© Copyright Ned Mohan 2006 1
Ned Mohan Oscar A. Schott Professor of Power Electronics and Systems Department of Electrical and Computer Engineering University of Minnesota Minneapolis, MN 55455 USA
First Course on
POWER SYSTEMS
© Copyright Ned Mohan 2006 2
Bus-1 Bus-3
Bus-2
1mP 1eP
2mP
2eP
P jQ+
200km
150km150km
A 345-kV Example System
© Copyright Ned Mohan 2006 3
TOPICS IN POWER SYSTEMS Week Book Chapters Laboratory
1 Chapter 1: Overview Chapter 2: Fundamentals
Lab 1: Visit to a local substation; otherwise a virtual substation
2 Chapter 3: Energy Sources Lab 2: Introduction to PSCAD/EMTDC; 3-phase circuits, vars, power-factor correction
3 Chapter 4: Transmission Lines Lab 3: Transmission Lines using PSCAD-EMTDC
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 Over-Voltages, Surge Arrestors, Insulation Coordination
Lab 13: Over-voltages and Surge Arrestors using PSCAD-EMTDC
© Copyright Ned Mohan 2006 4
Chapter 1
POWER SYSTEMS: A CHANGING LANDSCAPE
© Copyright Ned Mohan 2006 5
Fig. 1-1 Interconnected North American Power Grid [2].
NATURE OF POWER SYSTEMS
© Copyright Ned Mohan 2006 6
Fig. 1-2 NERC Interconnections [3]. Source: NERC.
Control Areas
© Copyright Ned Mohan 2006 7
Fig. 1-3 One-line diagram as an example.
13.8 kVTransmission line
Generator
Load
Feeder
Step up Transformer
One-line Diagram
© Copyright Ned Mohan 2006 8
CHANGING LANDSCAPE OF POWER SYSTEMS AND UTILITY DEREGULATION
Fig. 1-4 Changing landscape [4]. Source: ABB. ( )a ( )b( )a ( )b
© Copyright Ned Mohan 2006 9
CHAPTER 2
REVIEW OF BASIC ELECTRIC CIRCUITS AND ELECTROMAGNETIC CONCEPTS
© Copyright Ned Mohan 2006 10
Fig. 2-1 Convention for voltages and currents.
i
av
−
+
bv
ba abv
+−
−
+ i
av
−
+
bv
ba abv
+−
−
+
Symbols and Conventions
© Copyright Ned Mohan 2006 11
Fig. 2-2 Phasor diagram.
Imaginary
Real
I I φ= ∠ −
V V 0= ∠
φ−
positiveangles
Imaginary
Real
I I φ= ∠ −
V V 0= ∠
φ−
positiveangles
Phasors
© Copyright Ned Mohan 2006 12
Fig. 2-3 A circuit (a) in time-domain and (b) in phasor-domain; (c) impedance triangle.
Im
Z
cjX−
R
LjX
Re0
i( t )
L
R
C
+
−
v( t )2V cos( t )ω=
V V 0= ∠
I
Lj L j Xω =
R
C1j j XCω
⎛ ⎞− = −⎜ ⎟⎝ ⎠
+
−
( )a ( )b ( )c
Im
Z
cjX−
R
LjX
Re0
Im
Z
cjX−
R
LjX
Re0
i( t )
L
R
C
+
−
v( t )2V cos( t )ω=
V V 0= ∠
I
Lj L j Xω =
R
C1j j XCω
⎛ ⎞− = −⎜ ⎟⎝ ⎠
+
−
( )a ( )b ( )c
i( t )
L
R
C
+
−
v( t )2V cos( t )ω=
i( t )
L
R
C
+
−
v( t )2V cos( t )ω=
V V 0= ∠
I
Lj L j Xω =
R
C1j j XCω
⎛ ⎞− = −⎜ ⎟⎝ ⎠
+
−V V 0= ∠
I
Lj L j Xω =
R
C1j j XCω
⎛ ⎞− = −⎜ ⎟⎝ ⎠
+
−
( )a ( )b ( )c
Phasor Analysis
© Copyright Ned Mohan 2006 13
Fig. 2-4 Impedance network of Example 2-1.
5j− Ω
0.1j Ω
2Ω5j− Ω
0.1j Ω
2Ω
Example of Impedance Calculation
© Copyright Ned Mohan 2006 14
Fig. 2-5 Circuit of Example 2-2.
0.3Ω 0.5j Ω 0.2j Ω
15j Ω 7.0Ω
+
−
1V
1I
mI 2I
Example of Impedance Calculation
© Copyright Ned Mohan 2006 15
Figure 2-6 A generic circuit divided into two sub-circuits.
( ) ( ) ( )p t v t i t=−
+
Subcircuit 1 Subcircuit 2( )v t
Power Flow
© Copyright Ned Mohan 2006 16
Figure 2-7 Instantaneous power with sinusoidal currents and voltages.
( )i t( )v t
t
( )p t average power
0
( )i t
( )v t/φ ω
t
( )p t average power
0
( )a ( )b( )i t
( )v tt
( )p t average power
0
( )i t
( )v t/φ ω
t
( )p t average power
0
( )a ( )b
Real and Reactive Power
© Copyright Ned Mohan 2006 17
Fig. 2-8 (a) Circuit in phasor-domain; (b) phasor diagram; (c) power triangle.
V
I
S P jQ= +
−
+
Subcircuit 1 Subcircuit 2
vV V φ= ∠
iI I φ= ∠
φ
Im
Reφ
Q
P
SIm
Re
( a )
( b ) ( c )
V
I
S P jQ= +
−
+
Subcircuit 1 Subcircuit 2
vV V φ= ∠
iI I φ= ∠
φ
Im
Reφ
Q
P
SIm
Re
( a )
( b ) ( c )
P, Q and VA by Phasors
© Copyright Ned Mohan 2006 18
Fig. 2-9 Power factor correction in Example 2-5.
+
−
1V
LP P=
CjQ−L LP jQ+
13.963j− Ω
Example of Power Factor Correction
© Copyright Ned Mohan 2006 19
Fig. 2-10 One-line diagram of a three-phase transmission and distribution system.
Feeder
Step up Transformer
GeneratorTransmission
line13.8 kV
Load
One-line Diagram
© Copyright Ned Mohan 2006 20
Fig. 2-11 Three-phase voltages in time and phasor domain.
( )bnv t
tω0
23
π
( )cnv t( )anv t
23
π
120°
a b c− −positivesequence
bnV
cnV
anV120°
120°
( )a
( )b
Three-Phase Voltages
© Copyright Ned Mohan 2006 21
Fig. 2-12 Balanced wye-connected, three-phase circuit.
aI
bnVcnV −+
cI
LZ
bIc b
a
n N
anV −
+
−+
aI
bnVcnV −+
cI
LZ
bIc b
a
n N
anV−
+
−+
nI
(a) (b)
aI
bnVcnV −+
cI
LZ
bIc b
a
n N
anV −
+
−+
aI
bnVcnV −+
cI
LZ
bIc b
a
n N
anV −
+
−+
aI
bnVcnV −+
cI
LZ
bIc b
a
n N
anV−
+
−+
nI
(a) (b)
Balanced Three-Phase Circuit Analysis
© Copyright Ned Mohan 2006 22
Fig. 2-13 Per-phase circuit and the corresponding phasor diagram.
aIa
Nn
anV−
+
(Hypothetical)
a
aIbI
cI
anV
cnV
bnV
φ
( a ) ( b )
aIa
Nn
anV−
+
(Hypothetical)
aaI
a
Nn
anV−
+
(Hypothetical)
a
aIbI
cI
anV
cnV
bnV
φ
( a ) ( b )
Per-Phase Analysis
© Copyright Ned Mohan 2006 23
Fig. 2-14 Balanced three-phase network with mutual couplings.
a AaAZ
(Hypothetical)
selfZ
selfZ
selfZ
mutualZ
mutualZ
mutualZ
a
b
c
A
B
C
aI
bI
cI
( )a ( )b
Balanced Mutual Coupling
© Copyright Ned Mohan 2006 24
Fig. 2-15 Line-to-line voltages in a three-phase circuit.
o30
anV
bV−abV
cnV
caV
bcV
bnV
Line-Line Voltages
© Copyright Ned Mohan 2006 25
Fig. 2-16 Delta-wye transformation.
aI
ZΥ
cb
a
ZΔ
aI
bcIcaIabI
ZΥ ZΥc b
a
( b )( a )
ZΔ
ZΔ
aI
ZΥ
cb
a
ZΔ
aI
bcIcaIabI
ZΥ ZΥc b
a
( b )( a )
ZΔ
ZΔ
Wye-Delta Transformation
© Copyright Ned Mohan 2006 26
Fig. 2-17 Power transfer between two ac systems.
sVRV
+
−
+
−
I
jX
RV
sV
I
δφ
( )a
( )b
jXI
Power Flow in AC Systems
© Copyright Ned Mohan 2006 27
Fig. 2-18 Power as a function of δ .
max/P P
δ0
0.5
090 0180
1.0
Power-Angle Diagram
© Copyright Ned Mohan 2006 28
Per Unit Quantities , , base
base base basebase
VR X ZI
= (in Ω) (2-48)
, , basebase base base
base
IG B YV
= (in ) (2-49)
( ), ,base base base basebaseP Q VA V I= (in Watt, VAR, or VA) (2-50)
In terms of these base quantities, the per-unit quantities can be specified as
actual valuePer-UnitValue = base value
(2-51)
© Copyright Ned Mohan 2006 29
Fig. 2-19 Energy Efficiency /o inP Pη = .
inP oP
lossP
Power SystemApparatus
Energy Efficiency of Apparatus
© Copyright Ned Mohan 2006 30
Fig. 2-20 Ampere’s Law.
