First Course on POWER SYSTEMS - Oakland...

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

1mP 1eP

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

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.

ba abv

Symbols and Conventions

Fig. 2-2 Phasor diagram.

Imaginary

I I φ= ∠ −

V V 0= ∠

positiveangles

Imaginary

I I φ= ∠ −

V V 0= ∠

positiveangles

Phasors

Fig. 2-3 A circuit (a) in time-domain and (b) in phasor-domain; (c) impedance triangle.

cjX−

i( t )

v( t )2V cos( t )ω=

V V 0= ∠

Lj L j Xω =

C1j j XCω

⎛ ⎞− = −⎜ ⎟⎝ ⎠

( )a ( )b ( )c

cjX−

i( t )

V V 0= ∠

Lj L j Xω =

C1j j XCω

⎛ ⎞− = −⎜ ⎟⎝ ⎠

( )a ( )b ( )c

i( t )

V V 0= ∠

Lj L j Xω =

C1j j XCω

⎛ ⎞− = −⎜ ⎟⎝ ⎠

−V V 0= ∠

Lj L j Xω =

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

Example of Impedance Calculation

Fig. 2-5 Circuit of Example 2-2.

0.3Ω 0.5j Ω 0.2j Ω

15j Ω 7.0Ω

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

( )p t average power

( )i t

( )v t/φ ω

( )a ( )b( )i t

( )v tt

( )i t

( )v t/φ ω

( )a ( )b

Real and Reactive Power

Fig. 2-8 (a) Circuit in phasor-domain; (b) phasor diagram; (c) power triangle.

S P jQ= +

Subcircuit 1 Subcircuit 2

vV V φ= ∠

iI I φ= ∠

( b ) ( c )

S P jQ= +

Subcircuit 1 Subcircuit 2

vV V φ= ∠

iI I φ= ∠

( b ) ( c )

P, Q and VA by Phasors

Fig. 2-9 Power factor correction in Example 2-5.

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

One-line Diagram

Fig. 2-11 Three-phase voltages in time and phasor domain.

( )bnv t

( )cnv t( )anv t

a b c− −positivesequence

anV120°

Three-Phase Voltages

Fig. 2-12 Balanced wye-connected, three-phase circuit.

bnVcnV −+

anV −

bnVcnV −+

anV−

(a) (b)

bnVcnV −+

anV −

bnVcnV −+

anV −

bnVcnV −+

anV−

(a) (b)

Balanced Three-Phase Circuit Analysis

Fig. 2-13 Per-phase circuit and the corresponding phasor diagram.

anV−

(Hypothetical)

( a ) ( b )

anV−

(Hypothetical)

anV−

(Hypothetical)

( a ) ( b )

Per-Phase Analysis

Fig. 2-14 Balanced three-phase network with mutual couplings.

a AaAZ

(Hypothetical)

mutualZ

( )a ( )b

Balanced Mutual Coupling

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

bV−abV

Line-Line Voltages

Fig. 2-16 Delta-wye transformation.

bcIcaIabI

ZΥ ZΥc b

( b )( a )

bcIcaIabI

ZΥ ZΥc b

( b )( a )

Wye-Delta Transformation

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

Power Flow in AC Systems

Fig. 2-18 Power as a function of δ .

max/P P

090 0180

Power-Angle Diagram

Per Unit Quantities , , base

base base basebase

VR X ZI

= (in Ω) (2-48)

, , basebase 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

Power SystemApparatus

Energy Efficiency of Apparatus

Fig. 2-20 Ampere’s Law.

(a) (b) (c)

Electro-Magnetic Concepts:Ampere’s Law

Fig. 2-21 Example 2-9.

(a) (b)

Example of a Toroid

Fig. 2-22 B-H characteristic of ferromagnetic materials.

(a) (b)

B-H Curves in Ferromagnetic Materials

Fig. 2-23 Toroid with flux mφ .

