copyright Ned Mohan 2006
30
Example of a Toroid
i
rm
ID
ID
OD
OD
(a)
(b)
Fig. 2-21 Example 2-9.
© Copyright Ned Mohan 2006
31
B-H Curves in
Ferromagnetic Materials
Bm
Bm
μo
Bsat
μm
μo
Hm
(a)
Hm
(b)
Fig. 2-22 B-H characteristic of ferromagnetic materials.
© Copyright Ned Mohan 2006
32
Flux and Flux-Density
Am
φm
Fig. 2-23 Toroid with flux φm .
© Copyright Ned Mohan 2006
33
Inductance
i
φm
Am
i
⎛N⎞
× ⎜⎜ ⎟⎟
⎝ Am ⎠
Hm
×(μm)
N
Lm =
μm
(a)
Bm
×(Am)
φm
×(N)
λm
N2
Am
μmAm
(b)
Fig. 2-24 Coil inductance.
© Copyright Ned Mohan 2006
34
Example of a Toroid
w
r
h
Fig. 2-25 Rectangular toroid.
© Copyright Ned Mohan 2006
35
Faraday’s Law
φ (t )
i (t )
+
e (t )
−
N
Fig. 2-26 Voltage polarity and direction of flux and current.
© Copyright Ned Mohan 2006
36
Plot of time-varying Flux
and Voltage
e(t )
φ (t )
0
t
Fig. 2-27 Example 2-11.
© Copyright Ned Mohan 2006
37
Leakage Flux
φm
i
i
+
+
e
−
e
−
(a)
φl
(b)
Fig. 2-28 Including leakage flux.
© Copyright Ned Mohan 2006
38
Representation of Leakage
Flux by Leakage
Inductance
φm
i (t )
+
+
Ll
Ll
e(t )
di
dt
−
+
em (t )
−
Ll
R
+
v(t )
Lm
−
+
e(t )
−
i (t )
+
em (t )
−
−
(a)
(b)
Fig. 2-29 Analysis including the leakage flux.
© Copyright Ned Mohan 2006
39
CHAPTER 3
ELECTRIC ENERGY AND
THE ENVIRONMENT
© Copyright Ned Mohan 2006
40
Energy Consumption and
Production in the U.S.
(a)
(b)
Fig. 3-1 Production and consumption of energy in the United States in 2004 [1].
© Copyright Ned Mohan 2006
41
Power Generation by
Various Fuel Types in the
U.S.
Fig. 3-2 Electric power generation by various fuel types in the U.S. in 2005 [1].
© Copyright Ned Mohan 2006
42
Hydro Power Generation
Water
H
Penstock
Generator
Turbine
Fig. 3-3 Hydro power (Source: www.bpa.gov).
© Copyright Ned Mohan 2006
43
Rankine Thermodynamic
Cycle in Coal Plants
Steam at High pressure
Heat in
Boiler
Pump
Turbine
Condenser
Gen
Heat out
Fig. 3-4 Rankine thermodynamic cycle in coal-fired power plants.
Visit the following website for Power Plant Animations:
http://www.cf.missouri.edu/energy/?fun=1&flash=ppmap
© Copyright Ned Mohan 2006
44
Brayton Cycle in Gas
Turbines
Fuel in
Compressor
Air in
Combustion
Chamber
Turbine
Exhaust
Fig. 3-5 Brayton thermodynamic cycle in natural-gas power plants.
© Copyright Ned Mohan 2006
45
Nuclear Power Plant
Types
(a )
( b)
Fig. 3-6 (a) BWR and (b) PWR reactors [5].
Visit the following websites for Nuclear Power Plant Animations:
PWR: http://www.nrc.gov/reading-rm/basic-ref/students/animated-pwr.html
BWR: http://www.nrc.gov/reading-rm/basic-ref/students/animated-bwr.html
© Copyright Ned Mohan 2006
46
Wind Resources in the
U.S.
Fig. 3-7 Wind-resource map of the United States [6].
© Copyright Ned Mohan 2006
47
Coefficient of
Performance
Fig. 3-8 c p as a function of λ [7]; these would vary based on the turbine design.
© Copyright Ned Mohan 2006
48
Wind Generation using an
Induction Generator
Connected Directly to the
AC Grid
Induction
Generator
Wind
Turbine
Utility
Fig. 3-9 Induction generator directly connected to the grid [8].
© Copyright Ned Mohan 2006
49
Wind Generation using a Doubly-Fed
Induction Generator
Wound rotor
Induction Generator
AC
Wind
Turbine
DC
DC
Generator-side
Converter
AC
Grid-side
Converter
Fig. 3-10 Doubly-fed, wound-rotor induction generator [9].
© Copyright Ned Mohan 2006
50
Wind Generation using an AC
Generator Connected through Power
Electronics
Power Electronics Interface
Gen
Conv1
Conv 2
Utility
Fig. 3-11 Power Electronics connected generator [10].
© Copyright Ned Mohan 2006
51
Photovoltaics
Fig. 3-12 PV cell characteristics [11].
© Copyright Ned Mohan 2006
52
Interfacing PV with AC
Grid
Isolated
DC-DC
Converter
PWM
Converter
Max. Powerpoint Tracker
Utility
1φ
Fig. 3-13 Photovoltaic systems.
© Copyright Ned Mohan 2006
53
Fuel Cells
Maximum Theoretical Voltage
E=
Activation
Losses
Cell Voltage ( VC in Volts )
1.2 -
- Δgƒ
- 1200
2F
- 1000
1-
Ohmic
0.8 -
- 800
Losses
- 600
0.6 -
Cell Power
PC= VC x i
Mass
Transport
Losses
0.4 -
- 400
- 200
0.2 -
0 -|
0
Cell Power ( PC in mW )
Open 1.4 Circuit
Voltage
|
|
|
|
500
1000
1500
2000
Current Density ( i in mA/cm2 )
-0
Fig. 3-14 Fuel cell v-i relationship and cell power [12].
© Copyright Ned Mohan 2006
54
Greenhouse Effect
Fig. 3-15 Greenhouse effect [13].
© Copyright Ned Mohan 2006
55
Resource mix
XcelEnergy
6
5
1
4
3
2
1
2
3
4
5
6
Fig. 3-16 Resource mix at XcelEnergy [14].
© Copyright Ned Mohan 2006
56
Fuel Costs in the U.S. in
2005
Fig. 3-17 Electric power industry fuel costs in the U.S. in 2005 [1].
© Copyright Ned Mohan 2006
57
CHAPTER 4
AC TRANSMISSION LINES
AND UNDERGROUND
CABLES
© Copyright Ned Mohan 2006
58
Transmission Tower,
Conductor and Bundling
(b)
© Copyright Ned Mohan 2006 (a )
(c)
Fig. 4-1 500-kV transmission line (Source: University of Minnesota EMTP course).
59
Transposition
a
D2
D1
b
D3
c
1 cycle
(a )
(b)
Fig. 4-2 Transposition of transmission lines.
© Copyright Ned Mohan 2006
60
Distributed Parameters
line
line
R
L
C
neutral (zeroimpedance)
Fig. 4-3 Distributed parameter representation on a per-phase basis.
© Copyright Ned Mohan 2006
61
Calculation of
Transmission Line
Resistance: Skin Effect
J
T
D
(a )
surface
( b)
towards center
Fig. 4-4 (a) Cross-section of ACSR conductors, (b) skin-effect in a solid conductor.
© Copyright Ned Mohan 2006
62
Calculation of
Transmission Line
Inductance
c
c
c
ic
a
ia
a
b
r
ib
ia
r
x
dx
D
a
D
(a )
b
b
x
ib
(b)
D
dx
(c)
Fig. 4-5 Flux linkage with conductor-a.