3i
2i
1iH
dl
(a) (b) (c)
3i
2i
1iH
dl
3i
2i
1iH
dl
(a) (b) (c)
Electro-Magnetic Concepts:Ampere’s Law
© Copyright Ned Mohan 2006 31
Fig. 2-21 Example 2-9.
i
OD
ID
i
OD
ID
OD
ID
mr
OD
ID
mr
(a) (b)
Example of a Toroid
© Copyright Ned Mohan 2006 32
Fig. 2-22 B-H characteristic of ferromagnetic materials.
mB
mH
satB
oμ
oμ
mμ
mB
mH
(a) (b)
mB
mH
mB
mH
satB
oμ
oμ
mμ
mB
mH
satB
oμ
oμ
mμ
mB
mH
(a) (b)
B-H Curves in Ferromagnetic Materials
© Copyright Ned Mohan 2006 33
Fig. 2-23 Toroid with flux mφ .
mA
mφ
mA
mφ
Flux and Flux-Density
© Copyright Ned Mohan 2006 34
Inductance
Fig. 2-24 Coil inductance.
2
mm
m m
NL
Aμ
=
mλ( )N×
mφ( )mA×( )mμ×
mBmHm
N⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠i
2
mm
m m
NL
Aμ
=
mλ( )N×
mφ( )mA×( )mμ×
mBmHm
N⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠i
(a) (b)
mμ
mφmA
i
N
mμ
mφmA
i
N
© Copyright Ned Mohan 2006 35
Fig. 2-25 Rectangular toroid.
h
w r
h
w r
Example of a Toroid
© Copyright Ned Mohan 2006 36
Fig. 2-26 Voltage polarity and direction of flux and current.
( )i t
( )tφ
( )e t+
−N
( )i t
( )tφ
( )e t+
−N
Faraday’s Law
© Copyright Ned Mohan 2006 37
Fig. 2-27 Example 2-11.
( )tφ( )e t
t0
( )tφ( )e t
t0
Plot of time-varying Flux and Voltage
© Copyright Ned Mohan 2006 38
Fig. 2-28 Including leakage flux. (a) (b)
i
−
+
e
i
−
+
e
i
−
+e
mφ
lφ
i
−
+e
mφ
lφ
Leakage Flux
© Copyright Ned Mohan 2006 39
Fig. 2-29 Analysis including the leakage flux.
( )v t+
−
R
mφ
lL ( )i t
( )me t( )e t+
−
+
−( )v t+
−
R
mφ
lL ( )i t
( )me t( )e t+
−
+
−
ldiLdt
( )me t( )e t
+
−
+
−
+ −( )i t
mL
lL
ldiLdt
( )me t( )e t
+
−
+
−
+ −( )i t
mL
lL
(a) (b)
Representation of Leakage Flux by Leakage Inductance
© Copyright Ned Mohan 2006 40
CHAPTER 3
ELECTRIC ENERGY AND THE ENVIRONMENT
© Copyright Ned Mohan 2006 41
Fig. 3-1 Production and consumption of energy in the United States in 2004 [1]. ( )a ( )b
Energy Consumption and Production in the U.S.
© Copyright Ned Mohan 2006 42
Fig. 3-2 Electric power generation by various fuel types in the U.S. in 2005 [1].
Power Generation by Various Fuel Types in the U.S.
© Copyright Ned Mohan 2006 43
Fig. 3-3 Hydro power (Source: www.bpa.gov).
HGenerator
Penstock
Turbine
Water
Hydro Power Generation
© Copyright Ned Mohan 2006 44
Fig. 3-4 Rankine thermodynamic cycle in coal-fired power plants.
Steam at High pressure
Pump
Heat in
Heat out
TurbineBoiler
Condenser
Gen
Rankine Thermodynamic Cycle in Coal Plants
Visit the following website for Power Plant Animations:
http://www.cf.missouri.edu/energy/?fun=1&flash=ppmap
© Copyright Ned Mohan 2006 45
Fig. 3-5 Brayton thermodynamic cycle in natural-gas power plants.
Air in
Compressor Turbine
Exhaust
Fuel in
CombustionChamber
Brayton Cycle in Gas Turbines
© Copyright Ned Mohan 2006 46
Fig. 3-6 (a) BWR and (b) PWR reactors [5]. ( )a ( )b
Nuclear Power Plant Types
Visit the following websites for Nuclear Power Plant Animations:PWR: http://www.nrc.gov/reading-rm/basic-ref/students/animated-pwr.htmlBWR: http://www.nrc.gov/reading-rm/basic-ref/students/animated-bwr.html
© Copyright Ned Mohan 2006 47
Fig. 3-7 Wind-resource map of the United States [6].
Wind Resources in the U.S.
© Copyright Ned Mohan 2006 48
Fig. 3-8 pc as a function of λ [7]; these would vary based on the turbine design.
Coefficient of Performance
© Copyright Ned Mohan 2006 49
Fig. 3-9 Induction generator directly connected to the grid [8].
Utility
InductionGenerator
WindTurbine
Wind Generation using an Induction Generator Connected Directly to the AC Grid
© Copyright Ned Mohan 2006 50
Fig. 3-10 Doubly-fed, wound-rotor induction generator [9].
AC
DC
DC
AC
Wound rotor Induction Generator
Generator-sideConverter
Grid-sideConverter
Wind Turbine
AC
DC
DC
AC
Wound rotor Induction Generator
Generator-sideConverter
Grid-sideConverter
Wind Turbine
Wind Generation using a Doubly-Fed Induction Generator
© Copyright Ned Mohan 2006 51
Fig. 3-11 Power Electronics connected generator [10].
Gen
Utility
Power Electronics Interface
1Conv 2Conv
Wind Generation using an AC Generator Connected through Power Electronics
© Copyright Ned Mohan 2006 52
Fig. 3-12 PV cell characteristics [11].
Photovoltaics
© Copyright Ned Mohan 2006 53
Fig. 3-13 Photovoltaic systems.
Isolated DC-DC
Converter
PWM Converter
Max. Power-point Tracker
Utility1φ
Isolated DC-DC
Converter
PWM Converter
Max. Power-point Tracker
Utility1φ
Interfacing PV with AC Grid
© Copyright Ned Mohan 2006 54
Fig. 3-14 Fuel cell v-i relationship and cell power [12].
1.4 -
1.2 -
1 -
0.8 -
0.6 -
0.4 -
0.2 -
0 - - 0
- 200
- 400
- 600
- 800
- 1000
- 1200
|
0|
500|
1000|
1500|
2000
Maximum Theoretical Voltage
Current Density ( i in mA/cm2 )
ActivationLosses
Ohmic Losses
MassTransport
Losses
OpenCircuitVoltage
Cell P
ower
( P
Cin
mW
)
Cell V
olta
ge (
V Cin
Volts
)- ΔgƒE =2 F
Cell PowerPC= VC x i
1.4 -
1.2 -
1 -
0.8 -
0.6 -
0.4 -
0.2 -
0 -
1.4 -
1.2 -
1 -
0.8 -
0.6 -
0.4 -
0.2 -
0 - - 0
- 200
- 400
- 600
- 800
- 1000
- 1200
- 0
- 200
- 400
- 600
- 800
- 1000
- 1200
|
0|
500|
1000|
1500|
2000|
0|
500|
1000|
1500|
2000
Maximum Theoretical Voltage
Current Density ( i in mA/cm2 )
ActivationLosses
Ohmic Losses
MassTransport
Losses
OpenCircuitVoltage
Cell P
ower
( P
Cin
mW
)
Cell V
olta
ge (
V Cin
Volts
)- ΔgƒE =2 F- ΔgƒE =2 F
Cell PowerPC= VC x i
Fuel Cells
© Copyright Ned Mohan 2006 55
Fig. 3-15 Greenhouse effect [13].
Greenhouse Effect
© Copyright Ned Mohan 2006 56
Fig. 3-16 Resource mix at XcelEnergy [14].
1
12
2
3
3445
5
6
6
1
12
2
3
3445
5
6
6
Resource mix XcelEnergy
© Copyright Ned Mohan 2006 57
Fig. 3-17 Electric power industry fuel costs in the U.S. in 2005 [1].
Fuel Costs in the U.S. in 2005
© Copyright Ned Mohan 2006 58
CHAPTER 4
AC TRANSMISSION LINES AND UNDERGROUND CABLES
© Copyright Ned Mohan 2006 59 Fig. 4-1 500-kV transmission line (Source: University of Minnesota EMTP course).
( )a ( )c
( )b
( )a ( )c
( )b
Transmission Tower, Conductor and Bundling
© Copyright Ned Mohan 2006 60
Fig. 4-2 Transposition of transmission lines.
1D2D
3D
1 cycle
( )a ( )b
a
b
c
Transposition
© Copyright Ned Mohan 2006 61
Fig. 4-3 Distributed parameter representation on a per-phase basis.
line
neutral (zeroimpedance)
line R L
C
Distributed Parameters
© Copyright Ned Mohan 2006 62
Fig. 4-4 (a) Cross-section of ACSR conductors, (b) skin-effect in a solid conductor.