Flux and Flux-Density

Inductance

Fig. 2-24 Coil inductance.

mλ( )N×

mφ( )mA×( )mμ×

N⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠i

mλ( )N×

mφ( )mA×( )mμ×

N⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠i

(a) (b)

Fig. 2-25 Rectangular toroid.

Example of a Toroid

Fig. 2-26 Voltage polarity and direction of flux and current.

( )i t

( )tφ

( )e t+

( )i t

( )tφ

( )e t+

Faraday’s Law

Fig. 2-27 Example 2-11.

( )tφ( )e t

Plot of time-varying Flux and Voltage

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

Leakage Flux

Fig. 2-29 Analysis including the leakage flux.

( )v t+

lL ( )i t

( )me t( )e t+

−( )v t+

lL ( )i t

( )me t( )e t+

ldiLdt

( )me t( )e t

+ −( )i t

ldiLdt

( )me t( )e t

+ −( )i t

(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

Hydro Power Generation

Fig. 3-4 Rankine thermodynamic cycle in coal-fired power plants.

Steam at High pressure

Heat in

Heat out

TurbineBoiler

Condenser

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].

Wound rotor Induction Generator

Generator-sideConverter

Grid-sideConverter

Wind Turbine

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].

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].

0 - - 0

- 1000

- 1200

Maximum Theoretical Voltage

Current Density ( i in mA/cm2 )

ActivationLosses

Ohmic Losses

MassTransport

Losses

OpenCircuitVoltage

Cell P

Cell V

)- ΔgƒE =2 F

Cell PowerPC= VC x i

0 - - 0

- 1000

- 1200

- 1000

- 1200

Maximum Theoretical Voltage

Current Density ( i in mA/cm2 )

ActivationLosses

Ohmic Losses

MassTransport

Losses

OpenCircuitVoltage

Cell P

Cell V

)- 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].

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

( )a ( )c

Transmission Tower, Conductor and Bundling

Fig. 4-2 Transposition of transmission lines.

1 cycle

( )a ( )b

Transposition

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

neutral (zeroimpedance)

line R L

Distributed Parameters

Fig. 4-4 (a) Cross-section of ACSR conductors, (b) skin-effect in a solid conductor.

towards centersurface( )a ( )b

Calculation of Transmission Line Resistance: Skin Effect

Fig. 4-5 Flux linkage with conductor-a. ( )a ( )b ( )c

Drai bi

rai x dx

Calculation of Transmission Line Inductance

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

q1 21x

Electric Field Due to Transmission Line Voltage

Fig. 4-7 Shunt capacitances.

( )a ( )b

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

( )RI s( )SI s R sL

( )xI s

Distributed-Parameter Representation

Fig. 4-10 Per-phase transmission line terminated with a resistance equal to cZ .

SI j Lω

−0R RV V= ∠

( )a ( )b

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

Long-Line Representation

Fig. 4-12 Per-phase transmission line representation based on length.

RIseriesZ

2shuntY

RIlinej Lω

( )a ( )b ( )c

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

Slack Bus

PV Bus

PQ BusP jQ+

150km 150km

Bus 1 Bus 3

Slack Bus

PV Bus

PQ BusP jQ+

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

12Z 23Z1V 3V

Calculating Y-Bus in the Example Power System

Fig. 5-3 Plot of 24 x− as a function of x .

24 x−4

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.

mφmi+

(a) (b)

Transformer Principle: Generation of Flux

Fig. 6-2 B-H characteristics of ferromagnetic materials.

(a) (b)

Core in Transformers

Fig. 6-3 Transformer with the open-circuited second coil.

Transformer

(a) (b)

Flux Coupling

Fig. 6-4 Transformer with load connected to the secondary winding.

( )1i t

( )2i t

( )1i t

( )2i t

( )2i t′( )1i t

Transformer

( )2i t( )2i t′( )1i t

Transformer

( )2i t

(a) (b)

Transformer with Load Connected to the Secondary

Fig. 6-5 Transformer equivalent circuit including leakage impedances and core losses.