© Copyright Ned Mohan 2006
63
Electric Field Due to
Transmission Line Voltage
x
1
x1
2
q
x2
Fig. 4-6 Electric field due to a charge.
© Copyright Ned Mohan 2006
64
Calculation of Transmission Line
Capacitance
c
c
qc
C
qa
qb
a
C
n
hypothetical
neutral
C
b
D
(a )
b
a
( b)
Fig. 4-7 Shunt capacitances.
© Copyright Ned Mohan 2006
65
Typical Parameters for various
Voltage Transmission Lines
Table 4-1
Transmission Line Parameters with Bundled Conductors (except at 230 kV)
at 60 Hz [2, 6]
Nominal Voltage
R (Ω / km )
ω L (Ω / km )
ωC ( μ / km )
230 kV
0.055
0.489
3.373
345 kV
0.037
0.376
4.518
500 kV
0.029
0.326
5.220
765 kV
0.013
0.339
4.988
© Copyright Ned Mohan 2006
66
Calculating Transmission
Line Parameters using
EMTDC
Fig. 4-8 A 345-kV, single-conductor per phase, transmission system.
© Copyright Ned Mohan 2006
67
Distributed-Parameter
Representation
I S ( s)
+
I x ( s)
+
VS ( s )
−
Vx ( s )
−
x
R
1
sC
sL
I R (s)
+
VR ( s )
−
0
Fig. 4-9 Distributed per-phase transmission line ( G not shown).
© Copyright Ned Mohan 2006
68
Voltage Profile under
SIL
+
jω L
IS
1
−j
ωC
VS
−
(a )
+
Zc
IR
V R = V R ∠0
−
VS
VR
x
( b)
0
Fig. 4-10 Per-phase transmission line terminated with a resistance equal to Z c .
© Copyright Ned Mohan 2006
69
Typical Surge Impedances and SIL
for various Voltage Transmission
Lines
Table 4-2
Surge Impedance and Three-Phase Surge Impedance Loading [2, 6]
Nominal Voltage
Z c (Ω)
SIL ( MW )
230 kV
375
140 MW
345 kV
280
425 MW
500 kV
250
1000 MW
765 kV
255
2300 MW
© Copyright Ned Mohan 2006
70
Loadability of
Transmission Lines
Table 4-3
Loadability of Transmission Lines [6]
Line Length (km)
Limiting Factor
Multiple of SIL
0 - 80
Thermal
>3
80 - 240
5% Voltage Drop
1.5 - 3
240 - 480
Stability
1.0 – 1.5
© Copyright Ned Mohan 2006
71
Long-Line
Representation
I S ( s)
Z series
+
VS ( s )
−
I R (s)
+
Yshunt
2
Yshunt
2
VR ( s )
−
Fig. 4-11 Long line representation.
© Copyright Ned Mohan 2006
72
Transmission Line
Representations
Z series
IS
+
VS
−
IR
+
Yshunt
2
Yshunt
2
IS
+
VR
VS
−
−
(a )
jω Lline
Rline
−j
−j
2
ωCline
( b)
2
ωCline
IR
+
IS
jω Lline
Rline
+
IR
+
VR
VS
VR
−
−
−
(c )
Fig. 4-12 Per-phase transmission line representation based on length.
© Copyright Ned Mohan 2006
73
Underground Cables
Fig. 4-13 Underground cable.
© Copyright Ned Mohan 2006
74
CHAPTER 5
POWER FLOW IN
POWER SYSTEM
NETWORKS
© Copyright Ned Mohan 2006
75
Three-Bus Example Power
System
Bus 1
Bus 3
200km
150km
150km
Slack Bus
P + jQ
PQ Bus
Bus 2
PV Bus
Fig. 5-1 A three-bus 345-kV example system.
© Copyright Ned Mohan 2006
76
Transmission Lines in
Example Power System
Table 5-1 Per-Unit Values in the Example System
Total Susceptance B in μ
Line
Series Impedance Z in Ω (pu)
1-2
Z12 = (5.55 + j56.4) Ω = (0.0047 + j 0.0474) pu
BTotal = 675μ
= (0.8034) pu
1-3
Z13 = (7.40 + j 75.2) Ω = (0.0062 + j 0.0632) pu
BTotal = 900μ
= (1.0712) pu
2-3
Z 23 = (5.55 + j56.4) Ω = (0.0047 + j 0.0474) pu
BTotal = 675μ
= (0.8034) pu
© Copyright Ned Mohan 2006
(pu)
77
Calculating Y-Bus in the
Example Power System
Bus 1
Bus 3
Z13
V3
V1
Z12
Z 23
I1
I3
Bus 2
V2
I2
Fig. 5-2 Example system of Fig. 5-1 for assembling Y-bus matrix.
© Copyright Ned Mohan 2006
78
Newton-Raphson
Procedure
4 − x2
4
2
0
−2
x (2)
0.5
1.0
1.5
2
x (1)
x (0)
3.0
3.5
4.0
x
−4
−6
−8
−10
−12
Fig. 5-3 Plot of 4 − x 2 as a function of x .
© Copyright Ned Mohan 2006
79
Power Flow Results in the
Example Power System
V1 = 1∠00 pu
V3 = 0.978∠-8.79 0 pu
( 2.39 + j 0.29 ) pu
( 0.69 - j1.11) pu
( 5.0 +
P1 + jQ1 = (3.08 - j 0.82) pu
V2 = 1.05∠-2.07 0 pu
( 2.68 +
j1.0 ) pu
j1.48) pu
P2 + jQ2 = ( 2.0
+ j 2.67) pu
Fig. 5-4 Power-Flow results of Example 5-4.
© Copyright Ned Mohan 2006
80
CHAPTER 6
TRANSFORMERS IN POWER
SYSTEMS
© Copyright Ned Mohan 2006
81
Transformer Principle:
Generation of Flux
φm
+
e1
−
+
im
N1
e1
Lm
−
(a)
(b)
Fig. 6-1 Principle of transformers, beginning with just one coil.
© Copyright Ned Mohan 2006
82
Core in Transformers
Bm
Bm
μo
Bsat
μm
μo
Hm
(a)
Hm
(b)
Fig. 6-2 B-H characteristics of ferromagnetic materials.
© Copyright Ned Mohan 2006
83
Flux Coupling
φm
+
+
e1
N1
−
e1
N2
+
e2
im
+
Lm
e2
−
−
−
(a)
N1
N2
Ideal
Transformer
(b)
Fig. 6-3 Transformer with the open-circuited second coil.
© Copyright Ned Mohan 2006
84
Transformer with Load
Connected to the
Secondary
+
e1
φm
i1 (t )
+
N1
−
i2 (t )
e1
N2
i2′ (t )
i1 (t )
i2 (t )
im
+
Lm
e2
−
−
+
N1
N2
Ideal
Transformer
e2 −
(a)
(b)
Fig. 6-4 Transformer with load connected to the secondary winding.
© Copyright Ned Mohan 2006
85
Transformer Model
I1
R1
I 2'
jX l1
+
+
Rhe
V1
−
E1
jX l 2
im
jX m
−
N1
Real Transformer
R2
I2
+
+
E2
V2
−
−
N2
Ideal Transformer
Fig. 6-5 Transformer equivalent circuit including leakage impedances and core losses.
© Copyright Ned Mohan 2006
86
Eddy Current and
Hysteresis Losses
φm
circulating
currents
circulating
currents
i
φm
(a)
(b)
Fig. 6-6 Eddy currents in the transformer core.
© Copyright Ned Mohan 2006
87
Transformer Simplified
Model
Ip
+
Zp
Is
1: n
Zs
+
Vp
Vs′ n p
−
−
ns
+
Vs
−
Fig. 6-7 Simplified transformer model.
