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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The electrical power supply is about to change; future generation will increasingly take place in and near local neighborhoods with diminishing reliance on distant power plants. The existing grid is not adapted for this purpose as it is largely a remnant from the 20th century. Can the grid be transformed into an intelligent and flexible grid that is future proof?
This revised edition of Electrical Power System Essentials contains not only an accessible, broad and up-to-date overview of alternating current (AC) power systems, but also end-of-chapter exercises in every chapter, aiding readers in their understanding of the material introduced.
With an original approach the book covers the generation of electric energy from thermal power plants as from renewable energy sources and treats the incorporation of power electronic devices and FACTS. Throughout there are examples and case studies that back up the theory or techniques presented.
The authors set out information on mathematical modelling and equations in appendices rather than integrated in the main text. This unique approach distinguishes it from other text books on Electrical Power Systems and makes the resource highly accessible for undergraduate students and readers without a technical background directly related to power engineering.
After laying out the basics for a steady-state analysis of the three-phase power system, the book examines:
- generation, transmission, distribution, and utilization of electric energy
- wind energy, solar energy and hydro power
- power system protection and circuit breakers
- power system control and operation
- the organization of electricity markets and the changes currently taking place
- system blackouts
- future developments in power systems, HVDC connections and smart grids
The book is supplemented by a companion website from which teaching materials can be downloaded.
https://www.wiley.com//legacy/wileychi/powersystem/material.htmlSie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 562
Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Copyright
Preface
List of Abbreviations
List of Symbols
Text Symbols
Graphical Symbols
Chapter 1: Introduction to Power System Analysis
1.1 Introduction
1.2 Scope of the Material
1.3 General Characteristics of Power Systems
1.4 Phasors
1.5 Equivalent Line-to-neutral Diagrams
1.6 Power in Single-phase Circuits
1.7 Power in Three-phase Circuits
1.8 Per-unit Normalization
1.9 Power System Structure
Problems
References
Chapter 2: The Generation of Electric Energy
2.1 Introduction
2.2 Thermal Power Plants
2.3 Nuclear Power Plants
2.4 Renewable Energy
2.5 The Synchronous Machine
References
Chapter 3: The Transmission of Electric Energy
3.1 Introduction
3.2 Transmission and Distribution Network
3.3 Network Structures
3.4 Substations
3.5 Substation Concepts
3.6 Protection of Transmission and Distribution Networks
3.7 Surge Arresters
3.8 Transformers
3.9 Power Carriers
3.10 High-Voltage Direct Current Transmission
Problems
References
Chapter 4: The Utilization of Electric Energy
4.1 Introduction
4.2 Types of Load
4.3 Classification of Grid Users
Problems
Reference
Chapter 5: Power System Control
5.1 Introduction
5.2 Basics of Power System Control
5.3 Active Power and Frequency Control
5.4 Voltage Control and Reactive Power
5.5 Control of Transported Power
5.6 Flexible AC Transmission Systems (FACTS)
Problems
References
Chapter 6: Energy Management Systems
6.1 Introduction
6.2 Load Flow or Power Flow Computation
6.3 Optimal Power Flow
6.4 State Estimator
Problems
References
Chapter 7: Electricity Markets
7.1 Introduction
7.2 Electricity Market Structure
7.3 Market Clearing
7.4 Social Welfare
7.5 Market Coupling
7.6 Allocation Mechanism and Zonal/Nodal Markets
References
Chapter 8: Future Power Systems
8.1 Introduction
8.2 Renewable Energy
8.3 Decentralized or Distributed Generation
8.4 Power-Electronic Interfaces
8.5 Energy Storage
8.6 Blackouts and Chaotic Phenomena
References
Appendix A: Maxwell's Laws
A.1 Introduction
A.2 Power Series Approach to Time-Varying Fields
A.3 Quasi-static Field of a Parallel-plate Capacitor
A.4 Quasi-static Field of a Single-turn Inductor
A.5 Quasi-static Field of a Resistor
A.6 Circuit Modeling
Reference
Appendix B: Power Transformer Model
B.1 Introduction
B.2 The Ideal Transformer
B.3 Magnetically Coupled Coils
B.4 The Nonideal Transformer
B.5 Three-Phase Transformer
Appendix C: Synchronous Machine Model
C.1 Introduction
C.2 The Primitive Synchronous Machine
C.3 The Single-Phase Synchronous Machine
C.4 The Three-Phase Synchronous Machine
C.5 Synchronous Generator in the Power System
Appendix D: Induction Machine Model
D.1 Introduction
D.2 The Basic Principle of the Induction Machine
D.3 The Magnetic Field in the Air Gap
D.4 A Simple Circuit Model for the Induction Machine
D.5 Induction Motor in the Power System
Appendix E: The Representation of Lines and Cables
E.1 Introduction
E.2 The Long Transmission Line
E.3 The Medium-Length Transmission Line
E.4 The Short Transmission Line
E.5 Comparison of the Three Line Models
E.6 The Underground Cable
Solutions
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
Further Reading
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction to Power System Analysis
Figure 1.1 The Earth's city lights, indicating the most urbanized areas. The Visible Earth, NASA.
Figure 1.2 Transmission line–transformer–transmission line–load: the energy is stored in the electromagnetic field.
Figure 1.3 Break-even distance for HVDC [4].
Figure 1.4 Alternating voltages: triangular, sinusoidal, and block.
Figure 1.5 The definition of RMS values of sinusoidal quantities.
Figure 1.6 Mean value of a squared sine.
Figure 1.7 Phase voltages in a balanced three-phase power system (50 Hz).
Figure 1.8 A balanced three-phase power system.
Figure 1.9 Magnetic field generated by a three-phase coil system [9].
Figure 1.10 Magnetic field generated by a single-phase coil system.
Figure 1.11 Magnetic field generated by a two-phase coil system.
Figure 1.12 Two- (a) and three- (b) conductor system.
Figure 1.13 Reducing losses by increasing the voltage level.
Figure 1.14 Voltage levels and transformation steps in the Dutch power system;
*
this voltage level can be 20 kV as well.
Figure 1.15 Line-to-line and line-to-neutral voltages.
Figure 1.16 Relation between a counterclockwise rotating radius and a sinusoidal signal.
Figure 1.17 Relation between the sinusoidal voltage and current and the corresponding phasors for a resistance, inductance, and capacitance.
Figure 1.18 The phasor as a vector in the complex plane.