1E mjX

Ideal Transformer

2I1R l1jX 2Rl2jX

Real Transformer

1E mjX

Ideal Transformer

2I1R l1jX 2Rl2jX

Real Transformer

Transformer Model

Fig. 6-6 Eddy currents in the transformer core.

circulatingcurrents

(a) (b)

Eddy Current and Hysteresis Losses

Fig. 6-7 Simplified transformer model.

pI sIpZ sZ1: n

pn sn+

sV ′

Transformer Simplified Model

pI sIpsZ 1: n

pI sIspZ1: n

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

Bus 3Bus 1

Including Nominal Turns-Ratio Transformer in Power Flow Studies

Fig. 6-12 Auto-transformer.

( )1 2V V+

( )1 2I I+

Auto-Transformer

Fig. 6-13 Phase-shift in Δ -Y connected transformers.

03012:

3jn e n

Phase-Shift Due to Wye-Delta Transformers

Fig. 6-14 Transformer for phase-angle control.

( )a ( )b

aV ′

bV ′

cV ′

a bV ′ ′

b cV ′ ′

c aV ′ ′

( )a ( )b

aV ′

bV ′

cV ′

a bV ′ ′

b cV ′ ′

c aV ′ ′

Phase-Shift Control by Transformers

Fig. 6-15 Three-winding auto-transformer.

( )HZ Ω

( )LZ Ω

( )TZ Ω

( )a( )b

a′ a

( )HZ Ω

( )LZ Ω

( )TZ Ω

( )a( )b

a′ a

Three-Winding Auto-Transformers

Fig. 6-16 General representation of an auto-transformer and a phase-shifter.

1I 2I+

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.

1/Y Z=

( )a ( )b

⎛ ⎞−⎜ ⎟⎝ ⎠

1 1 Yt t

⎛ ⎞−⎜ ⎟⎝ ⎠

1 1 Yt t

⎛ ⎞−⎜ ⎟⎝ ⎠

PU Representation of Off-Nominal Turns-Ratio Transformers

Fig. 6-18 Transformer of Example 6-3.

0.1j pu

( )a ( )b

1 0.909Y j pu= −

2 0.826Y j pu=

0.11sZ j pu=

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

MOSFET

Switching Frequency (Hz)

Thyristor IGBT MOSFETIGCT(a)

101 102 103 104

MOSFET

101 102 103 104

MOSFET

101 102 103 104

MOSFET

Symbols and Capabilities of Power Semiconductor Devices

Figure 7-2 Power semiconductor devices: (a) ratings (source: Siemens), (b) variousapplications (source: ABB).

Device blocking voltage [V]

104103102101

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

104103102101

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

( )b( )a

104103102101

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

104103102101

HVDCTraction

MotorDrive

PowerSupply

Auto-motive

Lighting

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

1000MW500MW

1000MW

312MW370MW

1620MW

2000MW

2250MW

2138MW

3100MW

1920MW

1000MW500MW

1000MW

312MW370MW

1620MW

2000MW

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.

(a) (b)

Thyristors

Fig. 7-8 Thyristor circuit with a resistive load and a series inductance.

0tω =

dvsv R

0tω =

dvsv R( )a

dvsv R

Primitive Thyristor Circuits

Fig. 7-9 Three-phase Full-Bridge thyristor converter. (a)

cnv− +

− +anv

bnv sL

− +anv ai

cnv− +

− +anv

bnv sL

− +anv ai

cnv− +

− +anv

bnv sL

− +anv ai

cnv− +

− +anv

bnv sL

− +anv ai

Three-Phase Thyristor Converter

Fig. 7-10 Waveforms in a three-phase rectifier with 0sL = and 0α = .

cvbvav

o60 tω

LL2Vdv

cvbvav

o60 tω

LL2Vdv

Three-Phase Diode Rectifier Waveforms

anv bnv cnvPnv

Three-Phase Thyristor Converter Waveforms with zero AC-Side Inductance

anvbnv

Three-Phase Inverter Waveforms

Fig. 7-13 Average dc-side voltage as a function of α . ( )b( )a

α090 0160

Rectifierd dP V I= = +

Inverterd dP V I= = −

DC-Side Voltage as a Function of Delay Angle

Fig. 7-14 Waveforms with sL .

anv bnv cnvPnv

Thyristor Converter Waveforms in the Presence of AC-Side Inductance

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

1aI1aI

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.