D
TJ
towards centersurface( )a ( )b
Calculation of Transmission Line Resistance: Skin Effect
© Copyright Ned Mohan 2006 63
Fig. 4-5 Flux linkage with conductor-a. ( )a ( )b ( )c
Drai bi
ci
rai x dx
a b
c
a b
c
D
D
bidx
xa
b
c
Calculation of Transmission Line Inductance
© Copyright Ned Mohan 2006 64
Fig. 4-6 Electric field due to a charge.
x
q1 21x
2x
Electric Field Due to Transmission Line Voltage
© Copyright Ned Mohan 2006 65
Fig. 4-7 Shunt capacitances.
aq bq
cq
a b
c
D
aq bq
cq
a b
c
D a b
c
a b
c
( )a ( )b
CC
C
n
hypotheticalneutral
Calculation of Transmission Line Capacitance
© Copyright Ned Mohan 2006 66
Table 4-1 Transmission Line Parameters with Bundled Conductors (except at 230 kV)
at 60 Hz [2, 6]
Nominal Voltage ( / )R kmΩ ( / )L kmω Ω ( / )C kmω μ
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
Typical Parameters for various Voltage Transmission Lines
© Copyright Ned Mohan 2006 67
Fig. 4-8 A 345-kV, single-conductor per phase, transmission system.
Calculating Transmission Line Parameters using EMTDC
© Copyright Ned Mohan 2006 68
Fig. 4-9 Distributed per-phase transmission line ( G not shown).
+
−( )RV s
+
−( )SV s
+
−( )xV s
0x
( )RI s( )SI s R sL
1sC
( )xI s
Distributed-Parameter Representation
© Copyright Ned Mohan 2006 69
Fig. 4-10 Per-phase transmission line terminated with a resistance equal to cZ .
+
−SV
SI j Lω
1jCω
−cZ
+
−0R RV V= ∠
0x
SV RV
( )a ( )b
RI
Voltage Profile under SIL
© Copyright Ned Mohan 2006 70
Table 4-2 Surge Impedance and Three-Phase Surge Impedance Loading [2, 6]
Nominal Voltage ( )cZ Ω ( )SIL MW
230 kV 375 140 MW
345 kV 280 425 MW
500 kV 250 1000 MW
765 kV 255 2300 MW
Typical Surge Impedances and SIL for various Voltage Transmission Lines
© Copyright Ned Mohan 2006 71
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
Loadability of Transmission Lines
© Copyright Ned Mohan 2006 72
Fig. 4-11 Long line representation.
+
−
( )SV s
( )SI s
+
−
( )RV s
( )RI sseriesZ
2shuntY
2shuntY
Long-Line Representation
© Copyright Ned Mohan 2006 73
Fig. 4-12 Per-phase transmission line representation based on length.
+
−
SV
SI
+
−
RV
RIseriesZ
2shuntY
2shuntY
+
−
SV
SI
+
−
RV
RIlinej Lω
2
line
jCω
−
2
line
jCω
−
lineR
( )a ( )b ( )c
+
−
SV
SI
+
−
RV
RIlinej LωlineR
Transmission Line Representations
© Copyright Ned Mohan 2006 74
Fig. 4-13 Underground cable.
Underground Cables
© Copyright Ned Mohan 2006 75
CHAPTER 5
POWER FLOW IN POWER SYSTEM NETWORKS
© Copyright Ned Mohan 2006 76
Fig. 5-1 A three-bus 345-kV example system.
Bus 1 Bus 3
Bus 2
Slack Bus
PV Bus
PQ BusP jQ+
200km
150km 150km
Bus 1 Bus 3
Bus 2
Slack Bus
PV Bus
PQ BusP jQ+
200km
150km 150km
Three-Bus Example Power System
© Copyright Ned Mohan 2006 77
Table 5-1 Per-Unit Values in the Example System
Line Series Impedance Z in Ω (pu) Total Susceptance B in μ (pu)
1-2 12 (5.55 56.4)Z j= + Ω = (0.0047 0.0474)j+ pu 675TotalB μ= = (0.8034) pu
1-3 13 (7.40 75.2)Z j= + Ω = (0.0062 0.0632)j+ pu 900TotalB μ= = (1.0712) pu
2-3 23 (5.55 56.4)Z j= + Ω = (0.0047 0.0474)j+ pu 675TotalB μ= = (0.8034) pu
Transmission Lines in Example Power System
© Copyright Ned Mohan 2006 78
Fig. 5-2 Example system of Fig. 5-1 for assembling Y-bus matrix.
Bus 1 Bus 3
Bus 2
1I
2I
3I
13Z
12Z 23Z1V 3V
2V
Calculating Y-Bus in the Example Power System
© Copyright Ned Mohan 2006 79
Fig. 5-3 Plot of 24 x− as a function of x .
x
24 x−4
2
0
2−
4−
6−
8−
10−
12−
0.5 1.0 1.5 2 3.0 3.5 4.0
(0)x(1)x(2)x
Newton-Raphson Procedure
© Copyright Ned Mohan 2006 80
Fig. 5-4 Power-Flow results of Example 5-4.
01 1 0V pu= ∠
02 1.05 -2.07V pu= ∠
03 0.978 -8.79V pu= ∠
( )0.69 - 1.11j pu
( )2.39 0.29j pu+
( )2.68 1.48j pu+
( )5.0 1.0j pu+1 1 (3.08 - 0.82)P jQ j pu+ =
2 2 ( 2.0 2.67)P jQ j pu+ = +
Power Flow Results in the Example Power System
© Copyright Ned Mohan 2006 81
CHAPTER 6
TRANSFORMERS IN POWER SYSTEMS
© Copyright Ned Mohan 2006 82
Fig. 6-1 Principle of transformers, beginning with just one coil.
+
−1e
1N
mφ+
−1e
1N
mφmi+
−
1e mL
mi+
−
1e mL
(a) (b)
Transformer Principle: Generation of Flux
© Copyright Ned Mohan 2006 83
Fig. 6-2 B-H characteristics of ferromagnetic materials.
mB
mH
mB
mH
satB
oμ
oμ
mμ
mB
mH
satB
oμ
oμ
mμ
mB
mH
(a) (b)
Core in Transformers
© Copyright Ned Mohan 2006 84
Fig. 6-3 Transformer with the open-circuited second coil.
+
−
mi+
−
1e mL
Ideal
Transformer
1N 2N
2e
+
−
mi+
−
1e mL
Ideal
Transformer
1N 2N
2e
+
−1e
1N
mφ
2e
2N
−+
+
−1e
1N
mφ
2e
2N
−+
(a) (b)
Flux Coupling
© Copyright Ned Mohan 2006 85
Fig. 6-4 Transformer with load connected to the secondary winding.
+
−1e
1N
mφ
2e
2N
−+
( )1i t
( )2i t
+
−1e
1N
mφ
2e
2N
−+
( )1i t
( )2i t
( )2i t′( )1i t
+
−
mi+
−
1e mL
Ideal
Transformer
1N 2N
2e
( )2i t( )2i t′( )1i t
+
−
mi+
−
1e mL
Ideal
Transformer
1N 2N
2e
( )2i t
(a) (b)
Transformer with Load Connected to the Secondary
© Copyright Ned Mohan 2006 86
Fig. 6-5 Transformer equivalent circuit including leakage impedances and core losses.
'2I1I
+
−
mi+
−
1E mjX
Ideal Transformer
1N 2N
2E
2I1R l1jX 2Rl2jX
2V
+
−
1V
+
−
Real Transformer
heR
'2I1I
+
−
mi+
−
1E mjX
Ideal Transformer
1N 2N
2E
2I1R l1jX 2Rl2jX
2V
+
−
1V
+
−
Real Transformer
heR
Transformer Model
© Copyright Ned Mohan 2006 87
Fig. 6-6 Eddy currents in the transformer core.
mφ
circulatingcurrents
imφ
circulatingcurrents
i
mφ
circulatingcurrents
mφ
circulatingcurrents
(a) (b)
Eddy Current and Hysteresis Losses
© Copyright Ned Mohan 2006 88
Fig. 6-7 Simplified transformer model.
+
−
pV+
−
sV
pI sIpZ sZ1: n
pn sn+
−
sV ′
Transformer Simplified Model
© Copyright Ned Mohan 2006 89Fig. 6-8 Transferring leakage impedances across the ideal transformer of the model.
+
−
pV+
−
sV
pI sIpsZ 1: n
pn sn
+
−
pV+
−
sV
pI sIspZ1: n
pn sn
( )a
( )b
Transferring Leakage Impedances from One Side to Another
© Copyright Ned Mohan 2006 90
Fig. 6-9 Transformer equivalent circuit in per unit (pu).
+
−
(pu)pV+
−
(pu)sV
(pu)I (pu)I(pu)trZ
Transformer Equivalent Circuit in Per Unit
© Copyright Ned Mohan 2006 91
Fig. 6-10 Winding connections in a three-phase system. ( )a ( )b
Connection of Transformer Windings
© Copyright Ned Mohan 2006 92
Fig. 6-11 Including nominal-voltage transformers in per-unit.
345 / 500kV 500 / 345kV
500kV
Bus 3Bus 1
Including Nominal Turns-Ratio Transformer in Power Flow Studies
© Copyright Ned Mohan 2006 93
Fig. 6-12 Auto-transformer.
+
−
1V
+
−
2V
1I
1n2n
( )a
2I
+
−
1V
( )1 2V V+
1I
1n
2n 2I
( )1 2I I+
+
−2V
( )b
+
−
Auto-Transformer
© Copyright Ned Mohan 2006 94
Fig. 6-13 Phase-shift in Δ -Y connected transformers.
( )c
+
−
AV
+
−
aV
03012:
3jn e n
( )a
a
bc
A
B
C( )b
aVAV
1n2n
Phase-Shift Due to Wye-Delta Transformers
© Copyright Ned Mohan 2006 95
Fig. 6-14 Transformer for phase-angle control.
a
bc
a′
b′
c′
( )a ( )b
aV
bVcV
abV
bcV
caV
( )c
aV ′
bV ′
cV ′
a bV ′ ′
b cV ′ ′
c aV ′ ′
φ
aV
a
bc
a′
b′
c′
a
bc
a′
b′
c′
( )a ( )b
aV
bVcV
abV
bcV
caV
( )c
aV ′
bV ′
cV ′
a bV ′ ′
b cV ′ ′
c aV ′ ′
φ
aV
Phase-Shift Control by Transformers
© Copyright Ned Mohan 2006 96
Fig. 6-15 Three-winding auto-transformer.