Figure 1.19 Two basic operations on vectors: addition and subtraction.
Figure 1.20 The complex power, impedance, and admittance as vectors in the complex plane.
Figure 1.21 Vector diagrams of various powers of the
a
operator.
Figure 1.22 The relation between the line-to-line and the line-to-neutral voltage.
Figure 1.23 Power transport over a short single-phase transmission line.
Figure 1.24 Phasor diagram.
Figure 1.25 Conversion of a delta-connected load to a wye-connected load.
Figure 1.26 Phasors in Figure 1.24, obtained from single-phase computations (solid), are rotated counterclockwise with 120 (dashed) and 240 (dotted) degrees.
Figure 1.27 An inductive load split up into a resistor in parallel with an inductor.
Figure 1.28 Voltage, current, and instantaneous power of an inductive load.
Figure 1.29 A simple series (a) and parallel (b) circuit.
Figure 1.30 Phasor diagram of a single-phase load.
Figure 1.31 Quadrant diagram.
Figure 1.32 The complex power (consumed by an inductive load).
Figure 1.33 The complex power consumed by the inductive load.
Figure 1.34 An ideal (single-phase) transformer.
Figure 1.35 Single-phase circuit with ideal transformers.
Figure 1.36 Per-unit diagram of the single-phase circuit in Figure 1.35.
Figure 1.37 One-line diagram of a section of a power system [10].
Chapter 2: The Generation of Electric Energy
Figure 2.1 A thermal power plant. Reproduced with permission of TenneT TSO B.V.
Figure 2.2
pV
diagram of an ideal gas with two states (
p
1
,
V
1
) and (
p
2
,
V
2
).
Figure 2.3 Schematic representation of a heat engine working between a hot reservoir (
T
1
) and a cold reservoir (
T
2
).
Figure 2.4 The
pV
diagram of the Carnot cycle.
Figure 2.5 The principal components of a thermal power plant with the Rankine cycle.
Figure 2.6 The Rankine cycle.
Figure 2.7 A typical fission reaction.
Figure 2.8 The boiling water reactor.
Figure 2.9 The pressurized water reactor.
Figure 2.10 A typical fusion reaction.
Figure 2.11 A wind park. Reproduced with permission of TenneT TSO B.V.
Figure 2.12 Typical power curve of a wind turbine.
Figure 2.13 The Danish wind turbine concept.
Figure 2.14 A wind turbine with a doubly fed induction generator.
Figure 2.15 A direct drive wind turbine.
Figure 2.16 The Kaplan water turbine. Reproduced with permission of Voith Siemens Hydro Power Generation GmbH & Co. KG.
Figure 2.17 The Francis water turbine. Reproduced with permission of Voith Siemens Hydro Power Generation GmbH & Co. KG.
Figure 2.18 The Pelton wheel turbine. Reproduced with permission of Voith Siemens Hydro Power Generation GmbH & Co. KG.
Figure 2.19 Power tower. Reproduced with permission of DOE/NREL.
Figure 2.20 Solar panels at Aspen Mountain. Reproduced with permission of DOE/NREL.
Figure 2.21 A dry steam power plant.
Figure 2.22 A flash steam power plant.
Figure 2.23 A binary-cycle power plant.
Figure 2.24 Cross sections of an elementary three-phase generator with (a) a two-pole cylindrical rotor and (b) a four-pole salient-pole rotor. The black dot indicates that the positive current is directed out of plane of the paper. The cross indicates that the positive current is directed into plane of the paper.
Figure 2.25 The equivalent circuit of a synchronous generator connected to an infinite bus with the corresponding phasor diagram; the generator is coupled to the grid, but there is no power exchange.
Figure 2.27 The equivalent circuit of a synchronous generator connected to an infinite bus with the corresponding phasor diagram; the generator injects power into the grid.
Figure 2.26 Generator and grid represented as two objects tied together by an elastic spring; (a) no power exchange between the generator and the grid (the situation of Figure 2.25); (b) the generator injects power into the grid (the situation of Figure 2.27).
Figure 2.28 “Active and reactive axis” projected in the phasor diagram.
Figure 2.29 “Phasor diagram” projected in the three-phase active and reactive power coordinate system.
Figure 2.30 Heating limits.
Figure 2.31 Active power of a synchronous generator as a function of the power angle δ.
Figure 2.32 Example of a loading capability curve of a synchronous generator.
Chapter 3: The Transmission of Electric Energy
Figure 3.1 Selection of rated voltage for three-phase AC power transmission [2].
Figure 3.2 Voltage levels and transformation steps in the Dutch power system;
*
this voltage level can be 20 kV as well.
Figure 3.3 The Dutch high-voltage network. Reproduced with permission of TenneT TSO B.V.
Figure 3.4 Network structures with single-point feeding; (a) radial structure; (b) loop structure; (c) multi-loop structure.
Figure 3.5 Restoration of energy supply in a faulted, radially operated system.
Figure 3.6 Network structure with multiple-point feeding.
Figure 3.7 An open-air substation. Reproduced with permission of TenneT TSO B.V.
Figure 3.8 Three single-phase circuit breakers. Reproduced with permission of TenneT TSO B.V.
Figure 3.9 Pantograph disconnector in open position. Reproduced with permission of TenneT TSO B.V.
Figure 3.10 A feeder of an SF
6
-insulated substation (E-SEP 245 kV). Reproduced with permission of Eaton Holec.
Figure 3.11 Single bus system.
Figure 3.12 Double bus system.
Figure 3.13 Substation layout according to the polygon concept.
Figure 3.14 The one-and-a-half circuit breaker concept.
Figure 3.15 Division of power system into protection zones.
Figure 3.16 Overlapping zones of protection systems.
Figure 3.17 Locations of the current transformers.
Figure 3.18 A simple radial line.
Figure 3.19 Differential comparison protection applied to a generator winding.
Figure 3.20 Current-limiting fuse link.
Figure 3.21 Degree of thermal ionization for some metal vapors and atomic gases.
Figure 3.22 Cross section of an oil circuit breaker.
Figure 3.23 Operating principle of an SF
6
puffer circuit breaker.
Figure 3.24 Vacuum interrupter with slits in the contacts to bring the arc in a spiraling motion.
Figure 3.25 The use of horseshoe magnets as is done in the Eaton Holec interrupters. Reproduced with permission of Eaton Holec.