2dvAC 1 AC 2AC 1 AC 2

HVDC System Representation for Control

Fig. 7-19 Control of an HVDC system [3].

0dI,d refI

Inverter characteristicwith γ γ=

Rectifier characteristicin a current-control mode

Control of HVDC Converters

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.

busvLi+

busvLi +

−convV

−busV

+ −L LjX I

( )a ( )b ( )c

L LjX I

Phasor Diagram on the Ac-Side of the Voltage-Link Converter

Fig. 7-23 Synthesis of sinusoidal voltages.

1: ad 1: bd 1: cd 1: ad

−aNv

( )a ( )b

Representation of Voltage-Link Converter with Ideal Transformers

Fig. 7-24 Sinusoidal variation of turns-ratio ad .

0.5 dV

Synthesis of “Average”Sinusoidal Voltages

Fig. 7-25 Three-phase synthesis.

av bv cvac-side

0.5 dV

aNv bNv cNv

Converter Output Voltages and Voltages across the Load

Fig. 7-26 Realization of the ideal transformer functionality.

daiaia

−aNv

(a) (b)

Buck Boost

−aNv

Switching Power-Pole of Voltage-Link Converters

Fig. 7-27 PWM to synthesize sinusoidal waveform.

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±

1House

2House

3House

Residential Distribution System

Fig. 8-2 System load.

Load(MW)

percentage of the time 100%0

Load(MW)

percentage of the time

TimeAM NOON PM

12 6 12 6 12

(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.

−Utility

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

mj Lω

Induction Motor Per-Phase Diagram

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

1f LoadTorque

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

oV60Hz ac

inputrectifier

topology to convertdc to dc with isolation

Feedbackcontroller

HF transformer

dc to HF ac

OutputinV

Switch-Mode DC Power Supplies

Fig. 8-8 Uninterruptible power supply.

Rectifier Inverter Filter Critical Load

Energy Storage

Uninterruptible Power Supplies (UPS)

Fig. 8-9 Alternate feeder.

Feeder 1

Feeder 2

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

Figure 8-14 Voltage and current phasors in simple R-L circuit.

( )a( )b

Linear Load

Figure 8-15 Current drawn by power electronics equipment without PFC.

( )distortion s s1i i i= −

t0/1φ ω

s1iisvs

( )distortion s s1i i i= −

t0/1φ ω

s1iisvs

Waveforms Associated with Power Electronics-Based Load

/4I π

distortioni

Figure 5-4 Example 5-1.

/4I π

distortioni

/4I π

distortioni

/4I π

distortioni

Example of Distorted Current

Fig. 8-17 Relation between PF/DPF and THD.

%THD0 50 100 150 200 250 300

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

Distortion(%)Harmonic

100 1000−

50 100−

20 50−

1/scI I

35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11

100 1000−

50 100−

20 50−

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

100 1000−

50 100−

20 50−

1/scI I

35 ≤ h23 35h≤ <17 23h≤ <11 17h≤ <h < 11

100 1000−

50 100−

20 50−

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

IEEE Harmonic Limits

(a) (b)Figure 5-6 (a) Utility supply; (b) short circuit current.