H
L T H
L
T
( )HZ Ω
( )LZ Ω
( )TZ Ω
1n
2n
3n
( )a( )b
a
bc
A
B
C
a′ a
a′
A
C
H
L T H
L
T
( )HZ Ω
( )LZ Ω
( )TZ Ω
1n
2n
3n
( )a( )b
a
bc
A
B
C
a′ a
a′
A
C
Three-Winding Auto-Transformers
© Copyright Ned Mohan 2006 97
Fig. 6-16 General representation of an auto-transformer and a phase-shifter.
+
−1V
1: t
1I 2I+
−
2V
+
−
2Vt
1/Y Z=
General Representation of Auto- and Phase-Shift Transformers
© Copyright Ned Mohan 2006 98
Fig. 6-17 Transformer with an off-nominal turns-ratio or taps in per unit; t is real.
+
−
1V
1/Y Z=
1: t
1I 2I
+
−
2V
( )a ( )b
11 Yt
⎛ ⎞−⎜ ⎟⎝ ⎠
2
1 1 Yt t
⎛ ⎞−⎜ ⎟⎝ ⎠
/Y t
+
−
1V
+
−
2V
1I 2I
11 Yt
⎛ ⎞−⎜ ⎟⎝ ⎠
2
1 1 Yt t
⎛ ⎞−⎜ ⎟⎝ ⎠
/Y t
+
−
1V
+
−
2V
1I 2I
PU Representation of Off-Nominal Turns-Ratio Transformers
© Copyright Ned Mohan 2006 99
Fig. 6-18 Transformer of Example 6-3.
+
−
1V
0.1j pu
1: t
1I 2I
+
−
2V
( )a ( )b
1 0.909Y j pu= −
2 0.826Y j pu=
0.11sZ j pu=
+
−
1V
+
−
2V
1I 2I
Example of Off-Nominal Turns-Ratio Transformers
© Copyright Ned Mohan 2006 100
CHAPTER 7
HIGH VOLTAGE DC (HVDC) TRANSMISSION SYSTEMS
© Copyright Ned Mohan 2006 101
Fig. 7-1 Power semiconductor devices.
Thyristor IGBT MOSFETIGCT(a)
101 102 103 104
102
104
106
108
Thyr
istor
IGBT
MOSFET
Pow
er (V
A)
Switching Frequency (Hz)
IGCT
(b)
Thyristor IGBT MOSFETIGCT(a)
Thyristor IGBT MOSFETIGCT(a)
101 102 103 104
102
104
106
108
Thyr
istor
IGBT
MOSFET
Pow
er (V
A)
Switching Frequency (Hz)
IGCT
(b)
101 102 103 104
102
104
106
108
Thyr
istor
IGBT
MOSFET
Pow
er (V
A)
Switching Frequency (Hz)
IGCT
101 102 103 104
102
104
106
108
Thyr
istor
IGBT
MOSFET
Pow
er (V
A)
Switching Frequency (Hz)
IGCT
(b)
Symbols and Capabilities of Power Semiconductor Devices
© Copyright Ned Mohan 2006 102
Figure 7-2 Power semiconductor devices: (a) ratings (source: Siemens), (b) variousapplications (source: ABB).
( )a
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
HVDCTraction
MotorDrive
PowerSupply
Auto-motive
Lighting
FACTS
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
HVDCTraction
MotorDrive
PowerSupply
Auto-motive
Lighting
FACTS
( )b( )a
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
HVDCTraction
MotorDrive
PowerSupply
Auto-motive
Lighting
FACTS
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
Device blocking voltage [V]
Dev
ice
curr
ent [
A]
104103102101
104
103
102
101
100
HVDCTraction
MotorDrive
PowerSupply
Auto-motive
Lighting
FACTS
( )b
Power Semiconductor Devices and Applications
© Copyright Ned Mohan 2006 103
Fig. 7-3 HVDC system – one-line diagram. 1AC 2AC
HVDC Line
HVDC System
© Copyright Ned Mohan 2006 104
Fig. 7-4 HVDC systems: (a) Current-Link, and (b) Voltage-Link.
AC1 AC2−
+
AC1 AC2AC1 AC2AC1 AC2−
+
AC1 AC2AC1 AC2AC1 AC2AC1 AC2
( )a ( )b
HVDC Systems: Voltage-Link and Current-Link
© Copyright Ned Mohan 2006 105
Fig. 7-5 HVDC projects, mostly current-link systems, in North America [source: ABB]
3100MW
1920MW
200MW
200MW
200MW
200MW
200MW
210MW
150MW
200MW
600MW
1000MW500MW
1000MW
36MW
312MW370MW
320MW
100MW
200MW
350MW
330MW
1620MW
2000MW
2000MW
690MW
2250MW
2138MW
3100MW
1920MW
200MW
200MW
200MW
200MW
200MW
210MW
150MW
200MW
600MW
1000MW500MW
1000MW
36MW
312MW370MW
320MW
100MW
200MW
350MW
330MW
1620MW
2000MW
2000MW
690MW
2250MW
2138MW
HVDC Projects in North America
© Copyright Ned Mohan 2006 106
Fig. 7-6 Block diagram of a current-link HVDC system.
Current-Link HVDC System
© Copyright Ned Mohan 2006 107
Fig. 7-7 Thyristors.
K
A
G
P
N
P
N
A
G
pn1
pn2
pn3
K
(a) (b)
K
A
GK
A
G
P
N
P
N
A
G
pn1
pn2
pn3
K
P
N
P
N
A
G
pn1
pn2
pn3
K
(a) (b)
Thyristors
© Copyright Ned Mohan 2006 108
Fig. 7-8 Thyristor circuit with a resistive load and a series inductance.
( )b
tω
tω
tω
0
0
0
α α
dvdV
svsi
Gi
0tω =
( )a
si
sL+
−
+
−
dvsv R
( )b
tω
tω
tω
0
0
0
α α
dvdV
svsi
Gi
0tω =
( )b
tω
tω
tω
0
0
0
α α
dvdV
svsi
Gi
0tω =
( )a
si
sL+
−
+
−
dvsv R( )a
si
sL+
−
+
−
dvsv R
Primitive Thyristor Circuits
© Copyright Ned Mohan 2006 109
Fig. 7-9 Three-phase Full-Bridge thyristor converter. (a)
di
6
5
4 2
1 3
cnv− +
− +
− +anv
bnv sL
ai
+
−
− +anv ai
dI+
−dv
N
P1
35
462
(b)
n n
(a)
di
6
5
4 2
1 3
cnv− +
− +
− +anv
bnv sL
ai
+
−
− +anv ai
dI+
−dv
N
P1
35
462
(b)
n n
+
−
dv
(a)
di
6
5
4 2
1 3
cnv− +
− +
− +anv
bnv sL
ai
+
−
− +anv ai
dI+
−dv
N
P1
35
462
(b)
n n
(a)
di
6
5
4 2
1 3
cnv− +
− +
− +anv
bnv sL
ai
+
−
− +anv ai
dI+
−dv
N
P1
35
462
(b)
n n
+
−
dv
Three-Phase Thyristor Converter
© Copyright Ned Mohan 2006 110
Fig. 7-10 Waveforms in a three-phase rectifier with 0sL = and 0α = .
t0
cvbvav
(a)
ai
0o120
o60 tω
bi
0
ci
0
Nv
Pv
(c)t
doV
LL2Vdv
0(b)
tω
tω
t0
cvbvav
(a)
ai
0o120
o60 tω
bi
0
ci
0
Nv
Pv
(c)t
doV
LL2Vdv
0(b)
tω
tω
Three-Phase Diode Rectifier Waveforms
© Copyright Ned Mohan 2006 111Fig. 7-11 Waveforms with 0sL = .
0
0
0
0
tω
tω
tω
tω
anv bnv cnvPnv
Nnv
Aα
ai
bi
ci
1
4
3
66
1
5 5
2
4
α
0
0
0
0
tω
tω
tω
tω
anv bnv cnvPnv
Nnv
Aα
ai
bi
ci
1
4
3
66
1
5 5
2
4
α
Three-Phase Thyristor Converter Waveforms with zero AC-Side Inductance
© Copyright Ned Mohan 2006 112Fig. 7-12 Waveforms in the inverter mode.
0
0
0
0
tω
tω
tω
tω
anvbnv
cnv
Pnv
Nnv
ai
bi
ci
1
4
3
6
1
5
2
4
α
3
2
0
0
0
0
tω
tω
tω
tω
anvbnv
cnv
Pnv
Nnv
ai
bi
ci
1
4
3
6
1
5
2
4
α
3
2
Three-Phase Inverter Waveforms
© Copyright Ned Mohan 2006 113
Fig. 7-13 Average dc-side voltage as a function of α . ( )b( )a
00
dV
α090 0160
0180
dV
dI
Rectifierd dP V I= = +
Inverterd dP V I= = −
DC-Side Voltage as a Function of Delay Angle
© Copyright Ned Mohan 2006 114
Fig. 7-14 Waveforms with sL .