© Copyright Ned Mohan 2006
88
Transferring Leakage
Impedances from One Side
to Another
Ip
Z ps
+
Vp
Is
1: n
+
np
ns
Vs
−
−
(a)
Ip
1: n
Z sp
+
Vp
np
ns
−
Is
+
Vs
−
( b)
Fig. 6-8 Transferring leakage impedances across the ideal transformer of the model.
© Copyright Ned Mohan 2006
89
Transformer Equivalent
Circuit in Per Unit
I (pu)
+
I (pu)
Z tr (pu)
+
V p (pu)
Vs (pu)
−
−
Fig. 6-9 Transformer equivalent circuit in per unit (pu).
© Copyright Ned Mohan 2006
90
Connection of Transformer
Windings
(a )
(b)
Fig. 6-10 Winding connections in a three-phase system.
© Copyright Ned Mohan 2006
91
Including Nominal TurnsRatio Transformer in
Power Flow Studies
Bus 1
Bus 3
500 kV
345 / 500 kV
500 / 345kV
Fig. 6-11 Including nominal-voltage transformers in per-unit.
© Copyright Ned Mohan 2006
92
Auto-Transformer
n2
I1
+
V2
V1
−
−
+
V2
−
I2
+
n2
n1
(a)
I2
+
( I1 + I 2 )
+
(V1 + V2 )
I1
V1
−
n1
−
( b)
Fig. 6-12 Auto-transformer.
© Copyright Ned Mohan 2006
93
Phase-Shift Due to WyeDelta Transformers
n1 j 300
e : n2
3
a
VA
A
n1
C
Va
n2
+
+
VA
Va
−
−
B
(a )
c
b
(b)
(c)
Fig. 6-13 Phase-shift in Δ -Y connected transformers.
© Copyright Ned Mohan 2006
94
Phase-Shift Control by
Transformers
a′
Va
Vab
a
φ
Va
Vc′a′
c′
c
Vca
b
(a )
b′
Va′
Vc′
Vc
Vb′
Vb
(b)
Va ′b′
Vbc
( c ) Vb′c′
Fig. 6-14 Transformer for phase-angle control.
© Copyright Ned Mohan 2006
95
Three-Winding AutoTransformers
Z L (Ω)
a
H
a′
A
a
L
T
C
c
(a )
B
H
Z H (Ω)
n2
L
n1
n3 Z T (Ω)
b
© Copyright Ned Mohan 2006
a′
A
T
C
( b)
Fig. 6-15 Three-winding auto-transformer.
96
General Representation of
Auto- and Phase-Shift
Transformers
I2
I1
+
V1
−
YA = 1/ Z A
+
V2
t
−
+
V2
−
1: t
Fig. 6-16 General representation of an auto-transformer and a phase-shifter.
© Copyright Ned Mohan 2006
97
PU Representation of OffNominal Turns-Ratio
Transformers
I1 Y = 1/ Z
A
A
I2
I1
+
+
+
V1
V2
V1
−
−
−
1: t
(a )
YA / t
⎛ 1⎞
⎜ 1 − ⎟ YA
⎝ t⎠
⎛ 1 1⎞
⎜ 2 − ⎟ YA
t⎠
⎝t
I2
+
V2
−
(b)
Fig. 6-17 Transformer with an off-nominal turns-ratio or taps in per unit; t is real.
© Copyright Ned Mohan 2006
98
Example of Off-Nominal Turns-Ratio
Transformers
I1 j 0.1 pu
I2
I1
+
+
V1
V2
−
−
1: t
+
V1
−
Z s = j 0.11 pu
I2
+
Y1 = − j 0.909 pu
Y2 = j 0.826 pu
(a )
V2
−
(b)
Fig. 6-18 Transformer of Example 6-3.
© Copyright Ned Mohan 2006
99
CHAPTER 7
HIGH VOLTAGE DC (HVDC)
TRANSMISSION SYSTEMS
© Copyright Ned Mohan 2006
100
Power (VA)
108
Thyristor
IGCT
IGBT
(a)
MOSFET
106
Thyristor
Symbols and Capabilities
of Power Semiconductor
Devices
IGCT
IGBT
104
102
MOSFET
101 102 103 104
Switching Frequency (Hz)
(b)
Fig. 7-1 Power semiconductor devices.
© Copyright Ned Mohan 2006
101
Device current [A]
Power Semiconductor
Devices and Applications
104
Traction
103
102
101
HVDC
FACTS
Motor
Drive
Power
Supply
Automotive
Lighting
100
101
102
103
104
Device blocking voltage [V]
(a )
( b)
Figure 7-2 Power semiconductor devices: (a) ratings (source: Siemens), (b) various
applications (source: ABB).
© Copyright Ned Mohan 2006
102
HVDC System
HVDC Line
AC1
AC2
Fig. 7-3 HVDC system – one-line diagram.
© Copyright Ned Mohan 2006
103
HVDC Systems: VoltageLink and Current-Link
+
AC1
AC2
(a )
AC1
−
AC2
(b)
Fig. 7-4 HVDC systems: (a) Current-Link, and (b) Voltage-Link.
© Copyright Ned Mohan 2006
104
HVDC Projects in North
America
2250MW
320MW
2000MW
312MW
150MW
350MW
1620MW
2138MW
370MW
500MW
200MW
200MW
1000MW
690MW
2000MW
1000MW
330MW
200MW
3100MW
100MW
200MW
1920MW
210MW
200MW
200MW
200MW
600MW
36MW
Fig. 7-5 HVDC projects, mostly current-link systems, in North America [source: ABB]
© Copyright Ned Mohan 2006
105
Current-Link HVDC
System
Fig. 7-6 Block diagram of a current-link HVDC system.
© Copyright Ned Mohan 2006
106
Thyristors
A
A
(a)
(b)
G
G
P
pn1
N
pn2
P
K
pn3
N
K
Fig. 7-7 Thyristors.
© Copyright Ned Mohan 2006
107
Primitive Thyristor
Circuits
is
+
+
vs
(a )
Ls
vd
R
−
−
vd
Vd
0
α
( b)
is
0
iG
0
ωt = 0
vs
α
ωt
ωt
ωt
Fig. 7-8 Thyristor circuit with a resistive load and a series inductance.
© Copyright Ned Mohan 2006
108
Three-Phase Thyristor
Converter
id
+
van
− +
n −
−
vbn
vcn
ia
1
van
− +
5
3
ia
1
P
3
Ls
5
+
n
vd
+
4
6
+
4
2
6
−
(a)
+
2
−
Id
−
vd
N
(b)
Fig. 7-9 Three-phase Full-Bridge thyristor converter.
© Copyright Ned Mohan 2006
109
Three-Phase Diode
Rectifier Waveforms
va
vb
vc
vP
0
ia
t
0
120 o
60 o
ωt
ib
vN
0
(a)
vd
2VLL
ωt
ic
Vdo
0
ωt
t
0
(b)
(c)
Fig. 7-10 Waveforms in a three-phase rectifier with Ls = 0 and α = 0 .
© Copyright Ned Mohan 2006
110
Three-Phase Thyristor
Converter Waveforms with
zero AC-Side Inductance
v Pn
van
vcn
vbn
Aα
ωt
0
α
v Nn
ia
0
1
1
4
ωt
4
ib
3
0
ωt
6
ic
0
6
5
5
2
© Copyright Ned Mohan 2006
Fig. 7-11 Waveforms with Ls = 0 .
ωt
111
Three-Phase Inverter
Waveforms
v Nn
van
0
vbn
vcn
ωt
α
vPn
ia
1
1
ωt
0
4
ib
4
3
3
ωt
0
6
ic
5
ωt
0
2
© Copyright Ned Mohan 2006Fig.