Figure 3.26 Porcelain-housed metal oxide surge arrester.
Figure 3.27 An ideal transformer.
Figure 3.28 Two extreme transformer “designs.”
Figure 3.29 Single-phase transformer arrangements.
Figure 3.30 Construction of a three-phase transformer from three single-phase transformers (view from above).
Figure 3.31 Three-phase transformer arrangements.
Figure 3.32 A three-phase transformer under construction. Reproduced with permission of TenneT TSO B.V.
Figure 3.33 A single-phase three-winding transformer.
Figure 3.34 A Yy-4 transformer (the terminals of the secondary side of the transformer are labeled as c/a/b). When the terminals of the secondary side of the transformer are labeled as a/b/c, a Yy-0 transformer results. When the terminals of the secondary side of the transformer are labeled as b/c/a, a Yy-8 transformer results.
Figure 3.35 A Yd-11 transformer (the terminals of the secondary side of the transformer are labeled as a/b/c). When the terminals of the secondary side of the transformer are labeled as c/a/b, a Yd-3 transformer results. When the terminals of the secondary side of the transformer are labeled as b/c/a, a Yd-7 transformer results.
Figure 3.36 Transformer core characteristics: (a) nonlinear
B–H
characteristic, (b) nonlinear
B–H
characteristic with hysteresis, and (c) nonlinear Φ
–i
m
characteristic with hysteresis.
Figure 3.37 Construction of the magnetization current and a comparable wave shape built up by a first and third harmonic (inset).
Figure 3.38 Remanent flux and transformer inrush current.
Figure 3.39 Voltage, flux, and current values after energization of a power transformer.
Figure 3.40 Transformer equivalent circuit with the secondary circuit elements referred tothe primary side of the ideal transformer.
Figure 3.41 Overhead transmission lines. Reproduced with permission of TenneT TSO B.V.
Figure 3.42 Cross section of an aluminum conductor with a steel core (ACSR: aluminum conductor steel reinforced).
Figure 3.43 A 150 kV double-circuit transmission line tower (distances are given in meters). (a) Ground wires or shield wires. (b) A bundle of two conductors per phase.
Figure 3.44 A single disk of an insulator string.
Figure 3.45 Three disks of an insulator string and a three-conductor bundle suspended from a tower by an insulator string. Reproduced with permission of TenneT TSO B.V.
Figure 3.46 Arcing horns protect the insulator string from a high-current arc. Reproduced with permission of TenneT TSO B.V.
Figure 3.47 A locally strong electric field. (a) A parallel-plate capacitor with a sharp point on one of the plates. (b) A transmission line with a rain drop.
Figure 3.48 Negative corona [8].
Figure 3.49 Front view of a transmission line conductor above the Earth's surface.
Figure 3.50 Reducing the electric field strength at the surface of the conductor (a) by increasing the diameter of the conductor or (b) by using a conductor bundle with 4 conductors.
Figure 3.51 Current-carrying bundled conductors attract each other. For the sake of clarity, only the magnetic field surrounding the right conductor that causes an electromagnetic force on the left conductor is shown in the drawing; the interaction of the left conductor on the right one is not illustrated here. But if you hold your book upside down, you can see this effect.
Figure 3.52 A spacer for a four-conductor bundle. Reproduced with permission of Alcoa Conductor Accessories.
Figure 3.53 Galloping lines. The arrows indicate the minimum distance (when there is a risk of a short circuit) and the maximum distance between the conductors. Reproduced with permission of M. Tunstall.
Figure 3.54 Snow and ice deposit on a conductor. The arrow indicates the spatial orientation of the line.
Figure 3.55 A conductor vibration damper. Reproduced with permission of Alcoa Conductor Accessories.
Figure 3.56 The Earth–ionosphere electric system [9]. Currents are in μA/km
2
; average of the total Earth, over a long time.
Figure 3.57 Charge separation in clouds. The highlighted box in the middle shows the interaction between a large polarized raindrop falling down and two smaller charged drops that are lifted up.
Figure 3.58 Lightning protection by means of shield wires;
r
s
1
corresponds to a large lightning current that hits either the ground or the shield wires;
r
s
2
corresponds to a smaller lightning current that hits either the ground or the phase conductor.
Figure 3.59 Transposition of overhead transmission lines.
Figure 3.60 A twisting pylon.
Figure 3.61 A three-core (a) and single-core (b) cable; (a) 6/10 kV with 3 × 240 mm
2
aluminum, circular solid, conductors and XLPE insulation; (b) 220/380 kV with 1 × 1600 mm
2
copper, circular stranded compacted, conductors and XLPE insulation. Reproduced with permission of Prysmian.
Figure 3.62 Belted cable (a) and a three-core cable where each of the three-phase conductors has its own sheath (b) and their equipotential lines.
Figure 3.63 Conductor construction.
Figure 3.64 The flux lines of a current-carrying cable and the induced eddy currents in the sheath of a neighboring cable (without bonding) [10].
Figure 3.65 Single-point bonding (a) and both-ends bonding (b) of a metallic cable sheath.
Figure 3.66 Cross-bonding of single-conductor cables.
Figure 3.67 Gas-insulated line. Reproduced with permission of Siemens.
Figure 3.68 (a) Mercury-arc valve. (b) Mutator for rectification of three-phase AC.
Figure 3.69 Principle diagram of a DC generator.
Figure 3.70 A 6-pulse converter.
Figure 3.71 The operation of the thyristor.
Figure 3.72 A 6-pulse thyristor bridge.
Figure 3.73 Monopolar and bipolar point-to point-system.
Chapter 4: The Utilization of Electric Energy
Figure 4.1 Example of a daily load curve in the Netherlands (Monday, May 15, 2006).
Figure 4.2 Magnetic field generated by a three-phase coil system [1].
Figure 4.3 Three-phase induction motor supplied by a single-phase source and the resulting rotating magnetic field.
Figure 4.4 The equivalent circuit of a synchronous motor connected to an infinite bus with the corresponding phasor diagram; the resistance of the stator coil is neglected.
Figure 4.5 A model for an induction motor as derived in Figure D.9.
Figure 4.6 The electromagnetic torque of an induction motor as a function of the angular rotor speed.
Figure 4.7 Voltage and current waveforms of an incandescent light bulb (a) without a dimmer and (b) with a dimmer (the grid voltage is shown as a dashed line).
Figure 4.8 Single-phase half-wave (a) and full-wave (b) rectification.