(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

Steam at High pressure

Heat in

Heat out

TurbineBoiler

Condenser

( )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)

Synchronous Generator Structure

Fig. 9-4 Three phase windings on the stator.

axisa −

axisb −

axisc −

2 / 3π

axisa −

axisb −

axisc −

2 / 3π

3 ' 4 ' 5 '

6 ' aiθ

ai7 '1

3 ' 4 ' 5 '

6 ' aiθ

(a) (b)

Sinusoidally-Distributed Windings

Fig. 9-5 Connection of three phase windings.

axisa −

axisb −

axisc − o240∠

o120∠

axisa −

axisb −

axisc − o240∠

o120∠

(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 .

a-axis

Synchronous Generator Rotor Field

Fig. 9-7 Current direction and voltage polarities; the rotor position shown induces maximum ae .

a-axis

Ssynω

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.

a-axis

Ssynω

( )a ( )b ( )c

a-axis

(at 0)fB t =

Representation of Induced Stator Voltage due to Rotor Field

Fig. 9-9 Armature reaction due to phase currents.

axisa −

axisb −

axisc −

2 / 3π

axisa −

axisb −

axisc −

2 / 3π

( )a ( )b ( )c

(at 0)ARB t =

a-axis

,a ARE

Armature Reaction Due to Three Stator Currents

Fig. 9-10 Phasor diagram and per-phase equivalent circuit.

( )a ( )b

,a ARE

m ajX IafE

+−,a ARE

+ −m ajX I +

Superposition of the two Induced Voltages and Per-Phase Representation

Fig. 9-11 Power output and synchronism.

stability limitsteady state

generatormode

motoringmode

δ0o90−

Pstability limit

steady state

oV V 0∞ ∞= ∠

af afE E δ= ∠

−( )a

Power Out as a function of rotor Angle

Fig. 9-12 Steady state stability limit.

δ0 o90( )a

1δ 2δ

e1Pe2P

m1Pm2P

( )bδ0 o90

eP,maxeP

Steady State Stability Limit

Fig. 9-13 Excitation control to supply reactive power.

aqI⎧⎪⎨⎪⎩ o90

aIaIaI

s ajX I s ajX Is ajX I

afEafE afE

δ δ δaV

( )a ( )b ( )c

aqI⎧⎪⎨⎪⎩ o90

aIaIaI

s ajX I s ajX Is ajX I

afEafE afE

δ δ δaV

( )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

+ −'

jX IjX IjX I

s ajX I'afE

's ajX I

''s ajX I

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+

(a) (b)

S SP jQ+ R RP jQ+LjX

A Radial System

Fig. 10-2 Phasor diagram and the equivalent circuit with 1puS RV V= = . (a)

S SP jQ+

Voltages and Current Phasors with Both-Side Voltages at 1 PU

Fig. 10-3 Voltage profile along the transmission line.

(a) (b)

(1pu)SV

(1pu)RV

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

SV RVLjX

R RP jQ+

/RP SIL

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.

Synchronous Generator Reactive Power Supply Capability

Fig. 10-6 Effect of leading and lagging currents due to the shunt compensating device.

ThjX +

ThjX IThV

ThjX I

ThjX I+ −

Effect of Current Power Factor on Bus Voltage

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

1j Cω

( )a ( )b

Static Var Compensators (SVC)

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

( )a ( )c0

090α ≤

090α >

Thyristor Controlled Reactors (TCR)

Fig. 10-9 Parallel combination of SVC and TCR.

1j Cω

( )b ( )c

busV busV

I0 0 Iinductivecapacitive inductivecapacitive

1V ′

2V2V ′

LinearRange

Voltage Control by SVC and TCR Combination

Fig. 10-10 STATCOM.

convVconvI

+ −convjXI

busV−

+ −convjXI

( )a ( )b

STATCOM

Fig. 10-11 STATCOM VI characteristic. convI0 inductivecapacitive

LinearRange

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

1VE δ′∠ 0BV ∠+

I'( )d trj X X+ / 2LjX

One-Machine Infinite-Bus System

Fig. 11-2 Power-angle characteristics.

post-faultPre-fault

Bus 1 0B BV V= ∠

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

Rotor-Angle Swing Following a Fault and a Line Taken Out

Fig. 11-4 Fault on one of the transmission lines.