0
0
tω
tω
anv bnv cnvPnv
Nnv
uA
ai 1
4
1
4
α
u
0
0
tω
tω
anv bnv cnvPnv
Nnv
uA
ai 1
4
1
4
α
u
Thyristor Converter Waveforms in the Presence of AC-Side Inductance
© Copyright Ned Mohan 2006 115
Fig. 7-15 Power-factor angle. ( )b( )a
aV aV
1aI1aI
1φ−
1φ−
Power Factor Angle in Rectifier and Inverter Modes
© Copyright Ned Mohan 2006 116
CU One-line Diagram
© Copyright Ned Mohan 2006 117
Fig. 7-17 Six-pulse and 12-pulse current and voltage waveforms [2]. ( )a ( )b
( )ai Y Y−
( )ai Y − Δ
12-Pulse Waveforms
© Copyright Ned Mohan 2006 118
Fig. 7-18 A pole of an HVDC system.
di
+
−
1dv
+
−
2dvAC 1 AC 2AC 1 AC 2
dR dL
di
+
−
1dv
+
−
2dvAC 1 AC 2AC 1 AC 2
dR dL
HVDC System Representation for Control
© Copyright Ned Mohan 2006 119
Fig. 7-19 Control of an HVDC system [3].
1dV
0dI,d refI
min
Inverter characteristicwith γ γ=
Rectifier characteristicin a current-control mode
Control of HVDC Converters
© Copyright Ned Mohan 2006 120Fig. 7-20 Voltage-link HVDC transmission system [source: ABB].
A Voltage-Link HVDC System in Northeastern U.S.
© Copyright Ned Mohan 2006 121
Fig. 7-21 Block diagram of voltage-link HVDC system.
AC1 AC2−
+
AC1 AC2AC1 AC2AC1 AC2−
+
1 1,P Q 2 2,P Q
Voltage-Link HVDC System Block Diagram
© Copyright Ned Mohan 2006 122
Fig. 7-22 Block diagram of a voltage-link converter and the phasor diagram.
+
−dV
convv
L
busvLi+
−dV
convv
L
busvLi +
−convV
LI
+
−busV
+ −L LjX I
( )a ( )b ( )c
LI
LI
L LjX I
convV
busV
Phasor Diagram on the Ac-Side of the Voltage-Link Converter
© Copyright Ned Mohan 2006 123
Fig. 7-23 Synthesis of sinusoidal voltages.
+
−dV
abc
1: ad 1: bd 1: cd 1: ad
ai
dV
dai
+
−aNv
( )a ( )b
Representation of Voltage-Link Converter with Ideal Transformers
© Copyright Ned Mohan 2006 124
Fig. 7-24 Sinusoidal variation of turns-ratio ad .
1
0.5
0
dV
0.5 dV
0
tω
tω
ad
aNv
ˆad
aV
Synthesis of “Average”Sinusoidal Voltages
© Copyright Ned Mohan 2006 125
Fig. 7-25 Three-phase synthesis.
a
b
c
N
2dV
2dV
2dV
av bv cvac-side
( )a
dV
0.5 dV
0
av
tω
( )a
aNv bNv cNv
Converter Output Voltages and Voltages across the Load
© Copyright Ned Mohan 2006 126
Fig. 7-26 Realization of the ideal transformer functionality.
aq
+
−
dV
daiaia
N
+
−aNv
(a) (b)
+
−
dV
aq
Buck Boost
aq−
ai
+
−aNv
Switching Power-Pole of Voltage-Link Converters
© Copyright Ned Mohan 2006 127
Fig. 7-27 PWM to synthesize sinusoidal waveform.
aNv
0 tω
dV
aNv
aNv
aNv
0
0sT
( )a
( )b
hV
1f sf 2 sf 3 sf
Switching in Sinusoidal “Average” Voltage Waveform
© Copyright Ned Mohan 2006 128
CHAPTER 8
Distribution System, Loads and Power Quality
© Copyright Ned Mohan 2006 129
Fig. 8-1 Residential distribution system.
13.8kV
Transformer
120V±
120V±
120V±
1House
2House
3House
Residential Distribution System
© Copyright Ned Mohan 2006 130
Fig. 8-2 System load.
100%0
Load(MW)
percentage of the time 100%0
Load(MW)
percentage of the time
kW
TimeAM NOON PM
12 6 12 6 12
peak
(a) (b)
Daily Load and Load-Duration Curves
© Copyright Ned Mohan 2006 131
Fig. 8-3 Utility loads.
Motors 51%HVAC 16%
IT 14%
Lighting 19%
Motors 51%HVAC 16%
IT 14%
Lighting 19%
Motors 51%HVAC 16%
IT 14%
Lighting 19%
36%Residential35%
Commercial
29%Industrial
36%Residential35%
Commercial
29%Industrial
( )a ( )b
Utility Load Distribution
© Copyright Ned Mohan 2006 132
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 1.0 2.0 0 Incandescent Lighting 1.0 1.5 0 Fluorescent Lighting 0.9 1.0 1.0
Motor Loads 0.8 – 0.9 0.05 – 0.5 1.0 – 3.0 Modern Power-
Electronics based Loads
1.0 0 0
Power Factor and Voltage Sensitivity of Power Systems Load
© Copyright Ned Mohan 2006 133
Fig. 8-4 Voltage-link-system for modern and future power-electronics based loads.
dV
+
−Utility
Load
Voltage-Link System used in Power Electronics Based Loads
© Copyright Ned Mohan 2006 134
Fig. 8-5 Per-phase, steady state equivalent circuit of a three-phase induction motor.
+
−
(at )ωaV
+
−
lsj Lω
maImaE
'raIaIsR
'lrj Lω
' synr
slipR
ωω
mj Lω
Induction Motor Per-Phase Diagram
© Copyright Ned Mohan 2006 135
Fig. 8-6 Torque-speed characteristic of induction motor at various applied frequencies.
emT
1f LoadTorque
0
2f3f4f5f
mω1synω
1slipω3synω
3slipω
Torque-Speed Characteristics
© Copyright Ned Mohan 2006 136
Fig. 8-7 Switch-mode dc power supply.
60Hz ac
inputrectifier
topology to convertdc to dc with isolation
Feedbackcontroller
HF transformer
dc to HF ac
OutputinV
+
−
*oV
oV60Hz ac
inputrectifier
topology to convertdc to dc with isolation
Feedbackcontroller
HF transformer
dc to HF ac
OutputinV
+
−
*oV
oV
Switch-Mode DC Power Supplies
© Copyright Ned Mohan 2006 137
Fig. 8-8 Uninterruptible power supply.
Rectifier Inverter Filter Critical Load
Energy Storage
Rectifier Inverter Filter Critical Load
Energy Storage
Rectifier Inverter Filter Critical Load
Energy Storage
Uninterruptible Power Supplies (UPS)
© Copyright Ned Mohan 2006 138
Fig. 8-9 Alternate feeder.
Load
Feeder 1
Feeder 2
Load
Feeder 1
Feeder 2
Static Power-Transfer Switch
© Copyright Ned Mohan 2006 139
Fig. 8-10 CBEMA curve.
CBEMA Curve Showing Acceptable Voltage-Time Region
© Copyright Ned Mohan 2006 140
Fig. 8-11 Dynamic Voltage Restorer (DVR).
Power Electronic Interface
Loadsv−
+
− +injv
Power Electronic Interface
LoadPower Electronic Interface
Loadsv−
+
− +injv
Dynamic Voltage Restorers (DVR)
© Copyright Ned Mohan 2006 141
Fig. 8-12 Three-Phase Voltage Regulator (Courtesy of Siemens) [5].
Voltage Regulating Transformers
© Copyright Ned Mohan 2006 142
Fig. 8-13 STATCOM [4].
Utility
STATCOM
jXUtility
STATCOM
jX
STATCOM
© Copyright Ned Mohan 2006 143
Figure 8-14 Voltage and current phasors in simple R-L circuit.
sI
sV
φ
is
vs
+
−
( )a( )b
sI
sV
φ
is
vs
+
−
is
vs
+
−
( )a( )b
Linear Load
© Copyright Ned Mohan 2006 144
Figure 8-15 Current drawn by power electronics equipment without PFC.
t
( )distortion s s1i i i= −
t0/1φ ω
s1iisvs
1T
0
( )a
( )b
t
( )distortion s s1i i i= −
t0/1φ ω
s1iisvs
1T
0
( )a
( )b
Waveforms Associated with Power Electronics-Based Load
© Copyright Ned Mohan 2006 145
1T
si
t
I
I−0
s1i
t0
I
I−
/4I π
t
distortioni
0
Figure 5-4 Example 5-1.
( )c
( )b
( )a
1T
si
t
I
I−0
s1i
t0
I
I−
/4I π
t
distortioni
0
Figure 5-4 Example 5-1.
( )c
( )b
( )a
Figure 8-16 Example 8-1.
1T
si
t
I
I−0
s1i
t0
I
I−
/4I π
t
distortioni
0
Figure 5-4 Example 5-1.
( )c
( )b
( )a
1T
si
t
I
I−0
s1i
t0
I
I−
/4I π
t
distortioni
0
Figure 5-4 Example 5-1.
( )c
( )b
( )a
Figure 8-16 Example 8-1.
Example of Distorted Current
© Copyright Ned Mohan 2006 146
Fig. 8-17 Relation between PF/DPF and THD.
PFDPF
%THD0 50 100 150 200 250 300
1
.0 9
.0 8
.0 7
.0 6
.0 5
.0 4
PFDPF
%THD0 50 100 150 200 250 300
1
.0 9
.0 8
.0 7
.0 6
.0 5
.0 4
Influence of Distortion on Power Factor
© Copyright Ned Mohan 2006 147
Table 5-1 Harmonic current distortion (Ih/I1)
1/scI I
( %)Odd Harmonic Order h in
35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11
15 0.
12 0.
10 0.
7 0.
4 0.
7 0.
5 5.
4 5.
35.
2 0.
6 0.
5 0.
4 0.
2 5.
15.
2 5.
2 0.
15.
10.
0 6.
14.
10.
0 7.
0 5.
0 3.
20 0.
15 0.
12 0.
8 0.