Figure 4.9 Single-phase full-wave rectifier during the positive (a) and the negative (b) half cycle of
V
in
.
Figure 4.10 Full-wave rectification with a capacitive filter.
Figure 4.11 Single-phase full-wave controlled rectification.
Figure 4.12 Three-phase full-wave uncontrolled rectification.
Figure 4.13 Wiring layout of four houses or buildings connected to a three-phase supply.
Figure 4.14 Line-to-neutral diagram of a load that is supplied by a simplified grid.
Figure 4.15 High-speed train. Reproduced with permission of TenneT TSO B.V.
Figure 4.16 Supply principle of a DC railway system.
Figure 4.17 Supply principle of an AC railway system.
Chapter 5: Power System Control
Figure 5.1 Generator connected to a variable load.
Figure 5.2 Phasor diagram: (a) reactive power consumption
Q
≈ 1 Mvar; (b) reactive power consumption
Q
= 1.5 Mvar.
Figure 5.3 Power transport through an impedance.
Figure 5.4 The basic principle of the speed governor control system of a generating unit.
Figure 5.5 Speed governor characteristics.
Figure 5.6 Speed governor characteristics of two generators in parallel.
Figure 5.7 LFC action.
Figure 5.8 Power exchange between three control areas: (a) original (scheduled) situation; (b) incremental generation after losing 400 MW of generation in control area B and the resulting flows.
Figure 5.9 Simplified AVR diagram.
Figure 5.10 AVR control: increased internal EMF resulting in a larger reactive power generation (dashed).
Figure 5.11 AVR characteristics.
Figure 5.12 Tap-changing transformer.
Figure 5.13 Generator connected to a variable load by means of a tap-changing transformer.
Figure 5.14 Capacitor banks. Reproduced with permission of TenneT TSO B.V.
Figure 5.15 Static var compensator (SVC). TSC, thyristor-switched capacitor; TCR, thyristor-controlled reactor.
Figure 5.16 Current (solid line) through a TCR as a function of the delay angle α and its first harmonic component (dotted line).
v
, the system voltage;
i
, the lagging current through the reactor.
Figure 5.17 SVC
V–I
characteristic.
V
, the phasor of the system voltage;
I
, the phasor of the first harmonic current.
Figure 5.18 Voltage source (PWM) converter: upper graph, pulse-width modulation (PWM) control signals; lower graph, the AC voltage (solid line) at the output and its first harmonic component (dotted line).
Figure 5.19 Principle layout of a STATCOM.
Figure 5.20 Reactive power exchange between the STATCOM and the grid.
Figure 5.21 The phase shifter [1].
Figure 5.22 Phasor diagram of the phase shifter.
Figure 5.23 Phase-shifter application in case of two parallel connections: (a) the single-line diagram; (b) the single-phase equivalent circuit.
Figure 5.24 A transmission line with a series capacitor.
V
r
, the voltage at the receiving end of the line without a series capacitor; , the voltage at the receiving end of the line with a series capacitor.
Figure 5.25 Thyristor-controlled series capacitor (TCSC).
Figure 5.26 The variation of the TCSC reactance as a function of the firing angle of the thyristor.
Figure 5.27 Static synchronous series compensator (SSSC).
Figure 5.28 Unified power flow controller (UPFC).
Figure 5.29 Phasor diagrams illustrating the operation of the UPFC.
Chapter 6: Energy Management Systems
Figure 6.1 Load flow computation: input data and computational results.
Figure 6.2 Example admittance network for building the admittance matrix
Y
.
Figure 6.3 Example load flow network.
Figure 6.4 The Newton–Raphson method.
Figure 6.5 Flowchart of the Newton–Raphson load flow computation.
Figure 6.6 The Newton–Raphson method with a fixed Jacobian.
Figure 6.7 State estimator: input data and computational results.
Figure 6.8 DC circuit with two voltmeters and one ammeter.
Figure 6.9 Probability density function of the chi-square distribution for
df
degrees of freedom. The black area is equal to . indicates the value from a chi-square distribution (with
df
degrees of freedom), which has a given area α above it; those critical values can be read from Table 6.5.
Figure 6.10 Flowchart of the state estimator computation.
Chapter 7: Electricity Markets
Figure 7.1 Organization of the electricity market; solid arrows, power flows; open arrows, commercial relations.
Figure 7.2 Market time frames.
Figure 7.3 Market clearing algorithm.
Figure 7.4 Market clearing example.
Figure 7.5 Consumer and producer surplus.
Figure 7.6 Two interconnected areas.
Figure 7.7 A low-price and a high-price area.
Figure 7.8 Import (area B) and export (area A). The dotted lines indicate the situation without import/export between the two areas.
Figure 7.9 Market clearing example with 50 MW export.
Figure 7.10 Net export curve (NEC).
Figure 7.11 NECs of a low-price and a high-price area.
Figure 7.12 NECs of a low-price and a high-price area combined in a single graph and a close-up. Hatched area: gain in social welfare.
Figure 7.13 Congestion of the interconnection capacity and a close-up. Hatched area: market surplus. Gray area: utility surplus (congestion rent). Black area: market efficiency loss (deadweight loss).
Figure 7.14 Transit flows (a) and loop flows (b) are indicated by the white arrows.
Chapter 8: Future Power Systems
Figure 8.1 Today's electricity grid: central generation, unidirectional distribution. Reproduced with permission of Eric Verdult, www.kennisinbeeld.nl. (
See color plate section for the color representation of this figure.
)
Figure 8.2 Future power grid: distributed generation, bidirectional distribution. Reproduced with permission of Eric Verdult, www.kennisinbeeld.nl. (
See color plate section for the color representation of this figure.
)
Figure 8.3 Power-electronic interfaces.
Figure 8.4 Example network.
Figure 8.5
PV
curve and load characteristic.
Figure 8.6
PV
curves at various power factors.
Figure 8.7 The area around the great lakes.
Figure 8.8 Cascade sequence [7]. The arrows indicate the power flows. Black lines represent separations between areas within the Eastern Interconnection. Regions affected by the blackout are highlighted by the dashed areas.
Figure 8.9 The Italian interconnectors [8].
Figure 8.10 Increased sag of overloaded lines can cause a short circuit; dashed, the “normally” loaded line; solid, the overloaded line.
Figure 8.11 The Italian frequency (line only) and total active power imbalance (line + area) [8]; the dotted line at a frequency of 47.5 Hz is the critical threshold.