2
7-12 Waveforms in the inverter mode.
112
DC-Side Voltage as a
Function of Delay Angle
Vd
Vd
Rectifier
P = Vd I d = +
1800
0
90
0
(a )
160
0
α
0
Id
( b)
Inverter
P = Vd I d = −
Fig. 7-13 Average dc-side voltage as a function of α .
© Copyright Ned Mohan 2006
113
Thyristor Converter
Waveforms in the
Presence of AC-Side
Inductance
v Pn
van
vcn
vbn
Au
ωt
0
α
v Nn
u
ia
0
1
4
1
4
ωt
Fig. 7-14 Waveforms with Ls .
© Copyright Ned Mohan 2006
114
Power Factor Angle in
Rectifier and Inverter
Modes
Va
Va
−φ1
−φ1
I a1
I a1
(a )
(b)
Fig. 7-15 Power-factor angle.
© Copyright Ned Mohan 2006
115
CU One-line Diagram
© Copyright Ned Mohan 2006
116
12-Pulse Waveforms
ia (Y − Y )
ia (Y − Δ )
(a )
( b)
Fig. 7-17 Six-pulse and 12-pulse current and voltage waveforms [2].
© Copyright Ned Mohan 2006
117
HVDC System
Representation for Control
id
+
vd 1
AC 1
−
Rd
Ld
−
vd 2
AC 2
+
Fig. 7-18 A pole of an HVDC system.
© Copyright Ned Mohan 2006
118
Control of HVDC
Converters
Vd 1
Inverter characteristic
with γ = γ min
Rectifier characteristic
in a current-control mode
0
I d , ref
Id
Fig. 7-19 Control of an HVDC system [3].
© Copyright Ned Mohan 2006
119
A Voltage-Link HVDC
System in Northeastern
U.S.
© Copyright Ned
2006
Fig.Mohan
7-20 Voltage-link
HVDC transmission system [source: ABB].
120
Voltage-Link HVDC
System Block Diagram
+
AC1
−
P1 , Q1
P2 , Q2
AC2
Fig. 7-21 Block diagram of voltage-link HVDC system.
© Copyright Ned Mohan 2006
121
Phasor Diagram on the AcSide of the Voltage-Link
Converter
IL
IL
+
Vd
−
iL
vconv
vbus
L
(a )
+
+ jX L I L
Vconv
−
−
Vconv
+
−
( b)
jX L I L
Vbus
Vbus
IL
(c)
Fig. 7-22 Block diagram of a voltage-link converter and the phasor diagram.
© Copyright Ned Mohan 2006
122
Representation of VoltageLink Converter with Ideal
Transformers
a
b
c
+
Vd
ida
ia
+
Vd
−
1: d a
1: d b
(a )
1: d c
vaN
1: d a
−
( b)
Fig. 7-23 Synthesis of sinusoidal voltages.
© Copyright Ned Mohan 2006
123
Synthesis of “Average”
Sinusoidal Voltages
da
1
0.5
dˆa
ωt
0
vaN
Vd
0.5Vd
Vˆa
ωt
0
Fig. 7-24 Sinusoidal variation of turns-ratio d a .
© Copyright Ned Mohan 2006
124
Converter Output Voltages
and Voltages across the
Load
a
Vd
b
c
va
vb
vc
Vd
2
Vd
2
Vd
2
N
(a )
vaN
vbN
vcN
va
ac-side
0.5Vd
ωt
0
(a )
Fig. 7-25 Three-phase synthesis.
© Copyright Ned Mohan 2006
125
Switching Power-Pole of
Voltage-Link Converters
Buck
Boost
+
ida
+
a
+
vaN
−
N
Vd
−
qa
(a)
ia
ia
Vd
+
vaN
−
−
qa
qa−
(b)
Fig. 7-26 Realization of the ideal transformer functionality.
© Copyright Ned Mohan 2006
126
Switching in Sinusoidal
“Average” Voltage
Waveform
Vd
vaN
0
vaN
vaN
0
0
vaN
ωt
Ts
(a )
Vh
f1
fs
2 fs
3 fs
( b)
Fig. 7-27 PWM to synthesize sinusoidal waveform.
© Copyright Ned Mohan 2006
127
CHAPTER 8
Distribution System, Loads
and Power Quality
© Copyright Ned Mohan 2006
128
Residential Distribution
System
±120V
13.8kV
±120V
House1
House2
Transformer
±120V
House 3
Fig. 8-1 Residential distribution system.
© Copyright Ned Mohan 2006
129
Daily Load and LoadDuration Curves
peak
Load
(MW)
kW
12
6
AM
12
NOON
Time
6
PM
12
0
(a)
percentage of the time
100%
(b)
Fig. 8-2 System load.
© Copyright Ned Mohan 2006
130
Utility Load Distribution
Lighting 19%
29%
Industrial
35%
Commercial
36%
Residential
IT
14%
HVAC 16%
(a )
Motors 51%
( b)
Fig. 8-3 Utility loads.
© Copyright Ned Mohan 2006
131
Power Factor and Voltage
Sensitivity of Power
Systems Load
Table 8-1 Power Factor and Voltage Sensitivity of Power Systems Load
Type of Load
Power Factor
a = ∂P / ∂V
b = ∂Q / ∂V
Electric Heating
Incandescent Lighting
Fluorescent Lighting
Motor Loads
Modern PowerElectronics based
Loads
1.0
1.0
0.9
0.8 – 0.9
1.0
2.0
1.5
1.0
0.05 – 0.5
0
0
0
1.0
1.0 – 3.0
0
© Copyright Ned Mohan 2006
132
Voltage-Link System used
in Power Electronics
Based Loads
+
Vd
Utility
Load
−
Fig. 8-4 Voltage-link-system for modern and future power-electronics based loads.
© Copyright Ned Mohan 2006
133
Induction Motor Per-Phase
Diagram
Rs
+
Va
(at ω )
I ra '
Ia
jω Lls
Ema
−
jω Llr '
+
−
I ma
jω Lm
Rr '
ω syn
ω slip
Fig. 8-5 Per-phase, steady state equivalent circuit of a three-phase induction motor.
© Copyright Ned Mohan 2006
134
Torque-Speed
Characteristics
Tem
f5
0
f4
f3
f2
ωslip ωsyn
3
3
f1
Load
Torque
ωslip ωsyn
1
ωm
1
Fig. 8-6 Torque-speed characteristic of induction motor at various applied frequencies.
© Copyright Ned Mohan 2006
135
Switch-Mode DC Power
Supplies
input
rectifier
60Hz
ac
+
dc to HF ac
Vin
−
topology to convert
dc to dc with isolation
Output
Vo
HF transformer
Feedback
controller
Vo*
Fig. 8-7 Switch-mode dc power supply.
© Copyright Ned Mohan 2006
136
Uninterruptible Power
Supplies (UPS)
Rectifier
Inverter
Filter
Critical
Load
Energy
Storage
Fig. 8-8 Uninterruptible power supply.
© Copyright Ned Mohan 2006
137
Static Power-Transfer
Switch
Feeder 1
Load
Feeder 2
Fig. 8-9 Alternate feeder.
© Copyright Ned Mohan 2006
138
CBEMA Curve Showing
Acceptable Voltage-Time
Region
Fig. 8-10 CBEMA curve.
© Copyright Ned Mohan 2006
139
Dynamic Voltage
Restorers (DVR)
−
vinj
+
+
vs
−
Power Electronic
Interface
Load
Fig. 8-11 Dynamic Voltage Restorer (DVR).