5 0.
Distortion(%)Harmonic
Total
>1000
100 1000−
50 100−
20 50−
< 20
Table 5-1 Harmonic current distortion (Ih/I1)
1/scI I
( %)Odd Harmonic Order h in
35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11
15 0.
12 0.
10 0.
7 0.
4 0.
7 0.
5 5.
4 5.
35.
2 0.
6 0.
5 0.
4 0.
2 5.
15.
2 5.
2 0.
15.
10.
0 6.
14.
10.
0 7.
0 5.
0 3.
20 0.
15 0.
12 0.
8 0.
5 0.
Distortion(%)Harmonic
Total
>1000
100 1000−
50 100−
20 50−
< 20
1Table 8-1 Harmonic current distortion ( / )hI ITable 5-1 Harmonic current distortion (Ih/I1)
1/scI I
( %)Odd Harmonic Order h in
35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11
15 0.
12 0.
10 0.
7 0.
4 0.
7 0.
5 5.
4 5.
35.
2 0.
6 0.
5 0.
4 0.
2 5.
15.
2 5.
2 0.
15.
10.
0 6.
14.
10.
0 7.
0 5.
0 3.
20 0.
15 0.
12 0.
8 0.
5 0.
Distortion(%)Harmonic
Total
>1000
100 1000−
50 100−
20 50−
< 20
Table 5-1 Harmonic current distortion (Ih/I1)
1/scI I
( %)Odd Harmonic Order h in
35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11
15 0.
12 0.
10 0.
7 0.
4 0.
7 0.
5 5.
4 5.
35.
2 0.
6 0.
5 0.
4 0.
2 5.
15.
2 5.
2 0.
15.
10.
0 6.
14.
10.
0 7.
0 5.
0 3.
20 0.
15 0.
12 0.
8 0.
5 0.
Distortion(%)Harmonic
Total
>1000
100 1000−
50 100−
20 50−
< 20
1Table 8-1 Harmonic current distortion ( / )hI I
IEEE Harmonic Limits
© Copyright Ned Mohan 2006 148
−
+
scIZs
Vs−
+
Zs
Vs
(a) (b)Figure 5-6 (a) Utility supply; (b) short circuit current.
−
+
scIZs
Vs−
+
Zs
Vs−
+
Zs
Vs
(a) (b)Figure 5-6 (a) Utility supply; (b) short circuit current.Figure 8-18 (a) Utility Supply, (b) Short-Circuit Current.
Short-Circuit Current
© Copyright Ned Mohan 2006 149
Fig. 8-19 Average retail price of electricity to ultimate customers [4].
Retail Price of Electricity in the U.S.
© Copyright Ned Mohan 2006 150
CHAPTER 9
SYNCHRONOUS GENERATORS
© Copyright Ned Mohan 2006 151
Fig. 9-1 Synchronous generators driven by (a) steam turbines, and (b) hydraulic turbines.
HGenerator
Penstock
Turbine
Water
Steam at High pressure
Pump
Heat in
Heat out
TurbineBoiler
Condenser
Gen
( )a ( )b
Application of Synchronous Generators
© Copyright Ned Mohan 2006 152
Fig. 9-2 Machine cross-section. (a) (b)
Air gap
Stator
Air gap
Stator
Cross-section of Synchronous Generators
© Copyright Ned Mohan 2006 153
Fig. 9-3 Machine structure. (a) (b) (c)
SN SN
S
N N
S
S
N N
S
S
N
N
S S
N
N
S
Synchronous Generator Structure
© Copyright Ned Mohan 2006 154
Fig. 9-4 Three phase windings on the stator.
axisa −
axisb −
axisc −
2 / 3π
2 / 3π
2 / 3π
bi
ai
ci
axisa −
axisb −
axisc −
2 / 3π
2 / 3π
2 / 3π
bi
ai
ci1
234
56
7
1'
2 '
3 ' 4 ' 5 '
6 ' aiθ
ai7 '1
234
56
7
1'
2 '
3 ' 4 ' 5 '
6 ' aiθ
ai7 '
(a) (b)
Sinusoidally-Distributed Windings
© Copyright Ned Mohan 2006 155
Fig. 9-5 Connection of three phase windings.
a'a
b
'b
c
'c
ai
bi
ci
axisa −
axisb −
axisc − o240∠
o120∠
o0∠
θ
a'a
b
'b
c
'c
ai
bi
ci
axisa −
axisb −
axisc − o240∠
o120∠
o0∠
θb
bi
ai
cic
a
bbi
ai
cic
a
(a) (b)
Three-Phase Winding Connection in a Wye
© Copyright Ned Mohan 2006 156
Fig. 9-6 Field winding on the rotor that is supplied by a dc current fI .
θ
a-axis
N
S
synω
Synchronous Generator Rotor Field
© Copyright Ned Mohan 2006 157
Fig. 9-7 Current direction and voltage polarities; the rotor position shown induces maximum ae .
θ
a-axis
N
Ssynω
+
−
ae
Voltage induced in the Stator Phase due to Rotating Rotor Field
© Copyright Ned Mohan 2006 158
Fig. 9-8 Induced emf afe due to rotating rotor field with the rotor.
θ
a-axis
N
Ssynω
+
−
afe
( )a ( )b ( )c
N
S
a-axis
(at 0)fB t =
afERe
Im
synω
Representation of Induced Stator Voltage due to Rotor Field
© Copyright Ned Mohan 2006 159
Fig. 9-9 Armature reaction due to phase currents.
axisa −
axisb −
axisc −
2 / 3π
2 / 3π
2 / 3π
bi
ai
ci
axisa −
axisb −
axisc −
2 / 3π
2 / 3π
2 / 3π
bi
ai
ci
( )a ( )b ( )c
0je
23
je
π
43
je
π
aI
Re
Im
θ
(at 0)ARB t =
a-axis
θ
,a ARE
090
Armature Reaction Due to Three Stator Currents
© Copyright Ned Mohan 2006 160
Fig. 9-10 Phasor diagram and per-phase equivalent circuit.
( )a ( )b
Re
Im
aI
afE
,a ARE
aE
m ajX IafE
+
−
+−,a ARE
+ −m ajX I +
−
aE
+
−
aV
sX
aI
sR
Superposition of the two Induced Voltages and Per-Phase Representation
© Copyright Ned Mohan 2006 161
Fig. 9-11 Power output and synchronism.
stability limitsteady state
generatormode
motoringmode
δ0o90−
o90
Pstability limit
steady state
+
−
oV V 0∞ ∞= ∠
aI
af afE E δ= ∠
TjX +
−( )a
( )b
Power Out as a function of rotor Angle
© Copyright Ned Mohan 2006 162
Fig. 9-12 Steady state stability limit.
δ0 o90( )a
eP
1δ 2δ
e1Pe2P
m1Pm2P
( )bδ0 o90
eP,maxeP
Steady State Stability Limit
© Copyright Ned Mohan 2006 163
Fig. 9-13 Excitation control to supply reactive power.
{aqI
aqI⎧⎪⎨⎪⎩ o90
o90
aIaIaI
s ajX I s ajX Is ajX I
afEafE afE
δ δ δaV
aV
aV
( )a ( )b ( )c
{aqI
aqI⎧⎪⎨⎪⎩ o90
o90
aIaIaI
s ajX I s ajX Is ajX I
afEafE afE
δ δ δaV
aV
aV
( )a ( )b ( )c
Reactive Power Control by Field Excitation
© Copyright Ned Mohan 2006 164
Fig. 9-14 Synchronous Condenser.
SynchronousCondenser
Synchronous Condenser
© Copyright Ned Mohan 2006 165
Fig. 9-15 Field exciter for automatic voltage regulation (AVR).
ac input
phase-controlledrectifier
slip rings
field winding
Generator
ac regulator
output
Automatic Voltage Regulation (AVR)
© Copyright Ned Mohan 2006 166
Fig. 9-16 Armature reaction flux in steady state.
Armature Reaction Flux in Steady State Leading to Synchronous Reactance
© Copyright Ned Mohan 2006 167
Fig. 9-17 Armature (a) and field current (b), after a sudden short circuit [source: 4]. ( )a ( )b
Simulation of a Short-Circuit Assuming a Constant-Flux Model
© Copyright Ned Mohan 2006 168
Fig. 9-18 Synchronous generator modeling for transient and sub-transient conditions.
( )b( )a
'
''
af
af
af
EEE
+
−
+ −'
''
s a
s a
s a
jX IjX IjX I
+
−
aE
aI
Re
Im
aI
afE
aE
s ajX I'afE
's ajX I
''s ajX I
''afE
Representation for Steady State, Transient Stability and Fault Analysis
© Copyright Ned Mohan 2006 169
CHAPTER 10
VOLTAGE REGULATION AND STABILITY IN POWER SYSTEMS
© Copyright Ned Mohan 2006 170
Fig. 10-1 A radial system.
R RP jQ+SV RV
LjXS SP jQ+
Load
(a) (b)
S SP jQ+ R RP jQ+LjX
SV RV
+
−
+
−
I
A Radial System
© Copyright Ned Mohan 2006 171
Fig. 10-2 Phasor diagram and the equivalent circuit with 1puS RV V= = . (a)
I
LjX I
RV
SV
δ
/ 2δ
(b)
S SP jQ+
SV
+
−
LjX I
RQRP
RV
+
−
Voltages and Current Phasors with Both-Side Voltages at 1 PU
© Copyright Ned Mohan 2006 172
Fig. 10-3 Voltage profile along the transmission line.
SV
+
−RV
+
−
xV
+
−x
(a) (b)
(1pu)SV
(1pu)RV
xV
RP SIL<
RP SIL>
Voltage Profile for Three Values of SIL
© Copyright Ned Mohan 2006 173
Fig. 10-4 Voltage collapse in a radial system (example of 345-kV line, 200 km long).