Appendix A: Maxwell's Laws
Figure A.1 Schematic outline of the Maxwell relations.
Figure A.2 Parallel-plate capacitor in air. All fringing in the resulting fields can be neglected as and .
Figure A.3 Single-turn inductor in air. All fringing in the resulting fields can be neglected as and .
Figure A.4 Resistor in air. All fringing in the resulting fields can be neglected as and .
Appendix B: Power Transformer Model
Figure B.1 The ideal transformer.
Figure B.2 The circuit representation of an ideal transformer.
Figure B.3 Transformation of a parallel-connected impedance.
Figure B.4 Transformation of a series-connected impedance.
Figure B.5 Two magnetically coupled coils.
Figure B.6 Magnetically coupled coils as a combination of an ideal transformer and a coil.
Figure B.7 Approach of representing magnetically coupled coils as a combination of an ideal transformer and a coil.
Figure B.8 Magnetically coupled coils as a combination of an ideal transformer and a coil.
Figure B.9 A transformer with leakage flux.
Figure B.10 Transformer equivalent circuit without core losses.
Figure B.11 Transformer equivalent circuit.
Figure B.12 Transformer equivalent circuit with the secondary circuit elements referred to the primary side of the ideal transformer.
Figure B.13 Simplified transformer equivalent circuit.
Figure B.14 Single-phase model of a three-phase, phase-shifting, transformer.
Appendix C: Synchronous Machine Model
Figure C.1 Three-dimensional view of a simple synchronous machine.
Figure C.2 Cross section of a simple synchronous machine and the field lines of the magnetic flux in the case that only the rotor current is present.
Figure C.3 The distribution of the magnetic flux density in the air gap for the simple synchronous machine of Figure C.2.
Figure C.4 The surface
S
as used to calculate the flux linkage with the stator winding.
Figure C.5 Flux linkage with a single stator turn.
Figure C.6 The induced voltage in a single stator turn.
Figure C.7 A distributed stator winding and the induced voltages.
Figure C.8 A sinusoidal distribution of the stator winding.
Figure C.9 The single-phase synchronous machine represented as two magnetically coupled coils.
Figure C.10 The lumped-element equivalent circuit for the single-phase synchronous machine.
Figure C.11 Cross section of a three-phase synchronous machine.
Figure C.12 Cross section of a single-phase machine to calculate the flux density due to the stator current.
Figure C.13 Single-phase model of a three-phase synchronous machine.
Figure C.14 The equivalent circuit of a synchronous generator connected to an infinite bus with the corresponding phasor diagram; the resistance of the stator winding is neglected.
Figure C.15 The phasor diagram of an underexcited (a) and an overexcited (b) synchronous machine.
Appendix D: Induction Machine Model
Figure D.1 Cross section of a simple induction machine with a single concentrated rotor winding (a) and the field lines of the magnetic flux at time
t
= 0 (b).
Figure D.2 The derived equations for the primitive induction machine with a single concentrated rotor winding as a function of time.
Figure D.3 The cross section of the primitive induction machine with two rotor windings.
Figure D.4 The electromagnetic torque of the primitive induction machine as a function of the angular velocity.
Figure D.5 The magnetic flux density distribution in the air gap due to the current
i
rd
. The dashed trace is the first harmonic of the flux density.
Figure D.6 A circuit model for the induction machine with two magnetically coupled coils.
Figure D.7 A circuit model for the induction machine with an ideal transformer.
Figure D.8 A practical circuit model for the induction machine.
Figure D.9 An induction machine model for power calculations.
Appendix E: The Representation of Lines and Cables
Figure E.1 Incremental length of a transmission line.
Figure E.2 Series connection of two transmission lines, represented as two ports.
Figure E.3 Equivalent circuit of a long transmission line.
Figure E.4 Equivalent circuit of a medium-length transmission line.
Figure E.5 Equivalent circuit of a short transmission line.
List of Tables
Chapter 1: Introduction to Power System Analysis
Table 1.1 Voltage–current relations
Table 1.2 Delta–wye transformation
Table 1.3 Power definitions
Table 1.4 Base quantities in the per-unit system
Chapter 2: The Generation of Electric Energy
Table 2.1 Net generation capacity in some European countries as of 31 December 2015 [3]
Chapter 3: The Transmission of Electric Energy
Table 3.1 The power carriers in the Dutch power system [7]
Table 3.2 Comparison of typical parameters of a 150 kV overhead line and underground cable
Chapter 6: Energy Management Systems
Table 6.1 Network node types
Table 6.2 Consecutive iterations with initial voltage:
V
2
= 1∠0
Table 6.3 Consecutive iterations with initial voltage:
V
2
= 0.1∠0
Table 6.4 Consecutive iterations of a decoupled load flow
Table 6.5 χ
2
critical values
Chapter 7: Electricity Markets
Table 7.1 Bilateral versus mediated market arrangements
Table 7.2 The producers' sale bids
Table 7.3 The consumers' purchase bids
Table 7.4 Aggregated supply and demand
Table 7.5 Trading volumes and revenues/expenses
Table 7.6 Aggregated supply and demand
Appendix A: Maxwell's Laws
Table 1.1 Electromagnetic field relations expressed in zero-, first- and
k
th
-order terms
Appendix B: Power Transformer Model
Table B.1 Single-phase equivalent models of three-phase transformers
Appendix E: The Representation of Lines and Cables
Table E.1 Line models for various line lengths
Table E.2 Sending end voltages computed with the three line models at various line lengths
Electrical Power System Essentials
Pieter Schavemaker
Principal Consultant The Netherlands
Lou van der Sluis
Professor emeritus Electrical Power Systems Delft University of Technology The Netherlands
Second Edition
This edition first published 2017
© 2017 John Wiley & Sons Ltd
First edition published 2008 by John Wiley & Sons Ltd.
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Library of Congress Cataloging-in-Publication Data
Names: Schavemaker, Pieter. | Van der Sluis, Lou.
Title: Electrical power system essentials / Pieter Schavemaker and Lou van der Sluis.
Description: Second edition. | Chichester, West Sussex : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016045881| ISBN 9781118803479 (cloth) | ISBN 9781118803462 (epub)
Subjects: LCSH: Electric power systems. | Electric power distribution. | Electric power production.