( )bπ δ

e mP P=

00δ cδ mδ

Pre-fault

during fault

post-fault

maxδ( )a

Bus 1 0B BV V= ∠

Power-Angle Characteristics

Fig. 11-5 Rotor oscillations after the fault is cleared.

e mP P=

01δ mδ

Pre-fault

post-fault

Rotor Oscillations After the Fault is Cleared

Fig. 11-6 Critical clearing angle. π δ

e mP P=

00δ critδ

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

40( )eP pu

e mP P=

Pre-fault

during fault

post-fault

00 22.47δ = 0115.28mδ =075cδ =

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

1mP 1eP

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

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

Regulator

Turbine

TurbineGovernor

Frequency

-SupplementaryControl

( )b mPΔ

Steam-ValvePosition

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

1 2( )m mP PΔ + Δ

unit1 unit 2

1mP2mP

unit 2

unit 1

2G1G ′

1G ′

Load Sharing

Fig. 12-5 Two control areas.

Area 1 Area 212P 21P

Synchronizing Torque between Two Control Areas

Fig. 12-6 Area Control Error (ACE) for Automatic Generation Control (AGC).

− −

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

1 2P−

1 3P−Bus-1 Bus-3

Bus-2M

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].

−− + −

−− − + −

1sPΔ 1

11GT s + 1

11ST s + 1 1

M s Dω+

1LoadPΔ

Regulator Governor Steam Turbine Area 1

1s δΔ

2sPΔ 2mPΔ

Regulator Governor Steam Turbine Area 22s δΔ

2LoadPΔ

11GT s + 2

11ST s + 2 2

M s Dω+

1 2( )δ δΔ −Δ12T12 12 1 2( )P T δ δΔ = Δ − Δ

11sT s +

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

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

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

[$ / ]

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

Load Sharing between Three Power Plants

CHAPTER 13

TRANSMISSION LINE FAULTS, RELAYING AND CIRCUIT BREAKERS

Fig. 13-1 Fault in power system.

( )a ( )b

Fault (Symmetric or Unsymmetric) on a Balanced Network

Fig. 13-2 Sequence components.

1cI2aI

0aI0bI

Symmetrical Components

Fig. 13-3 Sequence networks.

1aE−

1aV 2aV 0aV1aI 2aI 0aI

Sequence Networks: Per-Phase Representation of a Balanced Three-phase representation

Fig. 13-4 Three-phase symmetrical fault.

1aE−

+1 0aV =

Three-Phase Symmetrical Fault (ground may or may no be involved)

Fig. 13-5 Single line to ground fault.

1aE−

Single-Line to Ground (SLF) Fault through a Fault Impedance

Fig. 13-6 Double line to ground fault. ( )a ( )b

1aE−

1aVcI −

+2Z2aI

2aV−

+0Z0aI

Double-Line to Ground Fault

Fig. 13-7 Double line fault (ground not involved).

1aE−

+1Z1aI

1aV−

+2Z2aI

+ −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

−0aV

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-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=

0.10j pu0.12j pu 0.10j pu

0.10j pu0.06j pu 0.20j pu

/ 3 ( / 3)a faultI I=

Fig. 13-13 A SLG fault in the example 3-bus power system.

1mP 1eP

Bus-1Bus-3

An SLG Fault in the Example 3-Bus System

Fig. 13-14 Protection equipment.

Protection in Power System

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

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.

Differential Relays

Fig. 13-18 Directional over-current Relay.

Directional Over-Current Relays

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

instantaneous

Ground Directional Over-Current Relays for Ground Faults

Fig. 13-20 Impedance (distance) relay.

Impedance (Distance) Relays

Fig. 13-21 Microwave terminal [5].

Microwave Terminal for Pilot Relays

Fig. 13-22 Zones of protection.

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.

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.

Zone1Zone2

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

2 asymmetricsymmetric offset

−( )sv t

( )v t( )i t

( )a ( )b

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

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

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.

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.

1 10 100 1000 10000

Line-to-ground(Peak kV)

time in sμ

1175kVBIL

choppedwave Transformer Insulation

Withstand Capability Curve

Arrester Voltage, subjected to a 8 20 Lightning Current Impulsewith a peak of 20 kA

Transformer Insulation Protected by a ZnOArrester

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