First Course on POWER SYSTEMS - Oakland...

© 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


Bus-1 Bus-3

Bus-2

1mP 1eP

2mP

2eP

P jQ+

200km

150km150km

A 345-kV Example System


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


Chapter 1

POWER SYSTEMS: A CHANGING LANDSCAPE


Fig. 1-1 Interconnected North American Power Grid [2].

NATURE OF POWER SYSTEMS


Fig. 1-2 NERC Interconnections [3]. Source: NERC.

Control Areas


Fig. 1-3 One-line diagram as an example.

13.8 kVTransmission line

Generator

Load

Feeder

Step up Transformer

One-line Diagram


CHANGING LANDSCAPE OF POWER SYSTEMS AND UTILITY DEREGULATION

Fig. 1-4 Changing landscape [4]. Source: ABB. ( )a ( )b( )a ( )b


CHAPTER 2

REVIEW OF BASIC ELECTRIC CIRCUITS AND ELECTROMAGNETIC CONCEPTS


Fig. 2-1 Convention for voltages and currents.

i

av

−

+

bv

ba abv

+−

−

+ i

av

−

+

bv

ba abv

+−

−

+

Symbols and Conventions


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


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 V 0= ∠

I

Lj L j Xω =

R

C1j j XCω

⎛ ⎞− = −⎜ ⎟⎝ ⎠

+

−

( )a ( )b ( )c

i( t )

L

R

C

+

−


i( t )

L

R

C

+

−


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


Fig. 2-4 Impedance network of Example 2-1.

5j− Ω

0.1j Ω

2Ω5j− Ω

0.1j Ω

2Ω

Example of Impedance Calculation


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


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


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


0

( )a ( )b( )i t

( )v tt


0

( )i t

( )v t/φ ω

t


0

( )a ( )b

Real and Reactive Power


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


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


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


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


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


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


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


Fig. 2-15 Line-to-line voltages in a three-phase circuit.

o30

anV

bV−abV

cnV

caV

bcV

bnV

Line-Line Voltages


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


Fig. 2-17 Power transfer between two ac systems.

sVRV

+

−

+

−

I

jX

RV

sV

I

δφ

( )a

( )b

jXI

Power Flow in AC Systems


Fig. 2-18 Power as a function of δ .

max/P P

δ0

0.5

090 0180

1.0

Power-Angle Diagram


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)


Fig. 2-19 Energy Efficiency /o inP Pη = .

inP oP

lossP

Power SystemApparatus

Energy Efficiency of Apparatus


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


Fig. 2-21 Example 2-9.

i

OD

ID

i

OD

ID

OD

ID

mr

OD

ID

mr

(a) (b)

Example of a Toroid


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


Fig. 2-23 Toroid with flux mφ .

mA

mφ

mA

mφ

Flux and Flux-Density


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


Fig. 2-25 Rectangular toroid.

h

w r

h

w r

Example of a Toroid


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


Fig. 2-27 Example 2-11.

( )tφ( )e t

t0

( )tφ( )e t

t0

Plot of time-varying Flux and Voltage


Fig. 2-28 Including leakage flux. (a) (b)

i

−

+

e

i

−

+

e

i

−

+e

mφ

lφ

i

−

+e

mφ

lφ

Leakage Flux


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


CHAPTER 3

ELECTRIC ENERGY AND THE ENVIRONMENT


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.


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.


Fig. 3-3 Hydro power (Source: www.bpa.gov).

HGenerator

Penstock

Turbine

Water

Hydro Power Generation


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


Fig. 3-5 Brayton thermodynamic cycle in natural-gas power plants.

Air in

Compressor Turbine

Exhaust

Fuel in

CombustionChamber

Brayton Cycle in Gas Turbines


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


Fig. 3-7 Wind-resource map of the United States [6].

Wind Resources in the U.S.


Fig. 3-8 pc as a function of λ [7]; these would vary based on the turbine design.

Coefficient of Performance


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


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


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


Fig. 3-12 PV cell characteristics [11].

Photovoltaics


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


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


Fig. 3-15 Greenhouse effect [13].

Greenhouse Effect


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


Fig. 3-17 Electric power industry fuel costs in the U.S. in 2005 [1].

Fuel Costs in the U.S. in 2005


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


Fig. 4-2 Transposition of transmission lines.

1D2D

3D

1 cycle

( )a ( )b

a

b

c

Transposition


Fig. 4-3 Distributed parameter representation on a per-phase basis.

line

neutral (zeroimpedance)

line R L

C

Distributed Parameters


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


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


Fig. 4-6 Electric field due to a charge.

x

q1 21x

2x

Electric Field Due to Transmission Line Voltage


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


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


Fig. 4-8 A 345-kV, single-conductor per phase, transmission system.

Calculating Transmission Line Parameters using EMTDC


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


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


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


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


Fig. 4-11 Long line representation.

+

−

( )SV s

( )SI s

+

−

( )RV s

( )RI sseriesZ

2shuntY

2shuntY

Long-Line Representation


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


Fig. 4-13 Underground cable.