Classification: LCC TK1001 .S3555 2017 | DDC 621.319/13-dc23 LC record available at https://lccn.loc.gov/2016045881
A catalogue record for this book is available from the British Library.
Cover image: Reproduced by permission of TenneT TSO B.V.
Cover design by Wiley
Preface
In the field of power system analysis, an extensive amount of high-quality literature is available. Most of these textbooks follow more or less the same line and cover the same topics. This book differs from existing materials because the (steady-state) modeling of the power system components is covered in appendices. Therefore, the focus in the chapters itself is not on the modeling, but on the structure, functioning, and organization of the power system. The appendices contribute to the book by offering material that is not an integral part of the main text, but support and enhance it and as such are an integral part of the book. The book contains a large number of problems of which the extensive solutions are presented in a separate chapter.
The following is a short summary of the contents of the chapters and the appendices.
Chapter 1
(Introduction to Power System Analysis)
This first chapter describes the scope of the material and is an introduction to the steady-state analysis of power systems. Questions such as “why AC,” “why 50 or 60 Hz,” “why sinusoidally shaped AC,” “why a three-phase system” are addressed. The basics for a steady-state analysis of balanced three-phase power systems are outlined, such as phasors, single-line diagrams, active power, reactive power, complex power, power factor, and per-unit normalization.
Chapter 2
(The Generation of Electric Energy)
The conversion from a primary source of energy to electrical energy is the topic of
Chapter 2
. The primary source of energy can be fossil fuels such as gas, oil, and coal or uranium, but can come from renewable sources as well: wind energy, hydropower, solar power, or geothermal power. In order to understand the nature of a thermal power plant, which is still the main source of power in the system, the principles of thermodynamics are briefly discussed. The final conversion from mechanical energy to electrical energy is achieved by the synchronous machine. The coupling of the machine with the grid and the actual power injection is analyzed.
Chapter 3
(The Transmission of Electric Energy)
The transmission and distribution network is formed by the overhead lines, the underground cables, the transformers, and the substations between the points of power injection and power consumption. Various substation concepts are presented, together with substation components and the protection installed. The transformers, overhead transmission lines, underground cables, gas-insulated transmission lines, protective relay operating principles, surge arresters, fuses, and circuit breakers are then considered in more detail. The transformer design, possible phase shift, and specific properties due to the magnetic core are highlighted. As overhead transmission lines are the most visible part of the power system, they are discussed from the point of view of what may be seen and why it is like that. The underground cables are also considered, contrasting them with overhead transmission. The chapter ends with the principles of HVDC transmission.
Chapter 4
(The Utilization of Electric Energy)
The power system is designed and arranged in such a way that demand may be fulfilled: consumers are supplied with the requested amount of active and reactive power at constant frequency and with a constant voltage. A load actually transforms the AC electrical energy into another form of energy. The focus in this chapter is on the various types of loads that transform the AC electrical energy into mechanical energy (synchronous and induction motors), light, heat, DC electrical energy (rectifiers), and chemical energy. After that, the individual loads in the system are clustered and classified as grid users according to three categories: residential loads (mostly single-phase loads), commercial and industrial loads (often three-phase loads), and electric railways (either DC or single-phase AC).
Chapter 5
(Power System Control)
Continuous control actions are necessary in the system for the control of the voltage, to maintain the balance between the amount of generated and consumed electricity, and to keep the system frequency at either 50 or 60 Hz. It is demonstrated that, in transmission networks, there is more or less a “decoupling” between the active power and the voltage angles on one side and the reactive power and voltage magnitudes on the other, which is the basis for the control. The power balance is maintained (primary control), and the system frequency deviation minimized (secondary control), by controlling the active power output of the generators. Voltage is controlled locally either at generator buses by adjusting the generator voltage control or at fixed points in the system where tap-changing transformers, capacitor banks, or other reactive power consumers/producers are connected. Flexible AC transmission systems (FACTS) devices are large power-electronic devices; they are operated in a shunt configuration for reactive power and voltage control, or they are connected in series to control the power flow.
Chapter 6
(Energy Management Systems)
In the control center, the transmission and distribution of electrical energy are monitored, coordinated, and controlled. The energy management system (EMS) is the interface between the operator and the actual power system. The supervisory control and data acquisition (SCADA) system collects real-time measured data from the system and presents it to the computer screen of the operator, and it sends control signals from the control center to the actual components in the network. The EMS is in fact an extension of the basic functionality of the SCADA system and includes tools for the analysis and the optimal operation of the power system. The state estimator serves as a “filter” for the collected measurement data; it determines the state of the power system that matches best with the available measurements. This is necessary input for other analysis programs in the EMS, such as the load flow or power flow and the optimal power flow. The load flow computation is one of the most important power system computations, giving us insight into the steady-state behavior of the power system. Therefore, besides the well-known Newton–Raphson load flow, a decoupled load flow and the DC load flow are also presented.
Chapter 7
(Electricity Markets)
At a broad conceptual level, there exists such a thing as a “common market model” that provides for both spot market trading coordinated by a grid/market operator and for bilateral contract arrangements scheduled through the same entity. The spot market is based on a two-sided auction model: both the supply and demand bids are sent to the power exchange. Market equilibrium occurs when the economic balance among all participants is satisfied and the benefits for society, called “the social welfare,” are at their maximum value. The power system is a large interconnected system, so that multiple market areas are physically interconnected with each other: this facilitates the export of electricity from low-price areas to high-price areas.
Chapter 8
(Future Power Systems)
In this chapter some developments, originating from the complex technological, ecological, sociological, and political playing field and their possible consequences on the power system, are highlighted. A large-scale implementation of electricity generation based on renewable sources, for example, will cause structural changes in the existing distribution and transmission networks. Many of these units are decentralized generation units, rather small-scale units that are connected to the distribution networks often by means of a power-electronic interface. A transition from the current “vertically operated power system” into a “horizontally operated power system” in the future is not unlikely. Energy storage can be applied to level out large power fluctuations when the power is generated by renewable energy sources, driven by intermittent primary energy. The complexity of the system increases because of the use of FACTS devices, power-electronic interfaces, intermittent power production, and so on. Chaotic phenomena are likely to occur in the near future and large system blackouts will probably happen more often.