© Copyright Ned Mohan 2006
140
Voltage Regulating
Transformers
Fig. 8-12 Three-Phase Voltage Regulator (Courtesy of Siemens) [5].
© Copyright Ned Mohan 2006
141
STATCOM
jX
Utility
STATCOM
Fig. 8-13 STATCOM [4].
© Copyright Ned Mohan 2006
142
Linear Load
is
+
Vs
vs
φ
−
(b)
Is
(a)
Figure 8-14 Voltage and current phasors in simple R-L circuit.
© Copyright Ned Mohan 2006
143
Waveforms Associated
with Power ElectronicsBased Load
vs
0
idistortion (= is − is1 )
is1
is
t
φ1 / ω
T1
0
t
( b)
(a )
Figure 8-15 Current drawn by power electronics equipment without PFC.
© Copyright Ned Mohan 2006
144
Example of Distorted
Current
is
(a)
−I
I
t
0
T1
4I / π
is1
(b)
t
0
idistortion
I
(c)
t
0
−I
© Copyright Ned Mohan 2006
Figure 8-16
Example 8-1.
Figure
5-4 Example
5-1.
145
Influence of Distortion on
Power Factor
1
0.9
0.8
0.7
PF
DPF 0.6
0.5
0.4
0
50
100
150
200
250
300
%THD
Fig. 8-17 Relation between PF/DPF and THD.
© Copyright Ned Mohan 2006
146
IEEE Harmonic Limits
5-1 Harmoniccurrent
current distortion
(Ih/I(1I)h / I 1 )
TableTable
8-1 Harmonic
distortion
Odd Harmonic Order h (in %)
Total
Harmonic
Distortion(%)
h < 11
11 ≤ h < 17
17 ≤ h < 23
23 ≤ h < 35
35 ≤ h
< 20
4.0
2.0
15
.
0.6
0.3
5.0
20 − 50
7.0
3.5
2.5
10
.
0.5
8.0
50 − 100
10.0
4.5
4.0
15
.
0.7
12.0
100 − 1000
12.0
5.5
5.0
2.0
10
.
15.0
> 1000
15.0
7.0
6.0
2.5
14
.
20.0
I sc / I1
© Copyright Ned Mohan 2006
147
Short-Circuit Current
Zs
Zs
+
I sc
+
Vs
Vs
−
−
(a)
(b)
Figure
8-185-6
(a)(a)Utility
(b)short
Short-Circuit
Current.
Figure
UtilitySupply,
supply; (b)
circuit current.
© Copyright Ned Mohan 2006
148
Retail Price of Electricity in
the U.S.
Fig. 8-19 Average retail price of electricity to ultimate customers [4].
© Copyright Ned Mohan 2006
149
CHAPTER 9
SYNCHRONOUS
GENERATORS
© Copyright Ned Mohan 2006
150
Application of
Synchronous Generators
Steam at High pressure
Heat in
Boiler
Turbine
Water
Gen
Penstock
H
Pump
Condenser
(a )
Generator
Turbine
Heat out
(b)
Fig. 9-1 Synchronous generators driven by (a) steam turbines, and (b) hydraulic turbines.
© Copyright Ned Mohan 2006
151
Cross-section of
Synchronous Generators
Stator
Air gap
(a)
(b)
Fig. 9-2 Machine cross-section.
© Copyright Ned Mohan 2006
152
Synchronous Generator
Structure
S
N
N
S
N
N
S
S
N
S
(a)
(b)
(c)
Fig. 9-3 Machine structure.
© Copyright Ned Mohan 2006
153
Sinusoidally-Distributed
Windings
b − axis
ib
3'
2π / 3
2π / 3
ia
2π / 3
a − axis
4'
5'
1'
7'
1
7
ic
5
(a)
θ
ia
2
6
c − axis
ia
6'
2'
4
3
(b)
Fig. 9-4 Three phase windings on the stator.
© Copyright Ned Mohan 2006
154
Three-Phase Winding
Connection in a Wye
b − axis
∠120 o
b
ib
b
b'
ib
θ
a − axis
a ∠0 o
a ' ia
ic
ia
a
c'
c
ic
c
c − axis
∠240 o
(a)
(b)
Fig. 9-5 Connection of three phase windings.
© Copyright Ned Mohan 2006
155
Synchronous Generator
Rotor Field
θ
N
ωsyn
a-axis
S
Fig. 9-6 Field winding on the rotor that is supplied by a dc current I f .
© Copyright Ned Mohan 2006
156
Voltage induced in the
Stator Phase due to
Rotating Rotor Field
+
θ
N
ωsyn
a-axis
ea
S
−
Fig. 9-7 Current direction and voltage polarities; the rotor position shown induces
maximum ea .
© Copyright Ned Mohan 2006
157
Representation of Induced
Stator Voltage due to
Rotor Field
G
B f (at t = 0)
+
θ
N
ωsyn
a-axis
ωsyn
eaf
Im
N
Re
S
a-axis
−
(a )
Eaf
S
( b)
(c)
Fig. 9-8 Induced emf eaf due to rotating rotor field with the rotor.
© Copyright Ned Mohan 2006
158
Armature Reaction Due to
Three Stator Currents
e
b − axis
j
2π
3
Im
ib
2π / 3
2π / 3
2π / 3
ia
e j0
a − axis
Re
θ
900
ic
Ea , AR
e
4π
3
(a )
θ
Ia
c − axis
j
a-axis
(b )
G
B AR (at t = 0)
(c)
Fig. 9-9 Armature reaction due to phase currents.
© Copyright Ned Mohan 2006
159
Superposition of the two
Induced Voltages and PerPhase Representation
−
Ea , AR
+
+
jX m I a
−
Ia
Im
Eaf
Ea , AR
Re
jX m I a
+
Eaf
Ea
X As
Rs
+
Ea
Va
−
−
−
Ia
(a)
+
(b )
Fig. 9-10 Phasor diagram and per-phase equivalent circuit.
© Copyright Ned Mohan 2006
160
Power Out as a function of
rotor Angle
P
steady state
stability limit
Ia
+
jX T
generator
mode
+
−90 o
V∞ =V∞ ∠0 o
Eaf = Eaf ∠δ
−
0
90 o
δ
−
(a )
motoring
mode
steady state
stability limit
(b)
Fig. 9-11 Power output and synchronism.
© Copyright Ned Mohan 2006
161
Steady State Stability
Limit
Pe
Pe
Pe,max
Pe2
Pe1
0 δ1 δ 2
Pm2
Pm1
90 o
(a )
δ
0
90 o
(b)
δ
Fig. 9-12 Steady state stability limit.
© Copyright Ned Mohan 2006
162
Reactive Power Control by
Field Excitation
Eaf
Eaf
jX s I a
jX s I a
δ
Ia
Va
⎧
⎪
I aq ⎨
⎪⎩
(a )
Eaf
δ
Va
I aq
90 o
{
δ
jX s I a
Ia
Va
Ia
90 o
( b)
(c)
Fig. 9-13 Excitation control to supply reactive power.
© Copyright Ned Mohan 2006
163
Synchronous
Condenser
Synchronous
Condenser
Fig. 9-14 Synchronous Condenser.
© Copyright Ned Mohan 2006
164
Automatic Voltage
Regulation (AVR)
phase-controlled
rectifier
field winding
ac input
Generator
output
slip rings
ac regulator
Fig. 9-15 Field exciter for automatic voltage regulation (AVR).
© Copyright Ned Mohan 2006
165
Armature Reaction Flux in
Steady State Leading to
Synchronous Reactance
Fig. 9-16 Armature reaction flux in steady state.
© Copyright Ned Mohan 2006
166
Simulation of a ShortCircuit Assuming a
Constant-Flux Model
(a )
( b)
Fig. 9-17 Armature (a) and field current (b), after a sudden short circuit [source: 4].
