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Frequency Variations
in Power Systems
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Frequency Variations
in Power Systems
Modeling, State Estimation,
and Control
Federico Milano
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University College Dublin
Dublin, Ireland
´
Alvaro Ortega Manjavacas
Universidad Loyola Andaluc´ıa
Seville, Spain
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Thiseditionfirstpublished2020
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Library of Congress Cataloging-in-Publication Data
Names:Milano,Federico,author.—OrtegaManjavacas,A´lvaro,
author.
Title:Frequencyvariationsinpowersystems:modeling,stateestimation
andcontrol/FedericoMilano,A´lvaroOrtegaManjavacas.
Description:Hoboken,NJ:Wiley-IEEEPress,2020.—Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2019053575(print)—LCCN2019053576(ebook)—ISBN
9781119551843(hardback)—ISBN9781119551898(adobepdf)—ISBN
9781119551874(epub)
Subjects:LCSH:Electricpowersystems–Mathematicalmodels.—Electric
powersystems–Control.—Frequenciesofoscillating
systems–Measurement.—Frequencydividers.
Classification:LCCTK1005.M49752020(print)—LCCTK1005(ebook)—DDC
621.31–dc23
LCrecordavailableathttps://lccn.loc.gov/2019053575
LCebookrecordavailableathttps://lccn.loc.gov/2019053576
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10 9 8 7 6 5 4 3 2 1
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To my grandfather, Alessandro Cesare.
F.M.
To Alicia.
´
A.O.M.
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Space itself wobbles and rumbles like a drum...
Janna Levin
I should point out, however, that at first
some difficulty was experienced in observing the
phenomena predicted by the theory, owing
to the extreme smallness of the
variations in the period of oscillation.
Pieter Zeeman
If you want to find the secrets of the universe,
think in terms of energy, frequency and vibration.
Nikola Tesla
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The definition, measurement, and interpretation
of the frequency of a power system is a challenging problem
without (so far) a satisfying solution.
Harold Kirkham, William Dickerson, and Arun Phadke
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Contents
List of Figures xiii
List of Tables xix
Preface xxi
Acknowledgments xxvii
Acronyms and Abbreviations xxix
Notation xxxiii
Part I Background 1
1 Frequency in Power Systems 3
1.1 Conventional Definitions 3
1.2 Alternating Current 6
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1.3 Reference Frequency 8
1.4 Transforms 13
2 Power System Model 39
2.1 Time Scales 39
2.2 Quasi-Steady-State Model 41
2.3 Differential Algebraic Equations 45
2.4 Conventional Devices 48
3 Dynamic State Estimation 77
3.1 Basic Concepts 77
3.2 Introducing Dynamics 84
3.3 Estimation of Bus Frequencies 89
4 Frequency Control 105
4.1 Introduction 105
4.2 Power Balancing 106
4.3 Power Oscillation Damping 115
4.4 Nonsynchronous Devices 125
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x Contents
Part II Theory 137
5 Frequency Divider Formula 139
5.1 Rationale 139
5.2 Derivation 141
5.3 Equivalent Networks 155
5.4 Inclusion of Measurements 159
5.5 Frequency Participation Factors 163
6 Frequency Makers and Frequency Takers 175
6.1 Introduction 175
6.2 Derivation 176
6.3 Taxonomy 178
6.4 Examples 187
Part III Applications 205
7 Frequency Control 207
(cid:2) 7.1 Impact of Frequency Signals 207 (cid:2)
7.2 Synthesis of Frequency Signals 216
8 Dynamic State Estimation 227
8.1 Machine Rotor Speeds 227
8.2 Center of Inertia 247
8.3 Applications of the RoCoP 256
9 Power System Model 261
9.1 Introduction 261
9.2 Frequency Dependent Model 262
9.3 Example 265
10 Frequency in Power Systems 269
10.1 Definitions 269
10.2 Final Remarks 269
AppendixA Data 273
A.1 Three-Bus System 273
A.2 WSCC System 273
A.3 IEEE 14-Bus System 278
A.4 New England System 281
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Contents xi
AppendixB Irish Transmission System 291
Bibliography 293
Index 313
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List of Figures
1.1 How long is the coast of Britain? 5
1.2 Instantaneous power of nonisofrequential voltage and current 10
1.3 Instantaneous power of isofrequential voltage and current 11
1.4 Inst. power of almost isofrequential voltage and current (1) 12
1.5 Inst. power of almost isofrequential voltage and current (2) 12
1.6 Time dimensionality reduction in the phasor domain 15
1.7 Voltage with periodic frequency variations 19
1.8 DST of a voltage with periodic frequency variations 20
1.9 Graphical representation of symmetrical components 24
1.10 Examples of Clarke-transformed signals 27
(cid:2) 1.11 Graphical representation of the Park transform 29 (cid:2)
1.12 Examples of Park-transformed signals 32
2.1 Classification of power system models based on the time scale 40
2.2 Equivalent single-phase diagram of a balanced three-phase circuit 41
2.3 Voltage magnitude controller of the circuit example 42
2.4 Comparison of Park vectors and QSS phasors 44
2.5 Equivalent circuits of the synchronous machine 56
2.6 Approximated equivalent circuit of the synchronous machine 57
2.7 Circuit of a load 61
2.8 Circuit of the transmission line 63
2.9 Circuit of the transformer 65
2.10 Comparison of the dynamic response of EMT and QSS models 74
2.11 Effect of the CoI on system dynamics 75
3.1 Main components of a PMU 82
3.2 Dynamic response of the continuous EKF 88
3.3 Phase angle and its rough numerical derivative 91
3.4 Washout filter 92
3.5 Numerical derivative through a washout filter 93
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xiv List of Figures
3.6 Basic scheme of a PLL 94
3.7 Scheme of an SRF-PLL 96
3.8 Performance of the SRF-PLL and EKF-PLL 99
3.9 Performance of the STFT 102
4.1 Time scales of inertial response and frequency control 107
4.2 Scheme of a simple turbine governor model 108
4.3 Simplified automatic generation control scheme 110
4.4 Comparison of different values of machine inertia and damping 112
4.5 Comparison of different droop values of PFC (1) 113
4.6 Comparison of different droop values of PFC (2) 114
4.7 Comparison of different gain values of SFC 115
4.8 Control diagram of the IEEE Type DC1 exciter 116
4.9 PSS with washout filter and two lead-lead compensators 119
4.10 Effect of AVRs on system dynamics 121
4.11 Effect of different AVR parameters on system dynamics 123
(cid:2) 4.12 Effect of load models on system dynamics 124 (cid:2)
4.13 Effect of a PSS on system dynamics 125
4.14 Scheme of fast frequency/RoCoF and voltage controls of DERs 127
4.15 Models of fast frequency control for DERs 128
4.16 Models of fast frequency and RoCoF control for DERs 129
4.17 VSC and inner current-control in the dq-reference frame 131
4.18 Thermostatically controlled load with frequency control 132
4.19 Effect of frequency control of a WECS on system dynamics (1) 134
4.20 Effect of frequency control of a WECS on system dynamics (2) 135
5.1 Machine rotor speeds following a three-phase fault 140
5.2 Effect of inertia on the response of machine rotor speeds 140
5.3 Frequencies at the buses between two synchronous machines 141
5.4 Two-bus system 145
5.5 Three-bus system 147
5.6 Rotor speeds and bus frequencies 148
5.7 Frequency at bus 3 of the three-bus system 149
5.8 Frequency at bus 3 for a voltage and frequency dependent load 150
5.9 Comparison of bus frequencies at generator buses 152
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