Appendix A
(Maxwell's Laws)
Circuit theory can be regarded as describing a restricted class of solutions of Maxwell's equations. In this appendix, power series approximations will be applied to describe the electromagnetic field. It is shown that the zero- and first-order terms in these approximations (i.e., the quasi-static fields) form the basis for the lumped-circuit theory. By means of the second-order terms, the validity of the lumped-circuit theory at various frequencies can be estimated. It is the electrical size of the structure – its size in terms of the minimum wavelength of interest in the bandwidth over which the model must be valid – that dictates the sophistication and complexity of the required model. A criterion is derived that relates the dimensions of the electromagnetic structure to the smallest wavelength under consideration so that the validity of the lumped-element model can be verified.
Appendix B
(Power Transformer Model)
Transformers essentially consist of two coils around an iron core. The iron core increases the magnetic coupling between the two coils and ensures that almost all the magnetic flux created by one coil links the other coil. The central item of this appendix is the mathematical description of the voltage–current relations of the transformer. First, the voltage–current relation of an ideal transformer, including the impedance transformation, is given. After that, a more general description of the transformer by means of magnetically coupled coils is derived. In the next step, the nonideal behavior of the transformer, comprising leakage flux and losses in the windings and in the iron core, is taken into account, and a transformer equivalent circuit is derived. The appendix ends with an overview of single-phase equivalent models of three-phase transformers.
Appendix C
(Synchronous Machine Model)
A synchronous generator generates electricity by conversion of mechanical energy into electrical energy. The two basic parts of the synchronous machine are the rotor and the armature or stator. The iron rotor is equipped with a DC-excited winding, which acts as an electromagnet. When the rotor rotates and the rotor winding is excited, a rotating magnetic field is present in the air gap between the rotor and the armature. The armature has a three-phase winding in which the time-varying EMF is generated by the rotating magnetic field. For the analysis of the behavior of the synchronous machine in the power system, a qualitative description alone is not sufficient. The central item of this appendix is the mathematical description of the voltage–current relation of the synchronous generator. Based on the voltage–current relation, a circuit model is developed that is connected to an infinite bus to study the motor and generator behavior.
Appendix D
(Induction Machine Model)
The induction machine is an alternating current machine that is very well suited to be used as a motor when it is directly supplied from the grid. The stator of the induction machine has a three-phase winding; the rotor is equipped with a short-circuited rotor winding. When the rotor speed is different from the speed of the rotating magnetic field generated by the stator windings, we describe the rotor speed as being asynchronous, in which case the short-circuited rotor windings are exposed to a varying magnetic field that induces an EMF and currents in the short-circuited rotor windings. The induced rotor currents and the rotating stator field result in an electromagnetic torque that attempts to pull the rotor in the direction of the rotating stator field. The central item of this appendix is the mathematical description of the voltage–current relation and the torque–current relations of the induction machine. Based on the voltage–current relation, a circuit model is developed.
Appendix E
(The Representation of Lines and Cables)
When we speak of electricity, we think of current flowing through the conductors of overhead transmission lines and underground cables on its way from generator to load. This approach is valid because the physical dimensions of the power system are generally small compared to the wavelength of the currents and voltages in steady-state analysis. This enables us to apply Kirchhoff's voltage and current laws and use lumped elements in our modeling of overhead transmission lines and underground cables. We can distinguish four parameters for a transmission line: the series resistance (due to the resistivity of the conductor), the inductance (due to the magnetic field surrounding the conductors), the capacitance (due to the electric field between the conductors), and the shunt conductance (due to leakage currents in the insulation). Three different models are derived, which, depending on the line length, can be applied in power system analysis.
In the process of writing this book, we sometimes felt like working on a film script: we put the focus on selected topics and zoomed in or out whenever necessary, as there is always a delicate balance between the thing that you want to make clear and the depth of the explanation to reach this goal. We hope that we have reached our final goal and that this book provides you with a coherent and logical introduction to the interesting world of electrical power systems!
While writing this book we gratefully made use of the lecture notes that have been used over the years at the Delft University of Technology and the Eindhoven University of Technology in the Netherlands. The appendices on the modeling of the transformer, the synchronous machine, and the induction machine are based on the excellent Dutch textbook of Dr. Martin Hoeijmakers on the conversion of electrical energy. We are very grateful for the careful reading of the manuscript by Prof. Emeritus Koos Schot, Robert van Amerongen, and Jan Heijdeman. We would like to thank Ton Kokkelink and Rene Beune, both from TenneT TSO B.V., for their valuable comments on Chapters 5 and 7, respectively. We appreciate the contribution to the problems and their solutions of Romain Thomas, and Dr. Laura Ramirez Elizondo.
The companion website for the book is http://www.wiley.com/go/powersystem, where PowerPoint slides for classroom use can be downloaded.
Pieter H. Schavemaker Lou van der Sluis The Netherlands Spring 2017
List of Abbreviations
AC
alternating current
ACE
area control error
ACSR
aluminum conductor steel reinforced
ATC
available transmission capacity
AVR
automatic voltage regulator
BES
battery energy storage
CAES
compressed air energy storage
CHP
combined heat and power
CO
2
carbon dioxide
CT
current transformer
DAM
day-ahead market
DC
direct current
DG
decentralized generation, distributed generation, dispersed generation
EMF
electromotive force
EMS
energy management system
ENTSO-E
European network of transmission system operators for electricity
FACTS
flexible AC transmission systems
GIL
gas-insulated transmission line
GTO
gate turnoff thyristor
HVDC
high-voltage DC
ID
intraday
IEC
International Electrotechnical Commission
IEEE
Institute of Electrical and Electronics Engineers
IGBT
insulated gate bipolar transistor
IPP
independent power producer
ISO
independent system operator
LCC
line commutated converter
LED
light-emitting diode
LFC
load frequency control
LL
line-to-line
LN
line-to-neutral
LTI
linear time-invariant
MCP
market clearing price
MCV
market clearing volume
NEC
net export curve
OTC
over the counter
pu
per unit
PV
photovoltaic
PWM
pulse-width modulation
PX
power exchange
RMS
root mean square
SCADA
supervisory control and data acquisition
SF
6
sulfur hexafluoride
SIPL
switching impulse protective level
SMES
superconducting magnetic energy storage
SSSC
static synchronous series compensator
STATCOM
static synchronous compensator
SVC
static var compensator
TCR
thyristor-controlled reactor
TCSC
thyristor-controlled series capacitor
TSC
thyristor-switched capacitor
TSO
transmission system operator
UCTE
Union for the Coordination of Transmission of Electricity
UPFC
unified power flow controller
VSC
voltage source converter
XPLE
cross-linked polyethylene
List of Symbols
Text Symbols
Bold uppercase text symbols generally refer to matrices, for example, A.