© Copyright Ned Mohan 2006
167
Representation for Steady State,
Transient Stability and Fault
Analysis
Ia
Eaf
Eaf'
Im
Eaf''
jX s I a
jX s' I a
jX s'' I a
Eaf
Eaf'
Re
Ea
+
+
jX s I a
jX s' I a
jX s'' I a
−
+
Ea
Eaf''
−
−
(a )
Ia
(b)
Fig. 9-18 Synchronous generator modeling for transient and sub-transient conditions.
© Copyright Ned Mohan 2006
168
CHAPTER 10
VOLTAGE REGULATION
AND STABILITY IN
POWER SYSTEMS
© Copyright Ned Mohan 2006
169
A Radial System
VS PS + jQS
PS + jQS
PR + jQR VR
+
jX L
Load
(a)
PR + jQR
jX L
I
+
VS
VR
−
−
(b)
Fig. 10-1 A radial system.
© Copyright Ned Mohan 2006
170
Voltages and Current
Phasors with Both-Side
Voltages at 1 PU
VS
I
δ
δ /2
PS + jQS
jX L I
I
jX L
+
+
VS
VR
−
−
PR
QR
VR
(a)
(b)
Fig. 10-2 Phasor diagram and the equivalent circuit with VS = VR = 1pu .
© Copyright Ned Mohan 2006
171
Voltage Profile for Three
Values of SIL
+
+
VS
Vx
−
−
x
+
VR
VS
(1pu)
PR < SIL
PR > SIL
VR
(1pu)
−
(a)
Vx
(b)
Fig. 10-3 Voltage profile along the transmission line.
© Copyright Ned Mohan 2006
172
“Nose” Curves at Three
Power Factors as a
function of Loading
VS
VR
VS
VR
1.4
1.2
1
jX L
0.6
(a )
PF = 1
0.8
PR + jQR
PF = 0.9
(lagging)
PF = 0.9
(leading)
0.4
0.2
0
0
0.5
1
1.5
2
( b)
2.5
3
3.5
PR / SIL
Fig. 10-4 Voltage collapse in a radial system (example of 345-kV line, 200 km long).
© Copyright Ned Mohan 2006
173
Synchronous Generator
Reactive Power Supply
Capability
Q
B
A
0
P
C
Fig. 10-5 Reactive power supply capability of synchronous generators.
© Copyright Ned Mohan 2006
174
Effect of Current Power
Factor on Bus Voltage
I
+
+
jX Th I
jX Th
−
I
+
VTh
Vbus
VTh
jX Th I
Vbus
−
Vbus
jX Th I
−
VTh
(a)
(b)
I
Fig. 10-6 Effect of leading and lagging currents due to the shunt compensating device.
© Copyright Ned Mohan 2006
175
Static Var
Compensators (SVC)
Vbus
Vbus
1
jωC
IC
IC
(a )
0
( b)
Fig. 10-7 V-I characteristic of SVC.
© Copyright Ned Mohan 2006
176
Thyristor Controlled
Reactors (TCR)
Vbus
Vbus
vbus
IL
iL
iL
α ≤ 900
α > 900
0
(a )
( b)
(c )
IL
Fig. 10-8 Thyristor-Controlled Reactor (TCR).
© Copyright Ned Mohan 2006
177
Voltage Control by SVC
and TCR Combination
Vbus
Vbus
I
1
jωC
IL
IC
Vbus
A
B
V2
Linear
Range
capacitive
(a )
C
V1′
0
inductive
( b)
I
capacitive
V2′
V1
0
inductive
(c)
I
Fig. 10-9 Parallel combination of SVC and TCR.
© Copyright Ned Mohan 2006
178
STATCOM
I conv
I conv
Vbus
+
Vconv
jXI conv−
+
+
+
jXI conv
Vd
−
+
Vconv
Vbus
−
(a )
−
−
( b)
Fig. 10-10 STATCOM.
© Copyright Ned Mohan 2006
179
STATCOM V-I
Characteristic
Vbus
Linear
Range
capacitive
0
inductive I
conv
Fig. 10-11 STATCOM VI characteristic.
© Copyright Ned Mohan 2006
180
Thyristor-Controlled
Series Capacitor (TCSC)
(a )
(b)
(c)
Fig. 10-12 Thyristor-Controlled Series Capacitors (TCSC) [source: Siemens Corp.].
© Copyright Ned Mohan 2006
181
CHAPTER 11
TRANSIENT AND
DYNAMIC STABILITY OF
POWER SYSTEMS
© Copyright Ned Mohan 2006
182
One-Machine Infinite-Bus
System
V1
XL
VB = VB ∠0
j ( X d' + X tr )
E ′∠δ
Pm
Pe
XL
(a )
+
−
+
V1
−
I
jX L / 2
+
VB ∠0
−
( b)
Fig. 11-1 Simple one-generator system connected to an infinite bus.
© Copyright Ned Mohan 2006
183
Power-Angle
Characteristic in OneMachine Infinite-Bus
System
Pre-fault
post-fault
Pe
Bus 1
XL
VB = VB ∠0
Pm
during-fault
0
π
δ 0 δ1
(a )
δ
Pm
XL
Pe
(b)
Fig. 11-2 Power-angle characteristics.
© Copyright Ned Mohan 2006
184
Rotor-Angle Swing
Following a Fault and a
Line Taken Out
5 5
5 0
4 5
4 0
3 5
3 0
2 5
2 0
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
Fig. 11-3 Rotor-angle swing in Example 11-1.
© Copyright Ned Mohan 2006
185
Power-Angle
Characteristics
Pe
Bus 1
XL
Pre-fault
VB = VB ∠0
post-fault
B
Pe = Pm
A
Pm
during fault
XL
Pe
0
(a )
δ0
δ cA
δm
(b)
δ max
π
δ
Fig. 11-4 Fault on one of the transmission lines.
© Copyright Ned Mohan 2006
186
Rotor Oscillations After
the Fault is Cleared
Pe
Pre-fault
post-fault
C
Pe = Pm
D
δ2
0
δ1
δm
π
δ
Fig. 11-5 Rotor oscillations after the fault is cleared.
© Copyright Ned Mohan 2006
187
Critical Clearing Angle
using Equal-Area Criterion
Pe
Pre-fault
post-fault
B
Pe = Pm
A
0
δ 0 δ1
δ crit
δ max
π
δ
Fig. 11-6 Critical clearing angle.
© Copyright Ned Mohan 2006
188
Example using Equal-Area
Criterion
Pe ( pu ) 40
35
Pre-fault
30
25
20
Pe = Pm 15
A
10
during fault
5
0
0
post-fault
B
20
40
δ 0 = 22.470
60
δ cA =80750
100
1400
δ m 120
= 115.28
160
180
Fig. 11-7 Power angle curves and equal-area criterion in Example 11-2.
© Copyright Ned Mohan 2006
189
Transient Stability
Calculations in Large
Networks
Initial Power Flow
Calculate Pe and E ' ∠δ
for each generator
for k = 1,2,3....
Pm , k = Pe, k
Pm , k and Ek' held constant
Electro-dynamic
differential
Equations
ωk and δ k
Pe , k
Phasor Calculations
using Ek' ∠δ k
(load may be assumed
as a constant impedance)
Fig. 11-8 Block diagram of transient stability program for an n-generator case.
© Copyright Ned Mohan 2006
190
Example Power System for
Transient Stability
Analysis
Bus-1
Pm1
Bus-3
Pe1
Bus-2
Pe 2
Pm 2
Fig. 11-9 A 345-kV test example system.