Underground Cables


CHAPTER 5

POWER FLOW IN POWER SYSTEM NETWORKS


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


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



Transmission Lines in Example Power System


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


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


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


CHAPTER 6

TRANSFORMERS IN POWER SYSTEMS


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


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


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


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


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


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


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


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


Fig. 6-10 Winding connections in a three-phase system. ( )a ( )b

Connection of Transformer Windings


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


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


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


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


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


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


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


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


CHAPTER 7

HIGH VOLTAGE DC (HVDC) TRANSMISSION SYSTEMS


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)



101 102 103 104

102

104

106

108

Thyr

istor

IGBT

MOSFET

Pow

er (V

A)


IGCT

(b)

101 102 103 104

102

104

106

108

Thyr

istor

IGBT

MOSFET

Pow

er (V

A)


IGCT

101 102 103 104

102

104

106

108

Thyr

istor

IGBT

MOSFET

Pow

er (V

A)


IGCT

(b)

Symbols and Capabilities of Power Semiconductor Devices


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


Dev

ice

curr

ent [

A]

104103102101

104

103

102

101

100


Dev

ice

curr

ent [

A]

104103102101

104

103

102

101

100

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

FACTS

( )b( )a


Dev

ice

curr

ent [

A]

104103102101

104

103

102

101

100

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

FACTS


Dev

ice

curr

ent [

A]

104103102101

104

103

102

101

100


Dev

ice

curr

ent [

A]

104103102101

104

103

102

101

100

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

FACTS

( )b

Power Semiconductor Devices and Applications


Fig. 7-3 HVDC system – one-line diagram. 1AC 2AC

HVDC Line

HVDC System


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


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


Fig. 7-6 Block diagram of a current-link HVDC system.

Current-Link HVDC System


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


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


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


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


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


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


Fig. 7-15 Power-factor angle. ( )b( )a

aV aV

1aI1aI

1φ−

1φ−

Power Factor Angle in Rectifier and Inverter Modes


CU One-line Diagram


Fig. 7-17 Six-pulse and 12-pulse current and voltage waveforms [2]. ( )a ( )b

( )ai Y Y−

( )ai Y − Δ

12-Pulse Waveforms


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


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.


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


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


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


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


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


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


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


CHAPTER 8

Distribution System, Loads and Power Quality


Fig. 8-1 Residential distribution system.

13.8kV

Transformer

120V±

120V±

120V±

1House

2House

3House

Residential Distribution System


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


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


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


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


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


Fig. 8-6 Torque-speed characteristic of induction motor at various applied frequencies.

emT

1f LoadTorque

0

2f3f4f5f

mω1synω

1slipω3synω

3slipω

Torque-Speed Characteristics


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


Fig. 8-8 Uninterruptible power supply.

Rectifier Inverter Filter Critical Load

Energy Storage


Energy Storage


Energy Storage

Uninterruptible Power Supplies (UPS)


Fig. 8-9 Alternate feeder.

Load

Feeder 1

Feeder 2

Load

Feeder 1

Feeder 2

Static Power-Transfer Switch


Fig. 8-10 CBEMA curve.

CBEMA Curve Showing Acceptable Voltage-Time Region


Fig. 8-11 Dynamic Voltage Restorer (DVR).

Power Electronic Interface

Loadsv−

+

− +injv

Power Electronic Interface

LoadPower Electronic Interface

Loadsv−

+

− +injv

Dynamic Voltage Restorers (DVR)


Fig. 8-12 Three-Phase Voltage Regulator (Courtesy of Siemens) [5].

Voltage Regulating Transformers


Fig. 8-13 STATCOM [4].

Utility

STATCOM

jXUtility

STATCOM

jX

STATCOM


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


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


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


( )c

( )b

( )a


1T

si

t

I

I−0

s1i

t0

I

I−

/4I π

t

distortioni

0


( )c

( )b

( )a

1T

si

t

I

I−0

s1i

t0

I

I−

/4I π

t

distortioni

0


( )c

( )b

( )a


Example of Distorted Current


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


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


1/scI I


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.


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


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.


Total

>1000

100 1000−

50 100−

20 50−

< 20


1/scI I


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.


Total

>1000

100 1000−

50 100−

20 50−

< 20

1Table 8-1 Harmonic current distortion ( / )hI I

IEEE Harmonic Limits


−

+

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


Fig. 8-19 Average retail price of electricity to ultimate customers [4].

Retail Price of Electricity in the U.S.


CHAPTER 9

SYNCHRONOUS GENERATORS


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


Fig. 9-2 Machine cross-section. (a) (b)

Air gap

Stator

Air gap

Stator

Cross-section of Synchronous Generators


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


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


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


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


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


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


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


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


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


Fig. 9-12 Steady state stability limit.

δ0 o90( )a

eP

1δ 2δ

e1Pe2P

m1Pm2P

( )bδ0 o90

eP,maxeP

Steady State Stability Limit


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


Fig. 9-14 Synchronous Condenser.