Bold lowercase text symbols generally refer to vectors, for example, x.
Various notations of a voltage:
v
,
v
(
t
)
the sinusoidal time-varying quantity
V
the phasor representation of the sinusoidal time-varying quantity; the DC quantity
|
V
|
the effective or RMS value of the sinusoidal time-varying quantity; the length of the phasor representation of the sinusoidal time-varying quantity
The polarity of the voltage is indicated in circuit diagrams in one of the three following ways:
DC voltage source; the long plate indicates the positive terminal, and the short plate the negative terminal
AC voltage source; the plus sign indicates the positive terminal, and the minus sign the negative terminal
arrow: it specifies the voltage between two terminals/points in the circuit diagram; the arrowhead indicates the positive terminal, and the tail the negative terminal
Graphical Symbols
Graphical symbols in a circuit diagram:
inductance
capacitance
resistance
impedance, admittance, general load
fuse
transformer
magnetically coupled coils
DC voltage source
AC voltage source
current source
diode
power-electronic switching device (e.g., thyristor, GTO)
earth, neutral, reference
Graphical symbols in a single-line or one-line diagram:
transmission link, line, cable
circuit breaker
disconnector
busbar, node
load
synchronous generator
rotating machine
transformer
Chapter 1Introduction to Power System Analysis
1.1 Introduction
As electricity comes out of the alternating current (AC) outlet every day, and has already been doing so for more than 100 years, it may nowadays be regarded as a commodity. It is a versatile and clean source of energy; it is fairly cheap and “always available.” In the Netherlands, for instance, an average household encountered only 20 minutes' interruption to their supply in the year 2014 [1] out of a total of 8760 hours, resulting in an availability of 99.996195%!
Society's dependence on this commodity has become critical and the social impact of a failing power system is beyond imagination:
Cars would not be refueled as gas station pumps are driven by electricity.
The sliding doors of shops and shopping malls would not be able to open or close and people would therefore be locked out or in.
Electrified rail systems, such as subways and trains, would come to a standstill.
Traffic lights would not work.
Refrigerators would stop.
Heating/cooling installations would fail.
Cash dispensers would be offline.
Computers would serve us no longer.
Water supplies would stop or run out.
Many more examples may be given, but the message is clear: electric power systems are the backbone of modern society (see Figure 1.1), and chaos would result if the electricity supply failed for an extended period.
Figure 1.1 The Earth's city lights, indicating the most urbanized areas. The Visible Earth, NASA.
Our society needs engineers who know how to design, build, and operate an electrical power system. So let us discover what lies beyond the AC outlet and enter the challenging world of power system analysis.
1.2 Scope of the Material
Power system analysis is a broad subject, too broad to cover in a single textbook. The authors confine themselves to an overview of the structure of the power system (from generation via transmission and distribution to customers) and only take into account its steady-state behavior. This means that only the power frequency (50 or 60 Hz) is considered. An interesting aspect of power systems is that the modeling of the system depends on the time scale under review. Accordingly, the models for the power system components that are used in this book have a limited validity; they are only valid in the steady-state situation and for the analysis of low-frequency phenomena. In general, the time scales we are interested in are as follows:
Years, months, weeks, days, hours, minutes, and seconds for steady-state analysis at power frequency (50 or 60 Hz)
This is the time scale on which this book focuses. Steady-state analysis covers a variety of topics such as planning, design, economic optimization, load flow/power flow computations, fault calculations, state estimation, protection, stability, and control.
Milliseconds for dynamic analysis (kHz)
Understanding the dynamic behavior of electric networks and their components is important in predicting whether the system, or a part of the system, remains in a stable state after a disturbance. The ability of a power system to maintain stability depends heavily on the controls in the system to dampen the electromechanical oscillations of the synchronous generators.
Microseconds for transient analysis (MHz)
Transient analysis is of importance when we want to gain insight into the effect of switching actions, for example, when connecting or disconnecting loads or switching off faulty sections, or into the effect of atmospheric disturbances, such as lightning strokes, and the accompanying overvoltages and overcurrents in the system and its components.
Although the power system itself remains unchanged when different time scales are considered, components in the power system should be modeled in accordance with the appropriate time frame. An example to illustrate this is the modeling of an overhead transmission line. For steady-state computations at power frequency, the wavelength of the sinusoidal voltages and currents is 6000 km (in the case of 50 Hz):
1.1
λ
the wavelength [km]
v
the speed of light ≈ 300000 [km/s]
f
the frequency [Hz = 1/s]
Thus, the transmission line is, so to speak, of “electrically small” dimensions compared to the wavelength of the voltage. The Maxwell equations can therefore be approximated by a quasi-static approach, and the transmission line can accurately be modeled by lumped elements (see also Appendix A). Kirchhoff's laws may fruitfully be used to compute the voltages and currents. When the effects of a lightning stroke have to be analyzed, frequencies of 1 MHz and higher occur and the typical wavelength of the voltage and current waves is 300 m or less. In this case the transmission line is far from being “electrically small,” and it is not allowed to use the lumped-element representation anymore. The distributed nature of the transmission line has to be taken into account, and we have to calculate with traveling waves.
Despite the fact that we mainly use lumped-element models in our book, it is important to realize that the energy is mainly stored in the electromagnetic fields surrounding the conductors rather than in the conductors themselves as is shown in Figure 1.2. The Poynting vector, being the outer product of the electric field intensity vector and the magnetic field intensity vector, indicates the direction and intensity of the electromagnetic power flow [2, 3]:
1.2
S
the Poynting vector [W/m
2
]
E
the electric field intensity vector [V/m]
H
the magnetic field intensity vector [A/m]
Figure 1.2 Transmission line–transformer–transmission line–load: the energy is stored in the electromagnetic field.
Due to the finite conductivity of the conductor material and the finite permeability of the transformer core material, a small electric field component is present inside the conductor and a small magnetic field component results in the transformer core:
1.3
J
the current density vector [A/m
2
]
σ
the conductivity [S/m]
1.4
B
the magnetic flux density vector [T = A H/m
2
]
μ
the permeability [H/m]
This leads to small Poynting vectors pointing toward the conductor and the transformer core: the losses in the transmission line and the transformer are fed from the electromagnetic field, as is the power consumed by the load.