© Copyright Ned Mohan 2006
191
Rotor Angle Swings in the
Example Power System
Following a Fault
8 0 0
7 0 0
6 0 0
5 0 0
4 0 0
3 0 0
δ1
2 0 0
1 0 0
0
0
0 .2
0 .4
δ2
0 .6
0 . 8
1
1 .2
1 .4
1 .6
Fig. 11-10 Rotor-angle swings of δ1 and δ 2 in Example 11-3.
© Copyright Ned Mohan 2006
192
Importance of Dynamic
Stability
Fig. 11-11 Growing Power Oscillations: Western USA/Canada system, Aug 10, 1996 [4].
© Copyright Ned Mohan 2006
193
CHAPTER 12
CONTROL OF
INTERCONNECTED
POWER SYSTEM AND
ECONOMIC DISPATCH
© Copyright Ned Mohan 2006
194
Automatic Voltage
Regulation (AVR)
phase-controlled
rectifier
field winding
ac input
Generator
output
slip rings
ac regulator
Fig. 12-1 Field exciter for automatic voltage regulation (AVR).
© Copyright Ned Mohan 2006
195
Control Areas (Balancing
Authorities)
(a )
( b)
Fig. 12-2 (a) The Interconnections in North America, (b) Control Areas [Source: 2]
© Copyright Ned Mohan 2006
196
Load-Frequency Control
and Regulation
Frequency
G
f
1
R
Supplementary
Control
Regulator
Turbine
Governor
f0
Steam-Valve
Position
a
Δf
b
slope = − R
G
Turbine
Pm
Pm
(a )
Pe
PLoad
0
(b)
ΔPm
Pm
Fig. 12-3 Load-Frequency Control (ignore the supplementary control at present).
© Copyright Ned Mohan 2006
197
Load Sharing
unit 2
Pm 2
f
G′
G1 unit1 1
f0
Pe 2
unit 2
G2
a
Δf
b
e
c
d
( ΔPm1 + ΔPm 2 ) G1
unit 1
G1′
G2
Pm 2
Pm1
Pm1
Pe1
(a )
PLoad
0
ΔPm 2
ΔPm1
Pm
(b)
Fig. 12-4 Response of two generators to load-frequency control.
© Copyright Ned Mohan 2006
198
Synchronizing Torque
between Two Control
Areas
Area 1
P12
P21
jX 12
Area 2
Fig. 12-5 Two control areas.
© Copyright Ned Mohan 2006
199
Automatic Generation
Control (AGC) and Area
Control Error (ACE)
Δf (frequency deviation)
1
R
B
Supplementary
Controller
+
+
ACE
(Area Control Error)
k
s
−
−
Governor
Change in Steam Valve Position
ΔP (tie-line flow deviation)
Fig. 12-6 Area Control Error (ACE) for Automatic Generation Control (AGC).
© Copyright Ned Mohan 2006
200
Two Control Areas in the
Example Power System
Bus-1
Bus-3
P1−3
M
Pm1
Area 1
Pe1
P1− 2
Area 2
Load
M
Bus-2
Pe 2
Pm 2
Fig. 12-7 Two control areas in the example power system with 3 buses.
© Copyright Ned Mohan 2006
201
Power Flow on Tie-Lines
between Two Control
Areas Following a Load
Change
© Copyright Ned Mohan 2006
202
Electrical Equivalent of
Two Areas
E1∠δ1
+
−
jX 1
jX 12
jX 2
+
E2 ∠δ 2
−
Fig. 12-9 Electrical equivalent of two area interconnection.
© Copyright Ned Mohan 2006
203
Modeling of Two Control
Areas with AGC
1/ ωsyn
1
R1
B1
+
ACE1
+
1
K1
s Ts1s + 1
Regulator
ΔPLoad 1
−
−
ΔPs1
1
TG1s + 1
1
TS 1s + 1
ΔPv1
Governor
Steam Turbine
+
ΔPm1
−
−
ΔP12 = T12 ( Δδ1 − Δδ 2 )
ωsyn
sΔδ1
M 1s + D1
Area 1
T12
Δδ1
( Δδ 1 − Δ δ 2 )
1
s
−
Δδ 2
−
ACE 2
+
K2
1
s Ts 2 s + 1
ΔPs 2
−
−
Regulator
1
R2
B2
1
TG 2 s + 1
Governor
ΔPv 2
1
TS 2 s + 1
Steam Turbine
ΔPm 2
+
+
−
1
s
ωsyn
M 2 s + D2
+
s Δδ 2
Area 2
ΔPLoad 2
1/ ωsyn
Fig. 12-10 Two-area system with AGC. Source: adapted from [6].
© Copyright Ned Mohan 2006
204
Results of Simulink
Modeling Following a Step
Load Change in Control
Area 1
1.5
1
ΔPm1
0.5
ΔPm 2
0
-0.5
ΔP12
-1
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Fig. 12-11 Simulink results of the two-area system with AGC in Example 12-3.
© Copyright Ned Mohan 2006
205
Economic Dispatch: Heat
Rate of a Power Plant
Heat Rate
11.0
At this point, to produce 40 MW
Fuel Consumption = 400 MBTU-per-Hour
MBTU-per-Hour
10.0
MW
9.0
0
20
40
60
80
100 P [ MW ]
Fig. 12-12 Heat Rate at various generated power levels.
© Copyright Ned Mohan 2006
206
Cost Curve and Marginal
Cost of a Power Plant
Ci ( Pi )
∂Ci ( Pi )
∂Pi
[$ / hour ]
[$ / MWh ]
ΔPi
0
ΔCi
(a )
Pi [ MW ]
0
( b)
Pi [ MW ]
Fig. 12-13 (a) Fuel cost and (b) Marginal cost, as functions of the power output.
© Copyright Ned Mohan 2006
207
Load Sharing between
Three Power Plants
∂C1 ( P )
∂P
∂C2 ( P )
∂P
λ
0
P1
P
0
∂C3 ( P )
∂P
P2
P
0
P3
P
Fig. 12-14 Marginal costs for the three generators.
© Copyright Ned Mohan 2006
208
CHAPTER 13
TRANSMISSION LINE
FAULTS, RELAYING AND
CIRCUIT BREAKERS
© Copyright Ned Mohan 2006
209
Fault (Symmetric or
Unsymmetric) on a
Balanced Network
a
f
b
a
ia
c
ib
f
b
Ia
c
Ib
ic
g
(a )
Ic
g
( b)
Fig. 13-1 Fault in power system.
© Copyright Ned Mohan 2006
210
Symmetrical
Components
Ia
Ic
I c0 I
I c1
I a1
Ia2
I b2
b0
Ia0
Ib
Ic2
I b1
Fig. 13-2 Sequence components.
© Copyright Ned Mohan 2006
211
Sequence Networks: PerPhase Representation of a
Balanced Three-phase
representation
Z1
+
Ea1
−
Z2
+ I a1
Va1
−
Z0
+ Ia2
Va 2
−
+ Ia0
Va 0
−
Fig. 13-3 Sequence networks.
© Copyright Ned Mohan 2006
212
Three-Phase Symmetrical
Fault (ground may or may
no be involved)
f
a
Z1
b
Ia
c
Ib
−
Ic
g
+
Va1 = 0
−
+
Ea 1
I a1
(b)
(a )
Fig. 13-4 Three-phase symmetrical fault.
© Copyright Ned Mohan 2006
213
Single-Line to Ground
(SLF) Fault through a Fault
Impedance
+
f
Ea1
a
I a1
+
Z1
Va1
−
−
Ia2
b
+
Z2
Ia
c
3Z f
Va 2
−
Ia0
Zf
+
Z0
g
Va 0
−
(a )
( b)
Fig. 13-5 Single line to ground fault.