SynchronousCondenser

Synchronous Condenser


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)


Fig. 9-16 Armature reaction flux in steady state.

Armature Reaction Flux in Steady State Leading to Synchronous Reactance


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


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


CHAPTER 10

VOLTAGE REGULATION AND STABILITY IN POWER SYSTEMS


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


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


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


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


Fig. 10-5 Reactive power supply capability of synchronous generators.

A

B

C

0P

Q

Synchronous Generator Reactive Power Supply Capability


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


Fig. 10-7 V-I characteristic of SVC.

busV

1j Cω

CI

busV

0CI

( )a ( )b

Static Var Compensators (SVC)


Fig. 10-8 Thyristor-Controlled Reactor (TCR).

busV

LI

( )a ( )c0

LI

busV

090α ≤

090α >

( )b

busv

LiLi

Thyristor Controlled Reactors (TCR)


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


Fig. 10-10 STATCOM.

busV

convVconvI

+ −convjXI

+

dV−

+

busV−

+

−

convV

convI

+ −convjXI

( )a ( )b

STATCOM


Fig. 10-11 STATCOM VI characteristic. convI0 inductivecapacitive

LinearRange

busV

STATCOM V-I Characteristic


Fig. 10-12 Thyristor-Controlled Series Capacitors (TCSC) [source: Siemens Corp.]. ( )a ( )c( )b

Thyristor-Controlled Series Capacitor (TCSC)


CHAPTER 11

TRANSIENT AND DYNAMIC STABILITY OF POWER SYSTEMS


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


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


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


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


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


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


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


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


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


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


Fig. 11-11 Growing Power Oscillations: Western USA/Canada system, Aug 10, 1996 [4].

Importance of Dynamic Stability


CHAPTER 12

CONTROL OF INTERCONNECTED POWER SYSTEM AND ECONOMIC DISPATCH


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)


Fig. 12-2 (a) The Interconnections in North America, (b) Control Areas [Source: 2]

( )a ( )b

Control Areas (Balancing Authorities)


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


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


Fig. 12-5 Two control areas.

Area 1 Area 212P 21P

12jX

Synchronizing Torque between Two Control Areas


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)


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


Power Flow on Tie-Lines between Two Control Areas Following a Load Change


Fig. 12-9 Electrical equivalent of two area interconnection.

12jX 2jX1jX+

−1 1E δ∠

+

−2 2E δ∠

Electrical Equivalent of Two Areas


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


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


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


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


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


CHAPTER 13

TRANSMISSION LINE FAULTS, RELAYING AND CIRCUIT BREAKERS


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


Fig. 13-2 Sequence components.

2cI

1aI

1bI

1cI2aI

2bI

aI

bI

cI0cI

0aI0bI

Symmetrical Components


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


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)


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


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


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)


Fig. 13-8 Path for zero-sequence currents in transformers. ( )a ( )b ( )c

Path for Zero-Sequence Currents


Fig. 13-9 Neutral grounded through an impedance. ( )a

nZ

( )b

+

−0aV

0Z0aI

3 nZ

Neutral Grounded through an Neutral Impedance)


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)


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


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=


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


Fig. 13-14 Protection equipment.

CB

R

CT

PT

Protection in Power System


Fig. 13-15 Current Transformer (CT) [5]. (a)

(b)

CT

Burden

Current Transformers (CT)


Fig. 13-16 Capacitor-Coupled Voltage Transformer (CCVT) [5].

(a) (b)

Capacitor-Coupled Voltage Transformers (CCVT)


Fig. 13-17 Differential relay.

CT

CT

CT

Relay

Differential Relays


Fig. 13-18 Directional over-current Relay.

CB

R

CT

PT

Directional Over-Current Relays


Fig. 13-19 Ground directional over-current Relay.

Time

instantaneous

Ground Directional Over-Current Relays for Ground Faults


Fig. 13-20 Impedance (distance) relay.

X

R

Impedance (Distance) Relays


Fig. 13-21 Microwave terminal [5].

Microwave Terminal for Pilot Relays


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


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


Fig. 13-24 Relying in the example 3-bus power system.

AB

P jQ+

Zone1Zone2

Zone2

Relaying in the 3-Bus Example Power System


Fig. 13-25 6SF circuit breaker [5].

Circuit Breakers


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


CHAPTER 14

TRANSIENT OVER-VOLTAGES, SURGE PROTECTION AND INSULATION COORDINATION


Fig. 14-1 Lightening current impulse.

[ ]t sμ

0.5 peakI

peakIi

1t 2t

Lightning Current Impulse


Fig. 14-2 Lightening strike to the shield wire. ( )a ( )b

Lightening Strike to Shield Wire and Backflash


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


Fig. 14-4 Frequency dependence of the transmission line parameters [Source: 2].

Frequency Dependence of Transmission Line Parameters


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


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)


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

First Course on POWER SYSTEMS - Oakland...

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