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
( )a
( )b
SV RVLjX
R RP jQ+
/RP SIL
R
S
VV
1PF =
0.9(leading)PF =
0.9(lagging)PF =
“Nose” Curves at Three Power Factors as a function of Loading
© Copyright Ned Mohan 2006 174
Fig. 10-5 Reactive power supply capability of synchronous generators.
A
B
C
0P
Q
Synchronous Generator Reactive Power Supply Capability
© Copyright Ned Mohan 2006 175
Fig. 10-6 Effect of leading and lagging currents due to the shunt compensating device.
+
−
ThV
ThjX +
−
busV
I
(a)
busV
I
ThjX IThV
busV
I
ThjX I
ThV
(b)
ThjX I+ −
Effect of Current Power Factor on Bus Voltage
© Copyright Ned Mohan 2006 176
Fig. 10-7 V-I characteristic of SVC.
busV
1j Cω
CI
busV
0CI
( )a ( )b
Static Var Compensators (SVC)
© Copyright Ned Mohan 2006 177
Fig. 10-8 Thyristor-Controlled Reactor (TCR).
busV
LI
( )a ( )c0
LI
busV
090α ≤
090α >
( )b
busv
LiLi
Thyristor Controlled Reactors (TCR)
© Copyright Ned Mohan 2006 178
Fig. 10-9 Parallel combination of SVC and TCR.
busV
1j Cω
CI
( )a
LILI
I
( )b ( )c
busV busV
I0 0 Iinductivecapacitive inductivecapacitive
AB
C
1V
1V ′
2V2V ′
LinearRange
Voltage Control by SVC and TCR Combination
© Copyright Ned Mohan 2006 179
Fig. 10-10 STATCOM.
busV
convVconvI
+ −convjXI
+
dV−
+
busV−
+
−
convV
convI
+ −convjXI
( )a ( )b
STATCOM
© Copyright Ned Mohan 2006 180
Fig. 10-11 STATCOM VI characteristic. convI0 inductivecapacitive
LinearRange
busV
STATCOM V-I Characteristic
© Copyright Ned Mohan 2006 181
Fig. 10-12 Thyristor-Controlled Series Capacitors (TCSC) [source: Siemens Corp.]. ( )a ( )c( )b
Thyristor-Controlled Series Capacitor (TCSC)
© Copyright Ned Mohan 2006 182
CHAPTER 11
TRANSIENT AND DYNAMIC STABILITY OF POWER SYSTEMS
© Copyright Ned Mohan 2006 183
Fig. 11-1 Simple one-generator system connected to an infinite bus.
1V 0B BV V= ∠
( )amP
LX
LX
( )b
1VE δ′∠ 0BV ∠+
−
+
−
+
−
I'( )d trj X X+ / 2LjX
eP
One-Machine Infinite-Bus System
© Copyright Ned Mohan 2006 184
Fig. 11-2 Power-angle characteristics.
post-faultPre-fault
LX
LX
Bus 1 0B BV V= ∠
δπ0
eP
mP
0δ 1δ( )a ( )b
mP ePduring-fault
Power-Angle Characteristic in One-Machine Infinite-Bus System
© Copyright Ned Mohan 2006 185
Fig. 11-3 Rotor-angle swing in Example 11-1. 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4
2 0
2 5
3 0
3 5
4 0
4 5
5 0
5 5
Rotor-Angle Swing Following a Fault and a Line Taken Out
© Copyright Ned Mohan 2006 186
Fig. 11-4 Fault on one of the transmission lines.
( )bπ δ
eP
e mP P=
00δ cδ mδ
A
B
Pre-fault
during fault
post-fault
maxδ( )a
LX
LX
Bus 1 0B BV V= ∠
mP eP
Power-Angle Characteristics
© Copyright Ned Mohan 2006 187
Fig. 11-5 Rotor oscillations after the fault is cleared.
π δ
eP
e mP P=
01δ mδ
2δ
C
D
Pre-fault
post-fault
Rotor Oscillations After the Fault is Cleared
© Copyright Ned Mohan 2006 188
Fig. 11-6 Critical clearing angle. π δ
eP
e mP P=
00δ critδ
A
B
Pre-fault
post-fault
maxδ1δ
Critical Clearing Angle using Equal-Area Criterion
© Copyright Ned Mohan 2006 189
Fig. 11-7 Power angle curves and equal-area criterion in Example 11-2.
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40( )eP pu
e mP P=
Pre-fault
during fault
post-fault
00 22.47δ = 0115.28mδ =075cδ =
A
B
Example using Equal-Area Criterion
© Copyright Ned Mohan 2006 190
Fig. 11-8 Block diagram of transient stability program for an n-generator case.
Initial Power Flow'Calculate and eP E δ∠
for each generator
, ,m k e kP P=
', and held constantm k kP E
Electro-dynamicdifferentialEquations
for 1,2,3....k =
and k kω δ
,e kPPhasor Calculations
'using k kE δ∠(load may be assumedas a constant impedance)
Transient Stability Calculations in Large Networks
© Copyright Ned Mohan 2006 191
Fig. 11-9 A 345-kV test example system.
Bus-1 Bus-3
Bus-2
1mP 1eP
2mP
2eP
Example Power System for Transient Stability Analysis
© Copyright Ned Mohan 2006 192
Fig. 11-10 Rotor-angle swings of 1δ and 2δ in Example 11-3.
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 60
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
1δ2δ
Rotor Angle Swings in the Example Power System Following a Fault
© Copyright Ned Mohan 2006 193
Fig. 11-11 Growing Power Oscillations: Western USA/Canada system, Aug 10, 1996 [4].
Importance of Dynamic Stability
© Copyright Ned Mohan 2006 194
CHAPTER 12
CONTROL OF INTERCONNECTED POWER SYSTEM AND ECONOMIC DISPATCH
© Copyright Ned Mohan 2006 195
Fig. 12-1 Field exciter for automatic voltage regulation (AVR).
ac input
phase-controlledrectifier
slip rings
field winding
Generator
ac regulator
output
Automatic Voltage Regulation (AVR)
© Copyright Ned Mohan 2006 196
Fig. 12-2 (a) The Interconnections in North America, (b) Control Areas [Source: 2]
( )a ( )b
Control Areas (Balancing Authorities)
© Copyright Ned Mohan 2006 197
Fig. 12-3 Load-Frequency Control (ignore the supplementary control at present).
mP eP LoadP
( )a
Regulator
Turbine
TurbineGovernor
Frequency
-SupplementaryControl
( )b mPΔ
fΔ
f
ab
mP
0
0f
G
G
Steam-ValvePosition
mP
1R
slope R= −
Load-Frequency Control and Regulation
© Copyright Ned Mohan 2006 198
Fig. 12-4 Response of two generators to load-frequency control.
1mP 1eP
2mP 2eP
LoadP( )a ( )b
mP1mPΔ 2mPΔ
fΔ
f
1 2( )m mP PΔ + Δ
unit1 unit 2
ac d
1mP2mP
be
unit 2
unit 1
0
0f
1G
1G 2G
2G1G ′
1G ′
Load Sharing
© Copyright Ned Mohan 2006 199
Fig. 12-5 Two control areas.
Area 1 Area 212P 21P
12jX
Synchronizing Torque between Two Control Areas
© Copyright Ned Mohan 2006 200
Fig. 12-6 Area Control Error (ACE) for Automatic Generation Control (AGC).
+
+
− −
B1R
SupplementaryController
Governor
(frequency deviation)fΔ
(tie-line flow deviation)PΔ
(Area Control Error)ACE
Change in Steam Valve Positionks
Automatic Generation Control (AGC) and Area Control Error (ACE)
© Copyright Ned Mohan 2006 201
Fig. 12-7 Two control areas in the example power system with 3 buses.
1mP 1eP
2mP
2eP
1 2P−
1 3P−Bus-1 Bus-3
Bus-2M
M
Area 2Area 1Load
Two Control Areas in the Example Power System
© Copyright Ned Mohan 2006 202
Power Flow on Tie-Lines between Two Control Areas Following a Load Change
© Copyright Ned Mohan 2006 203
Fig. 12-9 Electrical equivalent of two area interconnection.
12jX 2jX1jX+
−1 1E δ∠
+
−2 2E δ∠
Electrical Equivalent of Two Areas
© Copyright Ned Mohan 2006 204
Modeling of Two Control Areas with AGC
Fig. 12-10 Two-area system with AGC. Source: adapted from [6].
+
+
−− + −
+
−− − + −
+
−
1B1
1R
1sPΔ 1
11GT s + 1
11ST s + 1 1
syn
M s Dω+
1s
1mPΔ
1LoadPΔ
Regulator Governor Steam Turbine Area 1
1s δΔ
1δΔ
2B2
1R
2sPΔ 2mPΔ
Regulator Governor Steam Turbine Area 22s δΔ
2LoadPΔ
2δΔ
2
11GT s + 2
11ST s + 2 2
syn
M s Dω+
1s
1 2( )δ δΔ −Δ12T12 12 1 2( )P T δ δΔ = Δ − Δ
1Ks
2Ks
1ACE
2ACE
1vPΔ
2vPΔ
1
11sT s +
2
11sT s +
1/ synω
+
−
1/ synω
© Copyright Ned Mohan 2006 205
Fig. 12-11 Simulink results of the two-area system with AGC in Example 12-3.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
-0.5
0
0.5
1
1.5
1mPΔ
2mPΔ
12PΔ
Results of SimulinkModeling Following a Step Load Change in Control Area 1
© Copyright Ned Mohan 2006 206
Economic Dispatch: Heat Rate of a Power Plant
Fig. 12-12 Heat Rate at various generated power levels. [ ]P MW0 20 40 60 80 100
9.0
10.0
11.0
MBTU-per-HourMW
At this point, to produce 40 MWFuel Consumption = 400 MBTU-per-Hour
Heat Rate
© Copyright Ned Mohan 2006 207
Cost Curve and Marginal Cost of a Power Plant
Fig. 12-13 (a) Fuel cost and (b) Marginal cost, as functions of the power output.