© Copyright Ned Mohan 2006
214
Double-Line to Ground
Fault
a
b
c
g
f
I a1
+
Ic
Ib
(a )
Ea1
−
Z1
+
Va1
−
Ia2
Z2
+
Va 2
−
Ia0
Z0
+
Va 0
−
( b)
Fig. 13-6 Double line to ground fault.
© Copyright Ned Mohan 2006
215
Double-Line Fault (ground
not involved)
+ Z f I a1 −
f
a
b
c
g
I a1
Ib
Ic
(a )
+
Ea1
−
Z1
Ia2
+
Va1
−
Z2
+
Va 2
−
(b)
Fig. 13-7 Double line fault (ground not involved).
© Copyright Ned Mohan 2006
216
Path for Zero-Sequence
Currents
(a)
( b)
(c)
Fig. 13-8 Path for zero-sequence currents in transformers.
© Copyright Ned Mohan 2006
217
Neutral Grounded through
an Neutral Impedance)
Ia0
Z0
Zn
(a )
3Z n
+
Va 0
−
( b)
Fig. 13-9 Neutral grounded through an impedance.
© Copyright Ned Mohan 2006
218
One-Line Diagram of a
Simple System)
V1 = 1.0∠0 pu
Bus-2
V3 = 0.98∠ − 11.790 pu
X Line1 = 0.10 pu
′′ 1 = 0.12 pu
X gen
X gen 2 = 0.12 pu
X gen 0 = 0.06 pu
Bus-1
X tr1 = 0.10 pu
X Line 2 = 0.10 pu
X Line 0 = 0.20 pu
Bus-3
PLoad = 1 pu
QLoad = 0
X tr 2 = 0.10 pu
X tr 0 = 0.10 pu
Fig. 13-10 (a) One-line diagram of a simple power system and bus voltages.
© Copyright Ned Mohan 2006
219
Thee-phase Fault on Bus-2
in the Simple System
j 0.12 pu
+
+
j 0.10 pu
I Load
I fault
V1 = 1.0∠0 pu
E ′′
j 0.10 pu
RLoad = 0.9604 pu
−
−
Fig. 13-11 Positive-sequence circuit for calculating a 3-phase fault on bus-2.
© Copyright Ned Mohan 2006
220
Single-Line to Ground
(SLG) Fault in the Simple
System
I a / 3 ( = I fault / 3)
I a1
+
E ′′
j 0.12 pu
+
+ j 0.10 pu +
V1 = 1.0∠0 pu
Va1
−
−
−
j 0.10 pu
RLoad = 0.9604 pu
Ia2
j 0.12 pu
+
j 0.10 pu
Va 2
−
j 0.10 pu
RLoad = 0.9604 pu
Va = 0
Ia0
j 0.06 pu
j 0.10 pu
+
Va 0
−
j 0.20 pu
RLoad = 0.9604 pu
−
Fig. 13-12 Sequence networks for calculating fault current due to SLG fault on bus-2.
© Copyright Ned Mohan 2006
221
An SLG Fault in the
Example 3-Bus System
Bus-1
Pm1
Bus-3
Pe1
Bus-2
Pe 2
Pm 2
Fig. 13-13 A SLG fault in the example 3-bus power system.
© Copyright Ned Mohan 2006
222
Protection in Power
System
CT
PT
CB
R
Fig. 13-14 Protection equipment.
© Copyright Ned Mohan 2006
223
Current Transformers
(CT)
CT
Burden
(b)
(a)
Fig. 13-15 Current Transformer (CT) [5].
© Copyright Ned Mohan 2006
224
Capacitor-Coupled Voltage
Transformers (CCVT)
(a)
(b)
Fig. 13-16 Capacitor-Coupled Voltage Transformer (CCVT) [5].
© Copyright Ned Mohan 2006
225
Differential Relays
CT
CT
CT
Relay
Fig. 13-17 Differential relay.
© Copyright Ned Mohan 2006
226
Directional Over-Current
Relays
CT
PT
CB
R
Fig. 13-18 Directional over-current Relay.
© Copyright Ned Mohan 2006
227
Ground Directional OverCurrent Relays for Ground
Faults
Time
instantaneous
Fig. 13-19 Ground directional over-current Relay.
© Copyright Ned Mohan 2006
228
Impedance (Distance)
Relays
X
R
Fig. 13-20 Impedance (distance) relay.
© Copyright Ned Mohan 2006
229
Microwave Terminal for
Pilot Relays
Fig. 13-21 Microwave terminal [5].
© Copyright Ned Mohan 2006
230
Zones of Protection
Zone 3: 1-1.5 sec
Time
Zone 2: 20-25 cycles
Zone 1: instantaneous
A
B
C
Fig. 13-22 Zones of protection.
© Copyright Ned Mohan 2006
231
Protection of Generator
and its Step-up
Transformer
Transformer
Relay
CT
Gen
Relay
F1 CT
F2
CT
Relay
Fig. 13-23 Protection of generator and the step-up transformer.
© Copyright Ned Mohan 2006
232
Relaying in the 3-Bus
Example Power System
Zone 2
Zone1
A
B
Zone 2
P + jQ
Fig. 13-24 Relying in the example 3-bus power system.
© Copyright Ned Mohan 2006
233
Circuit Breakers
Fig. 13-25 SF6 circuit breaker [5].
© Copyright Ned Mohan 2006
234
Illustration of Current
Offset in R-L Circuits
2
L
R
+
vs ( t )
+
offset
1
i (t )
v (t )
−
asymmetric
symmetric
1 .5
0 .5
00
- 0 .5
−
-1
0
0 .05
(a )
0.1
0 .15
0 .2
(b)
Fig. 13-26 Current in an RL circuit.
© Copyright Ned Mohan 2006
235
CHAPTER 14
TRANSIENT OVER-VOLTAGES,
SURGE PROTECTION AND
INSULATION COORDINATION
© Copyright Ned Mohan 2006
236
Lightning Current
Impulse
i
I peak
0.5I peak
t1
t2
t[ μ s ]
Fig. 14-1 Lightening current impulse.
© Copyright Ned Mohan 2006
237
Lightening Strike to Shield
Wire and Backflash
(a )
(b)
Fig. 14-2 Lightening strike to the shield wire.
© Copyright Ned Mohan 2006
238
Switching Surges
va
vb
L
L
A
B
500 kV Line
100 miles
vc
(open)
L
C
(a )
( b)
Fig. 14-3 Over-voltages due to switching of transmission lines.
© Copyright Ned Mohan 2006
239
Frequency Dependence of
Transmission Line
Parameters
Fig. 14-4 Frequency dependence of the transmission line parameters [Source: 2].
© Copyright Ned Mohan 2006
240
Calculation of Switching
Over-Voltages on Line 1-3
in the Example 3-Bus
Power System
Bus-1
Bus-3
Fig. 14-5 Calculation of switching over-voltages on a transmission line.
© Copyright Ned Mohan 2006
241
Standard Voltage Impulse
to Define Basic Insulation
Level (BIL)
i
V pea k
0.5 V pe ak
0 1 .2 μ s
40 μ s
t
Fig. 14-6 Standard Voltage Impulse Wave to define BIL.
© Copyright Ned Mohan 2006
242
Transformer Insulation
Protected by a ZnO
Arrester
1500
chopped
wave
Transformer Insulation
Withstand Capability Curve
BIL
1175kV
1300
Line-to-ground
(Peak kV) 1100
BSL
Arrester Voltage, subjected to a
8 × 20 μ s Lightning Current Impulse
with a peak of 20 kA
900
700
1
10
100
time in μ s
1000
10000
Fig. 14-7 A 345-kV transformer voltage insulation levels.
© Copyright Ned Mohan 2006
243