( )[$ / ]
i iC Phour
[ ]iP MW0
iCΔiPΔ
( )
[$ / ]
i i
i
C PPMWh
∂∂
[ ]iP MW0( )a ( )b
© Copyright Ned Mohan 2006 208
Fig. 12-14 Marginal costs for the three generators.
0 0 0P P P
1( )C PP
∂∂
2 ( )C PP
∂∂
3( )C PP
∂∂
1P 2P 3P
λ
Load Sharing between Three Power Plants
© Copyright Ned Mohan 2006 209
CHAPTER 13
TRANSMISSION LINE FAULTS, RELAYING AND CIRCUIT BREAKERS
© Copyright Ned Mohan 2006 210
Fig. 13-1 Fault in power system.
a
b
c
g
ai
f
bi
ci
a
b
c
g
aI
f
bI
cI
( )a ( )b
Fault (Symmetric or Unsymmetric) on a Balanced Network
© Copyright Ned Mohan 2006 211
Fig. 13-2 Sequence components.
2cI
1aI
1bI
1cI2aI
2bI
aI
bI
cI0cI
0aI0bI
Symmetrical Components
© Copyright Ned Mohan 2006 212
Fig. 13-3 Sequence networks.
1aE−
+
1Z
−
+
−
+
−
+
1aV 2aV 0aV1aI 2aI 0aI
2Z 0Z
Sequence Networks: Per-Phase Representation of a Balanced Three-phase representation
© Copyright Ned Mohan 2006 213
Fig. 13-4 Three-phase symmetrical fault.
a
b
c
g
aI
f
bI
cI
( )a
1aE−
+
1Z
−
+1 0aV =
1aI
( )b
Three-Phase Symmetrical Fault (ground may or may no be involved)
© Copyright Ned Mohan 2006 214
Fig. 13-5 Single line to ground fault.
( )a
( )b
1aE−
+
−
+
−
+
−
+
2aV
0aV
a
b
c
g
aI
f
fZ
1Z
2Z
0Z
1aI
2aI
0aI
3 fZ
1aV
Single-Line to Ground (SLF) Fault through a Fault Impedance
© Copyright Ned Mohan 2006 215
Fig. 13-6 Double line to ground fault. ( )a ( )b
1aE−
+
−
+
abc
gbI
f
1Z1aI
1aVcI −
+2Z2aI
2aV−
+0Z0aI
0aV
Double-Line to Ground Fault
© Copyright Ned Mohan 2006 216
Fig. 13-7 Double line fault (ground not involved).
( )a
abc
g
bI
f
cI
( )b
1aE−
+
−
+1Z1aI
1aV−
+2Z2aI
2aV
+ −1f aZ I
Double-Line Fault (ground not involved)
© Copyright Ned Mohan 2006 217
Fig. 13-8 Path for zero-sequence currents in transformers. ( )a ( )b ( )c
Path for Zero-Sequence Currents
© Copyright Ned Mohan 2006 218
Fig. 13-9 Neutral grounded through an impedance. ( )a
nZ
( )b
+
−0aV
0Z0aI
3 nZ
Neutral Grounded through an Neutral Impedance)
© Copyright Ned Mohan 2006 219
Fig. 13-10 (a) One-line diagram of a simple power system and bus voltages.
1LoadP pu=
0LoadQ =Bus-1
Bus-2
Bus-31 0.12genX pu′′ =
2 0.12genX pu=
0 0.06genX pu=1 0.10trX pu=
2 0.10trX pu=
0 0.10trX pu=
1 0.10LineX pu=
2 0.10LineX pu=
0 0.20LineX pu=
1 1.0 0V pu= ∠ 03 0.98 11.79V pu= ∠ −
One-Line Diagram of a Simple System)
© Copyright Ned Mohan 2006 220
Fig. 13-11 Positive-sequence circuit for calculating a 3-phase fault on bus-2.
0.12j pu
1 1.0 0V pu= ∠
+
−
+
−
LoadIfaultI
0.10j pu 0.10j pu
E ′′0.9604LoadR pu=
Thee-phase Fault on Bus-2 in the Simple System
© Copyright Ned Mohan 2006 221
Single-Line to Ground (SLG) Fault in the Simple System
Fig. 13-12 Sequence networks for calculating fault current due to SLG fault on bus-2.
1 1.0 0V pu= ∠
+
−
+
−
0.10j pu
E ′′0.9604LoadR pu=
0.12j pu 0.10j pu
0.9604LoadR pu=
0.9604LoadR pu=
1aV
+
−
2aV
+
−
0aV
+
−
0.10j pu0.12j pu 0.10j pu
0.10j pu0.06j pu 0.20j pu
1aI
2aI
0aI
0aV =
+
−
/ 3 ( / 3)a faultI I=
© Copyright Ned Mohan 2006 222
Fig. 13-13 A SLG fault in the example 3-bus power system.
1mP 1eP
2eP
Bus-1Bus-3
Bus-2
2mP
An SLG Fault in the Example 3-Bus System
© Copyright Ned Mohan 2006 223
Fig. 13-14 Protection equipment.
CB
R
CT
PT
Protection in Power System
© Copyright Ned Mohan 2006 224
Fig. 13-15 Current Transformer (CT) [5]. (a)
(b)
CT
Burden
Current Transformers (CT)
© Copyright Ned Mohan 2006 225
Fig. 13-16 Capacitor-Coupled Voltage Transformer (CCVT) [5].
(a) (b)
Capacitor-Coupled Voltage Transformers (CCVT)
© Copyright Ned Mohan 2006 226
Fig. 13-17 Differential relay.
CT
CT
CT
Relay
Differential Relays
© Copyright Ned Mohan 2006 227
Fig. 13-18 Directional over-current Relay.
CB
R
CT
PT
Directional Over-Current Relays
© Copyright Ned Mohan 2006 228
Fig. 13-19 Ground directional over-current Relay.
Time
instantaneous
Ground Directional Over-Current Relays for Ground Faults
© Copyright Ned Mohan 2006 229
Fig. 13-20 Impedance (distance) relay.
X
R
Impedance (Distance) Relays
© Copyright Ned Mohan 2006 230
Fig. 13-21 Microwave terminal [5].
Microwave Terminal for Pilot Relays
© Copyright Ned Mohan 2006 231
Fig. 13-22 Zones of protection.
Time
A B C
Zone 1: instantaneousZone 2: 20-25 cycles
Zone 3: 1-1.5 sec
Zones of Protection
© Copyright Ned Mohan 2006 232
Fig. 13-23 Protection of generator and the step-up transformer.
Gen
Relay RelayRelay
CT CT CTTransformer F1 F2
Protection of Generator and its Step-up Transformer
© Copyright Ned Mohan 2006 233
Fig. 13-24 Relying in the example 3-bus power system.
AB
P jQ+
Zone1Zone2
Zone2
Relaying in the 3-Bus Example Power System
© Copyright Ned Mohan 2006 234
Fig. 13-25 6SF circuit breaker [5].
Circuit Breakers
© Copyright Ned Mohan 2006 235
Fig. 13-26 Current in an RL circuit.
0 0 . 0 5 0 . 1 0 . 1 5 0 . 2- 1
- 0 . 5
0
0 . 5
1
1 . 5
2 asymmetricsymmetric offset
+
−( )sv t
+
−
( )v t( )i t
R L
( )a ( )b
0
Illustration of Current Offset in R-L Circuits
© Copyright Ned Mohan 2006 236
CHAPTER 14
TRANSIENT OVER-VOLTAGES, SURGE PROTECTION AND INSULATION COORDINATION
© Copyright Ned Mohan 2006 237
Fig. 14-1 Lightening current impulse.
[ ]t sμ
0.5 peakI
peakIi
1t 2t
Lightning Current Impulse
© Copyright Ned Mohan 2006 238
Fig. 14-2 Lightening strike to the shield wire. ( )a ( )b
Lightening Strike to Shield Wire and Backflash
© Copyright Ned Mohan 2006 239
Fig. 14-3 Over-voltages due to switching of transmission lines. ( )b
avbv
cv
L
( )a
L
L
AB
C
500kV Line100 miles (open)
Switching Surges
© Copyright Ned Mohan 2006 240
Fig. 14-4 Frequency dependence of the transmission line parameters [Source: 2].
Frequency Dependence of Transmission Line Parameters
© Copyright Ned Mohan 2006 241
Fig. 14-5 Calculation of switching over-voltages on a transmission line.
Bus-1 Bus-3
Calculation of Switching Over-Voltages on Line 1-3 in the Example 3-Bus Power System
© Copyright Ned Mohan 2006 242
Fig. 14-6 Standard Voltage Impulse Wave to define BIL.
i
0 t
peakV
0.5 peakV
1.2 sμ 40 sμ
Standard Voltage Impulse to Define Basic Insulation Level (BIL)
© Copyright Ned Mohan 2006 243
Fig. 14-7 A 345-kV transformer voltage insulation levels.
1500
1300
1100
900
700
1 10 100 1000 10000
Line-to-ground(Peak kV)
time in sμ
1175kVBIL
BSL
choppedwave Transformer Insulation
Withstand Capability Curve
Arrester Voltage, subjected to a 8 20 Lightning Current Impulsewith a peak of 20 kA
sμ×
Transformer Insulation Protected by a ZnOArrester